(2) fundamental theories in photo gramme try

8/4/2019 (2) Fundamental Theories in Photo Gramme Try

1/17

Fundamental Theories

in Photogrammetry

Mathematical model of

Correspondence between Imagepoint and Object point

Digital Photogrammetry



2/17

Coordinate Reference Frames

Image space coordinate system

Object space coordinate system



Figure 1. Object and image space

coordinate systems


3/17

Sensor Model (Interior Orientation)

Focal length or principal distance (f)

The location of principal point in theimage plane

The description of lens distortion



00 , yx


4/17

Image coordinate system

T

00 ),(2D yyxx

T00 ),,(3D fyyxx

Digital PhotogrammetryDigital Photogrammetry

Figure 2. Elements defining the image

coordinate system


5/17

Lens distortion


Radial component of lens distortion

Tangential component

Figure 3. Lens distortion


6/17

The elements of exterior orientation are described byposition of perspective center and thepose of the ray axis

Platform Model (Exterior Orientation)

T

LLL ZYXL ),,(

=

L

L

L

ZZ

YY

XX

kM

f

yy

xx

0

0

Where k: a scale factor

M: rotation matrix

Point correspondence between

image space and object space

(4)


),,(

Figure 4. Object and image

coordinate systems


7/17

M: Rotation Matrix

The standard approach to constructing M is by

using three sequential rotation:about the X-axis

about the once-rotated Y-axis

about the twice-rotated Z-axis

=

cossin-0sincos0

001

M

=

cos0sin

010

sin-0cos

M

=

100

0cossin-

0sincos

M



8/17

M: Rotation Matrix - continued

MMMM =

+

+

=

coscoscossinsin

sinsincoscossinsinsinsinscossincoscossincos-sinsincossinsinsincoscoscos

coM

=

333231

232221

131211

mmm

mmm

mmm

M

(5)

(6)



9/17

Derivation of Collinearity equations

=

L

L

L

ZZ

YY

XX

mmm

mmm

mmm

k

f

yy

xx

333231

232221

131211

0

0

=

L

L

L

ZZ

YY

XX

kM

f

yy

xx

0

0

[ ]

[ ]

[ ])()()(

)()()(

)()()(

333231

2322210

1312110

LLL

LLL

LLL

ZZmYYmXXmkf

ZZmYYmXXmkyy

ZZmYYmXXmkxx

++=

++=

++=

[ ][ ]

[ ]

[ ])()()(

)()()(

)()()()()()(

333231

2322210

333231

1312110

LLL

LLL

LLL

LLL

ZZmYYmXXm

ZZmYYmXXmfyy

ZZmYYmXXmZZmYYmXXmfxx

++

++=

++

++=

(7-1)

(7-2)

(7-3)

Further processes : dividing (7-1) by (7-3) and dividing (7-2) by (7-3)

(8-1)

(8-2)



10/17

Derivation of Collinearity equations-continued

Likewise, one can come to the following relationship:

[ ][ ]

[ ][ ])()()(

)()()()(

)()()(

)()()()(

33023013

32022012

33023013

31021011

fmyymxxm

fmyymxxmZZYY

fmyymxxm

fmyymxxmZZXX

LL

LL

++

++=

++

++

= (9-1)

(9-2)



11/17

The Interpretation of Collinearity Equations

Mathematically

Whether the balance between unknowns and

number of equations (measurements) would leadto the solution?

Geometrically

?

?

?



12/17

The Interpretation of Collinearity Equations-continued

Resection: Derive exterior orientation via photo

measurements and the known object points. Intersection: Determine the position

(coordinates) of object point by intersecting, atleast, two rays originated from differentperspective centers.

Back-Projection: Predict the photo coordinatesfor a known object point and exterior orientation.

Note: To get better understanding of the configurations,one should be able to distinguish observations from(unknown) parameters in each case.



13/17

From Collinearity equations toCoplanarity equations

=

=

12

12

12

LL

LL

LL

Z

Y

X

ZZ

YY

XX

b

b

b

b

2

0

0

2

2

2

2

2

1

0

0

1

1

1

1

1

=

=

=

=

f

yy

xx

M

w

v

u

a

f

yy

xx

M

w

v

u

a

T

T

coplanarareand, 21 aab 0)( 21 =

aab

Therefore, the determinant of thevector components ends up zero

0

222

111 ==

wvu

wvu

bbb

F

ZYX

(10)



14/17

From Collinearity equations toCoplanarity equations continued

Note that the object point coordinateshave been eliminated from the equations.



15/17

What if Exterior Orientation is not known?

How to solve exterior orientation parameters?;namely how to determine the position and thepose of the perspective center w.r.t. the

assumed object coordinate system.

?



16/17

Solutions of Exterior Orientation

Single Photo (Single Photo Resection)

Stereo model

Block Triangulation

Always to think about the balance between

number of parameters and the number of

equations; most likely one would deal with

redundant observations for increasing accuracy and

reliability purposes.



17/17

Computational concerns Collinearity equations are nonlinear

systems, linearization and iterativecomputation are needed for theconvergent solution.

Due to the linearization, the approximationof parameters need to be provided prior to

the computation. What are the alternatives?


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