spatial domain processing remove the additive noise

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Image Comm. Lab EE/NTHU 1

Chapter 5Image Restoration and Reconstruction

Chapter 5Image Restoration and Reconstruction

• Degraded digital image restoration• Degraded digital image restoration

– Spatial domain processing

Remove the additive noise

– Frequency domain

Hi hli ht th bl d iHighlight the blurred image


5.1 Model of Image degradation and restoration5.1 Model of Image degradation and restoration

•g(x, y)=h(x, y)f(x, y)+(x, y)

•G(u, v)=H(u, v)F(u, v)+N(u, v)

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5.2 Noise Model 5.2 Noise Model

• Spatial and frequency property of noise• Spatial and frequency property of noise– White noise (random noise)

– Gaussian noise22 2/)(

2

1)(

zezp

– : mean ; :variance

– 70% [(-), (+)]

– 95 % [(-2), (+2)]



• Rayleigh noise• Rayleigh noise

The PDF is

• =a+(b/4)1/2

azfor

azforeazbzp

baz

0

)(2

)(/)( 2

• =b(4-)/4

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5 2 Noise Model5 2 Noise Model5.2 Noise Model5.2 Noise Model


5.2 Noise Model

• Erlang (gamma) noise

• =b/a; =b/a2

E ti l i

00

0)!1()(

1

zfor

zforeb

zazp

azbb

0zforae az

• Exponential noise

• =1/a; =1/a2

00

0)(

zfor

zforaezp

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• Uniform noise bf

1

• =(a+b)/2; =(b-a)2/12

• Impulse noise (salt and pepper noise)

otherwise

bzaforabzp

0)(

• Impulse noise (salt and pepper noise)

otherwise

bzfor

azfor

P

P

zp b

a

0

)(


5.2 Noise Model5.2 Noise Model

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5.2.3 Noise Model5.2.3 Noise Model--periodic noiseperiodic noise

• Periodic noise is due to the electrical or• Periodic noise is due to the electrical or electromechanical interference during image acquisition.

• It can be estimated through the inspection of the Fourier spectrum of the image.


5.2.3 Noise Model-periodic noise5.2.3 Noise Model-periodic noise

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5.2.4 Noise 5.2.4 Noise Model Model -- Noise parameters estimationNoise parameters estimation

• Capture a set of images of “flat” environment.• Use a optical sensor to capture the image of a solid• Use a optical sensor to capture the image of a solid

gray board that is illuminated uniformly. • The resulting image is a good indicator of system

noise.• Find the mean and standard deviation of the

histogram p(zi) of the resulting image.g p( i) g g

• From and calculate a and b (the parameter of that specific noise distribution).

Sz

iii

zpzμ )(

Sz

ii

2

i

zpμ-zσ )()( 2


5.2.4 Noise Model -- Noise parameters estimationNoise parameters estimation5.2.4 Noise Model -- Noise parameters estimationNoise parameters estimation

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5.3 Restoration using spatial filter5.3 Restoration using spatial filter

• The only degradation is the additive noisey gg(x, y)=f(x, y)+(x, y)G(u, v)=F(u, v)+N(u, v)

• Mean filters– Arithmetic mean filter

Sts

tsgmn

yxf)(

),(1

),(ˆ

– Geometric mean filter

xyStsmn ),(

mn

Sts xy

tsgyxf

/1

),(

),(),(ˆ


5.3 Restoration using spatial 5.3 Restoration using spatial filterfilter

– Harmonic mean filter is good for salt noise not– Harmonic mean filter is good for salt noise not for pepper noise.

– Contra-harmonic filter: Q>0 reduce pepper i Q<0 d lt i

xySts tsg

mnyxf

),( ),(

1),(

ˆ

noise Q<0 reduce salt noise.

xy

xy

Sts

Q

Sts

Q

tsg

tsg

yxf

),(

),(

1

),(

),(

),(ˆ

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• Order-Statistics filters ˆMedian filter

Max filter: find the brightest points to reduce the pepper noise

Min filter: find the darkest point to reduce the salt

),(),(ˆ),(

tsgmedianyxfxySts

),(max),(ˆ),(

tsgyxfxySts

Min filter: find the darkest point to reduce the salt noise

Midpoint filter

),(min),(ˆ),(

tsgyxfxySts

),(max),(min

2

1),(ˆ

),(),(tsgtsgyxf

xyxy StsSts

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• Alpha-trimmed mean filterp f

Suppose delete d/2 lowest and d/2 highest gray-level value in the neighborhood of (s, t)and average the rest.

tsgyxf ),(1

),(ˆ

where d=0 ~ mn-1

xyStsr tsg

dmnyxf

),(

),(),(



2nd pass 3rd pass

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5.3 Restoration using spatial filter

5.3 Restoration using spatial filter

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5.3 Restoration using spatial filter-Adaptive local noise reduction filter

Measurements for local region Sxy centered at (x, y)(1) noisy image at (x, y): g(x, y) with variance 2

