restoration slides for class pdf

8/10/2019 Restoration Slides for Class PDF

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Chapter 5:Image Restoration


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(Images from Rafael C. Gonzalez and Richard E.

Wood, Digital Image Processing, 2ndEdition.

Concept of Image Restoration

Image restoration is to restore a degraded image back tothe original image while image enhancement is to

manipulate the image so that it is suitable for a specific

application.


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Concept of Image Restoration

Degradation model:

),(),(),(),( yxyxhyxfyxg +=

where h(x,y) is a system that causes image distortion and(x,y) is noise.


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PDF: Statistical Way to Describe Noise

PDF tells how mucheach z value occurs.


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Image Degradation with Additive Noise

Original image

Histogram

Degraded images

),(),(),( yxyxfyxg +=


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Original image

Histogram

Degraded images

Image Degradation with Additive Noise (cont.)

),(),(),( yxyxfyxg +=


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Periodic Noise

Periodic noise

looks like dots

In the frequency

domain


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Estimation of Noise

We cannot use the image

histogram to estimate

noise PDF.

It is better to use the

histogram of one areaof an image that has

constant intensity to

estimate noise PDF.


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Periodic Noise Reduction by Freq. Domain Filtering

Band reject filter Restored image

Degraded image DFT

Periodic noise

can be reduced by

setting frequency

componentscorresponding to

noise to zero.


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Band Reject Filters

Use to eliminate frequency components in some bands

Periodic noise from the

previous slide that isFiltered out.


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Notch Reject Filter:

Degraded image DFTNotch filter

(freq. Domain)

Restored imageNoise


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Example: Image Degraded by Periodic Noise

Degraded image

DFT

(no shift)

Restored imageNoiseDFT of noise

G


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Geometric Mean Filter: Example

Original

image

Image

corrupted

by AWGN

Image

obtained

using a 3x3geometric

mean filter

Image

obtained

using a 3x3arithmetic

mean filter

AWGN: Additive White Gaussian Noise

C t h i Filt E l


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Contraharmonic Filters: Example

Imagecorrupted

by pepper

noise with

prob. = 0.1

Imagecorrupted

by salt

noise with

prob. = 0.1

Imageobtained

using a 3x3

contra-harmonic

mean filter

With Q = 1.5

Imageobtained

using a 3x3

contra-harmonic

mean filter

With Q=-1.5


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Order-Statistic Filters: Revisit

subimage

Original image

Moving

window

Statistic parametersMean, Median, Mode,

Min, Max, Etc.

Output image

M di Filt H it k


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Median Filter : How it works

A median filter is good for removing impulse, isolated noise

Degraded image

Salt noise

Pepper noise

Moving

window

Sorted

array

Salt noisePepper noise

Median

Filter output

Normally, impulse noise has high magnitudeand is isolated. When we sort pixels in the

moving window, noise pixels are usually

at the ends of the array.

Therefore, its rare that the noise pixel will be a median value.

M di Filt E l


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Median Filter : Example

Imagecorrupted

by salt-

and-pepper

noise withpa=pb= 0.1

Images obtained using a 3x3 median filter

1

4

2

3

M d Mi Filt E l


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Max and Min Filters: Example

Imagecorrupted

by pepper

noise with

prob. = 0.1

Imagecorrupted

by salt

noise with

prob. = 0.1

Imageobtained

using a 3x3

max filter

Imageobtained

using a 3x3

min filter

Al h t i d M Filt E l


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Alpha-trimmed Mean Filter: Example

Image

corrupted

by additive

uniform

noise

Image

obtained

using a 5x5arithmetic

mean filter

Image

additionallycorrupted

by additive

salt-and-pepper

noise

1 2

2 Image

obtained

using a 5x5geometric

mean filter

2

Alpha trimmed Mean Filter E ample (cont )


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Alpha-trimmed Mean Filter: Example (cont.)

