Chapter 5 Ip

7/27/2019 Chapter 5 Ip

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Image Restoration

Chapter 5

Prof. Vidya Manian

De t. of Electrical and Com uter En ineerin

INEL 5327 ECE, UPRM1


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Overview A model of the Image Degradation/Restoration Process

Noise Models Restoration in the Presence of Noise Only-Spatial

Periodic Noise Reduction by Frequency Domain Filtering

Estimatin the De radation Functions

Inverse Filtering

Minimum Mean Square error(Wiener) Filtering

Constrained Least Squares Filtering Geometric Mean Filter



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A model of the Image

egra on es ora on rocess

f(x,y) : image function

g(x,y): degraded image

(x,y): additive noise term



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A model of the Image

egra on es ora on rocess

Is H is a linear, position-invariant process, then thedegraded image is given in the spatial domain

g(x,y) = h(x,y) * f(x,y) + (x,y) (5.1-1)

Frequency domain representation

u,v = u,v u,v u,v . -



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Noise Models - Gaussian Noise

(5.2-1)

z: gray level

: mean

: standard deviation

2: variance



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Noise Models - Gaussian Noise



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Noise Models - Ra lei h Noise

(5.2-2)for z>= a

for z < a



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Noise Models - Ra lei h Noise



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Noise Models Erlan Gamma Noise

(5.2-5)for z>= 0

for z < 0



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Noise Models - Erlan Gamma Noise



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Noise Models Ex onential Noise

(5.2-8)for z>= 0

for z < 0



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Noise Models - Ex onential Noise



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Noise Models Uniform Noise

(5.2-11)for a


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Noise Models - Uniform Noise



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Noise Models Impulse (salt-and-

pepper o se

(5.2-14)for z = a

for z = b

otherwise



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Noise Models - Impulse (salt-and-

pepper o se



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Noise Models Periodic Noise



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Restoration in the Presence of Noise

n y- pa a er ng

x = f x + x 5.3-1

Frequency domain representation

G(u,v) = F(u,v) + N(u,v) (5.1-2)



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Mean Filters Arithmetic mean filterThe simplest of mean filters.

(5.3-3)

window of size m x n, centered at point (x,y).

The value of the restored image at any point (x,y) issimply arithmetic mean computed using the pixels in theregion defined by Sxy.



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Mean Filters Geometric mean filter

(5.3-4)

Each restored pixel is given by the product of thepixels in the subimage window, raised to the power

mn.



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Mean Filters Harmonic mean filter

(5.3-5)

Works well for salt noise, but fails for pepper noise.



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Mean Filters Contraharmonic mean

er

(5.3-6)

Q: order of the filter.

Works well for reducing or virtually eliminating the

effects of salt-and-pepper noise.Q positive eliminates the pepper

Q negative eliminates the saltINEL 5327 ECE, UPRM22


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Mean Filters Exam les I



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Mean Filters Exam les II



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Order Statistics Filters

r er tat st cs ters xpress on

Median Filter

Max Filter

Min Filter

Mid Point Filter

Alpha-trimmed mean filter



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Order Statistics Filters Exam le I



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Order Statistics Filters Exam le II



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Ada tive Filters Adaptive, local noise reduction filter

g(x,y) the value of the noisy image at

(x,y)

variance of the noise corrupting

f(x,y) to form g(x,y)

the local mean of the pixels in Sxy

the local variance of the pixels inSxy



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Ada tive Filters Adaptive median filter

Level A: A1 = Zmed Zmin

if A1 > 0 AND A2 < 0, Go to level B

If window size 0 AND B2 < 0, output Zxy

Else output Zmed



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Periodic Noise Reduction

energy in the Fourier transform, at locationscorresponding to the frequencies of the periodicinterference

Bandreject filters General location of the noise component(s) in the

frequency domain is approximately known

Bandpass filter performs



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Periodic Noise Reduction by Frequency

oma n er ngBandre ect Filters

Ideal

Butterworth bandreject

Gaussian Filters

Bandpass Filters



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Periodic Noise Reduction by Frequency

oma n er ng xamp e



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the Fourier spectrum

Butterworth bandreject filter of order 4 with

the noise impulses

from the transform



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common because it removes too much image detail

