05-1 - Image Restoration and Reconstruction

8/2/2019 05-1 - Image Restoration and Reconstruction

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Image restoration and

preliminaries, basic

techniques, and additive

Spring2008 ELEN4304/5365DIP 1

noise onlybyGlebV.Tcheslavski:[email protected]

http://ee.lamar.edu/gleb/dip/index.htm

Preliminaries

Image Restoration includes processes that

attempt to remove degradations and restore an

image to a previous state.

Restoration techniques are oriented towardmodeling the degradation and applying theinverse process to recover the original image.


s s ngu s e rom en ancemenin that degradation can be considered an externalinfluence that is separate from the image signal.


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Preliminaries

Restoration usually involves formulating acriterion of goodness that will give an optimalestimate of the desired result.

Enhancement, on the other hand, usuallymanipulates with subjective criteria.


,removing the image blur when applying a

deblurring function is a restoration.

Model for imagedegradation/restoration process

We will assume that a de radation function exists, which, to ether


with additive noise, operates on the input imagef(x,y) to produce a

degraded image g(x,y).

The objective of restoration is to obtain an estimate for the original

image from its degraded version g(x,y) while having some knowledge

about the degradation functionHand the noise (x,y).


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Model for image

degradation/restoration processIf degradationHis a linear, position-invariant process, the degraded

image in the spatial domain is

( , ) ( , ) ( , ) ( , )g x y h x y f x y x y= +convolution

Therefore, in the frequency domain it is

( , ) ( , ) ( , ) ( , )G u v H u v F u v N u v= +


We will assume first thatHis an identity operator and

all degradation is due to additive noise.

Noise modelsMost of the noise degradation comes to digital images during their

acquisition and/or transmission. A wide variety of factors may affect

. ,

levels and a sensor temperature of a CCD camera are two major

factors determining the amount of noise in the resulting image.

We assume (unless otherwise stated) that noise is independent of

spatial coordinates and that it is uncorrelated with respect to the

image itself (no correlation between pixel values and values of


noise components).

The spatial noise descriptor is the statistical behavior of the intensity

values in the noise component. Noise intensity can be considered as

a random variable characterized by a certain pdf.


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Noise pdfsp(z)

1

2

p(z)

0.607

2


a

Noise pdfs1. Gaussian (normal) noise is very attractive from a mathematical

point of view since its DFT is another Gaussian process.

. .

( )2

221

( )2

z z

p z e

=

Herez represents intensity,is the mean

p(z)

1

2


,deviation. 2 is the variance ofz.

Electronic circuit noise, sensor noise due to

low illumination or high temperature.

0.607

2


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Noise pdfs

2. Rayleigh noise is specified as( )

2

2z a

( )

0

z a e z ap z b

z a

= 0 and b is a positive integer. The mean

and variance are given by

z b a=


2 2b a =

When the denominator is the gamma function, the pdf describes

the gamma distribution.


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Noise pdfs

4. Exponential noise is specified as0azae z

=

p(z)

a

0 0z 0. The mean and variance are given by

1z a=


1 a =

Exponential pdf is a special case of Erlang pdf with b =1.Used in laser imaging.

Noise pdfs5. Uniform noise is specified as

1 ( )

0

p z b a

otherwise

=

The mean and variance are given by

a bz

+=


( )2

2

12

b a

=


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Noise pdfs

6. Impulse (salt-and-pepper) noise (bipolar) is specified as

( )

0

a

b

z a

p z P z b

otherwise

=

= =

Ifb>a, intensity b will appear as a light dot on the image and a


. a b ,

unipolar. Frequently, a and b aresaturatedvalues, resulting in

positive impulses being white and negative impulses being black.

This noise shows up when quick transitions such as faulty

switching take place.

Test pattern



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Additive Noise patterns


Additive Noise patterns

It is very

hard to

v sua y

discriminate

between first

five noise

corruptions.


s ogramsare quite

handy


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Periodic noise

This noise typically cams fromelectrical or electromechanical

acquisition. We dont consider

any other spatially dependent

noise.

If the noise is strong enough, it

can be seen in the fre uenc


domain.

Estimation of noise parameters

The parameters of periodic noise are usually estimated in the

fre uenc domain.

The parameters of noise pdf can be partially known from the sensor

specifications. However, they usually need to be estimated for a

specific imaging system. If such a system is available, the easiest way

to estimate the noise is to capture a set of images of a flat

environments (for instance, images of a solid gray board illuminated


uniformly). The resulting images are usually good indicators of

system noise.


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Estimation of noise parameters

If only the images are available, frequently it is possible to estimatethe parameters of noise pdf from small patches of reasonably constant

.


