research article optical flow inversion for remote sensing...

Research ArticleOptical Flow Inversion for Remote SensingImage Dense Registration and Sensorrsquos Attitude MotionHigh-Accurate Measurement

Chong Wang12 Zheng You12 Fei Xing12 Borui Zhao12 Bin Li12

Gaofei Zhang12 and Qingchang Tao12

1 Department of Precision Instrument Tsinghua University Beijing 100084 China2The State Key Laboratory of Precision Measurement Technology and Instruments Tsinghua University Beijing 100084 China

Correspondence should be addressed to Fei Xing xingfeimailtsinghuaeducn

Received 18 September 2013 Revised 4 November 2013 Accepted 4 November 2013 Published 16 January 2014

Academic Editor Bo Shen

Copyright copy 2014 Chong Wang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

It has been discovered that image motions and optical flows usually become much more nonlinear and anisotropic in space-bornecameras with large field of view especially when perturbations or jitters existThe phenomenon arises from the fact that the attitudemotion greatly affects the image of the three-dimensional planet In this paper utilizing the characteristics an optical flow inversionmethod is proposed to treat high-accurate remote sensor attitude motion measurement The principle of the new method is thatangular velocities can be measured precisely by means of rebuilding some nonuniform optical flows Firstly to determine therelative displacements and deformations between the overlapped images captured by different detectors is the primary process ofthe method A novel dense subpixel image registration approach is developed towards this goal Based on that optical flow canbe rebuilt and high-accurate attitude measurements are successfully fulfilled In the experiment a remote sensor and its originalphotographs are investigated and the results validate that the method is highly reliable and highly accurate in a broad frequencyband

1 Introduction

For the remote sensors in dynamic imaging one importanttechnology is imagemotion compensation Actually to deter-mine image motion velocity precisely is a very hard problemIn [1 2] optical correlators are utilized to measure imagemotion in real time based on a sequence of mild smearedimages with low exposure This technique is appropriate tothe situations in which the whole image velocity field isuniform Some other blind motion estimation algorithmsin [3ndash5] have been used to image postprocessing whichcan roughly detect inhomogeneous image motion but lackreal-time performance because of complexity As for spaceimaging in order to avoid motion blurring image motionvelocity needs to be computed in real time according tothe current physical information about spacecraftrsquos orbitand attitude motion which can be obtained by the space-borne sensors such as star trackers gyroscopes and GPS

Wang et al developed a computational model for imagemotion vectors and presented error budget analysis in [6]They focused on the small field of view (FOV) space cameraswhich are used in push-broom imaging with small attitudeangles In that situation the nonlinearity of image motionvelocity field does not appear significantly However forothers with larger FOV image motion velocity fields aredefinitely nonlinear and anisotropic because the geometryof the planet will greatly modulate the moving imagesUnder the circumstances the detectors need to be controlledseparately to keep the time series synchronized with theinstantaneous image velocities

The time-phase relations between the photos belongingto different detectors are affected by optical flows whichare uniquely determined by the behavior of image velocityfield in a specific period Some phenomena about movingimage variation and distortion due to optical flow have beendiscovered [7ndash10] References [7 8] reported the camera on

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 432613 16 pageshttpdxdoiorg1011552014432613

2 Mathematical Problems in Engineering

Mars Reconnaissance Orbiter (MRO) in High ResolutionImaging Science Experiment (HiRISE) missions of NASAIt takes pictures of Mars with resolutions of 03mpixelresolving objects Fourteen staggered parallel CCDs areoverlapped with 48 pixels at each end to fulfill the entirefield of view Although adjacent detectors overlap with equalphysical pixels yet their lapped image pixels are not equal andvarying with time because spacecraft jitters cause undulatingoptical flows within the interlaced areas [8] In additionwe found that when large FOV remote sensors performstereoscopic imaging with large pitch angles the lappedimages belonging to marginal detectors are bound to exceedor lose several hundred pixels compared to their physicaloverlaps Furthermore the unexpected quantity decreasessignificantly for the detectors mounted at the central regionof the focal plane

Although nonuniform optical flow brings many troublesin image processing such as registration resample intercon-nection and geometrical rectification it permits us to mea-sure the spacecraft attitudemotionwith very high accuracy ina broad bandwidth It is nearly impossible for conventionalspace-borne sensors to realize the target Precision attitudemotionmeasurement is very useful for remote sensing imageprocessing especially for image restoration from motionblurring as studied in [11 12] Associating the measurementand optical flow models the dynamic point spread functions(PSF) are able to be estimated to be set as the convolutionkernels in nonblinded deconvolution algorithms

The behavior of optical flow characterizes the entire two-dimensional flow field for an imagersquos motion and variationIn [13] optical flow estimation based on image sequencesof the same aurora to determine the flow field will provideaccess to the phase space the important information forunderstanding the physical mechanism of the aurora For thepurpose to improve the accuracy of optical flow estimationa two-step matching paradigm for optical flow estimation isapplied in [14] firstly the coarse distribution measuring ofmotion vectors is done with a simple frame-to-frame corre-lation technique also known as the digital symmetric phase-only filter (SPOF) and after that subpixel accuracy estimationresult is achieved by using the sequential tree-reweightedmax-product message passing (TRW-S) optimization Sim-ilarly Sakaino overcame the disadvantages in optical flowdetermination when moving objects with different shapesand sizes move against a complicated background the imageintensity between frames may violate the common situationimage brightness constancy to image brightness 5 changemodels as constraints in regular situations [15] Howeverunlike continuous image sequences if we merely obtainedseveral images of the identical moving objects captured bydifferent detectors with long intervals the former techniquesdo not work well for optical flow estimation for lacking of theinformation of imaging process of the instrument

In this paper a new optical flow inversion methodis proposed for precise attitude measurement Unlike thesituations in [13ndash15] the image sequences of video do notexist for the transmission type remote sensors instead of theimage pairs of the same earth scene which are captured bydifferent TDI CCD detectors in push-broom fashion The

time intervals between the independent image formationscorresponding to the overlapped detectors are much morethan the interval between sequential frames in video forwhich the frame rates usually exceed tens of frames persecond (fps) However we can model optical flows basedon the working mechanism of the instrument and imageprocessing techniques rather than estimating from framesequences of a specific detector The contents of this paperare organized as follows in Section 2 an analytical model ofimage motion velocity field is established which is applicableto dynamic imaging for three-dimensional planet surfaceby large FOV remote sensors The phenomenon of movingimage deformation due to optical flow is investigated inSection 3 Based on rough inversion of optical flow a novelmethod for dense image registration is developed to measurethe subpixel offsets between the lapped images which arecaptured by adjacent detectors In Section 4 an attitudemotion measuring method based on precise optical flowinversion is studied and the results of the experiment supportthe whole theory perfectly

2 Image Velocity Field Analysis

Suppose that a large FOV camera is performing push-broom imaging to the earth the scenario is illustrated inFigure 1 The planetrsquos surface cannot be regarded as a localplane but a three-dimensional ellipsoid since it may greatlyinfluence the image motion and time-varying deformationwhen complicated relative motion exists between the imagerand the earth

In order to set up the model of space imaging somecoordinate systems need to be defined as follows

(1) I(119874119890minus 119883119884119885) the inertial frame of the earth For

convenience here we choose 1198692000 frameThe origin119874119890is located at the earth center

(2) C(119900119904minus 1199091015840

11199091015840211990910158403) the frame of camera Axis 119900

11990411990910158403is the

optical axis and origin 119900119904is the center of the exit pupil

(3) O(119900119904minus 119906

111990621199063) the orbit frame Axis 119900

1199041199063passes

through the center of earth and axis 1199001199041199062is perpen-

dicular to the instant orbit plane(4) B(119900

119904minus 119909

119887119910119887119911119887) the body frame of the satellite

(5) P(119900 minus 119909119910119911) the frame of photographyThe origin 119900 isthe center of the photo Axis 119900119909 points to the columndirection and axis 119900119910 points to the row direction

(6) F(1199001015840 minus 119909101584011991010158401199111015840) the frame of focal plane Axes 11990010158401199091015840and 11990010158401199101015840 lie in the focal plane They are respectivelyparallel to 119900

11990411990910158402and 119900

11990411990910158401 Axis 11990010158401199111015840 coincides with the

optical axis(7) E(119874

119890minus 119909

119890119910119890119911119890) the frame of Terrestrial Reference

Frame (TRF) Axis 119874119890119910119890points to the North Pole

and axis 119874119890119909119890passes through the intersection of

Greenwich meridian and the equator

According to Figure 1 1205970is the ground track of the

satellite and 1205971and 120597

2are the ground traces corresponding to

two fixed boresights in FOV which are far away from 1205970if the

Mathematical Problems in Engineering 3

pminusrarr120591

Γ

q1205972

12059711205970

u1x9984001 u2

x9984002

u3

Os

p998400 rarr120591 998400

x9984003

Figure 1 The analysis of dynamic imaging for the three-dimensional planet

imager holds a large attitude angle Obviously the shapes andlengths of 120597

1and 120597

2also have notable differences during push

broom which implies that the geometrical structure of theimage is time varying as well as nonuniform Furthermoreit can be discovered later that the deforming rates mainlydepend on the planetrsquos apparent motion observed by thecamera

Considering an object point 119901 on the earth its positionvector relative to 119874

119890is denoted as 120588

119901 As a convention in the

following discussions I120588119901represents the vectormeasured in

frame I and accordingly C120588119901is the same vector measured

in frame C We select one unit vector 120591 which is tangent tothe surface of the earth at 119901 Let 119903(119909

1 119909

2 119909

3) be the position

vector of 119901 relative to 119900119904 then C 119903 and C 119903 characterize the

apparent motion of 119901 Assume that the image point 1199011015840 isformed on the focal plane with coordinates (1199091015840

1 1199091015840

2 1199091015840

3) in

frameC Generally the optical systems of space cameras arewell designed and are free from optical aberrations and thestatic PSF is approximate to the diffraction limit [16 17] thusfollowing [18] we have

1199091015840

119894= 120573119909

119894 (119894 = 1 2)

1199091015840

3= 119891

1015840

(1)

where1198911015840 is the effective focal length the lateral magnificationof 1199011015840 120573 = (minus1)

119898minus1

(1198911015840( 119903 sdot 1198903)) 119898 is the number of

intermediate images in the optical system and 119890119894(119894 = 1 2 3)

is the base ofCLet 119903

119904be the position vector of satellite relative to119874

119890 then

119903 = 120588 minus 119903119904 In imaging the flight trajectory of the satellite

platform inI can be treated as Keplerian orbit as illustratedin Figure 2 According to the orbit elements 119894

0 inclination

Ω longitude of ascending node 120596 argument of perigee 119886semimajor axis 119890 eccentricity119872

119905 mean anomaly at epoch

we implement Newton-Raphson method to solve (2) and getthe eccentric anomaly 119864 from the given mean anomaly119872

119905=

1198720+ 119899(119905 minus 119905

0) where 119899 = 2120587119875 119875 is the orbit period [11]

119872119905minus (119864 minus 119890 sin119864) = 0 (2)


Orbit plane

The equatorial plane

Perigee

u3

osu1

u2 rarrrsY

2a

i

Oe

Ω X

Υ

120596

rarrs

Figure 2 Orbital motion of remote sensor

In frame O

O119903119904= (

119886 (cos119864 minus 119890)119887 sin1198640

) OV

119904=(

minus119886 sin119864119887 cos1198640

)119899

1 minus 119890 cos119864

(3)

The coordinate transform matrix between O andI is

TOI

= (

119862120596119862Ω minus 1198781205961198621198940119878Ω minus119878120596119862Ω minus 119862120596119862119894

0119878Ω 119878119894

0119878Ω

119862120596119878Ω + 1198781205961198621198940119862Ω minus119878120596119878Ω + 119862120596119862119894

0119862Ω minus119878119894

0119862Ω

1198781205961198781198940

1198621205961198781198940

1198621198940

)

(4)

For simplicity we write 119862120572 = cos120572 119878120572 = sin120572In engineering the coordinate transfer matrix TOI also

can be derived from the real-time measurements of GPSSince the base vectors of frame O in I

3= minus

I119903119904| 119903

119904|

2

= (IV

119904times

I119903119904)|V

119904times 119903

119904| and

1=

2times

3then TOI =

(123)minus1

I119903119904= TOIsdot

O119903119904

IV119904= TOIsdot

OV119904 (5)

Associating the equation of boresight with the ellipsoidsurface of the earth inC yields

1198832 + 1198852

1198602

119890

+1198842

1198612119890

= 1

119883 minus 119883119904

1199041

=119883 minus 119884

119904

1199042

=119883 minus 119885

119904

1199043

(6)

Here 119860119890= 6378137 km and 119861

119890= 6356752 km being the

length of earthrsquos semimajor axis and semiminor axis 119904119894(119894 =

1 2 3) are the unit vectors of I 119903 We write the solution of (7)as I 120588 = (119883 119884 119885)

119879 Hence I119903 =

I120588 minus

I119903119904 C 119903 = M sdot A sdot

Tminus1

OI sdotI119903 where M is the coordinate transformation matrix

from frame B to frame C it is a constant matrix for fixedinstallation A is the attitude matrix of satellite according to1-2-3 rotating order we have

A = R120595sdot R

120579sdot R

120593(7)

in which

R120595= (

cos120595119905

sin1205951199050

minus sin120595119905cos120595

1199050

0 0 1

)

R120579= (

cos 1205791199050 minus sin 120579

119905

0 1 0

sin 1205791199050 cos 120579

119905

)

R120593= (

1 0 0

0 cos120593119905

sin120593119905

0 minus sin120593119905cos120593

119905

)

(8)

where 120593119905 120579

119905 and 120595

119905are in order the real-time roll angle

pitch angle and yaw angle at moment 119905 The velocity of 119901 inC can be written in the following scalar form

119894=

C 119903 sdot 119890119894

(119894 = 1 2 3) (9)

Thus the velocity of image point of 1199011015840 will be

1015840

119894= 120573119909

119894+ 120573

119894= (minus1)

1198981198911015840

( 119903 sdot 1198903)23119909119894

+ (minus1)119898minus1

1198911015840

119903 sdot 1198903

119894

(119894 = 1 2)

(10)

Substituting (2)ndash(9) into (10) the velocity vector of imagepoint V1015840 = (1015840

1 1015840

2)119879 can be expressed as the explicit function

of several variables that is

V1015840 = V (1198940 Ω 120596 119890119872

1199050 120593

119905 120579

119905 120595

119905

119905 120579

119905

119905 119909

1015840

1 119909

1015840

2) (11)

For conciseness this analytical expression of V1015840 is omittedhere

The orbit elements can be determined according toinstantaneous GPS data Besides they also can be calculatedwith sufficient accuracy in celestial mechanics [19] On theother hand the attitude angles 120593

119905 120579

119905 and 120595

119905can be roughly

measured by the star trackers andGPSMeanwhile their timerates

119905 120579

119905 and

119905have the following relations

(

1205961

1205962

1205963

) = R120595

(

0

0

119905

) + R120579

[

[

(

0120579119905

0

) + R120593(

0

0

119905

)]

]

(12)

1205961 120596

2 and 120596

3are the three components of the remote sen-

sorrsquos angular velocity C119904relative to orbital frame O which

is calibrated in frame C Those can be roughly measured byspace-borne gyroscopes or other attitude sensors

It is easy to verify from (11) that the instantaneous imagevelocity field on the focal plane appears significantly nonlin-ear and isotropic for large FOV remote sensors especially


when they are applied to perform large angle attitudemaneu-vering for example in sidelooking by swing or stereoscopiclooking by pitching and so forth Under these circumstancesin order to acquire photos with high spatial temporal andspectral resolution image motion velocity control strategiesshould be executed in real time [20] based on auxiliary datawhich measured by reliable space-borne sensors [21 22] Indetail for TDI CCD cameras the line rates of the detectorsmust be controlled synchronizing to the local image velocitymodules during exposure so as to avoid along-track motionblurring the attitude of remote sensor should be regulated intime to maintain the detectors push-broom direction aimingat the direction of image motion to avoid cross-track motionblurring

3 Optical Flow Rough Inversion andDense Image Registration

Optical flow is another important physical model carryingthe whole energy and information of moving images indynamic imaging A specific optical flow trajectory is anintegral curve which is always tangent to the image velocityfield thus we have

1199091015840

1(119879) = int

119879

0

1015840

1(119909

1015840

1 119909

1015840

2 119905) 119889119905

1199091015840

2(119879) = int

119879

0

1015840

2(119909

1015840

1 119909

1015840

2 119905) 119889119905

(13)

Since (13) are coupled nonlinear integral equations weconvert them to numerical forms and solve them iteratively

1199091015840

i (0) = 1199091015840

i (119905)10038161003816100381610038161003816119905=0

1199091015840

j (119899) = 1199091015840

j (119899 minus 1) +1

2

1015840

119895[119909

1015840

1(119899 minus 1) 119909

1015840

2(119899 minus 1) 119899]

+ 1015840

119895[119909

1015840

1(119899 minus 1) 119909

1015840

2(119899 minus 1)

119899 minus 1] Δ119905

(119895 = 1 2 119899 isin Z+

)

(14)

It is evident that the algorithm has enough precision solong as the step-size of time interval Δ119905 is small enough Itcan be inferred from (13) that strong nonlinear image velocityfield may distort optical flows so much that the geometricalstructure of image may have irregular behaviors Thereforeif we intend to inverse the information of optical flow tomeasure the attitude motion the general formula of imagedeformation due to the optical flows should be deduced

31 Time-Varying Image Deformation in Dynamic ImagingFirstly we will investigate some differential characteristics ofthe moving image of an extended object on the earth surfaceAs shown in Figure 1 considering a microspatial variation of119901 along 120591 on the curved surface can be expressed as 120575 120588

119901= 120575119897 120591

Its conjugated image is

1205751199091015840

119894= 120575120573119909

119894+ 120573120575119909

119894 (15)

We expand the term of 120575120573

120575120573 = (minus1)119898

1198911015840

( 119903 + 120575 119903) sdot 1198903

minus1198911015840

119903 sdot 1198903

= (minus1)119898minus1

1198911015840

119903 sdot 1198903

infin

sum119896=1

(minus1)119896

(120575 119903 sdot 119890

3

119903 sdot 1198903

)

119896

asymp (minus1)1198981198911015840 120591 sdot 119890

3120575119897

( 119903 sdot 1198903)2

(16)

Taking derivatives with respect to variable 119905 for either part of(15) we have

1205751015840

119894= 120575 120573119909

119894+ 120575120573

119894+ 120573120575119909

119894+ 120573120575

119894 (17)

According to (16) we know that 120575 120573 asymp 0 On the otherhand the variation of 119903 can be expressed through a series ofcoordinate transformations that is

C(120575 119903) = 120575119897 [MATminus1

OITEIE120591] (18)

Notice that E 120591 is a fixed tangent vector of earth surfaceat object point 119901 which is time-invariant and specifies anorientation of motionless scene on the earth

Consequently

(

C120575 119903

120575119897)

120591

= (MATminus1

OITEI +MATminus1

OITEI

+MATminus1

OITEI +MATminus1

OITEI)E120591

(19)

where the coordinate transformmatrix from frameE toI is

TEI = (

cos1198671199010 minus sin119867

119901

0 1 0

sin1198671199010 cos119867

119901

) (20)

Let 120596119890be the angular rate of the earth and 120572

119901the longitude of

119901 on the earth then the hour angle of 119901 at time 119905 is 119867119901(119905) =

GST+120572119901+120596

119890119905 in which GST represents Greenwich sidereal

timeThe microscale image deformation of the extended scene

on the earth along the direction of 120591 during 1199051sim 119905

2can be

written as

[1205751199091015840

119894]1199052

120591

minus [1205751199091015840

119894]1199051

120591

= int1199052

1199051

(1205751015840

119894)

120591

119889119905 (21)

From (17) we have

(1205751015840119894)

120591

120575119897=120575120573

120575119897119894+ 120573

120575119909119894

120575119897+ 120573

120575119894

120575119897 (22)

According to (16) (18) and (19) we obtain the terms in (22)

120575120573

120575119897= (minus1)

1198981198911015840 C 120591 sdot 119890

3

( 119903 sdot 1198903)2

120575119909119894

120575119897= MATminus1

OITEI 119890119894sdotE120591

120575119894

120575119897= (

C120575 119903

120575119897)

120591

sdot 119890119894+ (

C120575 119903

120575119897)

120591

sdot 119890119894

(23)


Furthermore if the camera is fixed to the satellite platformthen M = 0 119890

119894= 0

Consequently (22) becomes

F119894(119905 120591) =

(1205751015840119894)

120591

120575119897

= (minus1)1198981198911015840 C 120591 sdot 119890

3

( 119903 sdot 1198903)2119894

+ (minus1)1198981198911015840 ( 119903 sdot 119890

119894)

( 119903 sdot 1198903)2MATminus1

OITEI 119890119894sdotE120591


1198911015840

119903 sdot 1198903

(MATminus1

OITEI

+MATminus1

OITEI

+MATminus1

OITEI)E120591 sdot 119890

119894

(24)

For the motionless scene on the earth surface E120591 is a time-

independent but space-dependent unit tangent vector whichmeanwhile represents a specific orientation on the groundMoreover the physical meaning of function F

119894(119905 120591) is the

image deformation of unit-length curve on the curved surfacealong the direction of E

120591 in unit time interval That is theinstantaneous space-time deforming rate of the image of theobject along E

120591Consequently in dynamic imaging macroscopic defor-

mation on themoving image can be derived from the integralofF

119894(119905 120591) in space and time Referring to Figure 1 let Γ be an

arbitrary curve of the extended object on the earth let Γ1015840 be itsimage let two arbitrary points 119901 119902 isin Γ and let their Gaussianimages1199011015840 1199021015840 isin Γ1015840 Let E 120591 = T(119904) be a vector-valued functionwith variable 119904 (the length of the arc) which is time-invariantin frame E and gives the tangent vectors along the curve

So the image deformation taking place during 1199051sim 119905

2is

able to be described as

[(1199091015840

119901)119894

]1199052

1199051

minus [(1199091015840

119902)119894

]1199052

1199051

= intΓ

int1199052

1199051

F119894∘ T119889119905 119889119904 (25)

in whichF119894∘ T = F

119894[119905 T(119904)]

Now in terms of (24) and (25) we can see that the imagedeformation is also anisotropic and nonlinear which dependsnot only on optical flowrsquos evolution but also on the geometryof the scene

32 Dense Image Registration throughOptical Flow PredictionAs mentioned in the preceding sections optical flow is themost precise model in describing image motion and time-varying deformation On the contrary it is possible to inverseoptical flow with high accuracy if the image motion anddeformation can be detected As we know the low frequencysignal components of angular velocity are easier to be sensedprecisely by attitude sensors such as gyroscopes and startrackers but the higher frequency components are hard to

be measured with high accuracy However actually pertur-bations from high frequency jittering are the critical reasonfor motion blurring and local image deformations since theinfluences brought by low components of attitude motion areeasier to be restrained in imaging through regulating remotesensors

Since (13) and (25) are very sensitive to the attitudemotion the angular velocity is able to be measured with highresolution as well as broad frequency bandwidth so long asthe image motion and deformation are to be determinedwith a certain precision Fortunately the lapped images ofthe overlapped detectors meet the needs because they werecaptured in turn as the same parts of the optical flow passthrough these adjacent detectors sequentiallyWithout losinggenerality we will investigate the most common form ofCCD layout for which two rows of detectors are arrangedin parallel The time-phase relations of image formation dueto optical flow evolution are illustrated in Figure 3 wherethe moving image elements 120572

1 120572

2 (in the left gap)

1205731 120573

2 (in the right gap) are captured firstly at the same

time since their optical flows pass through the prior detectorsHowever because of nonuniform optical flows they willnot be captured simultaneously by the posterior detectorsTherefore the geometrical structures of photographs willbe time varying and nonlinear It is evident from Figure 3that the displacements and relative deformations in frameCbetween the lapped images can be determined by measuringthe offsets of the sample image element pairs in frameP

Let Δ1199101015840 = Δ11990910158401 Δ1199091015840 = Δ1199091015840

2be the relative offsets of the

same objectrsquos image on the two photos they are all calibratedinC orF We will measure them by image registration

As far as image registration method is concerned one ofthe hardest problems is complex deformation which is proneto weaken the similarity between the referenced images andsensed images so that itmight introduce large deviations fromthe true values or even lead to algorithm failure Some typicalmethods have been studied in [23ndash25] Generally most ofthem concentrated on several simple deforming forms suchas affine shear translation rotation or their combinationsinstead of investigating more sophisticated dynamic deform-ing models In [26ndash30] some effective approaches havebeen proposed to increase the accuracy and the robust ofalgorithms according to the respective reasonable modelsaccording to the specific properties of objective images

For conventional template based registration methodsonce a template has been extracted from the referencedimage the information about gray values shape and fre-quency spectrum does not increase since no additionalphysical information resources would be offered But actuallysuch information has changed when the optical flows arriveat the posterior detectors Therefore the cross-correlationsbetween the templates and sensed images certainly reduceSo in order to detect the minor image motions and com-plex deformations between the lapped images high-accurateregistration is indispensable which means that more pre-cise model should be implemented We treat it using thetechnique called template reconfiguration In summary themethod is established on the idea of keeping the completionof the information about optical flows


y998400

x998400

Posterior CCD

12057211205722

1205731

1205732

13998400

Prior CCD

Δx998400120578

Figure 3Nonlinear image velocity field and optical flow trajectoriesinfluence the time-phase relations between the lapped imagescaptured by the adjacent overlapped detectors