(2) The local mean mL in Sxy

(3) The local variance 2L in Sxy

(4) The local variance of noise is high 2L ≥ 2

Conditions: Assume additive noise

(a) Zero-noise case: If 2 =0 then f(x, y)=g(x, y)

(b) Edges: If 2L > 2

then f(x, y)≈g(x, y) (edge preserved)(c) If 2

L = 2 then f(x, y)= mL,

(d) Adaptive filter (assume 2L ≥ 2

because Sxy is a subset of g(x, y)))

In practice, 2 is unknown, so a test is built before filtering

]),([),(),(ˆ2

2

LL

myxgyxgyxf



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• Adaptive median filter can handle impulse noise with larger probability (P and P are large)with larger probability (Pa and Pb are large).Increasing(change) window size during operation

• Notations:zmin=minimum gray-level value in Sxy

zmax=maximum gray-level value in Sxy

di l l l i Szmed=median gray-level value in Sxy

zxy= gray-level at (x, y)Smax=maximum allowed size of Sxy



• The adaptive median filter algorithm works in two levels:• Level A: A1=z d zLevel A: A1 zmed zmin

A2=zmed zmax

If A1>0 AND A2<0 goto level Belse increase the window sizeIf window size Smax repeat level Aelse output zelse output zxy

Level B: B1=zxy zmin

B2=zxy zmax

If B1>0 AND B2<0, output zxy

Else output zmed

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5.4 Restoration using frequency5.4 Restoration using frequency--domain filterdomain filter5.4.1 5.4.1 BandBand--reject reject filterfilter

• Ideal band reject filterIdeal band reject filter

• Butterworth band reject filter

2/),(

2/),(2/

2/),(

1

0

1

),(

0

00

0

WDvuDif

WDvuDWDif

WDvuDif

vuH

• Butterworth band reject filter

n

DvuDWvuD

vuH 2

20

2 ),(),(

1

1),(

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5.4.1 5.4.1 BandBand--reject reject filter filter

• Gaussian band reject filter• Gaussian band reject filter

220

2

),(

),(

2

1

1),(

WvuD

DvuD

evuH


5.4.1 BandBand--reject reject filter filter 5.4.1 BandBand--reject reject filter filter

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5.4.1 BandBand--reject reject filter filter 5.4.1 BandBand--reject reject filter filter


5.4.2 5.4.2 BandpassBandpass filterfilter

• Obtained form band reject filter• Obtained form band reject filter

Hbp(u, v)=1-Hbr(u, v)

• To isolate an image of certain frequency band.

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5.4.3 Restoration using frequency-domain filterBandpass filter

5.4.3 Restoration using frequency-domain filterBandpass filter


5.4.4 Restoration using frequency-domain filter- Notch filtering

• Notch filter rejects (passes) frequencies in d fi d i hb h d bpredefined neighborhoods about a center

frequency. For notch rejects filter as

where

otherwise

DvuDorDvuDifvuH

1

),(),(0),( 0201

2/120

201 )2/()2/(),( vNvuMuvuD

2/120

202 )2/()2/(),( vNvuMuvuD

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5.4 Restoration using frequency-domain filter-Notch filtering

• Butterworth notch reject filter• Butterworth notch reject filter

• Gaussian notch reject filter

n

vuDvuDD

vuH

),(),(1

1),(

21

20

20

21 ),(),(

2

1

1)( D

vuDvuD

evuH

Hnp(u, v)=1-Hnr(u, v)

• It becomes a low-pass filter when u0=v0=0

01),( evuH


5.4 Restoration using frequency-domain filterNotch filtering

5.4 Restoration using frequency-domain filterNotch filtering

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5.4.4 Notch filtering5.4.4 Notch filtering

Use 1-D Notch pass filter to find the horizontal ripple noise


5.4.4 Optimal Notch filtering

• Clearly defined interference is not common.

• Images from electro-optical scanner are corrupted by periodic degradation.

• Several interference components are present.

• Place a notch pass filter H(u, v) at the location of each spike, i.e., N(u, v)=H(u, v)G(u, v), where G(u, v)is the corrupted image.

(x, y)=F-1{H(u, v)G(u, v)}.

• The effect of components not present in the estimate of (x, y) can be minimized.

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5.4.4 Optimal Notch filtering5.4.4 Optimal Notch filtering


5.4.4 Optimal Notch filter

• The restored image is ),(),(),(),(ˆ yxyxwyxgyxf

• The weighting function w(x, y) is found so that the variance of is minimized for a selected neighborhood.

• One way is to select w(x, y) so that the variance of the estimate is minimized over a small local neighborhood of size (2a+1, 2b+1).