Notch filters

neighborhoods about a center frequency

Shape of the notch can be arbitrary (eg.,

INEL 5327 ECE, UPRM


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Notch FiltersNotch Filters RejectRadius D0 with centers at (u0, v0), by simemetry at (-u0, -v0)

Ideal

Butterworth bandreject

Gaussian Filters

These three filters become highpass filters if u0 = v0 = 0

Notch Filters Pass



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spatial domain after passing through the notch filter

Fig 5.20 shows starlike components caused by

Optimum method to minimize local variance of the

Extract principal frequency component of

=

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Formation of HNP(u,v) requires judgment about

Corresponding pattern in spatial domain after

n(x,y)=F-1{HNP(u,v)G(u,v)}

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Notch Filters Exam le



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Periodic interference39

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Estimatin the de radation function By observation, experimentation and mathematical modeling Blind deconvolution: rocess of restorin an ima e b usin a

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degradation function that bas been estimated in some way

By observation deduce complete degradation function H(u,v)ase on assumption o position invariance

Perform experiment with the equipment used to acquire the

Obtain impulse response of the degradation. A linear, space-invariant system is characterized completely by its impulse

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

u,v s t e o t e o serve mage an s a constant

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Estimation b modelin

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turbulence

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Estimatin the De radation Function

Because the true degradation function is seldom known completely, the process of restoringimage is ca e in econvo ution .

Let assume that noise is negligible, so our principal equation (5-1.1) will be:

g(x,y) = h(x,y) * f(x,y)

By Image Observation

By Experimentation

By Modeling

For example a model for atmospheric turbulence:


E ti ti th D d ti F ti


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Estimating the Degradation Function

xamp e



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Inverse Filterin

A simple approach is direct inverse filtering to restore the image. Just dividingt e trans orm egra e unction G(u,v), y t e egra ation unction:

If we replace G(u,v) by F(u,v)H(u,v)+ N(u,v), we obtain:

As ou see we cannot recover the unde raded ima e exactl because N u v

is a random function whose Fourier transform is not known. If H(u,v) is zero orvery small values, then the ratio N(u,v)/H(u,v) would dominate the estimate



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By limiting the analysis to frequencies near origin,

Fig. 5.27: Inverse fitlering

G(u,v)/H(u,v) outside a radius of 40, 70 and 85

Cutoff is implemented by applying to the ratio a

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Inverse Filterin Exam le


Minimum Mean Square Error (Wiener)


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Minimum Mean Square Error (Wiener)

er ng

noise into the restoration process.

degradation function =

com lex

conjugate of =

=

power spectrum of the

noise =

If noise is zero, the power spectrum vanishes and Wiener filter reduces to the inverse

power spectrum o t e

undegraded image =

filter. If we consider a as a constant, we have the expression reduces to:

K is a specified constant.



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Wiener Filterin Exam le I



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Wiener Filterin Exam le II



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by the inverse Fourier transform of the frequencydomain estimate

A number of useful measures are based on the

power spectra of noise and of the undegradedimage: Signal to-noise ratio, mean square error

Root mean square error

Root mean square signal to noise ratio

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Constrained Least S uares Filterin

.

With Constrained Least Squares Filtering is required to know the mean and

variance of the noise and we can calculate these variables from a given degraded

ima e.

We can express our main expression in vector-matrix form, as follows:

Subject to the constraint:

The frequency domain solution is given by the next expression:


Constrained Least Squares Filtering -


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Constrained Least Squares Filtering

xamp e



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n ro uc on o ompu er

Tomo ra h De t. of Homeland

Security

INEL 5327 ECE, UPRM

Image Reconstruction from


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g

ro ec ons Reconstructing an image from a series of

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projections, X-ray computed tomography (CT)

Imaging a single object on a uniform background

3-D region of a human body round object is a

tumor with higher absorption characteristics

Pass a thin flat beam of X-rays from left to right,energy absorbed more by object than background

Any point in the signal is the sum of the absorptionvalues across the single ray in the beam

correspon ing spatia y to t at point in ormation is

the 1-D absorption signal)INEL 5327 ECE, UPRM


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B k ti


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Back ro ection

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which the beam came-

called smearing

direction of the beam

Reconstruction by adding the of backprojection

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s num er o pro ec ons ncreases , e s reng othe non-intersecting backprojections decreases60

projections intersects,

becomes circular

Projections 180 apart are mirror images of each

,

circle to generate all projections required forreconstruction

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Principles of computed


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omograp y

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of an object by X-raying the object from manydifferent directions