In operating as indicated in Figure 3 take the lappedimages captured by the detectors in prior array as thereferenced images and the images captured by posteriordetectors as the sensed images Firstly we will rebuild theoptical flows based on the rough measurements of the space-borne sensors and then reconfigure the original templates toconstruct the new templates whose morphologies are moreapproximate to the corresponding parts on the sensed imagesWith this process the information about imaging proceduresis able to be added into the new templates so as to increase thedegree of similarity to the sensed images The method maydramatically raise the accuracy of dense registration such thatthe high-accurate offsets between the lapped image pairs areable to be determined

In the experiment we examined Mapping Satellite-1 aChinese surveying satellite operating in 500 km height sunsynchronous orbit which is used for high-accurate pho-togrammetry [31] whose structure is shown in Figure 4 Oneof the effective payload three-line-array panchromatic CCDcameras has good geometrical accuracy whose ground pixelresolution is superior to 5m spectral range is 051 120583m sim

069 120583m and the swath is 60 km Another payload is that thehigh resolution camera is designed possessing Cook-TMAoptical system which gives a wide field of view [16 17] andthe panchromatic spatial resolution can reach 2m

In engineering for the purpose to improve the imagequality and surveying precision the high-accuracy measure-ments of jitter and attitude motion are very essential for pos-terior processing Thus here we investigate the images andthe auxiliary data of the large FOV high resolution camera todeal with the problem The experimental photographs werecaptured with 10∘ side looking The focal plane of the camera

High resolutionpanchromatic camera

Optical axis

Mapping satellite-01

O998400

x9984001x9984003

x9984002

Figure 4 The structure of Mapping Satellite-1 and its effectivepayloads

consists of 8 panchromatic TDI CCD detectors and there are120578 = 96 physical lapped pixels between each other

The scheme of the processing in registering one imageelement 120594 is illustrated in Figure 5

Step 1 Set the original lapped image strips (the images whichwere acquired directly by the detectors and without anypostprocessing) in frameC

Step 2 Compute the deformations of all image elementson referenced template with respect to their optical flowtrajectories

We extract the original template from the referencedimage denoted as 119879

1 which consists of 1198732 square elements

that is dim(1198791) = 119873 times 119873 Let 120594 be its central element and

119908 the width of each element here 119908 = 875 120583m Beforethe moving image was going to be captured by the posteriordetector in terms of (25) their current shapes and energydistribution can be predicted by the optical flow based on theauxiliary data of the remote sensor

In order to simplify the algorithm first order approxima-tion is allowed without introducing significant errors Thisapproximation means that the shape of every image elementis always quadrilateral Linear interpolations are carried outto determine the four sides according to the deformationsalong the radial directions of the vertexes as showed inFigure 5 The unit radial vectors are denoted by 120591

1015840

1sim 1205911015840

4in

frameC

1205911015840

1=radic2

21198901minusradic2

21198902 120591

1015840

3= minus

radic2

21198901+radic2

21198902

1205911015840

2=radic2

21198901+radic2

21198902 120591

1015840

4= minus

radic2

21198901minusradic2

21198902

(26)

Suppose image point 1199011015840 is the center of an arbitrary elementΣ1015840 in 119879

1 Let Σ be the area element on the earth surface which

is conjugate to Σ1015840 The four unit radial vectors of the vertexes


1

3

2

1

4

T0

T1 T9984001

T2 Ts

Referenced image of prior CCD Sensed image of posterior CCD

Figure 5 Optical flow prediction and template reconfiguration

on Σ 1205911sim 120591

4are conjugate to 1205911015840

1sim 1205911015840

4and tangent to the earth

surface at 119901 From the geometrical relations we have

C120591119894= (minus1)

119898

1199031015840 times 1205911015840119894times

C119899119901

100381610038161003816100381610038161199031015840 times 1205911015840

119894times

C119899119901

10038161003816100381610038161003816

E120591119894= Tminus1

EITOIAminus1Mminus1 C

120591119894

C119899119901= MATminus1

OITEIE119899119901

(27)

where E 119899119901is the unit normal vector of Σ at 119901 We predict

the deformations along 1205911sim 120591

4during 119905

1sim 119905

2according to

the measurements of GPS star trackers and gyroscopes asexplained in Figure 6 119905

1is the imaging time on prior detector

and 1199052is the imaging time on the posterior detector

[1205751199091015840

1]Δ119905

120591119896

= [1205751199091015840

1]1199052

120591119896

minus [1205751199091015840

1]1199051

120591119896

[1205751199091015840

2]Δ119905

120591119896

= [1205751199091015840

2]1199052

120591119896

minus [1205751199091015840

2]1199051

120591119896

(119896 = 1 sim 4)

(28)

The shape of deformed image Σ10158401199052can be got through linear

interpolation with

[120575 1199031015840

]Δ119905

120591119896

= ([1205751199091015840

1]Δ119905

120591119896

[12057511990910158402]Δ119905

120591119896

) (29)

Step 3 Reconfigure referenced template 1198791according to

optical flow prediction and then get a new template 1198792

Let 11987910158401be the deformed image of 119879

1computed in Step 2

Let 120594 = 119861119894119895be the central element of 1198791015840

1 integers 119894 and 119895 are

respectively the row number and column number of 119861119894119895The

gray value 119897119894119895of each element in 1198791015840

1is equal to its counterpart

in 1198791with the same indexes In addition we initialize a null

template 1198790whose shape and orientation are identical to 119879

1

the central element of 1198790is denoted by 119879

119894119895

[120575rarrr 998400]Δtminusrarr1205911[120575rarrr 998400]Δtminusrarr1205912


1 2

34

1998400

2998400

39984004998400

Σ998400t2

Σ998400t1

p998400rarr120591 998400

1rarr120591 998400

2

rarr120591 9984003

rarr120591 9984004

Figure 6 Deformation of single element

Then we cover 1198790upon 1198791015840

1and let their centers coincide

that is 119879119894119895= 119861

119894119895 as shown in Figure 7 Denote the vertexes

of 11987910158401as 119881119896

119894119895(119896 = 1 sim 4) Therefore the connective relation

for adjacent elements can be expressed by 1198811

119894119895= 119881

2

119894119895minus1=

1198813

119894minus1119895minus1= 1198814

119894minus1119895

Next we will reassign the gray value ℎ1015840119894119895to 119879

119894119895(119894 =

1 sdot sdot sdot 119873 119895 = 1 sdot sdot sdot 119873) in sequence to construct a new template1198792 The process is just a simulation of image resample when

optical flow arrives at the posterior detector as indicated inFigure 3

That is

ℎ1015840

119894119895=

119894+1

sum119898=119894minus1

119895+1

sum119899=119895minus1

120578119898119899119897119898119899 (30)

Weight coefficient 120578119898119899

= 1198781198981198991199082 where 119878

119898119899is the area of the

intersecting polygon of 119861119898119899

with 119879119894119895


V1iminus1jminus1

Biminus1jminus1

Bijminus1

Bi+1jminus1

V4i+1jminus1

Biminus1j

V1ij

Bij

Tij

V4ij V3

ij

Bi+1j

T9984001

T0

Biminus1j+1

V2ij

Bij+1

Bi+1j+1

V2iminus1j+1

V3i+1j+1

Figure 7 Template reconfiguration

Step 4 Computenormalized cross-correlation coefficientsbetween 119879

2and the sensed image and then determine the

subpixel offset of 1198792relative to the sensed image in frameP

Firstly for this method the search space on the sensedimage can be contracted so much since the optical flowtrajectories for the referenced elements have been predictedin Step 2 Assuming that the search space is 119879

119904 dim(119879

119904) =

119872 times 119872 When 119879119894119895

moves to the pixel (1198991 119899

2) on 119879

119904 the

normalized cross-correlation (NCC) coefficient is given by

120574 (1198991 119899

2)

=sum119909119910

[119892 (119909 119910) minus 119892119909119910] [ℎ (119909 minus 119899

1 119910 minus 119899

2) minus ℎ]

sum119909119910

[119892 (119909 119910) minus 119892119909119910]2

sum119909119910

[ℎ (119909 minus 1198991 119910 minus 119899

2) minus ℎ]

2

05

(31)

where 119892119909119910

is the mean gray value of the segment of 119879119904

that is masked by 1198792and ℎ is the mean of 119879

2 Equation

(31) requires approximately 1198732(119872 minus 119873 + 1)2 additions and

1198732(119872 minus 119873 + 1)2 multiplications whereas the complexity of

FFT algorithm needs about 121198722log2119872 real multiplications

and 181198722log2119872 real additionssubtractions [32 33]

At the beginning we take119872 = 101119873 = 7 and computethe NCC coefficient When 119872 is much larger than 119873 thecalculation in spatial domain will be efficient Suppose thatthe peak value 120574max is taken at the coordinate (119896119898) 119896119898 isin Z

in the sensed window Hence we will reduce search space intoa smaller one with dimension of 47 times 47 which centered on119879119904(119896119898) Next the subpixel registration is realized by phase

correlation algorithm with larger 119872 and 119873 to suppress thesystem errors owing to the deficiencies of detailed textures

on the photo Here we take119872 = 47119873 = 23 Let the subpixeloffset between the two registering image elements be denotedas 120575

119909and 120575

119910in frameP

The phase correlation algorithm in the frequency domainbecomes more efficient as 119873 approaches 119872 and both havelarger scales [28] Moreover the Fourier coefficients are nor-malized to unitmagnitude prior to computing the correlationso that the correlation is based only on phase information andbeing insensitive to changes in image intensity [27 29]

LetG(119906 V) be the 2D Discrete Fourier Transforms (DFT)of the sensed window then we have

G (119906 V) =(119873minus1)2

sum119909=minus(119873minus1)2

(119873minus1)2


119892 (119909 119910)119882119906119909

119872119882

V119910119872

H (119906 V) =(119873minus1)2


(119873minus1)2


ℎ (119909 119910)119882119906119909

119873119882

V119910119873

(32)

Here

119882119873= exp(minus1198952120587

119873) (33)

Cross-phase spectrum is given by

R (119906 V) =G (119906 V)Hlowast

(119906 V)|G (119906 V)Hlowast (119906 V)|

= exp (119895120601 (119906 V)) (34)

whereHlowast is the complex conjugate ofH By inverse DiscreteFourier Transform (IDFT) we have

120574 (1198991 119899

2) =

1

1198732

(119873minus1)2


(119873minus1)2

sumV=minus(119873minus1)2

R (119906 V)119882minus1199061198991

119873119882

minusV1198992

119873

(35)


Figure 8 Dense image registration for lapped image strips CCD1versus CCD2 (Gap 1 the left two) and CCD3 versus CCD4 (Gap 3the right two)

Suppose that the new peak 120574max appears at (1198961015840 1198981015840) 1198961015840 1198981015840 isin

Z referring to [27] we have the following relation

120574max (1198961015840

1198981015840

)

asymp120582

1198732

sin [120587 (1198961015840 + 120575119909)] sin [120587 (1198981015840 + 120575

119910)]

sin [(120587119873) (1198961015840 + 120575119909)] sin [(120587119873) (1198981015840 + 120575

119910)]

(36)

The right side presents the spatial distribution of the normal-ized cross-correlation coefficientsTherefore (120575

119909 120575

119910) are able

to be measured based on that In practice constant 120582 le 1which tends to decrease when small noise exists and equalsunity in ideal cases

Step 5 Dense registration is executed for the lapped imagestrips

Repeating Step 1simStep 4 we register the along-track sam-ple images selected from the referenced images to the sensedimageThemaximal sample rate can reach up to line-by-lineThe continuous procedure is shown in Figure 8 in which theimage pairs are marked

The curves of relative offsets inP are shown in Figures 9and 10

Let col119903 row

119903be the column and row indexes of image

elements on the referenced image and let col119904 row

119904be the

indexes of the same elements on the sensed image The totalcolumns of each detector 119876 = 4096 pix and the verticaldistance between the two detector arrays 119863 = 184975mmAccording to the results of registration we get the offsets

50 100 150 200 250 300 350 400 450 500

minus28minus26minus24 X 258

Y minus2515

Image rows (pixels)

Cros

s tra

ck(p

ixel

s)

CCD1 versus CCD2

50 100 150 200 250 300 350 400 450 500


Y minus5393

Image rows (pixels)

Alo

ng tr

ack

(pix

els)

X 423Y minus7363

S11S22

S22

S11

X 423Y minus2378

Figure 9Theoffsets of lapped images captured byCCD1 andCCD2

50 100 150 200 250 300 350 400 450 500minus17minus16minus15minus14minus13minus12

X 266Y minus1285 X 436

Y minus1297

Image rows (pixels)Cr

oss t

rack

(p

ixel

s)

CCD3 versus CCD4

50 100 150 200 250 300 350 400 450 500minus9minus8minus7minus6minus5

X 436Y minus6869

Image rows (pixels)

Alo

ng tr

ack

(pix

els)

X 266Y minus7663

S31

S31

S32

S32

Figure 10 The offsets of lapped images captured by CCD3 andCCD4

of images at 119899th gap 120575119899119909(cross track) 120575119899

119910(along track) in

frameP and Δ1199091015840119899 Δ1199101015840

119899(mm) in frameF

120575119899119909= col

119903+ col

119904minus 119876 minus 120578

119899

Δ1199091015840

119899= Δ(119909

1015840

2)119899

= 120575119899119909sdot 119908

120575119899119910= row

119904minus row

119903minus119863

119908

Δ1199101015840

119899= Δ(119909

1015840

1)119899

= 120575119899119910sdot 119908 + 119863

(37)

Four pixels S11 S12 S31 and S32 are examinedTheir data arelisted in Table 1

S11 and S31 are the images of the same object which wascaptured in order by CCD1 and CCD2 (Gap 1) S12 and S32were captured respectively by CCD3 and CCD4 (Gap 3)Referring to the auxiliary data S11 and S31 were capturedat same time and S12 and S32 were captured at differenttime which means that the along-track speeds of the twomoving images were quite different Moreover the cross-track image offsets in Gap 1 and Gap 3 vary so much whichsays that the optical flows were also distorted unevenly anddeflects away from the along-track directionOn the other


Table 1 The offsets between overlapped images

Sample Row no(pixel)

120575119899119909

(pixel)Δ119909

1015840

119899

(mm)120575119899

119910

(pixel)Δ119910

1015840

119899

(mm)

S11 258 minus2515 minus02200625 minus539 184503




hand it is has been discovered in Figures 9 and 10 that thefluctuation of image offsets taking place in Gap 1 is greaterin magnitude than in Gap 3 All the facts indicate that thedistorted optical flows can be detected from a plenty of imageoffsets We will see later that the nonlinear distribution of thedata strengthens the well-posedness of optical flow inversionalgorithm

4 Remote Sensor AttitudeMotion Measurement

In this section the attitude velocity of the remote sensor isgoing to be resolved by using optical flow inversion methodThe results of dense registration are applied to produceconditions of fixed solution for optical flow equations

41 The Principle of Optical Inversion For clarity in frameC the two coordinate components of image displacementof 119896th sample element belonging to 119899th lapped strip pair arewritten as Δ1199091015840

119899119896 Δ1199101015840

119899119896 From (13) and (25) it is easy to show

that the contributions to optical flow owing to orbital motionand earthrsquos inertial movement are of very slightly varying inshort term such that the corresponding displacements can beregarded as piecewise constants 119904

119909 119904119910

Let 120591119894119895 119905119894119895

be in order the two sequential imaging timeof the 119895th image sample on the overlapped detectors in 119895thgap They are usually recorded in the auxiliary data of theremote sensor Hence for every image element the quantityof discrete status in optical flow tracing will be

119873119894119895= [

119905119894119895minus 120591

119894119895

Δ119905] isin Z

+

(119894 = 1 sdot sdot sdot 119899 119895 = 1 sdot sdot sdot 119898) (38)

where 119899 is the amount of CCD gaps 119898 is the amount ofsample groups and Δ119905 is the time step We set samples withsame 119895 index into the same group in which the samples arecaptured by the prior detectors simultaneously

We expand (11) substitute it into (14) and (13) and thenarrange the scalar optical flow inversion equations in termsof the three axial angular velocity components 120596

1 120596

2 and 120596

3

(the variables in the inverse problem) yielding the linearoptical flow equations

Locus of optical flow

CCD

CCD

120575max

D

ci120583120581 = const

Figure 11 Coefficients Determination according to the CurrentLocation of the Image

For the 119897th group samples

1198731119897

sum119894=119897

119888119894

11198971120596119894

1+ 119888

119894

11198972120596119894

2+ 119888

119894

11198973120596119894

3= Δ119909

1015840

1119897minus 119904

1199091

1198731119897

sum119894=119897

119889119894

11198971120596119894

1+ 119889

119894

11198972120596119894

2+ 119889

119894

11198973120596119894

3= Δ119910

1015840

1119897minus 119904

1199101

119873119899119897

sum119894=119897

119888119894

1198991198971120596119894

1+ 119888

119894

1198991198972120596119894

2+ 119888

119894

1198991198973120596119894

3= Δ119909

1015840

119899119897minus 119904

119909119899

119873119899119897

sum119894=119897

119889119894

1198991198971120596119894

1+ 119889

119894

1198991198972120596119894

2+ 119889

119894

1198991198973120596119894

3= Δ119910

1015840

119899119897minus 119904

119910119899

(39)

Suppose that the sample process will stop until119898 groupshave been founded The coefficients are as follows

119888119894

120583]120581 = Ξ120581 (120583 lceil119894 minus ] + 1119873120583]

Nrceil)

119889119894

120583]120581 = Λ 120581(120583 lceil

119894 minus ] + 1119873120583]

Nrceil) (120581 = 1 2 3)

(40)


Here

Ξ119896= (

12058511119896

12058512119896

sdot sdot sdot 1205851N119896

12058521119896

12058522119896


sdot sdot sdot sdot sdot sdot

1205851198991119896

1205851198992119896

sdot sdot sdot 120585119899N119896

)

Λ119896= (

12058211119896

12058212119896


12058221119896

12058222119896



1205821198991119896

1205821198992119896

sdot sdot sdot 120582119899N119896

)

(41)

As for the algorithm to reduce the complexity all possiblevalues for the coefficients are stored in the matrixes Ξ

119896and

Λ119896 The accuracy is guaranteed because the coefficients for

the images moving into the same piece of region are almostequal to an identical constant in a short period which isexplained in Figure 11

It has beenmentioned that the optical flow is not sensitiveto satellitersquos orbit motion and earth rotation in a short term

namely the possible values are assigned by the followingfunctions

120585119894119895119896= 120585

119896(119886 119890 119894

0 Ω 120596 119909

1015840

119902 119910

1015840

119902 Δ119905)

120582119894119895119896= 120582

119896(119886 119890 119894

0 Ω 120596 119909

1015840

119902 119910

1015840

119902 Δ119905)

119894 = 1 sim 119899 119895 = 1 sim N 119902 = 1 sim N

(42)

HereN is the number of constant-valued segments in theregion encompassing all the possible optical flow trajectoriesThe orbital elements and integral step size Δ119905 are commonto all functions Furthermore when long termmeasurementsare executed Ξ

119896and Λ

119896only need to be renewed according

to the current parametersThe coefficientmatrix of the optical flow equations for 119895th

(1 le 119895 le 119898) group can be written as

C119895=

(((((((((((

(

1198881

11198951119888111198952

119888111198953

sdot sdot sdot 1198881198731119895

111989511198881198731119895

111989521198881198731119895

11198953sdot sdot sdot 0 0

119889111198951

119889111198952

119889111198953

sdot sdot sdot 1198891198731119895

111989511198891198731119895

111989521198891198731119895


sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot

11988811199021198951

11988811199021198952

11988811199021198953

sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot 119888119873119902119895

1199021198951119888119873119902119895

1199021198951119888119873119902119895

1199021198953

11988911199021198951

11988911199021198952

11988911199021198953


1199021198951119889119873119902119895

1199021198952119889119873119902119895

1199021198953


11988811198991198951

11988811198991198952

11988811198991198953

sdot sdot sdot sdot sdot sdot 119888119873119899119895

1198991198951119888119873119899119895

1198991198952119888119873119899119895

1198991198953sdot sdot sdot 0

1198891

11989911989511198891

11989911989521198891

1198991198953sdot sdot sdot sdot sdot sdot 119889

1

11989911989511198891

11989911989521198891


)))))))))))

)2119899times3119873119902119895

(43)

where119873119902119895= max119873

1119895 119873

119899119895 Consequently as we organize the equations for all groups

the global coefficient matrix will be given in the followingform

C =((

(

[C1]2119899times3119873

1199021

0 sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot 0

0 [C2]2119899times3119873

1199022

0 sdot sdot sdot sdot sdot sdot sdot sdot sdot 0

d sdot sdot sdot sdot sdot sdot

[C]2119899times3119873maxd

[C119898]2119899times3119873

119902119898

0

))

)2119899119898times3119873max

(44)

C is a quasidiagonal partitioned matrix every subblockhas 2119899 rows The maximal columns of C are 119873max =

max1198731199021 119873

119902119898

The unknown variables are as follows

[Ω]3119873maxtimes1

= [120596111205961212059613sdot sdot sdot 120596

119873max1

120596119873max2

120596119873max3

]119879

(45)

The constant are as followsΔu

2119898119899times1= [ Δ1199091015840

11Δ1199101015840

11sdot sdot sdot Δ1199091015840

1198991Δ1199101015840

1198991

sdot sdot sdot Δ11990910158401198981

Δ11991010158401198981

sdot sdot sdot Δ1199091015840119899119898

Δ1199101015840119899119898

]119879

s2119898119899times1

= [ 1199041199091

1199041199101

sdot sdot sdot 119904119909119899

119904119910119899

sdot sdot sdot 1199041199091

1199041199101

sdot sdot sdot 119904119909119899

119904119910119899]119879

(46)


Predicting the deformation anddisplacement of every element viaoptical flow prediction based on

auxiliary data and then reconstructing a

Reconfiguring the deformed image via image resampling process to

Using normalized cross-

on the sensed image captured by the posterior CCD

Measuring the relative offsets in

the sensed window

Computing the precise offset in

sensed window by adding the optical flow prediction

Utilizing the offsets data as the fixed solution conditions for optical

inversion equations and solving

The auxiliary data of the satellite

For validation andfurther usages

Preliminary information acquisition

Yes

No

Selecting the original template T1centered on the 120581th sampling pixelfrom referenced image captured by

the prior CCD 1

2

3

4 5

7

6

120581 = 120581 + 1

new deformed image T9984001

form a new template T2

Inverse problem solving

angular velocity minusrarr120596

photography frame between T2 and

120581 = Nmax

image frame between T1 and the

correlation algorithm to register T2

Figure 12 The flow chart of the attitude motion measurement

Δu has been measured by image dense registration scan be determined by auxiliary data of sensors The globalequations are expressed by

C2119898119899times3119873max

sdot [Ω]3119873maxtimes1

= Δu2119898119899times1

minus s2119898119899times1

(47)

As for this problem it is easy to be verified that conditions(1) 2119899119898 gt 3119873max (2) rank(C) = 3119873max easily meet

well in practical works To solve (44) well-posedness is thecritical issue for the inverse problem Strong nonlinearity andanisotropy of optical flow will greatly reduce the relevancebetween the coefficients inCmeanwhile it increases thewell-posedness of the solution The least-square solution of (47)can be obtained

[Ω] = (C119879

C)minus1

C119879

(Δu minus s) (48)

The well-posedness is able to be examined by SingularValue Decomposition (SVD) toC Consider the nonnegativedefinite matrix C119879C whose eigenvalues are given in order1205821ge 120582

2ge sdot sdot sdot ge 120582

3119873max

C = U [120590]V119879

(49)

where U2119898119899times2119898119899

and V3119873maxtimes3119873max

are unit orthogonal matri-ces and the singular values are 120590

119894= radic120582

119894 The well-posedness

of the solution is acceptable if condition number 120581(C) =

1205901120590

3119873maxle 119905119900119897

Associating the process of inverse problem solving inSection 4 with the process of preliminary information acqui-sition in Section 3 the whole algorithm for remote sensorrsquosattitude measurement is illustrated in the flow chart inFigure 12

42 Experimental Results and Validation In the experiment72940 samples on 7 image strip pairs were involved Consid-ering maintaining the values in Ξ and Λ nearly invariant weredistributed these samples into 20 subspaces and solved outthe three axial components of the angular velocity Accordingto Shannonrsquos sampling theorem the measurable frequency 119891

119888

is expected to reach up to the half of line rates of TDI CCDFor the experiment 119891

119888asymp 1749KHz The 120596

119894sim 119905 curves of

0 s sim 0148 s are shown in Figure 13In this period 120596

2max = 0001104∘s 120596

1max = 0001194∘s

The signal of 1205963(119905) is fluctuating around mean value 120596

3=

001752∘s It is not hard to infer that high frequency jitters


002 004 006 008 01 012 014minus1

01

002 004 006 008 01 012 014minus1

01

002 004 006 008 01 012 014001600170018

Imaging time (s)

1205961

(deg

s)

1205962

(deg

s)

1205963

(deg

s)

times10minus3

times10minus3

Figure 13 Solutions for the angular velocities of the remote sensor

were perturbing the remote sensor besides compared to thesignals of 120596

1(119905) and 120596

2(119905) the low frequency components

in 1205963(119905) are higher in magnitude Actually according to the

remote sensor satellite yaw angle is needed to be regulatedin real time to compensate for the image rotation on thefocal plane such that the detectors can always scan along thedirection of image motion Based on the auxiliary data theimagemotion velocity vector V of the central pixel in FOV canbe computed So the optimal yaw motion in principle will be

120595lowast

119905=V1199101015840

V1199091015840

120596lowast

3(119905) =

lowast

119905=V1199101015840V1199091015840 minus V

1199101015840 V1199091015840

V21199091015840

(50)