),(ˆ yxfg ( , )

• ∂σ2 (x, y) /∂ w(x, y)=0 → to select w(x, y)

22 ),(ˆ),(ˆ)12)(12(

1),(

b

bt

a

as

yxftysxfba

yx

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a=b=15



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),(),(),(),(ˆ yxyxwyxgyxf


5.5 Linear Position Invariant Degradation5.5 Linear Position Invariant Degradation

• The System Model η(x y)• The System Model

Hf(x, y)

η(x, y)

g(x, y)

g(x, y): the degraded image

f( ) h i i l if(x, y) : the original image

η(x, y): additive noise

H: System function which is a linear operator

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5.5 Linear Position Invariant Degradation

g(x y)=H[f(x y)]+(x y)g(x, y)=H[f(x, y)]+(x, y)

Assume H is linear and (x, y)=0

If g(x, y)=H[f(x, y)] is position invariant then

H[f(x-, y-)]=g(x-, y-)

( ) H[f( )]+ ( )g(x, y)=H[f(x, y)]+ (x, y)

= +(x, y)

=h(x, y)*f(x, y)+ (x, y) ddyxhf ),(),(


5.6 Estimating the Degradation Function

• To estimate the degradation functionTo estimate the degradation function

by Observation

by Experimentation

by Mathematical modeling

(Blind deconvolution)( )

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5.6 Estimating the Degradation Function

• By observation: construct an unblurredBy observation: construct an unblurred image of some strong signal content.

• Let the observed image be gs(x, y)

• The constructed image is

• Assume the noise is negligible),(

~yxfs

g g

• Find the degradation function H(u, v) which is similar to

),(~

),(),(

vuF

vuGvuH

s

ss


5.6 Estimating the Degradation Function5.6 Estimating the Degradation Function

• By experimentation: to obtain the impulse y p presponse of the degradation by imaging an impulse (small dot of light) using the same system setting.

• The Four Transform of an impulse is a constant A

• Let g(x, y) be the observed image.• Find the degradation function as

H(u, v)=G(u, v)/A

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5.6 Estimating the Degradation Function 5.6 Estimating the Degradation Function --by modelingby modeling

• By Modeling: through experience• The physical characteristic of turbulence as

H(u, v)=e-k(u2+v2)5/6

where k is a constant which depend on the nature of turbulence.

• It is similar to the Gaussian Low pass• It is similar to the Gaussian Low pass. k=0.0025 (severe turbulence)k=0.001k=0.00025 (low turbulence)

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5.6 Estimating the Degradation Function -by modeling

• Example:Example:

Image f(x, y) is blurred by uniform motion.

• x0(t) and y0(t) are the time varying component in x, y directions.

• The total exposure at any point of the film is p y pobtained by integrating the instantaneous exposure over the time interval during which the shutter is opened.

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5.6 Estimating the Degradation Function-by modeling

• Assume T is duration of exposure the blurred image

x-5 x-4 x-3 x-2 x-1 x

Horizontal Motion direction y0=0

exposure, the blurred image g(x, y) is

• Fourier Transform of g(x, y) is

T

dttyytxxfyxg0

00 )](),([),(

dxdyvyuxjyxgvuG )](2exp[)()(

Exposure pixel

x0(t)=at/T, T=constant a=velocity

dxdyvyuxjxpedttyytxxf

dxdyvyuxjyxgvuG

T

)](2[)](),([

)](2exp[),(),(

0

00



• Reverse the order of integrationT

• The term inside the outer brackets is Fourier transform of the displacement

• Therefore

0 0

0

( , ) [ ( ), ( )]exp[ 2 ( )]G u v f x x t y y t j ux vy dxdy dt

T

),(),())]()((2exp[),(),(

))]()((2exp[),(),(

0

00

0

00

vuHvuFdttvytuxjvuFvuG

dttvytuxjvuFvuG

T

T

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• Assume uniform motion in x direction only, i.e.,(t) t/T (t) 0 is l it T d tix0(t)=at/T, y0(t)=0, a is velocity, T=exposure duration

• Simplify as0 00

( , ) exp[ 2 ( ( ) ( ))]T

H u v j ux t vy t dt 00

( , ) exp[ 2 ( ))]

[ 2 / )] i ( )

T

T j ua

H u v j ux t dt

Tj t T dt

• Problem: when u=n/a, H(u, v)=0

0exp[ 2 / )] sin( ) j uaj uat T dt ua e

ua



• If we allow y component movement with• If we allow y-component movement, with the motion given by y0=bt/T then the degradation function is

)())(sin()(

),( vbuajevbuab

TvuH

)( vbua

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5.7 Inverse filtering5.7 Inverse filtering

• With degraded image: nfHg • With degraded image:

• Our goal is to find such that approximate g in a least square sense

nfHg

fHgn2

ˆ

fHˆf̂

)fH(g)fH(gfHgn T22 ˆˆˆ


5.7 Inverse filtering

• Minimizing by using)ˆ(fn2

J 0ˆ/)ˆ( ffJ

gHgH)(HHgHH)(Hf 1T1T1T1T ˆ

)(G

0)ˆ(2ˆ/)ˆ( fHgHff TJ

• Or we may have),(

),(),(ˆ

vu

vuvu

H

GF

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• Without noise ),(),(),( vuvuvu FHG