Image

corrupted

by additive

uniform

noise

Image

obtained

using a 5x5median filter

Image

additionallycorrupted

by additive

salt-and-pepper

noise

1 2

2Imageobtained

using a 5x5

alpha-trimmed

mean filter

with d= 5

2

Estimation by Image Observation


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Estimation by Image Observation

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

Subimage

Reconstructed

Subimage

),( vuGs ),( yxgs

),( yxfs

DFT

DFT

),( vuFs

Restorationprocess byestimation

Original image (unknown) Degraded image

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

vuF

vuGvuHvuHs

ss =

Estimated Transfer

function

Observation

This case is used when we

know only g(x,y) and cannot

repeat the experiment!

Estimation by Experiment


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Estimation by Experiment

Used when we have the same equipment set up and can repeat the

experiment.Input impulse image

System

H( )

Response image fromthe system

),( vuG

),( yxg),( yxA

{ } AyxADFT =),(

A

vuGvuH

),(),( =

DFTDFT



Estimation by Modeling


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Estimation by Modeling

Used when we know physical mechanism underlying the image

formation process that can be expressed mathematically.

AtmosphericTurbulence model

6/522 )(),( vukevuH +=

Example:Original image Severe turbulence

k= 0.00025k= 0.001

k= 0.0025

Low turbulenceMild turbulence

Motion Blurring Example


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Motion Blurring Example

For constant motion

)())(sin()(

),( vbuajevbuavbua

TvuH

++

+=

Original image Motion blurred image

a = b = 0.1, T= 1

Wiener Filter: Example


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Wiener Filter: Example

Original image

Blurred image

Due to Turbulence

Result of the

full inverse filterResult of the inverse

filter withD0=70

Result of the

full Wiener filter

Wiener Filter: Example (cont )


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Wiener Filter: Example (cont.)

Original image

Result of the inverse

filter withD0=70

Result of the

Wiener filter

Blurred image

Due to Turbulence

Example: Wiener Filter and Motion Blurring


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Example: Wiener Filter and Motion Blurring

Image

degradedby motion

blur +

AWGN

Result of the

inverse filter

Result of the

Wiener filter

2=650

2=325

2=130

Note: Kischosen

manually

Constrained Least Squares Filter: Example (cont )


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Constrained Least Squares Filter: Example (cont.)

Image

degradedby motion

blur +

AWGN

Result of the

ConstrainedLeast square

filter

Result of the

Wiener filter

2=650

2=325

2=130

Geometric Transformation


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Geometric Transformation

These transformations are often called rubber-sheet transformations:

Printing an image on a rubber sheet and then stretch this sheet accordingto some predefine set of rules.

A geometric transformation consists of 2 basic operations:

1. A spatial transformation :

Define how pixels are to be rearranged in the spatially

transformed image.

2. Gray level interpolation :

Assign gray level values to pixels in the spatiallytransformed image.

Geometric Transformation : Algorithm


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Geometric Transformation : Algorithm

Distorted image g

1. Select coordinate (x,y) infto be restored

2. Compute

),( yxrx =

),( yxsy =

3. Go to pixelin a distorted image g

),( yx

Imagefto be

restored

4. get pixel value atBy gray level interpolation

),( yxg

5. store that value in pixelf(x,y)

13

5

),( yx ),( yx

Geometric Distortion and Restoration Example (cont )


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Geometric Distortion and Restoration Example (cont.)

Original image and

tiepoints

Tiepoints of distorted

image

Distorted image Restored image

Use bilinearintepolation

(Images from Rafael C.

Gonzalez and Richard E.

Wood, Digital Image

Processing, 2ndEdition.

Example: Geometric Restoration


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Example: Geometric Restoration

(Images from Rafael C.

Gonzalez and Richard E.

Original image Geometrically distorted

image

Difference between2 above images

Restored image

Use the same

Spatial Trans.

as in the previousexample

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