The mean value of 120596lowast3(119905) 120596

lowast

3= 001198∘s We attribute

Δ120596lowast3= 120596

3minus 120596

lowast

3= 000554∘s to the error of satellite attitude

controlIn order to validate the measurement the technique of

template reconfiguration was implemented again to checkthe expected phenomenon that based on the high-accurateinformation the correlations between the new templates and119879119904should be further improved In addition the distribution

of 120574 near 120574max is going to become more compact which iseasy to be understood since much more useful informationabout remote sensorrsquos motion is introduced into templatereconstructions and increases the similarities between thelapped images

Unlike the processing in image dense registration in thevalidation phase larger original templates are selected Let 119879

1

be the referenced image template which centered at the exam-ining element 119879

2the new template reconfigured by rough

prediction of optical flow 2the new template reconfigured

based on precision attitude motion measurement and 119879119904the

template on sensed image which centered at the registrationpixel For all templates 119872 = 119873 = 101 The distributions ofthe normalized cross-correlation coefficients correspondingto the referenced template centered on the sampled selectedin 1198731199001000 row belonging to 1198731199007 CCD with sensed imagebelonging to1198731199008 CCD are illustrated in Figure 14

(a) shows the situation for1198791and119879

119904(b) for119879

2and119879

119904 and

(c) for 2and119879

119904The compactness of the data is characterized

by the peak value 120574max and the location variances 1205902119909 1205902

119910

1205902

119909=sum119872

119894=1sum119872

119895=1120574119894119895sdot (119894 minus 119909max)

2

sum119872

119894=1sum119872

119895=1120574119894119895

1205902

119910=sum119872

119894=1sum119872

119895=1120574119894119895sdot (119895 minus 119910max)

2

sum119872

119894=1sum119872

119895=1120574119894119895

(51)

where 119909max and 119910max are respectively the column and rownumber of the peak-valued location

In case (a) 120574max(119886) = 0893 standard deviation 120590119909(119886)

= 5653 and 120590119910(119886) = 8192 in case (b) 120574max(119887) =

0918 120590119909(119887) = 4839 and 120590

119910(119887) = 6686 in case (c) 120574max(119888)

= 0976 however the variance sharply shrinks to 120590119909(119888) =

327 120590119910(119888) = 406 In Table 2 some other samples with 1000

rows interval are also examinedThe samples can be regardedas independent to each other

Judging from the results the performances in case (c) arebetter than those in case (b) andmuchmore better than thosein case (a) since the precise attitude motion measurementsenhance the precision of optical inversion so as to improve thesimilarities between the new templates and sensed imagesNote that although in case (b) the variance decreases slightlyas we have analyzed in Section 32 compared to case (a) theoffsets of centroids from the peaks have been corrected wellby the use of the rough optical flow predictions

43 Summary and Discussions In terms of the precedingsections we can see that comparing to ordinary NCC theprecision of image registration is greatly improved since itis attributed to the assistance of the technique of templatereconfiguration Implementing the auxiliary data from thespace-borne sensors to optical flow prediction the relativedeformations between the lapped image pairs can be com-puted in considerable accuracy Afterwards it will be usedto estimate the gray values of the corresponding parts onsensed images and help us to construct a new template forregistration As we know the space-borne sensors may givemiddle and low frequency components of imagerrsquos attitudemotion in excellent precision Thus comparing to the clas-sical direct template based registration algorithms the simi-larity between the reconfigured template and sensed imagesmay greatly increase Furthermore the minor deformationsattributed to high frequency jitters can be detected by usingsubpixel registration between the reconfigured templates andsensed images This point of view is the exact basis of highfrequency jitters measurement with optical flow inversion

5 Conclusion

In this paper optical flows and time-varying image deforma-tion in space dynamic imaging are analyzed in detail Thenonlinear and anisotropic image motion velocity and opticalflows are utilized to strengthen the well-posedness of theinverse problem of attitude precise measurement by optical


0 20 40 60 80 1000

102030405060708090

100

Spatial domain X (pix)

Spat

ial d

omai

n Y

(pix

)

(a)

0 20 40 60 80 1000

102030405060708090

100


Spat

ial d

omai

n Y

(pix

)

(b)

0 20 40 60 80 1000

102030405060708090

100


Spat

ial d

omai

n Y

(pix

)

(c)

Figure 14 Normalized cross-correlations comparison ((a) shows the distribution of 120574 by applying direct NCC algorithm (b) shows thedistribution of 120574 after template reconfiguration with optical flow prediction (c) shows the distribution of 120574 derived from posterior templatereconfiguration with high-accurate senorrsquos attitude measurement It can be noticed that the values of 120574 tend to be distributed uniformlyaround the peak value location from left to right)

Table 2 Correlation coefficients distribution for registration templates

Row number 120574max (119886 119887 119888) 120590119909sim (119886 119887 119888) 120590

119910sim (119886 119887 119888)

No 1000 0893 0918 0976 5653 4839 327 8192 6686 406No 2000 0807 0885 0929 8704 6452 213 6380 7342 571No 3000 0832 0940 0988 4991 3023 155 7704 4016 193No 4000 0919 0935 0983 5079 3995 361 5873 5155 385No 5000 0865 0922 0951 5918 4801 237 6151 2371 257No 6000 0751 0801 0907 1257 9985 789 1466 8213 206No 7000 0759 0846 0924 1163 1084 714 1271 8267 490No 8000 0884 0900 0943 8125 3546 542 8247 6770 288

flow inversion method For the purpose of determiningthe conditions of fixed solutions of optical flow equationsinformation based image registration algorithms are pro-posed We apply rough optical flow prediction to improvethe efficiency and accuracy of dense image registration Basedon the results of registration the attitude motions of remotesensors in imaging are measured by using precise opticalflow inversion method The experiment on a remote sensorshowed that the measurements are achieved in very highaccuracy as well as with broad bandwidth This method canextensively be used in remote sensing missions such as imagestrips splicing geometrical rectification and nonblind imagerestoration to promote the surveying precision and resolvingpower

Conflict of Interests

The authors declare that they have no financial nor personalrelationships with other people or organizations that caninappropriately influence their work there is no professionalor other personal interest of any nature or kind in anyproduct service andor company that could be construed asinfluencing the position presented in or the review of thispaper

Acknowledgments

This work is supported by the National High TechnologyResearch andDevelopment Program of China (863 Program)(Grant no 2012AA121503 Grant no 2013AA12260 andGrantno 2012AA120603) and the National Natural Science Foun-dation of China (Grant no 61377012)

References

[1] V Tchernykh M Beck and K Janschek ldquoAn embedded opticalflow processor for visual navigation using optical correlatortechnologyrdquo in Proceedings of the IEEERSJ International Con-ference on Intelligent Robots and Systems (IROS rsquo06) pp 67ndash72Beijing China October 2006

[2] K Janschek and V Tchernykh ldquoOptical correlator for imagemotion compensation in the focal plane of a satellite camerardquo inProceedings of the 15th IFAC Symposium on Automatic Controlin Aerospace Bologna Italy 2001

[3] W Priedhorsky and J J Bloch ldquoOptical detection of rapidlymoving objects in spacerdquo Applied Optics vol 44 no 3 pp 423ndash433 2005

[4] T Brox and J Malik ldquoLarge displacement optical flow descrip-tor matching in variational motion estimationrdquo IEEE Transac-tions on Pattern Analysis andMachine Intelligence vol 33 no 3pp 500ndash513 2011


[5] B Feng P P Bruyant P H Pretorius et al ldquoEstimation ofthe rigid-body motion from three-dimensional images using ageneralized center-of-mass points approachrdquo IEEETransactionson Nuclear Science vol 53 no 5 pp 2712ndash2718 2006

[6] J Wang P Yu C Yan J Ren and B He ldquoSpace optical remotesensor image motion velocity vector computational modelingerror budget and synthesisrdquo Chinese Optics Letters vol 3 no 7pp 414ndash417 2005

[7] A SMcEwenM E BanksN Baugh et al ldquoThehigh resolutionimaging science experiment (HiRISE) during MROrsquos primaryscience phase (PSP)rdquo Icarus vol 205 no 1 pp 2ndash37 2010

[8] F Ayoub S Leprince R Binet K W Lewis O Aharonson andJ-P Avouac ldquoInfluence of camera distortions on satellite imageregistration and change detection applicationsrdquo in Proceedingsof the IEEE International Geoscience and Remote Sensing Sympo-sium (IGARSS rsquo08) pp II1072ndashII1075 BostonMass USA 2008

[9] S Leprince S Barbot F Ayoub and J-P Avouac ldquoAutomaticand precise orthorectification coregistration and subpixel cor-relation of satellite images application to ground deformationmeasurementsrdquo IEEE Transactions on Geoscience and RemoteSensing vol 45 no 6 pp 1529ndash1558 2007

[10] S Leprince PMuse and J-P Avouac ldquoIn-flight CCDdistortioncalibration for pushbroom satellites based on subpixel correla-tionrdquo IEEE Transactions on Geoscience and Remote Sensing vol46 no 9 pp 2675ndash2683 2008

[11] Y Yitzhaky RMilberg S Yohaev andN S Kopeika ldquoCompar-ison of direct blind deconvolution methods for motion-blurredimagesrdquo Applied Optics vol 38 no 20 pp 4325ndash4332 1999

[12] R C Hardie K J Barnard and R Ordonez ldquoFast super-resolutionwith affinemotion using an adaptivewiener filter andits application to airborne imagingrdquo Optics Express vol 19 no27 pp 26208ndash26231 2011

[13] E M Blixt J Semeter and N Ivchenko ldquoOptical flow analysisof the aurora borealisrdquo IEEE Geoscience and Remote SensingLetters vol 3 no 1 pp 159ndash163 2006

[14] M G Mozerov ldquoConstrained optical flow estimation as amatching problemrdquo IEEE Transactions on Image Processing vol22 no 5 pp 2044ndash2055 2013

[15] H Sakaino ldquoA semitransparency-based optical-flow methodwith a point trajectory model for particle-like videordquo IEEETransactions on Image Processing vol 21 no 2 pp 441ndash4502012

[16] D Korsch ldquoClosed form solution for three-mirror telescopescorrected for spherical aberration coma astigmatism and fieldcurvaturerdquo Applied Optics vol 11 no 12 pp 2986ndash2987 1972

[17] G Naletto V da Deppo M G Pelizzo R Ragazzoni and EMarchetti ldquoOptical design of the wide angle camera for theRosetta missionrdquo Applied Optics vol 41 no 7 pp 1446ndash14532002

[18] M Born EWolf A B Bhatia and P C Clemmow Principles ofOptics Electromagnetic Theory of Propagation Interference andDiffraction of Light 7th edition 1999

[19] H Schaub and J L Junkins Analytical Mechanics of SpaceSystems AIAA Education Series 2002

[20] CWang F Xing J HWang andZ You ldquoOptical flowsmethodfor lightweight agile remote sensor design and instrumenta-tionrdquo in International Symposium on Photoelectronic Detectionand Imaging vol 8908 of Proceeding of the SPIE 2013

[21] T Sun F Xing and Z You ldquoOptical system error analysis andcalibration method of high-accuracy star trackersrdquo Sensors vol13 no 4 pp 4598ndash4623 2013

[22] T Sun F Xing Z You and M Wei ldquoMotion-blurred staracquisition method of the star tracker under high dynamicconditionsrdquoOptics Express vol 21 no 17 pp 20096ndash20110 2013

[23] L Younes ldquoCombining geodesic interpolating splines and affinetransformationsrdquo IEEETransactions on Image Processing vol 15no 5 pp 1111ndash1119 2006

[24] B Zitova and J Flusser ldquoImage registration methods a surveyrdquoImage and Vision Computing vol 21 no 11 pp 977ndash1000 2003

[25] Z L Song S Li and T F George ldquoRemote sensing imageregistration approach based on a retrofitted SIFT algorithm andLissajous-curve trajectoriesrdquo Optics Express vol 18 no 2 pp513ndash522 2010

[26] V Arevalo and J Gonzalez ldquoImproving piecewise linear regis-tration of high-resolution satellite images through mesh opti-mizationrdquo IEEETransactions onGeoscience andRemote Sensingvol 46 no 11 pp 3792ndash3803 2008

[27] Z Levi and C Gotsman ldquoD-snake image registration by as-similar-as-possible template deformationrdquo IEEE Transactionson Visualization and Computer Graphics vol 19 no 2 pp 331ndash343 2013

[28] R J Althof M G J Wind and J T Dobbins III ldquoA rapid andautomatic image registration algorithmwith subpixel accuracyrdquoIEEE Transactions on Medical Imaging vol 16 no 3 pp 308ndash316 1997

[29] W Tong ldquoSubpixel image registrationwith reduced biasrdquoOpticsLetters vol 36 no 5 pp 763ndash765 2011

[30] Y Bentoutou N Taleb K Kpalma and J Ronsin ldquoAn automaticimage registration for applications in remote sensingrdquo IEEETransactions on Geoscience and Remote Sensing vol 43 no 9pp 2127ndash2137 2005

[31] L S Ming L Yan and L Jindong ldquoMapping satellite-1 trans-mission type photogrammetric and remote sensingrdquo Journal ofRemote Sensing vol 16 supplement pp 10ndash16 2012 (Chinese)

[32] J P Lewis ldquoFast template matchingrdquo Vision Interface vol 95pp 120ndash123 1995

[33] H Foroosh J B Zerubia and M Berthod ldquoExtension ofphase correlation to subpixel registrationrdquo IEEETransactions onImage Processing vol 11 no 3 pp 188ndash200 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


Mars Reconnaissance Orbiter (MRO) in High ResolutionImaging Science Experiment (HiRISE) missions of NASAIt takes pictures of Mars with resolutions of 03mpixelresolving objects Fourteen staggered parallel CCDs areoverlapped with 48 pixels at each end to fulfill the entirefield of view Although adjacent detectors overlap with equalphysical pixels yet their lapped image pixels are not equal andvarying with time because spacecraft jitters cause undulatingoptical flows within the interlaced areas [8] In additionwe found that when large FOV remote sensors performstereoscopic imaging with large pitch angles the lappedimages belonging to marginal detectors are bound to exceedor lose several hundred pixels compared to their physicaloverlaps Furthermore the unexpected quantity decreasessignificantly for the detectors mounted at the central regionof the focal plane

Although nonuniform optical flow brings many troublesin image processing such as registration resample intercon-nection and geometrical rectification it permits us to mea-sure the spacecraft attitudemotionwith very high accuracy ina broad bandwidth It is nearly impossible for conventionalspace-borne sensors to realize the target Precision attitudemotionmeasurement is very useful for remote sensing imageprocessing especially for image restoration from motionblurring as studied in [11 12] Associating the measurementand optical flow models the dynamic point spread functions(PSF) are able to be estimated to be set as the convolutionkernels in nonblinded deconvolution algorithms

The behavior of optical flow characterizes the entire two-dimensional flow field for an imagersquos motion and variationIn [13] optical flow estimation based on image sequencesof the same aurora to determine the flow field will provideaccess to the phase space the important information forunderstanding the physical mechanism of the aurora For thepurpose to improve the accuracy of optical flow estimationa two-step matching paradigm for optical flow estimation isapplied in [14] firstly the coarse distribution measuring ofmotion vectors is done with a simple frame-to-frame corre-lation technique also known as the digital symmetric phase-only filter (SPOF) and after that subpixel accuracy estimationresult is achieved by using the sequential tree-reweightedmax-product message passing (TRW-S) optimization Sim-ilarly Sakaino overcame the disadvantages in optical flowdetermination when moving objects with different shapesand sizes move against a complicated background the imageintensity between frames may violate the common situationimage brightness constancy to image brightness 5 changemodels as constraints in regular situations [15] Howeverunlike continuous image sequences if we merely obtainedseveral images of the identical moving objects captured bydifferent detectors with long intervals the former techniquesdo not work well for optical flow estimation for lacking of theinformation of imaging process of the instrument

In this paper a new optical flow inversion methodis proposed for precise attitude measurement Unlike thesituations in [13ndash15] the image sequences of video do notexist for the transmission type remote sensors instead of theimage pairs of the same earth scene which are captured bydifferent TDI CCD detectors in push-broom fashion The

time intervals between the independent image formationscorresponding to the overlapped detectors are much morethan the interval between sequential frames in video forwhich the frame rates usually exceed tens of frames persecond (fps) However we can model optical flows basedon the working mechanism of the instrument and imageprocessing techniques rather than estimating from framesequences of a specific detector The contents of this paperare organized as follows in Section 2 an analytical model ofimage motion velocity field is established which is applicableto dynamic imaging for three-dimensional planet surfaceby large FOV remote sensors The phenomenon of movingimage deformation due to optical flow is investigated inSection 3 Based on rough inversion of optical flow a novelmethod for dense image registration is developed to measurethe subpixel offsets between the lapped images which arecaptured by adjacent detectors In Section 4 an attitudemotion measuring method based on precise optical flowinversion is studied and the results of the experiment supportthe whole theory perfectly

2 Image Velocity Field Analysis

Suppose that a large FOV camera is performing push-broom imaging to the earth the scenario is illustrated inFigure 1 The planetrsquos surface cannot be regarded as a localplane but a three-dimensional ellipsoid since it may greatlyinfluence the image motion and time-varying deformationwhen complicated relative motion exists between the imagerand the earth

In order to set up the model of space imaging somecoordinate systems need to be defined as follows

(1) I(119874119890minus 119883119884119885) the inertial frame of the earth For

convenience here we choose 1198692000 frameThe origin119874119890is located at the earth center

(2) C(119900119904minus 1199091015840

11199091015840211990910158403) the frame of camera Axis 119900

11990411990910158403is the

optical axis and origin 119900119904is the center of the exit pupil

(3) O(119900119904minus 119906

111990621199063) the orbit frame Axis 119900

1199041199063passes

through the center of earth and axis 1199001199041199062is perpen-

dicular to the instant orbit plane(4) B(119900

119904minus 119909

119887119910119887119911119887) the body frame of the satellite

(5) P(119900 minus 119909119910119911) the frame of photographyThe origin 119900 isthe center of the photo Axis 119900119909 points to the columndirection and axis 119900119910 points to the row direction

(6) F(1199001015840 minus 119909101584011991010158401199111015840) the frame of focal plane Axes 11990010158401199091015840and 11990010158401199101015840 lie in the focal plane They are respectivelyparallel to 119900

11990411990910158402and 119900

11990411990910158401 Axis 11990010158401199111015840 coincides with the

optical axis(7) E(119874

119890minus 119909

119890119910119890119911119890) the frame of Terrestrial Reference

Frame (TRF) Axis 119874119890119910119890points to the North Pole

and axis 119874119890119909119890passes through the intersection of

Greenwich meridian and the equator

According to Figure 1 1205970is the ground track of the

satellite and 1205971and 120597

2are the ground traces corresponding to

two fixed boresights in FOV which are far away from 1205970if the


pminusrarr120591

Γ

q1205972

12059711205970

u1x9984001 u2

x9984002

u3

Os

p998400 rarr120591 998400

x9984003



1and 120597









1 119909

2 119909

3) be the position



1 1199091015840

2 1199091015840

3) in


1199091015840

119894= 120573119909

119894 (119894 = 1 2)

1199091015840

3= 119891

1015840

(1)


119898minus1





119890 then



0 inclination




119905=

1198720+ 119899(119905 minus 119905


119872119905minus (119864 minus 119890 sin119864) = 0 (2)


Orbit plane


Perigee

u3

osu1

u2 rarrrsY

2a

i

Oe

Ω X

Υ

120596

rarrs


In frame O

O119903119904= (

119886 (cos119864 minus 119890)119887 sin1198640

) OV

119904=(

minus119886 sin119864119887 cos1198640

)119899

1 minus 119890 cos119864

(3)


TOI

= (


0119878Ω 119878119894

0119878Ω

119862120596119878Ω + 1198781205961198621198940119862Ω minus119878120596119878Ω + 119862120596119862119894

0119862Ω minus119878119894

0119862Ω

1198781205961198781198940

1198621205961198781198940

1198621198940

)

(4)



3= minus

I119903119904| 119903

119904|

2

= (IV

119904times

I119903119904)|V

119904times 119903

119904| and

1=

2times

3then TOI =

(123)minus1

I119903119904= TOIsdot

O119903119904

IV119904= TOIsdot

OV119904 (5)


1198832 + 1198852

1198602

119890

+1198842

1198612119890

= 1

119883 minus 119883119904

1199041

=119883 minus 119884

119904

1199042

=119883 minus 119885

119904

1199043

(6)

Here 119860119890= 6378137 km and 119861

119890= 6356752 km being the



119879 Hence I119903 =

I120588 minus

I119903119904 C 119903 = M sdot A sdot

Tminus1



A = R120595sdot R

120579sdot R

120593(7)

in which

R120595= (

cos120595119905

sin1205951199050

minus sin120595119905cos120595

1199050

0 0 1

)

R120579= (

cos 1205791199050 minus sin 120579

119905

0 1 0

sin 1205791199050 cos 120579

119905

)

R120593= (

1 0 0

0 cos120593119905

sin120593119905

0 minus sin120593119905cos120593

119905

)

(8)

where 120593119905 120579

119905 and 120595



119894=

C 119903 sdot 119890119894

(119894 = 1 2 3) (9)


1015840

119894= 120573119909

119894+ 120573

119894= (minus1)

1198981198911015840

( 119903 sdot 1198903)23119909119894


1198911015840

119903 sdot 1198903

119894

(119894 = 1 2)

(10)


1 1015840



V1015840 = V (1198940 Ω 120596 119890119872

1199050 120593

119905 120579

119905 120595

119905

119905 120579

119905

119905 119909

1015840

1 119909

1015840

2) (11)



119905 120579

119905 and 120595



119905 120579

119905 and


(

1205961

1205962

1205963

) = R120595

(

0

0

119905

) + R120579

[

[

(

0120579119905

0

) + R120593(

0

0

119905

)]

]

(12)

1205961 120596

2 and 120596









1199091015840

1(119879) = int

119879

0

1015840

1(119909

1015840

1 119909

1015840

2 119905) 119889119905

1199091015840

2(119879) = int

119879

0

1015840

2(119909

1015840

1 119909

1015840

2 119905) 119889119905

(13)


1199091015840

i (0) = 1199091015840

i (119905)10038161003816100381610038161003816119905=0

1199091015840

j (119899) = 1199091015840

j (119899 minus 1) +1

2

1015840

119895[119909

1015840

1(119899 minus 1) 119909

1015840

2(119899 minus 1) 119899]

+ 1015840

119895[119909

1015840

1(119899 minus 1) 119909

1015840

2(119899 minus 1)

119899 minus 1] Δ119905

(119895 = 1 2 119899 isin Z+

)

(14)



119901= 120575119897 120591


1205751199091015840

119894= 120575120573119909

119894+ 120573120575119909

119894 (15)


120575120573 = (minus1)119898

1198911015840

( 119903 + 120575 119903) sdot 1198903

minus1198911015840

119903 sdot 1198903


1198911015840

119903 sdot 1198903

infin

sum119896=1

(minus1)119896

(120575 119903 sdot 119890

3

119903 sdot 1198903

)

119896

asymp (minus1)1198981198911015840 120591 sdot 119890

3120575119897

( 119903 sdot 1198903)2

(16)


1205751015840

119894= 120575 120573119909

119894+ 120575120573

119894+ 120573120575119909

119894+ 120573120575

119894 (17)


C(120575 119903) = 120575119897 [MATminus1

OITEIE120591] (18)


Consequently

(

C120575 119903

120575119897)

120591

= (MATminus1

OITEI +MATminus1

OITEI

+MATminus1

OITEI +MATminus1

OITEI)E120591

(19)


TEI = (

cos1198671199010 minus sin119867

119901

0 1 0

sin1198671199010 cos119867

119901

) (20)




GST+120572119901+120596




2can be

written as

[1205751199091015840

119894]1199052

120591

minus [1205751199091015840

119894]1199051

120591

= int1199052

1199051

(1205751015840

119894)

120591

119889119905 (21)

From (17) we have

(1205751015840119894)

120591

120575119897=120575120573

120575119897119894+ 120573

120575119909119894

120575119897+ 120573

120575119894

120575119897 (22)


120575120573

120575119897= (minus1)

1198981198911015840 C 120591 sdot 119890

3

( 119903 sdot 1198903)2

120575119909119894

120575119897= MATminus1

OITEI 119890119894sdotE120591

120575119894

120575119897= (

C120575 119903

120575119897)

120591

sdot 119890119894+ (

C120575 119903

120575119897)

120591

sdot 119890119894

(23)



119894= 0


F119894(119905 120591) =

(1205751015840119894)

120591

120575119897

= (minus1)1198981198911015840 C 120591 sdot 119890

3

( 119903 sdot 1198903)2119894

+ (minus1)1198981198911015840 ( 119903 sdot 119890

119894)

( 119903 sdot 1198903)2MATminus1

OITEI 119890119894sdotE120591


1198911015840

119903 sdot 1198903

(MATminus1

OITEI

+MATminus1

OITEI

+MATminus1


119894

(24)



119894(119905 120591) is the








2is


[(1199091015840

119901)119894

]1199052

1199051

minus [(1199091015840

119902)119894

]1199052

1199051

= intΓ

int1199052

1199051

F119894∘ T119889119905 119889119904 (25)


119894[119905 T(119904)]





1 120572

2 (in the left gap)

1205731 120573



Let Δ1199101015840 = Δ11990910158401 Δ1199091015840 = Δ1199091015840






y998400

x998400

Posterior CCD

12057211205722

1205731

1205732

13998400

Prior CCD

Δx998400120578







Optical axis


O998400

x9984001x9984003

x9984002











1015840

1sim 1205911015840

4in

frameC

1205911015840

1=radic2

21198901minusradic2

21198902 120591

1015840

3= minus

radic2

21198901+radic2

21198902

1205911015840

2=radic2

21198901+radic2

21198902 120591

1015840

4= minus

radic2

21198901minusradic2

21198902

(26)





1

3

2

1

4

T0

T1 T9984001

T2 Ts



on Σ 1205911sim 120591


1sim 1205911015840



C120591119894= (minus1)