• We have

• With noise

),(),(

),(),(ˆ vu

vu

vuvu F

H

GF

),(),(),( vuvuvu FHG

),(),(),(),( vuvuvuvu NFHG

• We have

• Restored image

),(

),(),(),(ˆ

vu

vuvuvu

H

NFF

)},(ˆ{),(ˆ vuyxf F-1F



F(u,v)

1/H(u,v)u, v

( )

u, v

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5.7 Inverse filtering5.7 Inverse filtering

Degradation function:

H(u,.v)=e-k(u2+v2)5/6

K=0.0025, M=N=480


5.8 Minimum Mean Square Error (Wiener) Filtering5.8 Minimum Mean Square Error (Wiener) Filtering

• Incorporate both the degradation function and statistical characteristics of noise into restoration

• Considering both the image and noise as random processes, and find an estimate of the uncorrupted image f (x y) such that the

),(ˆ yxfof the uncorrupted image f (x, y) such that the mean square error between them is minimized

• The error measure is e2=E{(f(x, y) – )2}),(ˆ yxf

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• Minimizing the error ∂e2 /∂f=0 we haveˆE{[f(x, y)– ]g(x’, y’)}=0

• For image coordinate pair (x, y) and (x’, y’), we assume the restoration filter is hR(x, y), and have

• Assume the ideal image and observed image are

),( yxf

( ) ( ) ( ) ( ) ( )x,y x',y' x,y x',y' x α,y β dα dβRE f g E g g h g g

jointly stationary, the expectation term can be expressed as covariance function as

ddyxhyxKyyxxK Rggfg ),()','()','(


5.8 Minimum Mean Square Error (Wiener) Filtering

• Take Fourier Transform we have• Take Fourier Transform we have

HR(u, v)=Wfg(u, v)Wgg-1(u, v)

with additive noise

Wfg(u, v)=H*(u, v)Sf(u, v)

Wgg(u, v)= |H(u, v)|2 Sf (u, v)+S(u, v)f

• H(u, v): the degradation function

• S(u, v)=|N(u, v)|2=power spectrum of the noise

• Sf(u, v)=|F(u, v)|2=power spectrum of the signal

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Assume the image and noise are uncorrelated

),(),(*

),(),(),(),(

),(),(*),(ˆ

2

2

vuGvuH

vuGvuSvuHvuS

vuSvuHvuF

f

f

),(),(/),(),(

),(

),(

1

),(),(/),(),(

2

2

2

vuGvuSvuSvuH

vuH

vuH

vuGvuSvuSvuH

f

f



• H(u, v): the degradation function( , ) g

• S(u, v)=|N(u, v)|2= power spectrum of the noise

• Sf(u, v)=|F(u, v)|2= power spectrum of the undegraded function

• If noise is zero, then it becomes an inverse filter.

• For a white noise |N(u, v)|2=constant then

),(),(

),(

),(

1),(ˆ

2

2

vuGKvuH

vuH

vuHvuF

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5.8 Minimum Mean Square Error

(Wiener) Filtering

5.8 Minimum Mean Square Error

(Wiener) Filtering

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5.9 Constraint Least Squares Filtering5.9 Constraint Least Squares Filtering

• Difficulty for Wiener filter: the power spectra of the undegraded image and the noise must be knownundegraded image and the noise must be known.

• Only the mean and variance of the noise are required

• Given a noisy image in vector form

with g(x, y) has a size MN, the matrix H has dimension MN MN and H is highly sensitive to noise.

• Base optimality of restoration on a measure of smoothness,

nfHg

p y ,such as the 2nd derivative of an image, i.e.,

• Find the minimum of C subject to the constraint

1

0

1

0

22 ),(M

x

N

y

yxfC

22ˆ ηfHg



• The frequency domain solution to this optimization problem isproblem is

where is a parameter that must be adjusted so that the constraint is satisfied.

)v,u(G)v,u(P)v,u(H

)v,u(*H)v,u(F 22

22ˆ ηfHg

• P(u,v) is Fourier transform of

010

141

010

),( yxp

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• To compute by iteration as follows:p y

Step 1. Define the residue vector r:

is a function of , so is r,

()=rTr=||r||2 is a monotonically increasing function of

Step 2 Adjust so that ||r||2 = || ||2 a

fHgr ˆ),(ˆ vuF

Step 2. Adjust so that ||r||2 = || ||2 a

where a is a accuracy factor

Step 3. If ||r||2 || ||2 then the best solution is found.

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• Because () is monotonic finding is not difficult• Because () is monotonic, finding is not difficult.

Step 1: Specify an initial

Step 2: Compute ||r||2

Step 3: if ||r||2 = || ||2 a is satisfied then stop

Step 4: if ||r||2 < || ||2–a increase and go to step 2Step 4: if ||r|| < || || a increase and go to step 2.