X-ray beam in the form of a cone

-point is proportional to the X-ray energy impingingon that point after it has passed through the subject

2-D equivalent of backprojection

volumeINEL 5327 ECE, UPRM


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the backprojectiosn would result in 3D rendition of

the structure of the chest cavity

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CT gets the same information by generating slicesthrough the body

3D representation can be obtained by stacking the

slices A CT is much more economical, number of detectors

required to get a high resolution slice is much

smaller than generating a complex 2D projection ofthe same resolution

X-ray levels and computation cost are less in the 1D

projection CT approachINEL 5327 ECE, UPRM


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object along parallel rays-

Cormack and Hounsfield shared the 1979 Nobel

tomography

X-ray beam and a single detector

pair is translated incrementally along the linear

directionINEL 5327 ECE, UPRM


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of the detector at each increment of translation

source/detector assembly is rotated and the

procedure is repeated to generate anotherprojection at a different angle. Procedure isrepeated for all angles [0,180]degrees

Image is generated by backprojection. A set of cross-sectional imaages(slices) is generated

by incrementally moving the subject pst the

source/detector planeINEL 5327 ECE, UPRM

Stackin these ima es com utationall roduces a


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Stackin these ima es com utationall roduces a

3D volume of a section of the body

Second generation (G2) CT scanners beam used is

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in the shape of a fan

Requires fewer translations of the source detector

pair

Third-generation (G3) scanners employ a bank ofdetectors long enough to cover the entire field ofview of a wider beam

Fourth generation (G4) scanners-employ a circularring of detection (5000) only the source has torotate

Advantage of G3 and G4 is speed.INEL 5327 ECE, UPRM

Disadvantage-X-ray scatter, higher doses than G1


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an o ac eve compara e Fifth generation (G5)CT scanners-electron beam CT

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e m na es a mec an ca mo on y emp oy ng

electron beams controlled electromagneticallyx genera on e ca - or scanner

is configured using slip rings that eliminate the need

source/detectors and the processing unit

360 deg while the patient is moved at a constants eed alon the axis er endicular to the scan

Results in a continuous helical volume of data that is

rocessed to obtain individual slice ima esINEL 5327 ECE, UPRM


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scanners thick fan beams are used in conjunctionwith parallel banks of detectors to collect volumetricCT data simultaneously

3D cross sectional slabs rather than single ones aregenerated per X ray burst

Uses Xray tubes more economically, reducing cost

and dosage

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INEL 5327 ECE, UPRM

The


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The71

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Geometric Mean Filter


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Geo e c Mea e

following expression:

,


Geometric Transformations


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Modif the s atial relationshi s between ixels in animage.

In this type of transformation we have to approach:

- Spatia trans ormation, which de ines therearrangement of pixels

assignment of gray levels to pixels in the spatiallytransformed image.


Exercises


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Explain why the filter is effective in eliminating

pepper noise when Q is positive.

Explain why the filter is effective in eliminating salt

noise when Q is negative. Discuss the behavior of the filter when Q=-1

intensity levels

geometric mean filtering is less blurredINEL 5327 ECE, UPRM


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currency exchange, finds that the photos of the coins

are blurred to the point where the date and othermarkings are not readable. The blurring is due tothe camera being out of focus when the pictures

were taken. Propose a step-by-steps solution torestore the images to the point that the Professor

can rea t e mar ngs, g ven t e camera use ortaking the photos and also the coins.

INEL 5327 ECE, UPRM

Restoring images degraded due to linear motion (in x-

-


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5.17) Linear motion. Estimate blurring function H(u,v)76

5.19) Images from a craft are blurred due to circular

motion (rapid rotation of the craft about its verticalaxis). The images are blurred by uniform rotational

motion. The craft rotation was limited to /8 radians

uring image acquisition. e image acquisitionprocess can be modeled as an ideal shutter that is

radians. Assuming vertical motion was negligible

restoring the images.INEL 5327 ECE, UPRM

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Chapter 5 Ip