119898

1199031015840 times 1205911015840119894times

C119899119901

100381610038161003816100381610038161199031015840 times 1205911015840

119894times

C119899119901

10038161003816100381610038161003816

E120591119894= Tminus1


120591119894

C119899119901= MATminus1

OITEIE119899119901

(27)



4during 119905

1sim 119905

2according to




[1205751199091015840

1]Δ119905

120591119896

= [1205751199091015840

1]1199052

120591119896

minus [1205751199091015840

1]1199051

120591119896

[1205751199091015840

2]Δ119905

120591119896

= [1205751199091015840

2]1199052

120591119896

minus [1205751199091015840

2]1199051

120591119896

(119896 = 1 sim 4)

(28)


interpolation with

[120575 1199031015840

]Δ119905

120591119896

= ([1205751199091015840

1]Δ119905

120591119896

[12057511990910158402]Δ119905

120591119896

) (29)




1computed in Step 2








1


119894119895



1 2

34

1998400

2998400

39984004998400

Σ998400t2

Σ998400t1

p998400rarr120591 998400

1rarr120591 998400

2

rarr120591 9984003

rarr120591 9984004




that is 119879119894119895= 119861


of 11987910158401as 119881119896



119894119895= 119881

2

119894119895minus1=

1198813

119894minus1119895minus1= 1198814

119894minus1119895


119894119895(119894 =



That is

ℎ1015840

119894119895=

119894+1

sum119898=119894minus1

119895+1

sum119899=119895minus1

120578119898119899119897119898119899 (30)


= 1198781198981198991199082 where 119878



with 119879119894119895


V1iminus1jminus1

Biminus1jminus1

Bijminus1

Bi+1jminus1

V4i+1jminus1

Biminus1j

V1ij

Bij

Tij

V4ij V3

ij

Bi+1j

T9984001

T0

Biminus1j+1

V2ij

Bij+1

Bi+1j+1

V2iminus1j+1

V3i+1j+1






119904 dim(119879

119904) =

119872 times 119872 When 119879119894119895


2) on 119879

119904 the


120574 (1198991 119899

2)

=sum119909119910

[119892 (119909 119910) minus 119892119909119910] [ℎ (119909 minus 119899

1 119910 minus 119899

2) minus ℎ]

sum119909119910

[119892 (119909 119910) minus 119892119909119910]2

sum119909119910

[ℎ (119909 minus 1198991 119910 minus 119899

2) minus ℎ]

2

05

(31)

where 119892119909119910



2 Equation









119909and 120575

119910in frameP



G (119906 V) =(119873minus1)2


(119873minus1)2


119892 (119909 119910)119882119906119909

119872119882

V119910119872

H (119906 V) =(119873minus1)2


(119873minus1)2


ℎ (119909 119910)119882119906119909

119873119882

V119910119873

(32)

Here

119882119873= exp(minus1198952120587

119873) (33)


R (119906 V) =G (119906 V)Hlowast

(119906 V)|G (119906 V)Hlowast (119906 V)|

= exp (119895120601 (119906 V)) (34)


120574 (1198991 119899

2) =

1

1198732

(119873minus1)2


(119873minus1)2


R (119906 V)119882minus1199061198991

119873119882

minusV1198992

119873

(35)





120574max (1198961015840

1198981015840

)

asymp120582

1198732

sin [120587 (1198961015840 + 120575119909)] sin [120587 (1198981015840 + 120575

119910)]

sin [(120587119873) (1198961015840 + 120575119909)] sin [(120587119873) (1198981015840 + 120575

119910)]

(36)


119909 120575

119910) are able





Let col119903 row



119904be the


50 100 150 200 250 300 350 400 450 500


Y minus2515

Image rows (pixels)

Cros

s tra

ck(p

ixel

s)

CCD1 versus CCD2

50 100 150 200 250 300 350 400 450 500


Y minus5393

Image rows (pixels)

Alo

ng tr

ack

(pix

els)

X 423Y minus7363

S11S22

S22

S11

X 423Y minus2378



X 266Y minus1285 X 436

Y minus1297


oss t

rack

(p

ixel

s)

CCD3 versus CCD4


X 436Y minus6869

Image rows (pixels)

Alo

ng tr

ack

(pix

els)

X 266Y minus7663

S31

S31

S32

S32




frameP and Δ1199091015840119899 Δ1199101015840


120575119899119909= col

119903+ col

119904minus 119876 minus 120578

119899

Δ1199091015840

119899= Δ(119909

1015840

2)119899

= 120575119899119909sdot 119908

120575119899119910= row

119904minus row

119903minus119863

119908

Δ1199101015840

119899= Δ(119909

1015840

1)119899

= 120575119899119910sdot 119908 + 119863

(37)






120575119899119909

(pixel)Δ119909

1015840

119899

(mm)120575119899

119910

(pixel)Δ119910

1015840

119899

(mm)









119899119896 Δ1199101015840



119909 119904119910

Let 120591119894119895 119905119894119895


119873119894119895= [

119905119894119895minus 120591

119894119895

Δ119905] isin Z

+




1 120596

2 and 120596

3



CCD

CCD

120575max

D

ci120583120581 = const



1198731119897

sum119894=119897

119888119894

11198971120596119894

1+ 119888

119894

11198972120596119894

2+ 119888

119894

11198973120596119894

3= Δ119909

1015840

1119897minus 119904

1199091

1198731119897

sum119894=119897

119889119894

11198971120596119894

1+ 119889

119894

11198972120596119894

2+ 119889

119894

11198973120596119894

3= Δ119910

1015840

1119897minus 119904

1199101

119873119899119897

sum119894=119897

119888119894

1198991198971120596119894

1+ 119888

119894

1198991198972120596119894

2+ 119888

119894

1198991198973120596119894

3= Δ119909

1015840

119899119897minus 119904

119909119899

119873119899119897

sum119894=119897

119889119894

1198991198971120596119894

1+ 119889

119894

1198991198972120596119894

2+ 119889

119894

1198991198973120596119894

3= Δ119910

1015840

119899119897minus 119904

119910119899

(39)


119888119894

120583]120581 = Ξ120581 (120583 lceil119894 minus ] + 1119873120583]

Nrceil)

119889119894

120583]120581 = Λ 120581(120583 lceil

119894 minus ] + 1119873120583]

Nrceil) (120581 = 1 2 3)

(40)


Here

Ξ119896= (

12058511119896

12058512119896


12058521119896

12058522119896



1205851198991119896

1205851198992119896

sdot sdot sdot 120585119899N119896

)

Λ119896= (

12058211119896

12058212119896


12058221119896

12058222119896



1205821198991119896

1205821198992119896

sdot sdot sdot 120582119899N119896

)

(41)


119896and





120585119894119895119896= 120585

119896(119886 119890 119894

0 Ω 120596 119909

1015840

119902 119910

1015840

119902 Δ119905)

120582119894119895119896= 120582

119896(119886 119890 119894

0 Ω 120596 119909

1015840

119902 119910

1015840

119902 Δ119905)

119894 = 1 sim 119899 119895 = 1 sim N 119902 = 1 sim N

(42)


119896and Λ




C119895=

(((((((((((

(

1198881

11198951119888111198952

119888111198953

sdot sdot sdot 1198881198731119895

111989511198881198731119895

111989521198881198731119895


119889111198951

119889111198952

119889111198953

sdot sdot sdot 1198891198731119895

111989511198891198731119895

111989521198891198731119895



11988811199021198951

11988811199021198952

11988811199021198953


1199021198951119888119873119902119895

1199021198951119888119873119902119895

1199021198953

11988911199021198951

11988911199021198952

11988911199021198953


1199021198951119889119873119902119895

1199021198952119889119873119902119895

1199021198953


11988811198991198951

11988811198991198952

11988811198991198953


1198991198951119888119873119899119895

1198991198952119888119873119899119895


1198891

11989911989511198891

11989911989521198891


1

11989911989511198891

11989911989521198891


)))))))))))

)2119899times3119873119902119895

(43)

where119873119902119895= max119873

1119895 119873



C =((

(

[C1]2119899times3119873

1199021


0 [C2]2119899times3119873

1199022



[C]2119899times3119873maxd

[C119898]2119899times3119873

119902119898

0

))

)2119899119898times3119873max

(44)


max1198731199021 119873

119902119898



= [120596111205961212059613sdot sdot sdot 120596

119873max1

120596119873max2

120596119873max3

]119879

(45)


2119898119899times1= [ Δ1199091015840

11Δ1199101015840


1198991Δ1199101015840

1198991

sdot sdot sdot Δ11990910158401198981

Δ11991010158401198981

sdot sdot sdot Δ1199091015840119899119898

Δ1199101015840119899119898

]119879

s2119898119899times1

= [ 1199041199091

1199041199101

sdot sdot sdot 119904119909119899

119904119910119899


1199041199101

sdot sdot sdot 119904119909119899

119904119910119899]119879

(46)








the sensed window








Yes

No


the prior CCD 1

2

3

4 5

7

6

120581 = 120581 + 1






120581 = Nmax





C2119898119899times3119873max


= Δu2119898119899times1


(47)



[Ω] = (C119879

C)minus1

C119879

(Δu minus s) (48)



3119873max

C = U [120590]V119879

(49)

where U2119898119899times2119898119899



119894= radic120582



1205901120590

3119873maxle 119905119900119897



119888


119888asymp 1749KHz The 120596



2max = 0001104∘s 120596

1max = 0001194∘s


3=



002 004 006 008 01 012 014minus1

01

002 004 006 008 01 012 014minus1

01

002 004 006 008 01 012 014001600170018

Imaging time (s)

1205961

(deg

s)

1205962

(deg

s)

1205963

(deg

s)

times10minus3

times10minus3



1(119905) and 120596




120595lowast

119905=V1199101015840

V1199091015840

120596lowast

3(119905) =

lowast

119905=V1199101015840V1199091015840 minus V

1199101015840 V1199091015840

V21199091015840

(50)


lowast


Δ120596lowast3= 120596

3minus 120596

lowast






1







119904(b) for119879

2and119879

119904 and

(c) for 2and119879



119910

1205902

119909=sum119872

119894=1sum119872

119895=1120574119894119895sdot (119894 minus 119909max)

2

sum119872

119894=1sum119872

119895=1120574119894119895

1205902

119910=sum119872

119894=1sum119872

119895=1120574119894119895sdot (119895 minus 119910max)

2

sum119872

119894=1sum119872

119895=1120574119894119895

(51)



= 5653 and 120590119910(119886) = 8192 in case (b) 120574max(119887) =

0918 120590119909(119887) = 4839 and 120590

119910(119887) = 6686 in case (c) 120574max(119888)






5 Conclusion



0 20 40 60 80 1000

102030405060708090

100


Spat

ial d

omai

n Y

(pix

)

(a)

0 20 40 60 80 1000

102030405060708090

100


Spat

ial d

omai

n Y

(pix

)

(b)

0 20 40 60 80 1000

102030405060708090

100


Spat

ial d

omai

n Y

(pix

)

(c)




119910sim (119886 119887 119888)





Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










pminusrarr120591

Γ

q1205972

12059711205970

u1x9984001 u2

x9984002

u3

Os

p998400 rarr120591 998400

x9984003



1and 120597









1 119909

2 119909

3) be the position



1 1199091015840

2 1199091015840

3) in


1199091015840

119894= 120573119909

119894 (119894 = 1 2)

1199091015840

3= 119891

1015840

(1)


119898minus1





119890 then



0 inclination




119905=

1198720+ 119899(119905 minus 119905


119872119905minus (119864 minus 119890 sin119864) = 0 (2)


Orbit plane


Perigee

u3

osu1

u2 rarrrsY

2a

i

Oe

Ω X

Υ

120596

rarrs


In frame O

O119903119904= (

119886 (cos119864 minus 119890)119887 sin1198640

) OV

119904=(

minus119886 sin119864119887 cos1198640

)119899

1 minus 119890 cos119864

(3)


TOI

= (


0119878Ω 119878119894

0119878Ω

119862120596119878Ω + 1198781205961198621198940119862Ω minus119878120596119878Ω + 119862120596119862119894

0119862Ω minus119878119894

0119862Ω

1198781205961198781198940

1198621205961198781198940

1198621198940

)

(4)



3= minus

I119903119904| 119903

119904|

2

= (IV

119904times

I119903119904)|V

119904times 119903

119904| and

1=

2times

3then TOI =

(123)minus1

I119903119904= TOIsdot

O119903119904

IV119904= TOIsdot

OV119904 (5)


1198832 + 1198852

1198602

119890

+1198842

1198612119890

= 1

119883 minus 119883119904

1199041

=119883 minus 119884

119904

1199042

=119883 minus 119885

119904

1199043

(6)

Here 119860119890= 6378137 km and 119861

119890= 6356752 km being the



119879 Hence I119903 =

I120588 minus

I119903119904 C 119903 = M sdot A sdot

Tminus1



A = R120595sdot R

120579sdot R

120593(7)

in which

R120595= (

cos120595119905

sin1205951199050

minus sin120595119905cos120595

1199050

0 0 1

)

R120579= (

cos 1205791199050 minus sin 120579

119905

0 1 0

sin 1205791199050 cos 120579

119905

)

R120593= (

1 0 0

0 cos120593119905

sin120593119905

0 minus sin120593119905cos120593

119905

)

(8)

where 120593119905 120579

119905 and 120595



119894=

C 119903 sdot 119890119894

(119894 = 1 2 3) (9)


1015840

119894= 120573119909

119894+ 120573

119894= (minus1)

1198981198911015840

( 119903 sdot 1198903)23119909119894


1198911015840

119903 sdot 1198903

119894

(119894 = 1 2)

(10)


1 1015840



V1015840 = V (1198940 Ω 120596 119890119872

1199050 120593

119905 120579

119905 120595

119905

119905 120579

119905

119905 119909

1015840

1 119909

1015840

2) (11)



119905 120579

119905 and 120595



119905 120579

119905 and


(

1205961

1205962

1205963

) = R120595

(

0

0

119905

) + R120579

[

[

(

0120579119905

0

) + R120593(

0

0

119905

)]

]

(12)

1205961 120596

2 and 120596









1199091015840

1(119879) = int

119879

0

1015840

1(119909

1015840

1 119909

1015840

2 119905) 119889119905

1199091015840

2(119879) = int

119879

0

1015840

2(119909

1015840

1 119909

1015840

2 119905) 119889119905

(13)


1199091015840

i (0) = 1199091015840

i (119905)10038161003816100381610038161003816119905=0

1199091015840

j (119899) = 1199091015840

j (119899 minus 1) +1

2

1015840

119895[119909

1015840

1(119899 minus 1) 119909

1015840

2(119899 minus 1) 119899]

+ 1015840

119895[119909

1015840

1(119899 minus 1) 119909

1015840

2(119899 minus 1)

119899 minus 1] Δ119905

(119895 = 1 2 119899 isin Z+

)

(14)



119901= 120575119897 120591


1205751199091015840

119894= 120575120573119909

119894+ 120573120575119909

119894 (15)


120575120573 = (minus1)119898

1198911015840

( 119903 + 120575 119903) sdot 1198903

minus1198911015840

119903 sdot 1198903


1198911015840

119903 sdot 1198903

infin

sum119896=1

(minus1)119896

(120575 119903 sdot 119890

3

119903 sdot 1198903

)

119896

asymp (minus1)1198981198911015840 120591 sdot 119890

3120575119897

( 119903 sdot 1198903)2

(16)


1205751015840

119894= 120575 120573119909

119894+ 120575120573

119894+ 120573120575119909

119894+ 120573120575

119894 (17)


C(120575 119903) = 120575119897 [MATminus1

OITEIE120591] (18)


Consequently

(

C120575 119903

120575119897)

120591

= (MATminus1

OITEI +MATminus1

OITEI

+MATminus1

OITEI +MATminus1

OITEI)E120591

(19)


TEI = (

cos1198671199010 minus sin119867

119901

0 1 0

sin1198671199010 cos119867

119901

) (20)




GST+120572119901+120596




2can be

written as

[1205751199091015840

119894]1199052

120591

minus [1205751199091015840

119894]1199051

120591

= int1199052

1199051

(1205751015840

119894)

120591

119889119905 (21)

From (17) we have

(1205751015840119894)

120591

120575119897=120575120573

120575119897119894+ 120573

120575119909119894

120575119897+ 120573

120575119894

120575119897 (22)


120575120573

120575119897= (minus1)

1198981198911015840 C 120591 sdot 119890

3

( 119903 sdot 1198903)2

120575119909119894

120575119897= MATminus1

OITEI 119890119894sdotE120591

120575119894

120575119897= (

C120575 119903

120575119897)

120591

sdot 119890119894+ (

C120575 119903

120575119897)

120591

sdot 119890119894

(23)



119894= 0


F119894(119905 120591) =

(1205751015840119894)

120591

120575119897

= (minus1)1198981198911015840 C 120591 sdot 119890

3

( 119903 sdot 1198903)2119894

+ (minus1)1198981198911015840 ( 119903 sdot 119890

119894)

( 119903 sdot 1198903)2MATminus1

OITEI 119890119894sdotE120591


1198911015840

119903 sdot 1198903

(MATminus1

OITEI

+MATminus1

OITEI

+MATminus1


119894

(24)



119894(119905 120591) is the








2is


[(1199091015840

119901)119894

]1199052

1199051

minus [(1199091015840

119902)119894

]1199052

1199051

= intΓ

int1199052

1199051

F119894∘ T119889119905 119889119904 (25)


119894[119905 T(119904)]





1 120572

2 (in the left gap)

1205731 120573



Let Δ1199101015840 = Δ11990910158401 Δ1199091015840 = Δ1199091015840






y998400

x998400

Posterior CCD

12057211205722

1205731

1205732

13998400

Prior CCD

Δx998400120578







Optical axis


O998400

x9984001x9984003

x9984002











1015840

1sim 1205911015840

4in

frameC

1205911015840

1=radic2

21198901minusradic2

21198902 120591

1015840

3= minus

radic2

21198901+radic2

21198902

1205911015840

2=radic2

21198901+radic2

21198902 120591

1015840

4= minus

radic2

21198901minusradic2

21198902

(26)





1

3

2

1

4

T0

T1 T9984001

T2 Ts



on Σ 1205911sim 120591


1sim 1205911015840



C120591119894= (minus1)

119898

1199031015840 times 1205911015840119894times

C119899119901

100381610038161003816100381610038161199031015840 times 1205911015840

119894times

C119899119901

10038161003816100381610038161003816

E120591119894= Tminus1


120591119894

C119899119901= MATminus1

OITEIE119899119901

(27)



4during 119905

1sim 119905

2according to




[1205751199091015840

1]Δ119905

120591119896

= [1205751199091015840

1]1199052

120591119896

minus [1205751199091015840

1]1199051

120591119896

[1205751199091015840

2]Δ119905

120591119896

= [1205751199091015840

2]1199052

120591119896

minus [1205751199091015840

2]1199051

120591119896

(119896 = 1 sim 4)

(28)


interpolation with

[120575 1199031015840

]Δ119905

120591119896

= ([1205751199091015840

1]Δ119905

120591119896

[12057511990910158402]Δ119905

120591119896

) (29)




1computed in Step 2








1


119894119895



1 2

34

1998400

2998400

39984004998400

Σ998400t2

Σ998400t1

p998400rarr120591 998400

1rarr120591 998400

2

rarr120591 9984003

rarr120591 9984004




that is 119879119894119895= 119861


of 11987910158401as 119881119896



119894119895= 119881

2

119894119895minus1=

1198813

119894minus1119895minus1= 1198814

119894minus1119895


119894119895(119894 =



That is

ℎ1015840

119894119895=

119894+1

sum119898=119894minus1

119895+1

sum119899=119895minus1

120578119898119899119897119898119899 (30)


= 1198781198981198991199082 where 119878



with 119879119894119895


V1iminus1jminus1

Biminus1jminus1

Bijminus1

Bi+1jminus1

V4i+1jminus1

Biminus1j

V1ij

Bij

Tij

V4ij V3

ij

Bi+1j

T9984001

T0

Biminus1j+1

V2ij

Bij+1

Bi+1j+1

V2iminus1j+1

V3i+1j+1






119904 dim(119879

119904) =

119872 times 119872 When 119879119894119895


2) on 119879

119904 the


120574 (1198991 119899

2)

=sum119909119910

[119892 (119909 119910) minus 119892119909119910] [ℎ (119909 minus 119899

1 119910 minus 119899

2) minus ℎ]

sum119909119910

[119892 (119909 119910) minus 119892119909119910]2

sum119909119910

[ℎ (119909 minus 1198991 119910 minus 119899

2) minus ℎ]

2

05

(31)

where 119892119909119910



2 Equation









119909and 120575

119910in frameP



G (119906 V) =(119873minus1)2


(119873minus1)2


119892 (119909 119910)119882119906119909

119872119882

V119910119872

H (119906 V) =(119873minus1)2


(119873minus1)2


ℎ (119909 119910)119882119906119909

119873119882

V119910119873

(32)

Here

119882119873= exp(minus1198952120587

119873) (33)


R (119906 V) =G (119906 V)Hlowast

(119906 V)|G (119906 V)Hlowast (119906 V)|

= exp (119895120601 (119906 V)) (34)


120574 (1198991 119899

2) =

1

1198732

(119873minus1)2


(119873minus1)2


R (119906 V)119882minus1199061198991

119873119882

minusV1198992

119873

(35)





120574max (1198961015840

1198981015840

)

asymp120582

1198732

sin [120587 (1198961015840 + 120575119909)] sin [120587 (1198981015840 + 120575

119910)]

sin [(120587119873) (1198961015840 + 120575119909)] sin [(120587119873) (1198981015840 + 120575

119910)]

(36)


119909 120575

119910) are able





Let col119903 row



119904be the


50 100 150 200 250 300 350 400 450 500


Y minus2515

Image rows (pixels)

Cros

s tra

ck(p

ixel

s)

CCD1 versus CCD2

50 100 150 200 250 300 350 400 450 500


Y minus5393

Image rows (pixels)

Alo

ng tr

ack

(pix

els)

X 423Y minus7363

S11S22

S22

S11

X 423Y minus2378



X 266Y minus1285 X 436

Y minus1297


oss t

rack

(p

ixel

s)

CCD3 versus CCD4


X 436Y minus6869

Image rows (pixels)

Alo

ng tr

ack

(pix

els)

X 266Y minus7663

S31

S31

S32

S32




frameP and Δ1199091015840119899 Δ1199101015840


120575119899119909= col

119903+ col

119904minus 119876 minus 120578

119899

Δ1199091015840

119899= Δ(119909

1015840

2)119899

= 120575119899119909sdot 119908

120575119899119910= row

119904minus row

119903minus119863

119908

Δ1199101015840

119899= Δ(119909

1015840

1)119899

= 120575119899119910sdot 119908 + 119863

(37)






120575119899119909

(pixel)Δ119909

1015840

119899

(mm)120575119899

119910

(pixel)Δ119910

1015840

119899

(mm)









119899119896 Δ1199101015840



119909 119904119910

Let 120591119894119895 119905119894119895


119873119894119895= [

119905119894119895minus 120591

119894119895

Δ119905] isin Z

+




1 120596

2 and 120596

3



CCD

CCD

120575max

D

ci120583120581 = const



1198731119897

sum119894=119897

119888119894

11198971120596119894

1+ 119888

119894

11198972120596119894

2+ 119888

119894

11198973120596119894

3= Δ119909

1015840

1119897minus 119904

1199091

1198731119897

sum119894=119897

119889119894

11198971120596119894

1+ 119889

119894

11198972120596119894

2+ 119889

119894

11198973120596119894

3= Δ119910

1015840

1119897minus 119904

1199101

119873119899119897

sum119894=119897

119888119894

1198991198971120596119894

1+ 119888

119894

1198991198972120596119894

2+ 119888

119894

1198991198973120596119894

3= Δ119909

1015840

119899119897minus 119904

119909119899

119873119899119897

sum119894=119897

119889119894

1198991198971120596119894

1+ 119889

119894

1198991198972120596119894

2+ 119889

119894

1198991198973120596119894

3= Δ119910

1015840

119899119897minus 119904

119910119899

(39)


119888119894

120583]120581 = Ξ120581 (120583 lceil119894 minus ] + 1119873120583]

Nrceil)

119889119894

120583]120581 = Λ 120581(120583 lceil

119894 minus ] + 1119873120583]

Nrceil) (120581 = 1 2 3)

(40)


Here

Ξ119896= (

12058511119896

12058512119896


12058521119896

12058522119896



1205851198991119896

1205851198992119896

sdot sdot sdot 120585119899N119896

)

Λ119896= (

12058211119896

12058212119896


12058221119896

12058222119896



1205821198991119896

1205821198992119896

sdot sdot sdot 120582119899N119896

)

(41)


119896and





120585119894119895119896= 120585

119896(119886 119890 119894

0 Ω 120596 119909

1015840

119902 119910

1015840

119902 Δ119905)

120582119894119895119896= 120582

119896(119886 119890 119894

0 Ω 120596 119909

1015840

119902 119910

1015840

119902 Δ119905)

119894 = 1 sim 119899 119895 = 1 sim N 119902 = 1 sim N

(42)


119896and Λ




C119895=

(((((((((((

(

1198881

11198951119888111198952

119888111198953

sdot sdot sdot 1198881198731119895

111989511198881198731119895

111989521198881198731119895


119889111198951

119889111198952

119889111198953

sdot sdot sdot 1198891198731119895

111989511198891198731119895

111989521198891198731119895



11988811199021198951

11988811199021198952

11988811199021198953


1199021198951119888119873119902119895

1199021198951119888119873119902119895

1199021198953

11988911199021198951

11988911199021198952

11988911199021198953


1199021198951119889119873119902119895

1199021198952119889119873119902119895

1199021198953


11988811198991198951

11988811198991198952

11988811198991198953


1198991198951119888119873119899119895

1198991198952119888119873119899119895


1198891

11989911989511198891

11989911989521198891


1

11989911989511198891

11989911989521198891


)))))))))))