Step 5: if ||r||2 > || ||2–a decrease and go to step 2.



• To compute the ||r||2 and || ||2

• The vector form can also be rewritten as

• Compute the Inverse Fourier transform of R(u, v) and have

• Consider the variance of the noise over the entire image, i h l h d

),(ˆ),(),(),( vuFvuHvuGvuR

1

0

1

0

22),(

M

x

N

y

yxrr

using the sample-average method

where

• So we have

1

0

1

0

22 ),(1 M

x

N

y

myxMN

1

0

1

0

),(1 M

x

N

y

yxMN

m

221

0

1

0

22),( mMNyx

M

x

N

y

η

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5.10 Geometric Mean Filter5.10 Geometric Mean Filter

1

Generalized form of Wiener filter:

),(

),(

),(),(

),(*

),(

),(*),(ˆ

22 vuG

vuS

vuSvuH

vuH

vuH

vuHvuF

f

, are positive real constant

=1 reduces to inverse filter

=0 becomes parametric Wiener filter

=0, =1 standard Wiener filter

=1/2, =1 spectrum equalization filter

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5.11 5.11 Image Reconstruction From ProjectionImage Reconstruction From Projection

• Introduction• Reconstructing an image from a series of projections.• Suppose image (2-D area) represents an cross-section

of human body.• Assume a single object on a uniform background.• The background in the image represents soft, uniform

tissue, whereas the round object is a tumor with higher absorption characteristicsabsorption characteristics.

e.g., X-ray Computed Tomography (CT)• Project the 1-D signal back across a 2-D area: duplicate

the 1-D signal in all column of the reconstructed image. backprojection



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• Rotate the source-detector pair by 90 degree.• A back-projection in vertical direction.• Assume the number of back-projections increase• Adding the all back-projections.



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5.11.2 Principal of Computed Tomography5.11.2 Principal of Computed Tomography

• X-ray Computed Tomography obtain a 3-D representation of the internal structure of an object byrepresentation of the internal structure of an object by X-raying the object from many directions.

• 2-D projections can be used to reconstruct the 3-D structure.

• The number of detectors the 1-D projection CT more practicalmore practical.

• Randon transform: in 1917, Johann Randon.

• Allan Cormack in 1963 applied the theory to build a CT prototype.

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• 1st generation CT scanner: a “pencil” X-ray beam and a single detector. The source-detector pair translates and then rotates as shown in Fig. 5.35 (a)

• 2nd generation CT scanner: a fan-shape beam, multiple detectors an fewer translation. i.e., Fig. 5.35 (b).

• 3rd generation CT scanner: a bank of detectors to cover the entire field of view of a wider beam (no translation required), i.e., Fig. 5.35 (c).

• 4th generation CT scanner: a circular ring of detectors to cover the entire field of view of a wider beam (no translation required), i.e., Fig. 5.35 (d).



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• 5th generation Electronic Beam CT (EBCT): no mechanical motion required. Tungstan anodes circles the patient which

t f b d it i f d t tgenerate a fan beam and excites a ring of detectors.

• The patient need to be stationary during the scanning time. The scanning is halt while the position of patient is incremented.

• 6th generation helical CT scanner : the source/detector pair rotates continuously 360 while patient is moving.

• 7th generation Multi-slice CT scanner: “thick” fan beam are used in conjunction to the parallel banks of detectors to collect volumetric CT data simultaneously.

• It generates 3-D cross-sectional “slabs” rather than single cross-sectional images. It utilizes X-ray tube more efficiently and less X-ray dosage.


5.11.3 Projections and Random Transform5.11.3 Projections and Random Transform

• CT imaging: X-ray, SPECT, PET, MRI• A straight line in Cartesian coordinate y=ax+b can beA straight line in Cartesian coordinate y ax b can be

described in polar coordinate as xcosθ +ysin θ =ρ

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5.11.3 Projections and 5.11.3 Projections and Random TransformRandom Transform

•A projection of a parallel beam can be modeled as a set of lines as Figure 5.37•An arbitrary point in the projection signal can be given by a•An arbitrary point in the projection signal can be given by a ray sum along the line xcosθk +ysin θk =ρj (i.e., L(j, k))



• The ray-sum is a line integral

dxdyyxyxfg jkkkj )sincos(),(),(

• The δ function indicates that the integral is computed along the line xcosθk +ysin θk =ρj. We consider all θ and ρ as

• In discrete case, it can be described as

yyyfg jkkkj )(),(),(

1 1

)i()()(M N

f

dxdyyxyxfg )sincos(),(),(

where x, y, θ, and ρ are discrete variable.

• Random transform: It generates the projection of f(x, y) along an arbitrary line in the xy-plane i.e., R{f(x, y)}=g(ρ, θ).