)2119899times3119873119902119895

(43)

where119873119902119895= max119873

1119895 119873



C =((

(

[C1]2119899times3119873

1199021


0 [C2]2119899times3119873

1199022



[C]2119899times3119873maxd

[C119898]2119899times3119873

119902119898

0

))

)2119899119898times3119873max

(44)


max1198731199021 119873

119902119898



= [120596111205961212059613sdot sdot sdot 120596

119873max1

120596119873max2

120596119873max3

]119879

(45)


2119898119899times1= [ Δ1199091015840

11Δ1199101015840


1198991Δ1199101015840

1198991

sdot sdot sdot Δ11990910158401198981

Δ11991010158401198981

sdot sdot sdot Δ1199091015840119899119898

Δ1199101015840119899119898

]119879

s2119898119899times1

= [ 1199041199091

1199041199101

sdot sdot sdot 119904119909119899

119904119910119899


1199041199101

sdot sdot sdot 119904119909119899

119904119910119899]119879

(46)








the sensed window








Yes

No


the prior CCD 1

2

3

4 5

7

6

120581 = 120581 + 1






120581 = Nmax





C2119898119899times3119873max


= Δu2119898119899times1


(47)



[Ω] = (C119879

C)minus1

C119879

(Δu minus s) (48)



3119873max

C = U [120590]V119879

(49)

where U2119898119899times2119898119899



119894= radic120582



1205901120590

3119873maxle 119905119900119897



119888


119888asymp 1749KHz The 120596



2max = 0001104∘s 120596

1max = 0001194∘s


3=



002 004 006 008 01 012 014minus1

01

002 004 006 008 01 012 014minus1

01

002 004 006 008 01 012 014001600170018

Imaging time (s)

1205961

(deg

s)

1205962

(deg

s)

1205963

(deg

s)

times10minus3

times10minus3



1(119905) and 120596




120595lowast

119905=V1199101015840

V1199091015840

120596lowast

3(119905) =

lowast

119905=V1199101015840V1199091015840 minus V

1199101015840 V1199091015840

V21199091015840

(50)


lowast


Δ120596lowast3= 120596

3minus 120596

lowast






1







119904(b) for119879

2and119879

119904 and

(c) for 2and119879



119910

1205902

119909=sum119872

119894=1sum119872

119895=1120574119894119895sdot (119894 minus 119909max)

2

sum119872

119894=1sum119872

119895=1120574119894119895

1205902

119910=sum119872

119894=1sum119872

119895=1120574119894119895sdot (119895 minus 119910max)

2

sum119872

119894=1sum119872

119895=1120574119894119895

(51)



= 5653 and 120590119910(119886) = 8192 in case (b) 120574max(119887) =

0918 120590119909(119887) = 4839 and 120590

119910(119887) = 6686 in case (c) 120574max(119888)






5 Conclusion



0 20 40 60 80 1000

102030405060708090

100


Spat

ial d

omai

n Y

(pix

)

(a)

0 20 40 60 80 1000

102030405060708090

100


Spat

ial d

omai

n Y

(pix

)

(b)

0 20 40 60 80 1000

102030405060708090

100


Spat

ial d

omai

n Y

(pix

)

(c)




119910sim (119886 119887 119888)





Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Orbit plane


Perigee

u3

osu1

u2 rarrrsY

2a

i

Oe

Ω X

Υ

120596

rarrs


In frame O

O119903119904= (

119886 (cos119864 minus 119890)119887 sin1198640

) OV

119904=(

minus119886 sin119864119887 cos1198640

)119899

1 minus 119890 cos119864

(3)


TOI

= (


0119878Ω 119878119894

0119878Ω

119862120596119878Ω + 1198781205961198621198940119862Ω minus119878120596119878Ω + 119862120596119862119894

0119862Ω minus119878119894

0119862Ω

1198781205961198781198940

1198621205961198781198940

1198621198940

)

(4)



3= minus

I119903119904| 119903

119904|

2

= (IV

119904times

I119903119904)|V

119904times 119903

119904| and

1=

2times

3then TOI =

(123)minus1

I119903119904= TOIsdot

O119903119904

IV119904= TOIsdot

OV119904 (5)


1198832 + 1198852

1198602

119890

+1198842

1198612119890

= 1

119883 minus 119883119904

1199041

=119883 minus 119884

119904

1199042

=119883 minus 119885

119904

1199043

(6)

Here 119860119890= 6378137 km and 119861

119890= 6356752 km being the



119879 Hence I119903 =

I120588 minus

I119903119904 C 119903 = M sdot A sdot

Tminus1



A = R120595sdot R

120579sdot R

120593(7)

in which

R120595= (

cos120595119905

sin1205951199050

minus sin120595119905cos120595

1199050

0 0 1

)

R120579= (

cos 1205791199050 minus sin 120579

119905

0 1 0

sin 1205791199050 cos 120579

119905

)

R120593= (

1 0 0

0 cos120593119905

sin120593119905

0 minus sin120593119905cos120593

119905

)

(8)

where 120593119905 120579

119905 and 120595



119894=

C 119903 sdot 119890119894

(119894 = 1 2 3) (9)


1015840

119894= 120573119909

119894+ 120573

119894= (minus1)

1198981198911015840

( 119903 sdot 1198903)23119909119894


1198911015840

119903 sdot 1198903

119894

(119894 = 1 2)

(10)


1 1015840



V1015840 = V (1198940 Ω 120596 119890119872

1199050 120593

119905 120579

119905 120595

119905

119905 120579

119905

119905 119909

1015840

1 119909

1015840

2) (11)



119905 120579

119905 and 120595



119905 120579

119905 and


(

1205961

1205962

1205963

) = R120595

(

0

0

119905

) + R120579

[

[

(

0120579119905

0

) + R120593(

0

0

119905

)]

]

(12)

1205961 120596

2 and 120596









1199091015840

1(119879) = int

119879

0

1015840

1(119909

1015840

1 119909

1015840

2 119905) 119889119905

1199091015840

2(119879) = int

119879

0

1015840

2(119909

1015840

1 119909

1015840

2 119905) 119889119905

(13)


1199091015840

i (0) = 1199091015840

i (119905)10038161003816100381610038161003816119905=0

1199091015840

j (119899) = 1199091015840

j (119899 minus 1) +1

2

1015840

119895[119909

1015840

1(119899 minus 1) 119909

1015840

2(119899 minus 1) 119899]

+ 1015840

119895[119909

1015840

1(119899 minus 1) 119909

1015840

2(119899 minus 1)

119899 minus 1] Δ119905

(119895 = 1 2 119899 isin Z+

)

(14)



119901= 120575119897 120591


1205751199091015840

119894= 120575120573119909

119894+ 120573120575119909

119894 (15)


120575120573 = (minus1)119898

1198911015840

( 119903 + 120575 119903) sdot 1198903

minus1198911015840

119903 sdot 1198903


1198911015840

119903 sdot 1198903

infin

sum119896=1

(minus1)119896

(120575 119903 sdot 119890

3

119903 sdot 1198903

)

119896

asymp (minus1)1198981198911015840 120591 sdot 119890

3120575119897

( 119903 sdot 1198903)2

(16)


1205751015840

119894= 120575 120573119909

119894+ 120575120573

119894+ 120573120575119909

119894+ 120573120575

119894 (17)


C(120575 119903) = 120575119897 [MATminus1

OITEIE120591] (18)


Consequently

(

C120575 119903

120575119897)

120591

= (MATminus1

OITEI +MATminus1

OITEI

+MATminus1

OITEI +MATminus1

OITEI)E120591

(19)


TEI = (

cos1198671199010 minus sin119867

119901

0 1 0

sin1198671199010 cos119867

119901

) (20)




GST+120572119901+120596




2can be

written as

[1205751199091015840

119894]1199052

120591

minus [1205751199091015840

119894]1199051

120591

= int1199052

1199051

(1205751015840

119894)

120591

119889119905 (21)

From (17) we have

(1205751015840119894)

120591

120575119897=120575120573

120575119897119894+ 120573

120575119909119894

120575119897+ 120573

120575119894

120575119897 (22)


120575120573

120575119897= (minus1)

1198981198911015840 C 120591 sdot 119890

3

( 119903 sdot 1198903)2

120575119909119894

120575119897= MATminus1

OITEI 119890119894sdotE120591

120575119894

120575119897= (

C120575 119903

120575119897)

120591

sdot 119890119894+ (

C120575 119903

120575119897)

120591

sdot 119890119894

(23)



119894= 0


F119894(119905 120591) =

(1205751015840119894)

120591

120575119897

= (minus1)1198981198911015840 C 120591 sdot 119890

3

( 119903 sdot 1198903)2119894

+ (minus1)1198981198911015840 ( 119903 sdot 119890

119894)

( 119903 sdot 1198903)2MATminus1

OITEI 119890119894sdotE120591


1198911015840

119903 sdot 1198903

(MATminus1

OITEI

+MATminus1

OITEI

+MATminus1


119894

(24)



119894(119905 120591) is the








2is


[(1199091015840

119901)119894

]1199052

1199051

minus [(1199091015840

119902)119894

]1199052

1199051

= intΓ

int1199052

1199051

F119894∘ T119889119905 119889119904 (25)


119894[119905 T(119904)]





1 120572

2 (in the left gap)

1205731 120573



Let Δ1199101015840 = Δ11990910158401 Δ1199091015840 = Δ1199091015840






y998400

x998400

Posterior CCD

12057211205722

1205731

1205732

13998400

Prior CCD

Δx998400120578







Optical axis


O998400

x9984001x9984003

x9984002











1015840

1sim 1205911015840

4in

frameC

1205911015840

1=radic2

21198901minusradic2

21198902 120591

1015840

3= minus

radic2

21198901+radic2

21198902

1205911015840

2=radic2

21198901+radic2

21198902 120591

1015840

4= minus

radic2

21198901minusradic2

21198902

(26)





1

3

2

1

4

T0

T1 T9984001

T2 Ts



on Σ 1205911sim 120591


1sim 1205911015840



C120591119894= (minus1)

119898

1199031015840 times 1205911015840119894times

C119899119901

100381610038161003816100381610038161199031015840 times 1205911015840

119894times

C119899119901

10038161003816100381610038161003816

E120591119894= Tminus1


120591119894

C119899119901= MATminus1

OITEIE119899119901

(27)



4during 119905

1sim 119905

2according to




[1205751199091015840

1]Δ119905

120591119896

= [1205751199091015840

1]1199052

120591119896

minus [1205751199091015840

1]1199051

120591119896

[1205751199091015840

2]Δ119905

120591119896

= [1205751199091015840

2]1199052

120591119896

minus [1205751199091015840

2]1199051

120591119896

(119896 = 1 sim 4)

(28)


interpolation with

[120575 1199031015840

]Δ119905

120591119896

= ([1205751199091015840

1]Δ119905

120591119896

[12057511990910158402]Δ119905

120591119896

) (29)




1computed in Step 2








1


119894119895



1 2

34

1998400

2998400

39984004998400

Σ998400t2

Σ998400t1

p998400rarr120591 998400

1rarr120591 998400

2

rarr120591 9984003

rarr120591 9984004




that is 119879119894119895= 119861


of 11987910158401as 119881119896



119894119895= 119881

2

119894119895minus1=

1198813

119894minus1119895minus1= 1198814

119894minus1119895


119894119895(119894 =



That is

ℎ1015840

119894119895=

119894+1

sum119898=119894minus1

119895+1

sum119899=119895minus1

120578119898119899119897119898119899 (30)


= 1198781198981198991199082 where 119878



with 119879119894119895


V1iminus1jminus1

Biminus1jminus1

Bijminus1

Bi+1jminus1

V4i+1jminus1

Biminus1j

V1ij

Bij

Tij

V4ij V3

ij

Bi+1j

T9984001

T0

Biminus1j+1

V2ij

Bij+1

Bi+1j+1

V2iminus1j+1

V3i+1j+1






119904 dim(119879

119904) =

119872 times 119872 When 119879119894119895


2) on 119879

119904 the


120574 (1198991 119899

2)

=sum119909119910

[119892 (119909 119910) minus 119892119909119910] [ℎ (119909 minus 119899

1 119910 minus 119899

2) minus ℎ]

sum119909119910

[119892 (119909 119910) minus 119892119909119910]2

sum119909119910

[ℎ (119909 minus 1198991 119910 minus 119899

2) minus ℎ]

2

05

(31)

where 119892119909119910



2 Equation









119909and 120575

119910in frameP



G (119906 V) =(119873minus1)2


(119873minus1)2


119892 (119909 119910)119882119906119909

119872119882

V119910119872

H (119906 V) =(119873minus1)2


(119873minus1)2


ℎ (119909 119910)119882119906119909

119873119882

V119910119873

(32)

Here

119882119873= exp(minus1198952120587

119873) (33)


R (119906 V) =G (119906 V)Hlowast

(119906 V)|G (119906 V)Hlowast (119906 V)|

= exp (119895120601 (119906 V)) (34)


120574 (1198991 119899

2) =

1

1198732

(119873minus1)2


(119873minus1)2


R (119906 V)119882minus1199061198991

119873119882

minusV1198992

119873

(35)





120574max (1198961015840

1198981015840

)

asymp120582

1198732

sin [120587 (1198961015840 + 120575119909)] sin [120587 (1198981015840 + 120575

119910)]

sin [(120587119873) (1198961015840 + 120575119909)] sin [(120587119873) (1198981015840 + 120575

119910)]

(36)


119909 120575

119910) are able





Let col119903 row



119904be the


50 100 150 200 250 300 350 400 450 500


Y minus2515

Image rows (pixels)

Cros

s tra

ck(p

ixel

s)

CCD1 versus CCD2

50 100 150 200 250 300 350 400 450 500


Y minus5393

Image rows (pixels)

Alo

ng tr

ack

(pix

els)

X 423Y minus7363

S11S22

S22

S11

X 423Y minus2378



X 266Y minus1285 X 436

Y minus1297


oss t

rack

(p

ixel

s)

CCD3 versus CCD4


X 436Y minus6869

Image rows (pixels)

Alo

ng tr

ack

(pix

els)

X 266Y minus7663

S31

S31

S32

S32




frameP and Δ1199091015840119899 Δ1199101015840


120575119899119909= col

119903+ col

119904minus 119876 minus 120578

119899

Δ1199091015840

119899= Δ(119909

1015840

2)119899

= 120575119899119909sdot 119908

120575119899119910= row

119904minus row

119903minus119863

119908

Δ1199101015840

119899= Δ(119909

1015840

1)119899

= 120575119899119910sdot 119908 + 119863

(37)






120575119899119909

(pixel)Δ119909

1015840

119899

(mm)120575119899

119910

(pixel)Δ119910

1015840

119899

(mm)









119899119896 Δ1199101015840



119909 119904119910

Let 120591119894119895 119905119894119895


119873119894119895= [

119905119894119895minus 120591

119894119895

Δ119905] isin Z

+




1 120596

2 and 120596

3



CCD

CCD

120575max

D

ci120583120581 = const



1198731119897

sum119894=119897

119888119894

11198971120596119894

1+ 119888

119894

11198972120596119894

2+ 119888

119894

11198973120596119894

3= Δ119909

1015840

1119897minus 119904

1199091

1198731119897

sum119894=119897

119889119894

11198971120596119894

1+ 119889

119894

11198972120596119894

2+ 119889

119894

11198973120596119894

3= Δ119910

1015840

1119897minus 119904

1199101

119873119899119897

sum119894=119897

119888119894

1198991198971120596119894

1+ 119888

119894

1198991198972120596119894

2+ 119888

119894

1198991198973120596119894

3= Δ119909

1015840

119899119897minus 119904

119909119899

119873119899119897

sum119894=119897

119889119894

1198991198971120596119894

1+ 119889

119894

1198991198972120596119894

2+ 119889

119894

1198991198973120596119894

3= Δ119910

1015840

119899119897minus 119904

119910119899

(39)


119888119894

120583]120581 = Ξ120581 (120583 lceil119894 minus ] + 1119873120583]

Nrceil)

119889119894

120583]120581 = Λ 120581(120583 lceil

119894 minus ] + 1119873120583]

Nrceil) (120581 = 1 2 3)

(40)


Here

Ξ119896= (

12058511119896

12058512119896


12058521119896

12058522119896



1205851198991119896

1205851198992119896

sdot sdot sdot 120585119899N119896

)

Λ119896= (

12058211119896

12058212119896


12058221119896

12058222119896



1205821198991119896

1205821198992119896

sdot sdot sdot 120582119899N119896

)

(41)


119896and





120585119894119895119896= 120585

119896(119886 119890 119894

0 Ω 120596 119909

1015840

119902 119910

1015840

119902 Δ119905)

120582119894119895119896= 120582

119896(119886 119890 119894

0 Ω 120596 119909

1015840

119902 119910

1015840

119902 Δ119905)

119894 = 1 sim 119899 119895 = 1 sim N 119902 = 1 sim N

(42)


119896and Λ




C119895=

(((((((((((

(

1198881

11198951119888111198952

119888111198953

sdot sdot sdot 1198881198731119895

111989511198881198731119895

111989521198881198731119895


119889111198951

119889111198952

119889111198953

sdot sdot sdot 1198891198731119895

111989511198891198731119895

111989521198891198731119895



11988811199021198951

11988811199021198952

11988811199021198953


1199021198951119888119873119902119895

1199021198951119888119873119902119895

1199021198953

11988911199021198951

11988911199021198952

11988911199021198953


1199021198951119889119873119902119895

1199021198952119889119873119902119895

1199021198953


11988811198991198951

11988811198991198952

11988811198991198953


1198991198951119888119873119899119895

1198991198952119888119873119899119895


1198891

11989911989511198891

11989911989521198891


1

11989911989511198891

11989911989521198891


)))))))))))

)2119899times3119873119902119895

(43)

where119873119902119895= max119873

1119895 119873



C =((

(

[C1]2119899times3119873

1199021


0 [C2]2119899times3119873

1199022



[C]2119899times3119873maxd

[C119898]2119899times3119873

119902119898

0

))

)2119899119898times3119873max

(44)


max1198731199021 119873

119902119898



= [120596111205961212059613sdot sdot sdot 120596

119873max1

120596119873max2

120596119873max3

]119879

(45)


2119898119899times1= [ Δ1199091015840

11Δ1199101015840


1198991Δ1199101015840

1198991

sdot sdot sdot Δ11990910158401198981

Δ11991010158401198981

sdot sdot sdot Δ1199091015840119899119898

Δ1199101015840119899119898

]119879

s2119898119899times1

= [ 1199041199091

1199041199101

sdot sdot sdot 119904119909119899

119904119910119899


1199041199101

sdot sdot sdot 119904119909119899

119904119910119899]119879

(46)








the sensed window








Yes

No


the prior CCD 1

2

3

4 5

7

6

120581 = 120581 + 1






120581 = Nmax





C2119898119899times3119873max


= Δu2119898119899times1


(47)



[Ω] = (C119879

C)minus1

C119879

(Δu minus s) (48)



3119873max

C = U [120590]V119879

(49)

where U2119898119899times2119898119899



119894= radic120582



1205901120590

3119873maxle 119905119900119897



119888


119888asymp 1749KHz The 120596



2max = 0001104∘s 120596

1max = 0001194∘s


3=



002 004 006 008 01 012 014minus1

01

002 004 006 008 01 012 014minus1

01

002 004 006 008 01 012 014001600170018

Imaging time (s)

1205961

(deg

s)

1205962

(deg

s)

1205963

(deg

s)

times10minus3

times10minus3



1(119905) and 120596




120595lowast

119905=V1199101015840

V1199091015840

120596lowast

3(119905) =

lowast

119905=V1199101015840V1199091015840 minus V

1199101015840 V1199091015840

V21199091015840

(50)


lowast


Δ120596lowast3= 120596

3minus 120596

lowast






1







119904(b) for119879

2and119879

119904 and

(c) for 2and119879



119910

1205902

119909=sum119872

119894=1sum119872

119895=1120574119894119895sdot (119894 minus 119909max)

2

sum119872

119894=1sum119872

119895=1120574119894119895

1205902

119910=sum119872

119894=1sum119872

119895=1120574119894119895sdot (119895 minus 119910max)

2

sum119872

119894=1sum119872

119895=1120574119894119895

(51)



= 5653 and 120590119910(119886) = 8192 in case (b) 120574max(119887) =

0918 120590119909(119887) = 4839 and 120590

119910(119887) = 6686 in case (c) 120574max(119888)






5 Conclusion



0 20 40 60 80 1000

102030405060708090

100


Spat

ial d

omai

n Y

(pix

)

(a)

0 20 40 60 80 1000

102030405060708090

100


Spat

ial d

omai

n Y

(pix

)

(b)

0 20 40 60 80 1000

102030405060708090

100


Spat

ial d

omai

n Y

(pix

)

(c)




119910sim (119886 119887 119888)





Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













1199091015840

1(119879) = int

119879

0

1015840

1(119909

1015840

1 119909

1015840

2 119905) 119889119905

1199091015840

2(119879) = int

119879

0

1015840

2(119909

1015840

1 119909

1015840

2 119905) 119889119905

(13)


1199091015840

i (0) = 1199091015840

i (119905)10038161003816100381610038161003816119905=0

1199091015840

j (119899) = 1199091015840

j (119899 minus 1) +1

2

1015840

119895[119909

1015840

1(119899 minus 1) 119909

1015840

2(119899 minus 1) 119899]

+ 1015840

119895[119909

1015840

1(119899 minus 1) 119909

1015840

2(119899 minus 1)

119899 minus 1] Δ119905

(119895 = 1 2 119899 isin Z+

)

(14)



119901= 120575119897 120591


1205751199091015840

119894= 120575120573119909

119894+ 120573120575119909

119894 (15)


120575120573 = (minus1)119898

1198911015840

( 119903 + 120575 119903) sdot 1198903

minus1198911015840

119903 sdot 1198903


1198911015840

119903 sdot 1198903

infin

sum119896=1

(minus1)119896

(120575 119903 sdot 119890

3

119903 sdot 1198903

)

119896

asymp (minus1)1198981198911015840 120591 sdot 119890

3120575119897

( 119903 sdot 1198903)2

(16)


1205751015840

119894= 120575 120573119909

119894+ 120575120573

119894+ 120573120575119909

119894+ 120573120575

119894 (17)


C(120575 119903) = 120575119897 [MATminus1

OITEIE120591] (18)


Consequently

(

C120575 119903

120575119897)

120591

= (MATminus1

OITEI +MATminus1

OITEI

+MATminus1

OITEI +MATminus1

OITEI)E120591

(19)


TEI = (

cos1198671199010 minus sin119867

119901

0 1 0

sin1198671199010 cos119867

119901

) (20)




GST+120572119901+120596




2can be

written as

[1205751199091015840

119894]1199052

120591

minus [1205751199091015840

119894]1199051

120591

= int1199052

1199051

(1205751015840

119894)

120591

119889119905 (21)

From (17) we have

(1205751015840119894)

120591

120575119897=120575120573

120575119897119894+ 120573

120575119909119894

120575119897+ 120573

120575119894

120575119897 (22)


120575120573

120575119897= (minus1)

1198981198911015840 C 120591 sdot 119890

3

( 119903 sdot 1198903)2

120575119909119894

120575119897= MATminus1

OITEI 119890119894sdotE120591

120575119894

120575119897= (

C120575 119903

120575119897)

120591

sdot 119890119894+ (

C120575 119903

120575119897)

120591

sdot 119890119894

(23)



119894= 0


F119894(119905 120591) =

(1205751015840119894)

120591

120575119897

= (minus1)1198981198911015840 C 120591 sdot 119890

3

( 119903 sdot 1198903)2119894

+ (minus1)1198981198911015840 ( 119903 sdot 119890

119894)

( 119903 sdot 1198903)2MATminus1

OITEI 119890119894sdotE120591


1198911015840

119903 sdot 1198903

(MATminus1

OITEI

+MATminus1

OITEI

+MATminus1


119894

(24)



119894(119905 120591) is the








2is


[(1199091015840

119901)119894

]1199052

1199051

minus [(1199091015840

119902)119894

]1199052

1199051

= intΓ

int1199052

1199051

F119894∘ T119889119905 119889119904 (25)


119894[119905 T(119904)]





1 120572

2 (in the left gap)

1205731 120573



Let Δ1199101015840 = Δ11990910158401 Δ1199091015840 = Δ1199091015840






y998400

x998400

Posterior CCD

12057211205722

1205731

1205732

13998400

Prior CCD

Δx998400120578







Optical axis


O998400

x9984001x9984003

x9984002











1015840

1sim 1205911015840

4in

frameC

1205911015840

1=radic2

21198901minusradic2

21198902 120591

1015840

3= minus

radic2

21198901+radic2

21198902

1205911015840

2=radic2

21198901+radic2

21198902 120591

1015840

4= minus

radic2

21198901minusradic2

21198902

(26)





1

3

2

1

4

T0

T1 T9984001

T2 Ts



on Σ 1205911sim 120591


1sim 1205911015840



C120591119894= (minus1)

119898

1199031015840 times 1205911015840119894times

C119899119901

100381610038161003816100381610038161199031015840 times 1205911015840

119894times

C119899119901

10038161003816100381610038161003816

E120591119894= Tminus1


120591119894

C119899119901= MATminus1

OITEIE119899119901

(27)



4during 119905

1sim 119905

2according to




[1205751199091015840

1]Δ119905

120591119896

= [1205751199091015840

1]1199052

120591119896

minus [1205751199091015840

1]1199051

120591119896

[1205751199091015840

2]Δ119905

120591119896

= [1205751199091015840

2]1199052

120591119896

minus [1205751199091015840

2]1199051

120591119896

(119896 = 1 sim 4)

(28)


interpolation with

[120575 1199031015840

]Δ119905

120591119896

= ([1205751199091015840

1]Δ119905

120591119896

[12057511990910158402]Δ119905

120591119896

) (29)




1computed in Step 2








1


119894119895



1 2

34

1998400

2998400

39984004998400

Σ998400t2

Σ998400t1

p998400rarr120591 998400

1rarr120591 998400

2

rarr120591 9984003

rarr120591 9984004




that is 119879119894119895= 119861


of 11987910158401as 119881119896



119894119895= 119881

2

119894119895minus1=

1198813

119894minus1119895minus1= 1198814

119894minus1119895


119894119895(119894 =



That is

ℎ1015840

119894119895=

119894+1

sum119898=119894minus1

119895+1

sum119899=119895minus1

120578119898119899119897119898119899 (30)


= 1198781198981198991199082 where 119878



with 119879119894119895


V1iminus1jminus1

Biminus1jminus1

Bijminus1

Bi+1jminus1

V4i+1jminus1

Biminus1j

V1ij

Bij

Tij

V4ij V3

ij

Bi+1j

T9984001

T0

Biminus1j+1

V2ij

Bij+1

Bi+1j+1

V2iminus1j+1

V3i+1j+1






119904 dim(119879

119904) =

119872 times 119872 When 119879119894119895


2) on 119879

119904 the


120574 (1198991 119899

2)

=sum119909119910

[119892 (119909 119910) minus 119892119909119910] [ℎ (119909 minus 119899

1 119910 minus 119899

2) minus ℎ]

sum119909119910

[119892 (119909 119910) minus 119892119909119910]2

sum119909119910

[ℎ (119909 minus 1198991 119910 minus 119899

2) minus ℎ]

2

05

(31)

where 119892119909119910



2 Equation









119909and 120575

119910in frameP



G (119906 V) =(119873minus1)2


(119873minus1)2


119892 (119909 119910)119882119906119909

119872119882

V119910119872

H (119906 V) =(119873minus1)2


(119873minus1)2


ℎ (119909 119910)119882119906119909

119873119882

V119910119873

(32)

Here

119882119873= exp(minus1198952120587

119873) (33)


R (119906 V) =G (119906 V)Hlowast

(119906 V)|G (119906 V)Hlowast (119906 V)|

= exp (119895120601 (119906 V)) (34)


120574 (1198991 119899

2) =

1

1198732

(119873minus1)2


(119873minus1)2


R (119906 V)119882minus1199061198991

119873119882

minusV1198992

119873

(35)





120574max (1198961015840

1198981015840

)

asymp120582

1198732

sin [120587 (1198961015840 + 120575119909)] sin [120587 (1198981015840 + 120575

119910)]

sin [(120587119873) (1198961015840 + 120575119909)] sin [(120587119873) (1198981015840 + 120575

119910)]

(36)


119909 120575

119910) are able





Let col119903 row



119904be the


50 100 150 200 250 300 350 400 450 500


Y minus2515

Image rows (pixels)

Cros

s tra

ck(p

ixel

s)

CCD1 versus CCD2

50 100 150 200 250 300 350 400 450 500


Y minus5393

Image rows (pixels)

Alo

ng tr

ack

(pix

els)

X 423Y minus7363

S11S22

S22

S11

X 423Y minus2378



X 266Y minus1285 X 436

Y minus1297


oss t

rack

(p

ixel

s)

CCD3 versus CCD4


X 436Y minus6869

Image rows (pixels)

Alo

ng tr

ack

(pix

els)

X 266Y minus7663

S31

S31

S32

S32




frameP and Δ1199091015840119899 Δ1199101015840


120575119899119909= col

119903+ col

119904minus 119876 minus 120578

119899

Δ1199091015840

119899= Δ(119909

1015840

2)119899

= 120575119899119909sdot 119908

120575119899119910= row

119904minus row

119903minus119863

119908

Δ1199101015840

119899= Δ(119909

1015840

1)119899

= 120575119899119910sdot 119908 + 119863

(37)






120575119899119909

(pixel)Δ119909

1015840

119899

(mm)120575119899

119910

(pixel)Δ119910

1015840

119899

(mm)









119899119896 Δ1199101015840



119909 119904119910

Let 120591119894119895 119905119894119895


119873119894119895= [

119905119894119895minus 120591

119894119895

Δ119905] isin Z

+




1 120596

2 and 120596

3



CCD

CCD

120575max

D

ci120583120581 = const



1198731119897

sum119894=119897

119888119894

11198971120596119894

1+ 119888

119894

11198972120596119894

2+ 119888

119894

11198973120596119894

3= Δ119909

1015840

1119897minus 119904

1199091

1198731119897

sum119894=119897

119889119894

11198971120596119894

1+ 119889

119894

11198972120596119894

2+ 119889

119894

11198973120596119894

3= Δ119910

1015840

1119897minus 119904

1199101

119873119899119897

sum119894=119897

119888119894

1198991198971120596119894

1+ 119888

119894

1198991198972120596119894

2+ 119888

119894

1198991198973120596119894

3= Δ119909

1015840

119899119897minus 119904

119909119899

119873119899119897

sum119894=119897

119889119894

1198991198971120596119894

1+ 119889

119894

1198991198972120596119894

2+ 119889

119894

1198991198973120596119894

3= Δ119910

1015840

119899119897minus 119904

119910119899

(39)


119888119894

120583]120581 = Ξ120581 (120583 lceil119894 minus ] + 1119873120583]

Nrceil)

119889119894

120583]120581 = Λ 120581(120583 lceil

119894 minus ] + 1119873120583]

Nrceil) (120581 = 1 2 3)

(40)


Here

Ξ119896= (

12058511119896

12058512119896


12058521119896

12058522119896



1205851198991119896

1205851198992119896

sdot sdot sdot 120585119899N119896

)

Λ119896= (

12058211119896

12058212119896


12058221119896

12058222119896



1205821198991119896

1205821198992119896

sdot sdot sdot 120582119899N119896

)

(41)


119896and





120585119894119895119896= 120585

119896(119886 119890 119894

0 Ω 120596 119909

1015840

119902 119910

1015840

119902 Δ119905)

120582119894119895119896= 120582

119896(119886 119890 119894

0 Ω 120596 119909

1015840

119902 119910

1015840

119902 Δ119905)

119894 = 1 sim 119899 119895 = 1 sim N 119902 = 1 sim N

(42)


119896and Λ




C119895=

(((((((((((

(

1198881

11198951119888111198952

119888111198953

sdot sdot sdot 1198881198731119895

111989511198881198731119895

111989521198881198731119895


119889111198951

119889111198952

119889111198953

sdot sdot sdot 1198891198731119895

111989511198891198731119895

111989521198891198731119895



11988811199021198951

11988811199021198952

11988811199021198953


1199021198951119888119873119902119895

1199021198951119888119873119902119895

1199021198953

11988911199021198951

11988911199021198952

11988911199021198953


1199021198951119889119873119902119895

1199021198952119889119873119902119895

1199021198953


11988811198991198951

11988811198991198952

11988811198991198953


1198991198951119888119873119899119895

1198991198952119888119873119899119895


1198891

11989911989511198891

11989911989521198891


1

11989911989511198891

11989911989521198891


)))))))))))

)2119899times3119873119902119895

(43)

where119873119902119895= max119873

1119895 119873



C =((

(

[C1]2119899times3119873

1199021


0 [C2]2119899times3119873

1199022



[C]2119899times3119873maxd

[C119898]2119899times3119873

119902119898

0

))

)2119899119898times3119873max

(44)


max1198731199021 119873

119902119898



= [120596111205961212059613sdot sdot sdot 120596

119873max1

120596119873max2

120596119873max3

]119879

(45)


2119898119899times1= [ Δ1199091015840

11Δ1199101015840


1198991Δ1199101015840

1198991

sdot sdot sdot Δ11990910158401198981

Δ11991010158401198981

sdot sdot sdot Δ1199091015840119899119898

Δ1199101015840119899119898

]119879

s2119898119899times1

= [ 1199041199091

1199041199101

sdot sdot sdot 119904119909119899

119904119910119899


1199041199101

sdot sdot sdot 119904119909119899

119904119910119899]119879

(46)








the sensed window








Yes

No


the prior CCD 1

2

3

4 5

7

6

120581 = 120581 + 1






120581 = Nmax





C2119898119899times3119873max


= Δu2119898119899times1


(47)



[Ω] = (C119879

C)minus1

C119879

(Δu minus s) (48)



3119873max

C = U [120590]V119879

(49)

where U2119898119899times2119898119899



119894= radic120582



1205901120590

3119873maxle 119905119900119897



119888


119888asymp 1749KHz The 120596



2max = 0001104∘s 120596

1max = 0001194∘s


3=



002 004 006 008 01 012 014minus1

01

002 004 006 008 01 012 014minus1

01

002 004 006 008 01 012 014001600170018

Imaging time (s)

1205961

(deg

s)

1205962

(deg

s)

1205963

(deg

s)

times10minus3

times10minus3



1(119905) and 120596




120595lowast

119905=V1199101015840

V1199091015840

120596lowast

3(119905) =

lowast

119905=V1199101015840V1199091015840 minus V

1199101015840 V1199091015840

V21199091015840

(50)


lowast


Δ120596lowast3= 120596

3minus 120596

lowast






1







119904(b) for119879

2and119879

119904 and

(c) for 2and119879



119910

1205902

119909=sum119872

119894=1sum119872

119895=1120574119894119895sdot (119894 minus 119909max)

2

sum119872

119894=1sum119872

119895=1120574119894119895

1205902

119910=sum119872

119894=1sum119872

119895=1120574119894119895sdot (119895 minus 119910max)

2

sum119872

119894=1sum119872

119895=1120574119894119895

(51)



= 5653 and 120590119910(119886) = 8192 in case (b) 120574max(119887) =

0918 120590119909(119887) = 4839 and 120590

119910(119887) = 6686 in case (c) 120574max(119888)






5 Conclusion



0 20 40 60 80 1000

102030405060708090

100


Spat

ial d

omai

n Y

(pix

)

(a)

0 20 40 60 80 1000

102030405060708090

100


Spat

ial d

omai

n Y

(pix

)

(b)

0 20 40 60 80 1000

102030405060708090

100


Spat

ial d

omai

n Y

(pix

)

(c)




119910sim (119886 119887 119888)





Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119894= 0


F119894(119905 120591) =

(1205751015840119894)

120591

120575119897

= (minus1)1198981198911015840 C 120591 sdot 119890

3

( 119903 sdot 1198903)2119894

+ (minus1)1198981198911015840 ( 119903 sdot 119890

119894)

( 119903 sdot 1198903)2MATminus1

OITEI 119890119894sdotE120591


1198911015840

119903 sdot 1198903

(MATminus1

OITEI

+MATminus1

OITEI

+MATminus1


119894

(24)



119894(119905 120591) is the








2is


[(1199091015840

119901)119894

]1199052

1199051

minus [(1199091015840

119902)119894

]1199052

1199051

= intΓ

int1199052

1199051

F119894∘ T119889119905 119889119904 (25)


119894[119905 T(119904)]





1 120572

2 (in the left gap)

1205731 120573



Let Δ1199101015840 = Δ11990910158401 Δ1199091015840 = Δ1199091015840






y998400

x998400

Posterior CCD

12057211205722

1205731

1205732

13998400

Prior CCD

Δx998400120578







Optical axis


O998400

x9984001x9984003

x9984002











1015840

1sim 1205911015840

4in

frameC

1205911015840

1=radic2

21198901minusradic2

21198902 120591

1015840

3= minus

radic2

21198901+radic2

21198902

1205911015840

2=radic2

21198901+radic2

21198902 120591

1015840

4= minus

radic2

21198901minusradic2

21198902

(26)





1

3

2

1

4

T0

T1 T9984001

T2 Ts



on Σ 1205911sim 120591


1sim 1205911015840



C120591119894= (minus1)

119898

1199031015840 times 1205911015840119894times

C119899119901

100381610038161003816100381610038161199031015840 times 1205911015840

119894times

C119899119901

10038161003816100381610038161003816

E120591119894= Tminus1


120591119894

C119899119901= MATminus1

OITEIE119899119901

(27)



4during 119905

1sim 119905

2according to




[1205751199091015840

1]Δ119905

120591119896

= [1205751199091015840

1]1199052

120591119896

minus [1205751199091015840

1]1199051

120591119896

[1205751199091015840

2]Δ119905

120591119896

= [1205751199091015840

2]1199052

120591119896

minus [1205751199091015840

2]1199051

120591119896

(119896 = 1 sim 4)

(28)


interpolation with

[120575 1199031015840

]Δ119905

120591119896

= ([1205751199091015840

1]Δ119905

120591119896

[12057511990910158402]Δ119905

120591119896

) (29)




1computed in Step 2








1


119894119895



1 2

34

1998400

2998400

39984004998400

Σ998400t2

Σ998400t1

p998400rarr120591 998400

1rarr120591 998400

2

rarr120591 9984003

rarr120591 9984004




that is 119879119894119895= 119861


of 11987910158401as 119881119896



119894119895= 119881

2

119894119895minus1=

1198813

119894minus1119895minus1= 1198814

119894minus1119895


119894119895(119894 =



That is

ℎ1015840

119894119895=

119894+1

sum119898=119894minus1

119895+1

sum119899=119895minus1

120578119898119899119897119898119899 (30)


= 1198781198981198991199082 where 119878



with 119879119894119895


V1iminus1jminus1

Biminus1jminus1

Bijminus1

Bi+1jminus1

V4i+1jminus1

Biminus1j

V1ij

Bij

Tij

V4ij V3

ij

Bi+1j

T9984001

T0

Biminus1j+1

V2ij

Bij+1

Bi+1j+1

V2iminus1j+1

V3i+1j+1






119904 dim(119879

119904) =

119872 times 119872 When 119879119894119895


2) on 119879

119904 the


120574 (1198991 119899

2)

=sum119909119910

[119892 (119909 119910) minus 119892119909119910] [ℎ (119909 minus 119899

1 119910 minus 119899

2) minus ℎ]

sum119909119910

[119892 (119909 119910) minus 119892119909119910]2

sum119909119910

[ℎ (119909 minus 1198991 119910 minus 119899

2) minus ℎ]

2

05

(31)

where 119892119909119910



2 Equation









119909and 120575

119910in frameP



G (119906 V) =(119873minus1)2


(119873minus1)2


119892 (119909 119910)119882119906119909

119872119882

V119910119872

H (119906 V) =(119873minus1)2


(119873minus1)2


ℎ (119909 119910)119882119906119909

119873119882

V119910119873

(32)

Here

119882119873= exp(minus1198952120587

119873) (33)


R (119906 V) =G (119906 V)Hlowast

(119906 V)|G (119906 V)Hlowast (119906 V)|

= exp (119895120601 (119906 V)) (34)


120574 (1198991 119899

2) =

1

1198732

(119873minus1)2


(119873minus1)2


R (119906 V)119882minus1199061198991

119873119882

minusV1198992

119873

(35)





120574max (1198961015840

1198981015840

)

asymp120582

1198732

sin [120587 (1198961015840 + 120575119909)] sin [120587 (1198981015840 + 120575

119910)]

sin [(120587119873) (1198961015840 + 120575119909)] sin [(120587119873) (1198981015840 + 120575

119910)]

(36)


119909 120575

119910) are able





Let col119903 row



119904be the


50 100 150 200 250 300 350 400 450 500


Y minus2515

Image rows (pixels)

Cros

s tra

ck(p

ixel

s)

CCD1 versus CCD2

50 100 150 200 250 300 350 400 450 500


Y minus5393

Image rows (pixels)

Alo

ng tr

ack

(pix

els)

X 423Y minus7363

S11S22

S22

S11

X 423Y minus2378



X 266Y minus1285 X 436

Y minus1297


oss t

rack

(p

ixel

s)

CCD3 versus CCD4


X 436Y minus6869

Image rows (pixels)

Alo

ng tr

ack

(pix

els)

X 266Y minus7663

S31

S31

S32

S32




frameP and Δ1199091015840119899 Δ1199101015840


120575119899119909= col

119903+ col

119904minus 119876 minus 120578

119899

Δ1199091015840

119899= Δ(119909

1015840

2)119899

= 120575119899119909sdot 119908

120575119899119910= row

119904minus row

119903minus119863

119908

Δ1199101015840

119899= Δ(119909

1015840

1)119899

= 120575119899119910sdot 119908 + 119863

(37)






120575119899119909

(pixel)Δ119909

1015840

119899

(mm)120575119899

119910

(pixel)Δ119910

1015840

119899

(mm)









119899119896 Δ1199101015840



119909 119904119910

Let 120591119894119895 119905119894119895


119873119894119895= [

119905119894119895minus 120591

119894119895

Δ119905] isin Z

+




1 120596

2 and 120596

3



CCD

CCD

120575max

D

ci120583120581 = const



1198731119897

sum119894=119897

119888119894

11198971120596119894

1+ 119888

119894

11198972120596119894

2+ 119888

119894

11198973120596119894

3= Δ119909

1015840

1119897minus 119904

1199091

1198731119897

sum119894=119897

119889119894

11198971120596119894

1+ 119889

119894

11198972120596119894

2+ 119889

119894

11198973120596119894

3= Δ119910

1015840

1119897minus 119904

1199101

119873119899119897

sum119894=119897

119888119894

1198991198971120596119894

1+ 119888

119894

1198991198972120596119894

2+ 119888

119894

1198991198973120596119894

3= Δ119909

1015840

119899119897minus 119904

119909119899

119873119899119897

sum119894=119897

119889119894

1198991198971120596119894

1+ 119889

119894

1198991198972120596119894

2+ 119889

119894

1198991198973120596119894

3= Δ119910

1015840

119899119897minus 119904

119910119899

(39)


119888119894

120583]120581 = Ξ120581 (120583 lceil119894 minus ] + 1119873120583]

Nrceil)

119889119894

120583]120581 = Λ 120581(120583 lceil

119894 minus ] + 1119873120583]

Nrceil) (120581 = 1 2 3)

(40)


Here

Ξ119896= (

12058511119896

12058512119896


12058521119896

12058522119896



1205851198991119896

1205851198992119896

sdot sdot sdot 120585119899N119896

)

Λ119896= (

12058211119896

12058212119896


12058221119896

12058222119896



1205821198991119896

1205821198992119896

sdot sdot sdot 120582119899N119896

)

(41)


119896and





120585119894119895119896= 120585

119896(119886 119890 119894

0 Ω 120596 119909

1015840

119902 119910

1015840

119902 Δ119905)

120582119894119895119896= 120582

119896(119886 119890 119894

0 Ω 120596 119909

1015840

119902 119910

1015840

119902 Δ119905)

119894 = 1 sim 119899 119895 = 1 sim N 119902 = 1 sim N

(42)


119896and Λ




C119895=

(((((((((((

(

1198881

11198951119888111198952

119888111198953

sdot sdot sdot 1198881198731119895

111989511198881198731119895

111989521198881198731119895


119889111198951

119889111198952

119889111198953

sdot sdot sdot 1198891198731119895

111989511198891198731119895

111989521198891198731119895



11988811199021198951

11988811199021198952

11988811199021198953


1199021198951119888119873119902119895

1199021198951119888119873119902119895

1199021198953

11988911199021198951

11988911199021198952

11988911199021198953


1199021198951119889119873119902119895

1199021198952119889119873119902119895

1199021198953


11988811198991198951

11988811198991198952

11988811198991198953


1198991198951119888119873119899119895

1198991198952119888119873119899119895


1198891

11989911989511198891

11989911989521198891


1

11989911989511198891

11989911989521198891


)))))))))))

)2119899times3119873119902119895

(43)

where119873119902119895= max119873

1119895 119873



C =((

(

[C1]2119899times3119873

1199021


0 [C2]2119899times3119873

1199022



[C]2119899times3119873maxd

[C119898]2119899times3119873

119902119898

0

))

)2119899119898times3119873max

(44)


max1198731199021 119873

119902119898



= [120596111205961212059613sdot sdot sdot 120596

119873max1

120596119873max2

120596119873max3

]119879

(45)


2119898119899times1= [ Δ1199091015840

11Δ1199101015840


1198991Δ1199101015840

1198991

sdot sdot sdot Δ11990910158401198981

Δ11991010158401198981

sdot sdot sdot Δ1199091015840119899119898

Δ1199101015840119899119898

]119879

s2119898119899times1

= [ 1199041199091

1199041199101

sdot sdot sdot 119904119909119899

119904119910119899


1199041199101

sdot sdot sdot 119904119909119899

119904119910119899]119879

(46)








the sensed window








Yes

No


the prior CCD 1

2

3

4 5

7

6

120581 = 120581 + 1






120581 = Nmax





C2119898119899times3119873max


= Δu2119898119899times1


(47)



[Ω] = (C119879

C)minus1

C119879

(Δu minus s) (48)



3119873max

C = U [120590]V119879

(49)

where U2119898119899times2119898119899



119894= radic120582



1205901120590

3119873maxle 119905119900119897



119888


119888asymp 1749KHz The 120596



2max = 0001104∘s 120596

1max = 0001194∘s


3=



002 004 006 008 01 012 014minus1

01

002 004 006 008 01 012 014minus1

01

002 004 006 008 01 012 014001600170018

Imaging time (s)

1205961

(deg

s)

1205962

(deg

s)

1205963

(deg

s)

times10minus3

times10minus3



1(119905) and 120596




120595lowast

119905=V1199101015840

V1199091015840

120596lowast

3(119905) =

lowast

119905=V1199101015840V1199091015840 minus V

1199101015840 V1199091015840

V21199091015840

(50)


lowast


Δ120596lowast3= 120596

3minus 120596

lowast






1







119904(b) for119879

2and119879

119904 and

(c) for 2and119879



119910

1205902

119909=sum119872

119894=1sum119872

119895=1120574119894119895sdot (119894 minus 119909max)

2

sum119872

119894=1sum119872

119895=1120574119894119895

1205902

119910=sum119872

119894=1sum119872

119895=1120574119894119895sdot (119895 minus 119910max)

2

sum119872

119894=1sum119872

119895=1120574119894119895

(51)



= 5653 and 120590119910(119886) = 8192 in case (b) 120574max(119887) =

0918 120590119909(119887) = 4839 and 120590

119910(119887) = 6686 in case (c) 120574max(119888)






5 Conclusion



0 20 40 60 80 1000

102030405060708090

100


Spat

ial d

omai

n Y

(pix

)

(a)

0 20 40 60 80 1000

102030405060708090

100


Spat

ial d

omai

n Y

(pix

)

(b)

0 20 40 60 80 1000

102030405060708090

100


Spat

ial d

omai

n Y

(pix

)

(c)




119910sim (119886 119887 119888)





Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










y998400

x998400

Posterior CCD

12057211205722

1205731

1205732

13998400

Prior CCD

Δx998400120578







Optical axis


O998400

x9984001x9984003

x9984002











1015840

1sim 1205911015840

4in

frameC

1205911015840

1=radic2

21198901minusradic2

21198902 120591

1015840

3= minus

radic2

21198901+radic2

21198902

1205911015840

2=radic2

21198901+radic2

21198902 120591

1015840

4= minus

radic2

21198901minusradic2

21198902

(26)





1

3

2

1

4

T0

T1 T9984001

T2 Ts



on Σ 1205911sim 120591


1sim 1205911015840



C120591119894= (minus1)

119898

1199031015840 times 1205911015840119894times

C119899119901

100381610038161003816100381610038161199031015840 times 1205911015840

119894times

C119899119901

10038161003816100381610038161003816

E120591119894= Tminus1


120591119894

C119899119901= MATminus1

OITEIE119899119901

(27)



4during 119905

1sim 119905

2according to




[1205751199091015840

1]Δ119905

120591119896

= [1205751199091015840

1]1199052

120591119896

minus [1205751199091015840

1]1199051

120591119896

[1205751199091015840

2]Δ119905

120591119896

= [1205751199091015840

2]1199052

120591119896

minus [1205751199091015840

2]1199051

120591119896

(119896 = 1 sim 4)

(28)


interpolation with

[120575 1199031015840

]Δ119905

120591119896

= ([1205751199091015840

1]Δ119905

120591119896

[12057511990910158402]Δ119905

120591119896

) (29)




1computed in Step 2








1


119894119895



1 2

34

1998400

2998400

39984004998400

Σ998400t2

Σ998400t1

p998400rarr120591 998400

1rarr120591 998400

2

rarr120591 9984003

rarr120591 9984004




that is 119879119894119895= 119861


of 11987910158401as 119881119896



119894119895= 119881

2

119894119895minus1=

1198813

119894minus1119895minus1= 1198814

119894minus1119895


119894119895(119894 =



That is

ℎ1015840

119894119895=

119894+1

sum119898=119894minus1

119895+1

sum119899=119895minus1

120578119898119899119897119898119899 (30)


= 1198781198981198991199082 where 119878



with 119879119894119895


V1iminus1jminus1

Biminus1jminus1

Bijminus1

Bi+1jminus1

V4i+1jminus1

Biminus1j

V1ij

Bij

Tij

V4ij V3

ij

Bi+1j

T9984001

T0

Biminus1j+1

V2ij

Bij+1

Bi+1j+1

V2iminus1j+1

V3i+1j+1






119904 dim(119879

119904) =

119872 times 119872 When 119879119894119895


2) on 119879

119904 the


120574 (1198991 119899

2)

=sum119909119910

[119892 (119909 119910) minus 119892119909119910] [ℎ (119909 minus 119899

1 119910 minus 119899

2) minus ℎ]

sum119909119910

[119892 (119909 119910) minus 119892119909119910]2

sum119909119910

[ℎ (119909 minus 1198991 119910 minus 119899

2) minus ℎ]