0 0

)sincos(),(),(x y

yxyxfg

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5.11.3 Projections and Random 5.11.3 Projections and Random TransformTransform

Example 5.17:

ryxA 222

where A is a constant, r is the radius of the object.

Since the object is circularly symmetry, th j ti i th f ll

otherwise

ryxAyxf

0),(

the projection is the same for all directions.

Let θ=0 we have

dyyfdxdyxyxfg ),()(),(),(



Example 5.17 (cont.)

• Since g(ρ, θ)=0 when ρ > rg(ρ, ) ρ

• The integral can be evaluated between r ρ r or

from to

• So we have

22 ry 22 ry

22

22

22

22),(),(

r

r

r

rAdydyyfg

rrA2 22 or

• g is independent of θ because the object is symmetry about the origin.

otherwise

rrAgg

0

2)(),(

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• The Random transform g(ρ, θ) is displayed as an image with ρand θ as rectangular coordinate, it is called a sinogram as h i Fi 5 39 ( i il t F i T f )shown in Figure 5.39. (similar to Fourier Transform).

• Sinogram is symmetric in both directions about the center of image object is symmetric and parallel to x and y axes.

• Sinogram is smooth object has a uniform intensity.

• CT Obtain a 3-D representation of a volume from its projections.

• Back-project each projection and then sum all the projections to generate the image (slice).

• Stacking all resulting images (slices) 3-D rendering of the volume.



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• A single point g(ρj, θk) of the complete projection g(ρ, θk ) with fixed θk as shown in Figure 5.37.

F i f i b b k j i hi i l i

5.11.3 Projections and 5.11.3 Projections and Random TransformRandom Transform

• Forming part of an image by back-projecting this single point g(ρj, θk) Copying the line L(ρj, θk) onto the image where each point in that line is g(ρj, θk).

• Repeat the process for all ρj (fixed θk) we have

fθk(x, y)=g(ρ, θk)=g(xcosθk+ysinθk , θk)

Th i f d f i l b k j i b i d• The image formed from a single back-projection obtained at an angle θ is fθ(x, y)=g(xcosθ+ysinθ, θ)

• We form the final image by integrating over all the back-projected images

• Or in discrete case

dyxfyxf ),(),(0

),(),(0

yxfyxf



Example 5.18

is used to generate the back-projected ),(),( yxfyxf

g p j

image from projections g(, ) as shown in Figures 5.32~5.34

and Figure 5.40.

),(),(0

yfyf

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5.11.4 The 5.11.4 The Fourier Slice TheoremFourier Slice Theorem

• 1-D transform of a projection with respective to ρ is

degG j 2),(),(

dxdydeyxyxfG j2)sincos(),(),(

dxdydeyxyxfG j 2)sincos()()( dxdydeyxyxfG j

)sincos(),(),(

dxdyeyxfG yxj

)sincos(2),(),(


5.11.4 The Fourier Slice Theorem5.11.4 The Fourier Slice Theorem

• By letting u=ωcosθ and v=ωsinθ we have

• Or

sin,cos

)(2),(),(

vu

vyuxj dxdyeyxfG

)sin,cos(),(),( sin,cos FvuFG vu

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5.11.4 The 5.11.4 The Fourier Slice TheoremFourier Slice Theorem

degG j2),(),(

)sin,cos(),(),( sin,cos FvuFG vu


5.11.5 Reconstruction 5.11.5 Reconstruction using Parallelusing Parallel--BeamBeam

• 2-D inverse Fourier transform

• By letting u=ωcosθ and v=ωsinθ and dudv= ωdωdθ

dudvevuFyxf vyuxj

)(2),(),(

ddeFyxf yxj

)sincos(2)sin,cos(),(

ddGf yxj )sincos(2)()(

• Since G(ω, θ+180o)=G(ω, θ), we have

ddeGyxf yxj

)sincos(2),(),(

ddeGyxf yxj

)sincos(2),(),(

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• Integral with respect to ω, the term xcosθ+ysinθ is a constant, treated as xcosθ+ysinθ =ρtreated as xcosθ+ysinθ ρ

• The added term ω is recognized as a 1-D filter function a ramp filter which is not integrable (as shown in Figure 5.42)

• Use a box (or window) to band limit the ramp filter

ddeGyxfyx

j

sincos

2),(),(

• Use a box (or window) to band-limit the ramp filter.

• Windowing the ramp so that it becomes zero outside a defined frequency interval (as shown in Figure 5.42(d) ).

• Filtered back-projection



otherwise

MM

cchwindow0

)1(01

2cos)1()(

otherwise0

h()( )

Hamming window: c=0.5

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• G(ω, θ) is the 1-D transform of g(ρ, θ) which is a single projection obtained at a fixed angle θ.projection obtained at a fixed angle θ.

• The complete back-projected image f(x, y) is obtained by the following steps:

1. Compute the 1-D Fourier Transform of each single projection at a fixed angle θ, i.e., g(ρ, θ).

2. Multiply each Fourier Transform G(ω,θ) by the filter function hi h i lti li d b it bl i d i f ti ω which is multiplied by a suitable windowing function.