2

05

(31)

where 119892119909119910



2 Equation









119909and 120575

119910in frameP



G (119906 V) =(119873minus1)2


(119873minus1)2


119892 (119909 119910)119882119906119909

119872119882

V119910119872

H (119906 V) =(119873minus1)2


(119873minus1)2


ℎ (119909 119910)119882119906119909

119873119882

V119910119873

(32)

Here

119882119873= exp(minus1198952120587

119873) (33)


R (119906 V) =G (119906 V)Hlowast

(119906 V)|G (119906 V)Hlowast (119906 V)|

= exp (119895120601 (119906 V)) (34)


120574 (1198991 119899

2) =

1

1198732

(119873minus1)2


(119873minus1)2


R (119906 V)119882minus1199061198991

119873119882

minusV1198992

119873

(35)





120574max (1198961015840

1198981015840

)

asymp120582

1198732

sin [120587 (1198961015840 + 120575119909)] sin [120587 (1198981015840 + 120575

119910)]

sin [(120587119873) (1198961015840 + 120575119909)] sin [(120587119873) (1198981015840 + 120575

119910)]

(36)


119909 120575

119910) are able





Let col119903 row



119904be the


50 100 150 200 250 300 350 400 450 500


Y minus2515

Image rows (pixels)

Cros

s tra

ck(p

ixel

s)

CCD1 versus CCD2

50 100 150 200 250 300 350 400 450 500


Y minus5393

Image rows (pixels)

Alo

ng tr

ack

(pix

els)

X 423Y minus7363

S11S22

S22

S11

X 423Y minus2378



X 266Y minus1285 X 436

Y minus1297


oss t

rack

(p

ixel

s)

CCD3 versus CCD4


X 436Y minus6869

Image rows (pixels)

Alo

ng tr

ack

(pix

els)

X 266Y minus7663

S31

S31

S32

S32




frameP and Δ1199091015840119899 Δ1199101015840


120575119899119909= col

119903+ col

119904minus 119876 minus 120578

119899

Δ1199091015840

119899= Δ(119909

1015840

2)119899

= 120575119899119909sdot 119908

120575119899119910= row

119904minus row

119903minus119863

119908

Δ1199101015840

119899= Δ(119909

1015840

1)119899

= 120575119899119910sdot 119908 + 119863

(37)






120575119899119909

(pixel)Δ119909

1015840

119899

(mm)120575119899

119910

(pixel)Δ119910

1015840

119899

(mm)









119899119896 Δ1199101015840



119909 119904119910

Let 120591119894119895 119905119894119895


119873119894119895= [

119905119894119895minus 120591

119894119895

Δ119905] isin Z

+




1 120596

2 and 120596

3



CCD

CCD

120575max

D

ci120583120581 = const



1198731119897

sum119894=119897

119888119894

11198971120596119894

1+ 119888

119894

11198972120596119894

2+ 119888

119894

11198973120596119894

3= Δ119909

1015840

1119897minus 119904

1199091

1198731119897

sum119894=119897

119889119894

11198971120596119894

1+ 119889

119894

11198972120596119894

2+ 119889

119894

11198973120596119894

3= Δ119910

1015840

1119897minus 119904

1199101

119873119899119897

sum119894=119897

119888119894

1198991198971120596119894

1+ 119888

119894

1198991198972120596119894

2+ 119888

119894

1198991198973120596119894

3= Δ119909

1015840

119899119897minus 119904

119909119899

119873119899119897

sum119894=119897

119889119894

1198991198971120596119894

1+ 119889

119894

1198991198972120596119894

2+ 119889

119894

1198991198973120596119894

3= Δ119910

1015840

119899119897minus 119904

119910119899

(39)


119888119894

120583]120581 = Ξ120581 (120583 lceil119894 minus ] + 1119873120583]

Nrceil)

119889119894

120583]120581 = Λ 120581(120583 lceil

119894 minus ] + 1119873120583]

Nrceil) (120581 = 1 2 3)

(40)


Here

Ξ119896= (

12058511119896

12058512119896


12058521119896

12058522119896



1205851198991119896

1205851198992119896

sdot sdot sdot 120585119899N119896

)

Λ119896= (

12058211119896

12058212119896


12058221119896

12058222119896



1205821198991119896

1205821198992119896

sdot sdot sdot 120582119899N119896

)

(41)


119896and





120585119894119895119896= 120585

119896(119886 119890 119894

0 Ω 120596 119909

1015840

119902 119910

1015840

119902 Δ119905)

120582119894119895119896= 120582

119896(119886 119890 119894

0 Ω 120596 119909

1015840

119902 119910

1015840

119902 Δ119905)

119894 = 1 sim 119899 119895 = 1 sim N 119902 = 1 sim N

(42)


119896and Λ




C119895=

(((((((((((

(

1198881

11198951119888111198952

119888111198953

sdot sdot sdot 1198881198731119895

111989511198881198731119895

111989521198881198731119895


119889111198951

119889111198952

119889111198953

sdot sdot sdot 1198891198731119895

111989511198891198731119895

111989521198891198731119895



11988811199021198951

11988811199021198952

11988811199021198953


1199021198951119888119873119902119895

1199021198951119888119873119902119895

1199021198953

11988911199021198951

11988911199021198952

11988911199021198953


1199021198951119889119873119902119895

1199021198952119889119873119902119895

1199021198953


11988811198991198951

11988811198991198952

11988811198991198953


1198991198951119888119873119899119895

1198991198952119888119873119899119895


1198891

11989911989511198891

11989911989521198891


1

11989911989511198891

11989911989521198891


)))))))))))

)2119899times3119873119902119895

(43)

where119873119902119895= max119873

1119895 119873



C =((

(

[C1]2119899times3119873

1199021


0 [C2]2119899times3119873

1199022



[C]2119899times3119873maxd

[C119898]2119899times3119873

119902119898

0

))

)2119899119898times3119873max

(44)


max1198731199021 119873

119902119898



= [120596111205961212059613sdot sdot sdot 120596

119873max1

120596119873max2

120596119873max3

]119879

(45)


2119898119899times1= [ Δ1199091015840

11Δ1199101015840


1198991Δ1199101015840

1198991

sdot sdot sdot Δ11990910158401198981

Δ11991010158401198981

sdot sdot sdot Δ1199091015840119899119898

Δ1199101015840119899119898

]119879

s2119898119899times1

= [ 1199041199091

1199041199101

sdot sdot sdot 119904119909119899

119904119910119899


1199041199101

sdot sdot sdot 119904119909119899

119904119910119899]119879

(46)








the sensed window








Yes

No


the prior CCD 1

2

3

4 5

7

6

120581 = 120581 + 1






120581 = Nmax





C2119898119899times3119873max


= Δu2119898119899times1


(47)



[Ω] = (C119879

C)minus1

C119879

(Δu minus s) (48)



3119873max

C = U [120590]V119879

(49)

where U2119898119899times2119898119899



119894= radic120582



1205901120590

3119873maxle 119905119900119897



119888


119888asymp 1749KHz The 120596



2max = 0001104∘s 120596

1max = 0001194∘s


3=



002 004 006 008 01 012 014minus1

01

002 004 006 008 01 012 014minus1

01

002 004 006 008 01 012 014001600170018

Imaging time (s)

1205961

(deg

s)

1205962

(deg

s)

1205963

(deg

s)

times10minus3

times10minus3



1(119905) and 120596




120595lowast

119905=V1199101015840

V1199091015840

120596lowast

3(119905) =

lowast

119905=V1199101015840V1199091015840 minus V

1199101015840 V1199091015840

V21199091015840

(50)


lowast


Δ120596lowast3= 120596

3minus 120596

lowast






1







119904(b) for119879

2and119879

119904 and

(c) for 2and119879



119910

1205902

119909=sum119872

119894=1sum119872

119895=1120574119894119895sdot (119894 minus 119909max)

2

sum119872

119894=1sum119872

119895=1120574119894119895

1205902

119910=sum119872

119894=1sum119872

119895=1120574119894119895sdot (119895 minus 119910max)

2

sum119872

119894=1sum119872

119895=1120574119894119895

(51)



= 5653 and 120590119910(119886) = 8192 in case (b) 120574max(119887) =

0918 120590119909(119887) = 4839 and 120590

119910(119887) = 6686 in case (c) 120574max(119888)






5 Conclusion



0 20 40 60 80 1000

102030405060708090

100


Spat

ial d

omai

n Y

(pix

)

(a)

0 20 40 60 80 1000

102030405060708090

100


Spat

ial d

omai

n Y

(pix

)

(b)

0 20 40 60 80 1000

102030405060708090

100


Spat

ial d

omai

n Y

(pix

)

(c)




119910sim (119886 119887 119888)





Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1

3

2

1

4

T0

T1 T9984001

T2 Ts



on Σ 1205911sim 120591


1sim 1205911015840



C120591119894= (minus1)

119898

1199031015840 times 1205911015840119894times

C119899119901

100381610038161003816100381610038161199031015840 times 1205911015840

119894times

C119899119901

10038161003816100381610038161003816

E120591119894= Tminus1


120591119894

C119899119901= MATminus1

OITEIE119899119901

(27)



4during 119905

1sim 119905

2according to




[1205751199091015840

1]Δ119905

120591119896

= [1205751199091015840

1]1199052

120591119896

minus [1205751199091015840

1]1199051

120591119896

[1205751199091015840

2]Δ119905

120591119896

= [1205751199091015840

2]1199052

120591119896

minus [1205751199091015840

2]1199051

120591119896

(119896 = 1 sim 4)

(28)


interpolation with

[120575 1199031015840

]Δ119905

120591119896

= ([1205751199091015840

1]Δ119905

120591119896

[12057511990910158402]Δ119905

120591119896

) (29)




1computed in Step 2








1


119894119895



1 2

34

1998400

2998400

39984004998400

Σ998400t2

Σ998400t1

p998400rarr120591 998400

1rarr120591 998400

2

rarr120591 9984003

rarr120591 9984004




that is 119879119894119895= 119861


of 11987910158401as 119881119896



119894119895= 119881

2

119894119895minus1=

1198813

119894minus1119895minus1= 1198814

119894minus1119895


119894119895(119894 =



That is

ℎ1015840

119894119895=

119894+1

sum119898=119894minus1

119895+1

sum119899=119895minus1

120578119898119899119897119898119899 (30)


= 1198781198981198991199082 where 119878



with 119879119894119895


V1iminus1jminus1

Biminus1jminus1

Bijminus1

Bi+1jminus1

V4i+1jminus1

Biminus1j

V1ij

Bij

Tij

V4ij V3

ij

Bi+1j

T9984001

T0

Biminus1j+1

V2ij

Bij+1

Bi+1j+1

V2iminus1j+1

V3i+1j+1






119904 dim(119879

119904) =

119872 times 119872 When 119879119894119895


2) on 119879

119904 the


120574 (1198991 119899

2)

=sum119909119910

[119892 (119909 119910) minus 119892119909119910] [ℎ (119909 minus 119899

1 119910 minus 119899

2) minus ℎ]

sum119909119910

[119892 (119909 119910) minus 119892119909119910]2

sum119909119910

[ℎ (119909 minus 1198991 119910 minus 119899

2) minus ℎ]

2

05

(31)

where 119892119909119910



2 Equation









119909and 120575

119910in frameP



G (119906 V) =(119873minus1)2


(119873minus1)2


119892 (119909 119910)119882119906119909

119872119882

V119910119872

H (119906 V) =(119873minus1)2


(119873minus1)2


ℎ (119909 119910)119882119906119909

119873119882

V119910119873

(32)

Here

119882119873= exp(minus1198952120587

119873) (33)


R (119906 V) =G (119906 V)Hlowast

(119906 V)|G (119906 V)Hlowast (119906 V)|

= exp (119895120601 (119906 V)) (34)


120574 (1198991 119899

2) =

1

1198732

(119873minus1)2


(119873minus1)2


R (119906 V)119882minus1199061198991

119873119882

minusV1198992

119873

(35)





120574max (1198961015840

1198981015840

)

asymp120582

1198732

sin [120587 (1198961015840 + 120575119909)] sin [120587 (1198981015840 + 120575

119910)]

sin [(120587119873) (1198961015840 + 120575119909)] sin [(120587119873) (1198981015840 + 120575

119910)]

(36)


119909 120575

119910) are able





Let col119903 row



119904be the


50 100 150 200 250 300 350 400 450 500


Y minus2515

Image rows (pixels)

Cros

s tra

ck(p

ixel

s)

CCD1 versus CCD2

50 100 150 200 250 300 350 400 450 500


Y minus5393

Image rows (pixels)

Alo

ng tr

ack

(pix

els)

X 423Y minus7363

S11S22

S22

S11

X 423Y minus2378



X 266Y minus1285 X 436

Y minus1297


oss t

rack

(p

ixel

s)

CCD3 versus CCD4


X 436Y minus6869

Image rows (pixels)

Alo

ng tr

ack

(pix

els)

X 266Y minus7663

S31

S31

S32

S32




frameP and Δ1199091015840119899 Δ1199101015840


120575119899119909= col

119903+ col

119904minus 119876 minus 120578

119899

Δ1199091015840

119899= Δ(119909

1015840

2)119899

= 120575119899119909sdot 119908

120575119899119910= row

119904minus row

119903minus119863

119908

Δ1199101015840

119899= Δ(119909

1015840

1)119899

= 120575119899119910sdot 119908 + 119863

(37)






120575119899119909

(pixel)Δ119909

1015840

119899

(mm)120575119899

119910

(pixel)Δ119910

1015840

119899

(mm)









119899119896 Δ1199101015840



119909 119904119910

Let 120591119894119895 119905119894119895


119873119894119895= [

119905119894119895minus 120591

119894119895

Δ119905] isin Z

+




1 120596

2 and 120596

3



CCD

CCD

120575max

D

ci120583120581 = const



1198731119897

sum119894=119897

119888119894

11198971120596119894

1+ 119888

119894

11198972120596119894

2+ 119888

119894

11198973120596119894

3= Δ119909

1015840

1119897minus 119904

1199091

1198731119897

sum119894=119897

119889119894

11198971120596119894

1+ 119889

119894

11198972120596119894

2+ 119889

119894

11198973120596119894

3= Δ119910

1015840

1119897minus 119904

1199101

119873119899119897

sum119894=119897

119888119894

1198991198971120596119894

1+ 119888

119894

1198991198972120596119894

2+ 119888

119894

1198991198973120596119894

3= Δ119909

1015840

119899119897minus 119904

119909119899

119873119899119897

sum119894=119897

119889119894

1198991198971120596119894

1+ 119889

119894

1198991198972120596119894

2+ 119889

119894

1198991198973120596119894

3= Δ119910

1015840

119899119897minus 119904

119910119899

(39)


119888119894

120583]120581 = Ξ120581 (120583 lceil119894 minus ] + 1119873120583]

Nrceil)

119889119894

120583]120581 = Λ 120581(120583 lceil

119894 minus ] + 1119873120583]

Nrceil) (120581 = 1 2 3)

(40)


Here

Ξ119896= (

12058511119896

12058512119896


12058521119896

12058522119896



1205851198991119896

1205851198992119896

sdot sdot sdot 120585119899N119896

)

Λ119896= (

12058211119896

12058212119896


12058221119896

12058222119896



1205821198991119896

1205821198992119896

sdot sdot sdot 120582119899N119896

)

(41)


119896and





120585119894119895119896= 120585

119896(119886 119890 119894

0 Ω 120596 119909

1015840

119902 119910

1015840

119902 Δ119905)

120582119894119895119896= 120582

119896(119886 119890 119894

0 Ω 120596 119909

1015840

119902 119910

1015840

119902 Δ119905)

119894 = 1 sim 119899 119895 = 1 sim N 119902 = 1 sim N

(42)


119896and Λ




C119895=

(((((((((((

(

1198881

11198951119888111198952

119888111198953

sdot sdot sdot 1198881198731119895

111989511198881198731119895

111989521198881198731119895


119889111198951

119889111198952

119889111198953

sdot sdot sdot 1198891198731119895

111989511198891198731119895

111989521198891198731119895



11988811199021198951

11988811199021198952

11988811199021198953


1199021198951119888119873119902119895

1199021198951119888119873119902119895

1199021198953

11988911199021198951

11988911199021198952

11988911199021198953


1199021198951119889119873119902119895

1199021198952119889119873119902119895

1199021198953


11988811198991198951

11988811198991198952

11988811198991198953


1198991198951119888119873119899119895

1198991198952119888119873119899119895


1198891

11989911989511198891

11989911989521198891


1

11989911989511198891

11989911989521198891


)))))))))))

)2119899times3119873119902119895

(43)

where119873119902119895= max119873

1119895 119873



C =((

(

[C1]2119899times3119873

1199021


0 [C2]2119899times3119873

1199022



[C]2119899times3119873maxd

[C119898]2119899times3119873

119902119898

0

))

)2119899119898times3119873max

(44)


max1198731199021 119873

119902119898



= [120596111205961212059613sdot sdot sdot 120596

119873max1

120596119873max2

120596119873max3

]119879

(45)


2119898119899times1= [ Δ1199091015840

11Δ1199101015840


1198991Δ1199101015840

1198991

sdot sdot sdot Δ11990910158401198981

Δ11991010158401198981

sdot sdot sdot Δ1199091015840119899119898

Δ1199101015840119899119898

]119879

s2119898119899times1

= [ 1199041199091

1199041199101

sdot sdot sdot 119904119909119899

119904119910119899


1199041199101

sdot sdot sdot 119904119909119899

119904119910119899]119879

(46)








the sensed window








Yes

No


the prior CCD 1

2

3

4 5

7

6

120581 = 120581 + 1






120581 = Nmax





C2119898119899times3119873max


= Δu2119898119899times1


(47)



[Ω] = (C119879

C)minus1

C119879

(Δu minus s) (48)



3119873max

C = U [120590]V119879

(49)

where U2119898119899times2119898119899



119894= radic120582



1205901120590

3119873maxle 119905119900119897



119888


119888asymp 1749KHz The 120596



2max = 0001104∘s 120596

1max = 0001194∘s


3=



002 004 006 008 01 012 014minus1

01

002 004 006 008 01 012 014minus1

01

002 004 006 008 01 012 014001600170018

Imaging time (s)

1205961

(deg

s)

1205962

(deg

s)

1205963

(deg

s)

times10minus3

times10minus3



1(119905) and 120596




120595lowast

119905=V1199101015840

V1199091015840

120596lowast

3(119905) =

lowast

119905=V1199101015840V1199091015840 minus V

1199101015840 V1199091015840

V21199091015840

(50)


lowast


Δ120596lowast3= 120596

3minus 120596

lowast






1







119904(b) for119879

2and119879

119904 and

(c) for 2and119879



119910

1205902

119909=sum119872

119894=1sum119872

119895=1120574119894119895sdot (119894 minus 119909max)

2

sum119872

119894=1sum119872

119895=1120574119894119895

1205902

119910=sum119872

119894=1sum119872

119895=1120574119894119895sdot (119895 minus 119910max)

2

sum119872

119894=1sum119872

119895=1120574119894119895

(51)



= 5653 and 120590119910(119886) = 8192 in case (b) 120574max(119887) =

0918 120590119909(119887) = 4839 and 120590

119910(119887) = 6686 in case (c) 120574max(119888)






5 Conclusion



0 20 40 60 80 1000

102030405060708090

100


Spat

ial d

omai

n Y

(pix

)

(a)

0 20 40 60 80 1000

102030405060708090

100


Spat

ial d

omai

n Y

(pix

)

(b)

0 20 40 60 80 1000

102030405060708090

100


Spat

ial d

omai

n Y

(pix

)

(c)




119910sim (119886 119887 119888)





Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










V1iminus1jminus1

Biminus1jminus1

Bijminus1

Bi+1jminus1

V4i+1jminus1

Biminus1j

V1ij

Bij

Tij

V4ij V3

ij

Bi+1j

T9984001

T0

Biminus1j+1

V2ij

Bij+1

Bi+1j+1

V2iminus1j+1

V3i+1j+1






119904 dim(119879

119904) =

119872 times 119872 When 119879119894119895


2) on 119879

119904 the


120574 (1198991 119899

2)

=sum119909119910

[119892 (119909 119910) minus 119892119909119910] [ℎ (119909 minus 119899

1 119910 minus 119899

2) minus ℎ]

sum119909119910

[119892 (119909 119910) minus 119892119909119910]2

sum119909119910

[ℎ (119909 minus 1198991 119910 minus 119899

2) minus ℎ]

2

05

(31)

where 119892119909119910



2 Equation









119909and 120575

119910in frameP



G (119906 V) =(119873minus1)2


(119873minus1)2


119892 (119909 119910)119882119906119909

119872119882

V119910119872

H (119906 V) =(119873minus1)2


(119873minus1)2


ℎ (119909 119910)119882119906119909

119873119882

V119910119873

(32)

Here

119882119873= exp(minus1198952120587

119873) (33)


R (119906 V) =G (119906 V)Hlowast

(119906 V)|G (119906 V)Hlowast (119906 V)|

= exp (119895120601 (119906 V)) (34)


120574 (1198991 119899

2) =

1

1198732

(119873minus1)2


(119873minus1)2


R (119906 V)119882minus1199061198991

119873119882

minusV1198992

119873

(35)





120574max (1198961015840

1198981015840

)

asymp120582

1198732

sin [120587 (1198961015840 + 120575119909)] sin [120587 (1198981015840 + 120575

119910)]

sin [(120587119873) (1198961015840 + 120575119909)] sin [(120587119873) (1198981015840 + 120575

119910)]

(36)


119909 120575

119910) are able





Let col119903 row



119904be the


50 100 150 200 250 300 350 400 450 500


Y minus2515

Image rows (pixels)

Cros

s tra

ck(p

ixel

s)

CCD1 versus CCD2

50 100 150 200 250 300 350 400 450 500


Y minus5393

Image rows (pixels)

Alo

ng tr

ack

(pix

els)

X 423Y minus7363

S11S22

S22

S11

X 423Y minus2378



X 266Y minus1285 X 436

Y minus1297


oss t

rack

(p

ixel

s)

CCD3 versus CCD4


X 436Y minus6869

Image rows (pixels)

Alo

ng tr

ack

(pix

els)

X 266Y minus7663

S31

S31

S32

S32




frameP and Δ1199091015840119899 Δ1199101015840


120575119899119909= col

119903+ col

119904minus 119876 minus 120578

119899

Δ1199091015840

119899= Δ(119909

1015840

2)119899

= 120575119899119909sdot 119908

120575119899119910= row

119904minus row

119903minus119863

119908

Δ1199101015840

119899= Δ(119909

1015840

1)119899

= 120575119899119910sdot 119908 + 119863

(37)






120575119899119909

(pixel)Δ119909

1015840

119899

(mm)120575119899

119910

(pixel)Δ119910

1015840

119899

(mm)









119899119896 Δ1199101015840



119909 119904119910

Let 120591119894119895 119905119894119895


119873119894119895= [

119905119894119895minus 120591

119894119895

Δ119905] isin Z

+




1 120596

2 and 120596

3



CCD

CCD

120575max

D

ci120583120581 = const



1198731119897

sum119894=119897

119888119894

11198971120596119894

1+ 119888

119894

11198972120596119894

2+ 119888

119894

11198973120596119894

3= Δ119909

1015840

1119897minus 119904

1199091

1198731119897

sum119894=119897

119889119894

11198971120596119894

1+ 119889

119894

11198972120596119894

2+ 119889

119894

11198973120596119894

3= Δ119910

1015840

1119897minus 119904

1199101

119873119899119897

sum119894=119897

119888119894

1198991198971120596119894

1+ 119888

119894

1198991198972120596119894

2+ 119888

119894

1198991198973120596119894

3= Δ119909

1015840

119899119897minus 119904

119909119899

119873119899119897

sum119894=119897

119889119894

1198991198971120596119894

1+ 119889

119894

1198991198972120596119894

2+ 119889

119894

1198991198973120596119894

3= Δ119910

1015840

119899119897minus 119904

119910119899

(39)


119888119894

120583]120581 = Ξ120581 (120583 lceil119894 minus ] + 1119873120583]

Nrceil)

119889119894

120583]120581 = Λ 120581(120583 lceil

119894 minus ] + 1119873120583]

Nrceil) (120581 = 1 2 3)

(40)


Here

Ξ119896= (

12058511119896

12058512119896


12058521119896

12058522119896



1205851198991119896

1205851198992119896

sdot sdot sdot 120585119899N119896

)

Λ119896= (

12058211119896

12058212119896


12058221119896

12058222119896



1205821198991119896

1205821198992119896

sdot sdot sdot 120582119899N119896

)

(41)


119896and





120585119894119895119896= 120585

119896(119886 119890 119894

0 Ω 120596 119909

1015840

119902 119910

1015840

119902 Δ119905)

120582119894119895119896= 120582

119896(119886 119890 119894

0 Ω 120596 119909

1015840

119902 119910

1015840

119902 Δ119905)

119894 = 1 sim 119899 119895 = 1 sim N 119902 = 1 sim N

(42)


119896and Λ




C119895=

(((((((((((

(

1198881

11198951119888111198952

119888111198953

sdot sdot sdot 1198881198731119895

111989511198881198731119895

111989521198881198731119895


119889111198951

119889111198952

119889111198953

sdot sdot sdot 1198891198731119895

111989511198891198731119895

111989521198891198731119895



11988811199021198951

11988811199021198952

11988811199021198953


1199021198951119888119873119902119895

1199021198951119888119873119902119895

1199021198953

11988911199021198951

11988911199021198952

11988911199021198953


1199021198951119889119873119902119895

1199021198952119889119873119902119895

1199021198953


11988811198991198951

11988811198991198952

11988811198991198953


1198991198951119888119873119899119895

1198991198952119888119873119899119895


1198891

11989911989511198891

11989911989521198891


1

11989911989511198891

11989911989521198891


)))))))))))

)2119899times3119873119902119895

(43)

where119873119902119895= max119873

1119895 119873



C =((

(

[C1]2119899times3119873

1199021


0 [C2]2119899times3119873

1199022



[C]2119899times3119873maxd

[C119898]2119899times3119873

119902119898

0

))

)2119899119898times3119873max

(44)


max1198731199021 119873

119902119898



= [120596111205961212059613sdot sdot sdot 120596

119873max1

120596119873max2

120596119873max3

]119879

(45)


2119898119899times1= [ Δ1199091015840

11Δ1199101015840


1198991Δ1199101015840

1198991

sdot sdot sdot Δ11990910158401198981

Δ11991010158401198981

sdot sdot sdot Δ1199091015840119899119898

Δ1199101015840119899119898

]119879

s2119898119899times1

= [ 1199041199091

1199041199101

sdot sdot sdot 119904119909119899

119904119910119899


1199041199101

sdot sdot sdot 119904119909119899

119904119910119899]119879

(46)








the sensed window








Yes

No


the prior CCD 1

2

3

4 5

7

6

120581 = 120581 + 1






120581 = Nmax





C2119898119899times3119873max


= Δu2119898119899times1


(47)



[Ω] = (C119879

C)minus1

C119879

(Δu minus s) (48)



3119873max

C = U [120590]V119879

(49)

where U2119898119899times2119898119899



119894= radic120582



1205901120590

3119873maxle 119905119900119897



119888


119888asymp 1749KHz The 120596



2max = 0001104∘s 120596

1max = 0001194∘s


3=



002 004 006 008 01 012 014minus1

01

002 004 006 008 01 012 014minus1

01

002 004 006 008 01 012 014001600170018

Imaging time (s)

1205961

(deg

s)

1205962

(deg

s)

1205963

(deg

s)

times10minus3

times10minus3



1(119905) and 120596




120595lowast

119905=V1199101015840

V1199091015840

120596lowast

3(119905) =

lowast

119905=V1199101015840V1199091015840 minus V

1199101015840 V1199091015840

V21199091015840

(50)


lowast


Δ120596lowast3= 120596

3minus 120596

lowast






1







119904(b) for119879

2and119879

119904 and

(c) for 2and119879



119910

1205902

119909=sum119872

119894=1sum119872

119895=1120574119894119895sdot (119894 minus 119909max)

2

sum119872

119894=1sum119872

119895=1120574119894119895

1205902

119910=sum119872

119894=1sum119872

119895=1120574119894119895sdot (119895 minus 119910max)

2

sum119872

119894=1sum119872

119895=1120574119894119895

(51)



= 5653 and 120590119910(119886) = 8192 in case (b) 120574max(119887) =

0918 120590119909(119887) = 4839 and 120590

119910(119887) = 6686 in case (c) 120574max(119888)






5 Conclusion



0 20 40 60 80 1000

102030405060708090

100


Spat

ial d

omai

n Y

(pix

)

(a)

0 20 40 60 80 1000

102030405060708090

100


Spat

ial d

omai

n Y

(pix

)

(b)

0 20 40 60 80 1000

102030405060708090

100


Spat

ial d

omai

n Y

(pix

)

(c)




119910sim (119886 119887 119888)





Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













120574max (1198961015840

1198981015840

)

asymp120582

1198732

sin [120587 (1198961015840 + 120575119909)] sin [120587 (1198981015840 + 120575

119910)]

sin [(120587119873) (1198961015840 + 120575119909)] sin [(120587119873) (1198981015840 + 120575

119910)]

(36)


119909 120575

119910) are able





Let col119903 row



119904be the


50 100 150 200 250 300 350 400 450 500


Y minus2515

Image rows (pixels)

Cros

s tra

ck(p

ixel

s)

CCD1 versus CCD2

50 100 150 200 250 300 350 400 450 500


Y minus5393

Image rows (pixels)

Alo

ng tr

ack

(pix

els)

X 423Y minus7363

S11S22

S22

S11

X 423Y minus2378



X 266Y minus1285 X 436

Y minus1297


oss t

rack

(p

ixel

s)

CCD3 versus CCD4


X 436Y minus6869

Image rows (pixels)

Alo

ng tr

ack

(pix

els)

X 266Y minus7663

S31

S31

S32

S32




frameP and Δ1199091015840119899 Δ1199101015840


120575119899119909= col

119903+ col

119904minus 119876 minus 120578

119899

Δ1199091015840

119899= Δ(119909

1015840

2)119899

= 120575119899119909sdot 119908

120575119899119910= row

119904minus row

119903minus119863

119908

Δ1199101015840

119899= Δ(119909

1015840

1)119899

= 120575119899119910sdot 119908 + 119863

(37)






120575119899119909

(pixel)Δ119909

1015840

119899

(mm)120575119899

119910

(pixel)Δ119910

1015840

119899

(mm)









119899119896 Δ1199101015840



119909 119904119910

Let 120591119894119895 119905119894119895


119873119894119895= [

119905119894119895minus 120591

119894119895

Δ119905] isin Z

+




1 120596

2 and 120596

3



CCD

CCD

120575max

D

ci120583120581 = const



1198731119897

sum119894=119897

119888119894

11198971120596119894

1+ 119888

119894

11198972120596119894

2+ 119888

119894

11198973120596119894

3= Δ119909

1015840

1119897minus 119904

1199091

1198731119897

sum119894=119897

119889119894

11198971120596119894

1+ 119889

119894

11198972120596119894

2+ 119889

119894

11198973120596119894

3= Δ119910

1015840

1119897minus 119904

1199101

119873119899119897

sum119894=119897

119888119894

1198991198971120596119894

1+ 119888

119894

1198991198972120596119894

2+ 119888

119894

1198991198973120596119894

3= Δ119909

1015840

119899119897minus 119904

119909119899

119873119899119897

sum119894=119897

119889119894

1198991198971120596119894

1+ 119889

119894

1198991198972120596119894

2+ 119889

119894

1198991198973120596119894

3= Δ119910

1015840

119899119897minus 119904

119910119899

(39)


119888119894

120583]120581 = Ξ120581 (120583 lceil119894 minus ] + 1119873120583]

Nrceil)

119889119894

120583]120581 = Λ 120581(120583 lceil

119894 minus ] + 1119873120583]

Nrceil) (120581 = 1 2 3)

(40)


Here

Ξ119896= (

12058511119896

12058512119896


12058521119896

12058522119896



1205851198991119896

1205851198992119896

sdot sdot sdot 120585119899N119896

)

Λ119896= (

12058211119896

12058212119896


12058221119896

12058222119896



1205821198991119896

1205821198992119896

sdot sdot sdot 120582119899N119896

)

(41)


119896and





120585119894119895119896= 120585

119896(119886 119890 119894

0 Ω 120596 119909

1015840

119902 119910

1015840

119902 Δ119905)

120582119894119895119896= 120582

119896(119886 119890 119894

0 Ω 120596 119909

1015840

119902 119910

1015840

119902 Δ119905)

119894 = 1 sim 119899 119895 = 1 sim N 119902 = 1 sim N

(42)


119896and Λ




C119895=

(((((((((((

(

1198881

11198951119888111198952

119888111198953

sdot sdot sdot 1198881198731119895

111989511198881198731119895

111989521198881198731119895


119889111198951

119889111198952

119889111198953

sdot sdot sdot 1198891198731119895

111989511198891198731119895

111989521198891198731119895



11988811199021198951

11988811199021198952

11988811199021198953


1199021198951119888119873119902119895

1199021198951119888119873119902119895

1199021198953

11988911199021198951

11988911199021198952

11988911199021198953


1199021198951119889119873119902119895

1199021198952119889119873119902119895

1199021198953


11988811198991198951

11988811198991198952

11988811198991198953


1198991198951119888119873119899119895

1198991198952119888119873119899119895


1198891

11989911989511198891

11989911989521198891


1

11989911989511198891

11989911989521198891


)))))))))))

)2119899times3119873119902119895

(43)

where119873119902119895= max119873

1119895 119873



C =((

(

[C1]2119899times3119873

1199021


0 [C2]2119899times3119873

1199022



[C]2119899times3119873maxd

[C119898]2119899times3119873

119902119898

0

))

)2119899119898times3119873max

(44)


max1198731199021 119873

119902119898



= [120596111205961212059613sdot sdot sdot 120596

119873max1

120596119873max2

120596119873max3

]119879

(45)


2119898119899times1= [ Δ1199091015840

11Δ1199101015840


1198991Δ1199101015840

1198991

sdot sdot sdot Δ11990910158401198981

Δ11991010158401198981

sdot sdot sdot Δ1199091015840119899119898

Δ1199101015840119899119898

]119879

s2119898119899times1

= [ 1199041199091

1199041199101

sdot sdot sdot 119904119909119899

119904119910119899


1199041199101

sdot sdot sdot 119904119909119899

119904119910119899]119879

(46)








the sensed window








Yes

No


the prior CCD 1

2

3

4 5

7

6

120581 = 120581 + 1






120581 = Nmax





C2119898119899times3119873max


= Δu2119898119899times1


(47)



[Ω] = (C119879

C)minus1

C119879

(Δu minus s) (48)



3119873max

C = U [120590]V119879

(49)

where U2119898119899times2119898119899



119894= radic120582



1205901120590

3119873maxle 119905119900119897



119888


119888asymp 1749KHz The 120596



2max = 0001104∘s 120596

1max = 0001194∘s


3=



002 004 006 008 01 012 014minus1

01

002 004 006 008 01 012 014minus1

01

002 004 006 008 01 012 014001600170018

Imaging time (s)

1205961

(deg

s)

1205962

(deg

s)

1205963

(deg

s)

times10minus3

times10minus3



1(119905) and 120596




120595lowast

119905=V1199101015840

V1199091015840

120596lowast

3(119905) =

lowast

119905=V1199101015840V1199091015840 minus V

1199101015840 V1199091015840

V21199091015840

(50)


lowast


Δ120596lowast3= 120596

3minus 120596

lowast






1







119904(b) for119879

2and119879

119904 and

(c) for 2and119879



119910

1205902

119909=sum119872

119894=1sum119872

119895=1120574119894119895sdot (119894 minus 119909max)

2

sum119872

119894=1sum119872

119895=1120574119894119895

1205902

119910=sum119872

119894=1sum119872

119895=1120574119894119895sdot (119895 minus 119910max)

2

sum119872

119894=1sum119872

119895=1120574119894119895

(51)



= 5653 and 120590119910(119886) = 8192 in case (b) 120574max(119887) =

0918 120590119909(119887) = 4839 and 120590

119910(119887) = 6686 in case (c) 120574max(119888)






5 Conclusion



0 20 40 60 80 1000

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)

(a)

0 20 40 60 80 1000

102030405060708090

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Spat

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omai

n Y

(pix

)

(b)

0 20 40 60 80 1000

102030405060708090

100


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(pix

)

(c)




119910sim (119886 119887 119888)





Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












120575119899119909

(pixel)Δ119909

1015840

119899

(mm)120575119899

119910

(pixel)Δ119910

1015840

119899

(mm)









119899119896 Δ1199101015840



119909 119904119910

Let 120591119894119895 119905119894119895


119873119894119895= [

119905119894119895minus 120591

119894119895

Δ119905] isin Z

+




1 120596

2 and 120596

3



CCD

CCD

120575max

D

ci120583120581 = const



1198731119897

sum119894=119897

119888119894

11198971120596119894

1+ 119888

119894

11198972120596119894

2+ 119888

119894

11198973120596119894

3= Δ119909

1015840

1119897minus 119904

1199091

1198731119897

sum119894=119897

119889119894

11198971120596119894

1+ 119889

119894

11198972120596119894

2+ 119889

119894

11198973120596119894

3= Δ119910

1015840

1119897minus 119904

1199101

119873119899119897

sum119894=119897

119888119894

1198991198971120596119894

1+ 119888

119894

1198991198972120596119894

2+ 119888

119894

1198991198973120596119894

3= Δ119909

1015840

119899119897minus 119904

119909119899

119873119899119897

sum119894=119897

119889119894

1198991198971120596119894

1+ 119889

119894

1198991198972120596119894

2+ 119889

119894

1198991198973120596119894

3= Δ119910

1015840

119899119897minus 119904

119910119899

(39)


119888119894

120583]120581 = Ξ120581 (120583 lceil119894 minus ] + 1119873120583]

Nrceil)

119889119894

120583]120581 = Λ 120581(120583 lceil

119894 minus ] + 1119873120583]

Nrceil) (120581 = 1 2 3)

(40)


Here

Ξ119896= (

12058511119896

12058512119896


12058521119896

12058522119896



1205851198991119896

1205851198992119896

sdot sdot sdot 120585119899N119896

)

Λ119896= (

12058211119896

12058212119896


12058221119896

12058222119896



1205821198991119896

1205821198992119896

sdot sdot sdot 120582119899N119896

)

(41)


119896and





120585119894119895119896= 120585

119896(119886 119890 119894

0 Ω 120596 119909

1015840

119902 119910

1015840

119902 Δ119905)

120582119894119895119896= 120582

119896(119886 119890 119894

0 Ω 120596 119909

1015840

119902 119910

1015840

119902 Δ119905)

119894 = 1 sim 119899 119895 = 1 sim N 119902 = 1 sim N

(42)


119896and Λ




C119895=

(((((((((((

(

1198881

11198951119888111198952

119888111198953

sdot sdot sdot 1198881198731119895

111989511198881198731119895

111989521198881198731119895


119889111198951

119889111198952

119889111198953

sdot sdot sdot 1198891198731119895

111989511198891198731119895

111989521198891198731119895



11988811199021198951

11988811199021198952

11988811199021198953


1199021198951119888119873119902119895

1199021198951119888119873119902119895

1199021198953

11988911199021198951

11988911199021198952

11988911199021198953


1199021198951119889119873119902119895

1199021198952119889119873119902119895

1199021198953


11988811198991198951

11988811198991198952

11988811198991198953


1198991198951119888119873119899119895

1198991198952119888119873119899119895


1198891

11989911989511198891

11989911989521198891


1

11989911989511198891

11989911989521198891


)))))))))))

)2119899times3119873119902119895

(43)

where119873119902119895= max119873

1119895 119873



C =((

(

[C1]2119899times3119873

1199021


0 [C2]2119899times3119873

1199022



[C]2119899times3119873maxd

[C119898]2119899times3119873

119902119898

0

))

)2119899119898times3119873max

(44)


max1198731199021 119873

119902119898



= [120596111205961212059613sdot sdot sdot 120596

119873max1

120596119873max2

120596119873max3

]119879

(45)


2119898119899times1= [ Δ1199091015840

11Δ1199101015840


1198991Δ1199101015840

1198991

sdot sdot sdot Δ11990910158401198981

Δ11991010158401198981

sdot sdot sdot Δ1199091015840119899119898

Δ1199101015840119899119898

]119879

s2119898119899times1

= [ 1199041199091

1199041199101

sdot sdot sdot 119904119909119899

119904119910119899


1199041199101

sdot sdot sdot 119904119909119899

119904119910119899]119879

(46)








the sensed window








Yes

No


the prior CCD 1

2

3

4 5

7

6

120581 = 120581 + 1






120581 = Nmax





C2119898119899times3119873max


= Δu2119898119899times1


(47)



[Ω] = (C119879

C)minus1

C119879

(Δu minus s) (48)



3119873max

C = U [120590]V119879

(49)

where U2119898119899times2119898119899



119894= radic120582



1205901120590

3119873maxle 119905119900119897



119888


119888asymp 1749KHz The 120596



2max = 0001104∘s 120596

1max = 0001194∘s


3=



002 004 006 008 01 012 014minus1

01

002 004 006 008 01 012 014minus1

01

002 004 006 008 01 012 014001600170018

Imaging time (s)

1205961

(deg

s)

1205962

(deg

s)

1205963

(deg

s)

times10minus3

times10minus3



1(119905) and 120596




120595lowast

119905=V1199101015840

V1199091015840

120596lowast

3(119905) =

lowast

119905=V1199101015840V1199091015840 minus V

1199101015840 V1199091015840

V21199091015840

(50)


lowast


Δ120596lowast3= 120596

3minus 120596

lowast






1







119904(b) for119879

2and119879

119904 and

(c) for 2and119879



119910

1205902

119909=sum119872

119894=1sum119872

119895=1120574119894119895sdot (119894 minus 119909max)

2

sum119872

119894=1sum119872

119895=1120574119894119895

1205902

119910=sum119872

119894=1sum119872

119895=1120574119894119895sdot (119895 minus 119910max)

2

sum119872

119894=1sum119872

119895=1120574119894119895

(51)



= 5653 and 120590119910(119886) = 8192 in case (b) 120574max(119887) =

0918 120590119909(119887) = 4839 and 120590

119910(119887) = 6686 in case (c) 120574max(119888)






5 Conclusion



0 20 40 60 80 1000

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n Y

(pix

)

(a)

0 20 40 60 80 1000

102030405060708090

100


Spat

ial d

omai

n Y

(pix

)

(b)

0 20 40 60 80 1000

102030405060708090

100


Spat

ial d

omai

n Y

(pix

)

(c)




119910sim (119886 119887 119888)





Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Here

Ξ119896= (

12058511119896

12058512119896


12058521119896

12058522119896



1205851198991119896

1205851198992119896

sdot sdot sdot 120585119899N119896

)

Λ119896= (

12058211119896

12058212119896


12058221119896

12058222119896



1205821198991119896

1205821198992119896

sdot sdot sdot 120582119899N119896

)

(41)


119896and





120585119894119895119896= 120585

119896(119886 119890 119894

0 Ω 120596 119909

1015840

119902 119910

1015840

119902 Δ119905)

120582119894119895119896= 120582

119896(119886 119890 119894

0 Ω 120596 119909

1015840

119902 119910

1015840

119902 Δ119905)

119894 = 1 sim 119899 119895 = 1 sim N 119902 = 1 sim N

(42)


119896and Λ




C119895=

(((((((((((

(

1198881

11198951119888111198952

119888111198953

sdot sdot sdot 1198881198731119895

111989511198881198731119895

111989521198881198731119895


119889111198951

119889111198952

119889111198953

sdot sdot sdot 1198891198731119895

111989511198891198731119895

111989521198891198731119895



11988811199021198951

11988811199021198952

11988811199021198953


1199021198951119888119873119902119895

1199021198951119888119873119902119895

1199021198953

11988911199021198951

11988911199021198952

11988911199021198953


1199021198951119889119873119902119895

1199021198952119889119873119902119895

1199021198953


11988811198991198951

11988811198991198952

11988811198991198953


1198991198951119888119873119899119895

1198991198952119888119873119899119895


1198891

11989911989511198891

11989911989521198891


1

11989911989511198891

11989911989521198891


)))))))))))

)2119899times3119873119902119895

(43)

where119873119902119895= max119873

1119895 119873



C =((

(

[C1]2119899times3119873

1199021


0 [C2]2119899times3119873

1199022



[C]2119899times3119873maxd

[C119898]2119899times3119873

119902119898

0

))

)2119899119898times3119873max

(44)


max1198731199021 119873

119902119898



= [120596111205961212059613sdot sdot sdot 120596

119873max1

120596119873max2

120596119873max3

]119879

(45)


2119898119899times1= [ Δ1199091015840

11Δ1199101015840


1198991Δ1199101015840

1198991

sdot sdot sdot Δ11990910158401198981

Δ11991010158401198981

sdot sdot sdot Δ1199091015840119899119898

Δ1199101015840119899119898

]119879

s2119898119899times1

= [ 1199041199091

1199041199101

sdot sdot sdot 119904119909119899

119904119910119899


1199041199101

sdot sdot sdot 119904119909119899

119904119910119899]119879

(46)








the sensed window








Yes

No


the prior CCD 1

2

3

4 5

7

6

120581 = 120581 + 1






120581 = Nmax





C2119898119899times3119873max


= Δu2119898119899times1


(47)



[Ω] = (C119879

C)minus1

C119879

(Δu minus s) (48)



3119873max

C = U [120590]V119879

(49)

where U2119898119899times2119898119899



119894= radic120582



1205901120590

3119873maxle 119905119900119897



119888


119888asymp 1749KHz The 120596



2max = 0001104∘s 120596

1max = 0001194∘s


3=



002 004 006 008 01 012 014minus1

01

002 004 006 008 01 012 014minus1

01

002 004 006 008 01 012 014001600170018

Imaging time (s)

1205961

(deg

s)

1205962

(deg

s)

1205963

(deg

s)

times10minus3

times10minus3



1(119905) and 120596




120595lowast

119905=V1199101015840

V1199091015840

120596lowast

3(119905) =

lowast

119905=V1199101015840V1199091015840 minus V

1199101015840 V1199091015840

V21199091015840

(50)


lowast


Δ120596lowast3= 120596

3minus 120596

lowast






1







119904(b) for119879

2and119879

119904 and

(c) for 2and119879



119910

1205902

119909=sum119872

119894=1sum119872

119895=1120574119894119895sdot (119894 minus 119909max)

2

sum119872

119894=1sum119872

119895=1120574119894119895

1205902

119910=sum119872

119894=1sum119872

119895=1120574119894119895sdot (119895 minus 119910max)

2

sum119872

119894=1sum119872

119895=1120574119894119895

(51)



= 5653 and 120590119910(119886) = 8192 in case (b) 120574max(119887) =

0918 120590119909(119887) = 4839 and 120590

119910(119887) = 6686 in case (c) 120574max(119888)






5 Conclusion



0 20 40 60 80 1000

102030405060708090

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n Y

(pix

)

(a)

0 20 40 60 80 1000

102030405060708090

100


Spat

ial d

omai

n Y

(pix

)

(b)

0 20 40 60 80 1000

102030405060708090

100


Spat

ial d

omai

n Y

(pix

)

(c)




119910sim (119886 119887 119888)





Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















the sensed window








Yes

No


the prior CCD 1

2

3

4 5

7

6

120581 = 120581 + 1






120581 = Nmax





C2119898119899times3119873max


= Δu2119898119899times1


(47)



[Ω] = (C119879

C)minus1

C119879

(Δu minus s) (48)



3119873max

C = U [120590]V119879

(49)

where U2119898119899times2119898119899



119894= radic120582



1205901120590

3119873maxle 119905119900119897



119888


119888asymp 1749KHz The 120596



2max = 0001104∘s 120596

1max = 0001194∘s


3=



002 004 006 008 01 012 014minus1

01

002 004 006 008 01 012 014minus1

01

002 004 006 008 01 012 014001600170018

Imaging time (s)

1205961

(deg

s)

1205962

(deg

s)

1205963

(deg

s)

times10minus3

times10minus3



1(119905) and 120596




120595lowast

119905=V1199101015840

V1199091015840

120596lowast

3(119905) =

lowast

119905=V1199101015840V1199091015840 minus V

1199101015840 V1199091015840

V21199091015840

(50)


lowast


Δ120596lowast3= 120596

3minus 120596

lowast






1







119904(b) for119879

2and119879

119904 and

(c) for 2and119879



119910

1205902

119909=sum119872

119894=1sum119872

119895=1120574119894119895sdot (119894 minus 119909max)

2

sum119872

119894=1sum119872

119895=1120574119894119895

1205902

119910=sum119872

119894=1sum119872

119895=1120574119894119895sdot (119895 minus 119910max)

2

sum119872

119894=1sum119872

119895=1120574119894119895

(51)



= 5653 and 120590119910(119886) = 8192 in case (b) 120574max(119887) =

0918 120590119909(119887) = 4839 and 120590

119910(119887) = 6686 in case (c) 120574max(119888)






5 Conclusion



0 20 40 60 80 1000

102030405060708090

100


Spat

ial d

omai

n Y

(pix

)

(a)

0 20 40 60 80 1000

102030405060708090

100


Spat

ial d

omai

n Y

(pix

)

(b)

0 20 40 60 80 1000

102030405060708090

100


Spat

ial d

omai

n Y

(pix

)

(c)




119910sim (119886 119887 119888)





Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










002 004 006 008 01 012 014minus1

01

002 004 006 008 01 012 014minus1

01

002 004 006 008 01 012 014001600170018

Imaging time (s)

1205961

(deg

s)

1205962

(deg

s)

1205963

(deg

s)

times10minus3

times10minus3



1(119905) and 120596




120595lowast

119905=V1199101015840

V1199091015840

120596lowast

3(119905) =

lowast

119905=V1199101015840V1199091015840 minus V

1199101015840 V1199091015840

V21199091015840

(50)


lowast


Δ120596lowast3= 120596

3minus 120596

lowast






1







119904(b) for119879

2and119879

119904 and

(c) for 2and119879



119910

1205902

119909=sum119872

119894=1sum119872

119895=1120574119894119895sdot (119894 minus 119909max)

2

sum119872

119894=1sum119872

119895=1120574119894119895

1205902

119910=sum119872

119894=1sum119872

119895=1120574119894119895sdot (119895 minus 119910max)

2

sum119872

119894=1sum119872

119895=1120574119894119895

(51)



= 5653 and 120590119910(119886) = 8192 in case (b) 120574max(119887) =

0918 120590119909(119887) = 4839 and 120590

119910(119887) = 6686 in case (c) 120574max(119888)






5 Conclusion



0 20 40 60 80 1000

102030405060708090

100


Spat

ial d

omai

n Y

(pix

)

(a)

0 20 40 60 80 1000

102030405060708090

100


Spat

ial d

omai

n Y

(pix

)

(b)

0 20 40 60 80 1000

102030405060708090

100


Spat

ial d

omai

n Y

(pix

)

(c)




119910sim (119886 119887 119888)





Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 20 40 60 80 1000

102030405060708090

100


Spat

ial d

omai

n Y

(pix

)

(a)

0 20 40 60 80 1000

102030405060708090

100


Spat

ial d

omai

n Y

(pix

)

(b)

0 20 40 60 80 1000

102030405060708090

100


Spat

ial d

omai

n Y

(pix

)

(c)




119910sim (119886 119887 119888)





Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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