3. Obtain the inverse 1-D Fourier Transform of each filtered transform.

4. Integral (sum) all the 1-D inverse transform from step 3.



Example

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Example



Let s()=F1{} and is the spatial convolution

ddyxsg

dgs

ddeGyxf

yx

yx

j

0

sincos0

sincos0

2

)sincos(),(

),()(

),(),(

ygyx

0 sincos

)()(

• Individual back-projection at angle can be obtained byconvolving the corresponding projection of g(, ) and s() .

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5.11.6 Reconstruction 5.11.6 Reconstruction using Fanusing Fan--BeamBeam

Let p(, ) denote the fan beam projection.

is the angular position of a particular detector measured with t t th t i th l di l t f threspect to the center ray. is the angular displacement of the

source measured with respect to the y-axis

L(ρ, θ) is a normal form projection ray of p(, ) .y p( , )

with θ= +and ρ = Dsin


5.11.6 Reconstruction 5.11.6 Reconstruction using using FanFan--Beam filtered Beam filtered backprojectionbackprojection

We assume that object encompass within a circular area of radius T about the origin of plane, g(, )=0 for >Tg p g( )

The convolution back-projection formula for the parallel-imaging becomes

ddyxsgyxfT

T

2

0)sincos(),(

2

1),(

Integrating with respect to and L t i d + i ( )Let x = r cos , y = r sin , and xcos +ysin = r cos( ),

We have

ddrsgyxfT

T

2

0)))(cos(),(

2

1),(

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Using θ= + and ρ = D sin , we change the integral from f(x y) to f(r ) based on x = r cos y = r sin

5.11.6 Reconstruction using Fan5.11.6 Reconstruction using Fan--Beam filtered Beam filtered backprojectionbackprojection

f(x, y) to f(r, ) based on x r cos , y r sin ,

ddD

DrsDgrfDT

DT

cos

sin)cos(),sin(2

1),(

2 )/(sin

)/(sin

1

1

Figure 5.46 shows that sin1(T/D) has maximum value m

ddD

Drsprfm

m

cos

sin)cos(),(2

1),(

2

0

p(, ) = g(ρ, θ) or p(, ) = g(Dsin , +)


5.11.6 Reconstruction 5.11.6 Reconstruction using using FanFan--Beam filtered Beam filtered backprojectionbackprojection

sin1(T/D) has maximumsin (T/D) has maximum value of m corresponding to>T

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5.11.6 5.11.6 Image Reconstruction Image Reconstruction using Fanusing Fan--Beam Beam filtered filtered backprojectionbackprojection

Rcos(+) Dsin = Rsin()

R and are determined by r R and are determined by r, ,


• By substituting


2

cos)'sin(),(1

),( ddDRsprfm

where and q( ) = p( )Dcos

2

0 2)'(),(

1

2

1),( ddhq

Rrf

m

m

)(sin

sin2

sR

Rs

2

1

h

0

)(),(2

),( pfm

where and q(, ) p(, )Dcos

• A fan-beam projection p taken at angle corresponding to parallel beam projection g taken at corresponding angle

p(, ) = g(, ) = g(D sin, +)

sin2

h

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p(,) = g(, ) = g(Dsin, +)

• is angular increment between two successive fan beam• is angular increment between two successive fan-beam

projections.

• is angular increment between two rays it determines the

number of samples in each projection.

• Let = = , = n , = mLet , n , m

p(n , m) = g[Dsin n , (m+n)]

• The nth ray in the mth radial projection

= the nth ray in the (m+n)the parallel projection



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5.12 Geometric Transformation5.12 Geometric Transformation

• It is also called Rubber sheet transform• It is also called Rubber sheet transform, which consists of two operations:

1) Spatial transformation: rearrangement of the pixels (locations) on the image.

2) Gray-level interpolation: assignment of ) G y p ggray levels to pixels in the spatially transformed image.

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5.12 Geometric Transformation-Geometric Distortion





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5.11 Geometric TransformationSpatial transformation

5.11 Geometric TransformationSpatial transformation

Mapping function T : maps a point from screen space to a point in texture spacescreen space to a point in texture space

Screen space Texture space


5.12 5.12 Geometric TransformationGeometric TransformationSpatial transformationSpatial transformation

• The transformation T may be expressed as

x’=r(x, y)=c1x+c2y+c3xy+c4

y’=s(x, y)=c5x+c6y+c7xy+c8

Find the tiepoints, which are subset of pixels whose locations in the distorted image (in

) d th t d i (iscreen space) and the corrected images (in texture space) are known precisely.

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5.12 Geometric Transformation-Spatial transformation

• General Transformation:• General Transformation:

h

x y w u v w T' ' ' 1

131211 SSS

where

333231

2322211

SSS

SSST



• T is said to be one-to-one if:• T1 is said to be one-to-one if:(a) The inverse transformation of T1 always exists.

(b) With T1, one point in screen space produces only one point in texture space

(c) Through T1-1, each point in texture space can find

th di i t ithe corresponding point in screen space.

=> that is the determinate of T1 is nonzero.

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• The 3×3 transformation matrix can be bestThe 3×3 transformation matrix can be bestunderstood by partitioning it into fourseparate sections.

• Rotation:

2221

12112 SS

SST

• Translation:• Perspective transformation:• Scaling:

S S31 32

S ST

13 23

S33


5.12 5.12 Geometric TransformationGeometric Transformation--Spatial transformationSpatial transformation

• The general representation of an Affine transformation is :

1

0

0

]1[]1[

3231

2221

1211

aa

aa

aa

vuyx

• It is a planar mapping, that can be used to map an input triangle to any arbitrary triangle at the output

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Affine Transformation




Perspective Transformation

333231

232221

131211

],,[],,[

aaa

aaa

aaa

wvuwyx

312111 avaua and let w=1

322212/avaua

332313

312111/avaua

avauawxx

332313

322212/avaua

avauawyy

If then it becomes Affine Transformation02313 aa

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5.12 5.12 Geometric TransformationGeometric Transformation--Spatial transformationSpatial transformation

• Perspective projectionp p j

X axis

Y axis

P(X,Y,Z)

YZ axis

(X ,Y ,0)P P

Z

D

xZ axis

Center of prjection

Projection Plane


5.12 5.12 Geometric TransformationGeometric Transformation--Spatial transformation

• Any position along the projection line is• Any position along the projection line is (α,β,γ) is

where δ=Z/(Z+D) and 0<δ<1

X X

Y Y

Z Z D( )

where δ Z/(Z+D), and 0<δ<1

)1

1(

D

ZXX p )

1

1(

D

ZYYp

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• Perspective Transformation



• Bilinear transformation is defined asf

where W=[u, v, 1] and Z=[x, y, 1]

ZT W: TWZ

0

0311

b

vbba

T

and are constant.

1

0

00

232

ba

buaaT

3210,3210 and,,,,,, bbbbaaaa

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Bilinear distortion for reference quadrangle and pointsBilinear distortion for reference quadrangle and points.

P

P

P

P

1

2

3

v

Y

T -1

P4

P1

P2 P3

P4' '

'(0,1) (1,1)

(1,0)(0,0)

u

X

T


5.12 Geometric Transformation-gray-level interpolation

5.12 Geometric Transformation-gray-level interpolation

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5.12 5.12 Geometric TransformationGeometric Transformation--gray-level interpolation

• Zero-order interpolation: the simplest one.• Bilinear interpolation: using four neighbors to do• Bilinear interpolation: using four neighbors to do

the interpolation for the gray level of non-integer point at (x’, y’):

v(x’, y’)=a+bx’+cy’+dx’y’• The four neighbors located at (xi, yi), with known

gray level vii i=1 and 2, then we have four different g y iiequations as

vii=a+bxi+cyi+d xi yi

• Solving the above four equations for the four parameters, a, b, c, and d.



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5.12 5.12 Geometric TransformationGeometric Transformation--graygray--level interpolationlevel interpolation

• resamplingInput Samples Output samples

Spatial Transformation

Output Grid

Image Reconstruction

Resampling Grid

Reconstructed Signal



• resamplingf(u)

uDiscrete Input Discrete output

g(x)

x

b( )Sample s(x)

f (u)c g (x)c

u xRecostructed Input

Warping Prefiltering

x=T(u) h(x)

Reconstructb(x)p ( )

Warped Input Continuous Output

g (x)c'

x

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Ideal warping resampling

c

cc

dttxhjtTbjf

dttxhtTf

Zxh(x(x)gxgxg

)())(()(

)())((

for ) )( )(

1

1

Zj

Zj

jxjf ),()(=

dttxhjtTbjx )())((),( 1where



Table : Elements of ideal resampling.

f u u Z( ),

f u f u b u f j b u jcj Z

( ) ( )* ( ) ( ) ( )

1

Stage Mathematical definition

Discrete Input

Reconstructed Input

Warped Signal g x f T xc c( ) ( ( )) 1

dttxhtgxhxgxg ccc )()()(*)()(

g x g x s xc( ) ' ( ) ( )

Warped Signal

Continuous Output

Discrete Output

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5.12 5.12 Geometric TransformationGeometric Transformation--graygray--level level interpolationinterpolation

• Cubic convolution interpolation:using much larger number of neighbors (i e 16) to obtain alarger number of neighbors (i.e., 16) to obtain a smoother interpolation.

• Ideal Interpolation:

Reconstruction

Filter

Warped

PrefilterSampler

f(u) f (u)c g (u)c' g(x)

b(u) h(u)

Filter Prefilter

WarpResampling Filter

s(T (x))-1

s(x)



spatial domain processing remove the additive noise

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