Gaussian Rayleigh Uniform

Estimation of noise parametersThe simplest use of the data from the image strips is to calculate the

mean and variance of intensity levels:

( )

1

0

12

0

( )

( )

L

i s i

i

L

i s i

i

z z p z

z z p z

=

=

=

=

If the sha e of df is Gaussian, the mean and variance com letel


specify it. Otherwise, these estimates are needed to derive a and b.

For impulse noise, we need to estimate actual probabilities Pa and Pbfrom the histogram.


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Restoration in the presence of

noise only Spatial filtering

When the only degradation in the image is noise:

( , ) ( , ) ( , )

( , ) ( , ) ( , )

g x y f x y x y

G u v F u v N u v

= +

= +

the noise terms are unknown, so, subtracting them from g(x,y) or

G(u,v) is not a realistic option. Therefore, spatial filtering is a good


.

We already discussed spatial filters before; here we discuss noise-

rejection capabilities of different spatial filters.

Restoration in the presence ofnoise only Spatial filtering1. Mean filters

xy

a rectangular neighborhood of size m x n, centered at the point (x,y).

The arithmetic mean filter computed the average value of the

corrupted image g(x,y) in the area defined by Sxy. The output is an

arithmetic mean:

1

x s t=


( ), xys t Smn

Such a filter smoothes local variations in an image; therefore, noise is

reduced as a result of blurring.


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noise only Spatial filteringb) Geometric mean filter: 1

mn

( ),

, ,xys t S

x y g s t

=

Geometric mean filter achieves smoothing comparable to the

arithmetic mean filter but it preserves more details.

c) Harmonic mean filter:


( ),

( , )1

( , )xys t S

f x y

g s t

=

Harmonic mean filter works well for salt noise and other types of

noise (such as Gaussian) but fails for pepper noise.

Restoration in the presence ofnoise only Spatial filteringd) Contraharmonic mean filter:

1Q+

( )

( )

,

,

,

( , )( , )

xy

xy

s t S

Q

s t S

f x yg s t

=

Here Q is the order of the filter. This filter is well suited for reducing


t e e ects o sa t-pepper no se. or pos t ve va ues o , t e m nates

paper noise; for negative values ofQ, it eliminates salt noise. This

filter cannot reduce both simultaneously.

Notice that contraharmonic filter reduces to the arithmetic mean filter

when Q = 0 and to the harmonic mean filter ifQ = -1.


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Ori inalImagecorru ted

image

Result of

byadditiveGaussiannoise

Result of


filteringwith a 3x3

arithmeticmeanfilter

filteringwith a 3x3

geometricmean filter

Restoration in the presence ofnoise only Spatial filtering

Imagecorru ted

Imagecorru ted

by saltnoise witha 0.1probability

by peppernoise witha 0.1probability

Result of Result of


filtering

with a 3x3contra-harmonicfilter Q=1.5

filtering

with a 3x3contra-harmonicfilter Q=-1.5


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noise only Spatial filteringSelecting an incorrect sign in contraharmonic filtering may lead tounpleasant results...


Pepper noise: result with Q = -1.5

instead of 1.5

Salt noise: result with Q = 1.5

instead of -1.5

Restoration in the presence ofnoise only Spatial filtering2. Order-statistic filters

They are filters, whose response is based on ordering (ranking) pixels

a) Median filter: replaces the pixel value by the median of the

intensity levels in the neighborhood of that pixel:

( ){ }

,

( , ) ( , )xys t S

f x y median g s t

=


Median filters provide excellent results for certain types of noise with

considerably less blurring than linear smoothing filters of the same

size. These filters are very effective against both bipolar and unipolar

noise contaminations (ifPa < 0.2 and Pb < 0.2). The same filter can be

applied more than once to yield better results.


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noise only Spatial filteringImagecorrupted

Result of asingle-pass

y sa -and-peppernoise with

Pa= Pb=

0.1

applicationof a 3x3medianfilter

Result of aResult of a


triple-passapplication

of a 3x3medianfilter

ou e-pass

applicationof a 3x3medianfilter

Restoration in the presence ofnoise only Spatial filteringb) Max and Min filters:

( ),

, max ,xys t S

x y g s t

=

( , ) min ( , )f x y g s t=

This filter is useful for finding the brightest points in an image;

therefore, its effective against pepper noise.


, xys t S

This filter is useful for finding the darkest points in an image;

therefore, its effective against salt noise.


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Imagecorru ted

Imagecorru ted

by saltnoise witha 0.1probability

by peppernoise witha 0.1probability


esu o

filtering

with a 3x3max filter

esu o

filtering

with a 3x3min filter


c) Midpoint filter: computes the midpoint between the maximum and

( ){ }

( ){ }

,,

1( , ) max ( , ) min ( , )2 xyxy s t Ss t S

f x y g s t g s t

= +

This filter is a combination of order statistics and averaging and


wor s est or auss an an un orm no se contam nat ons.


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noise only Spatial filteringd) Alpha-trimmed mean filter: if we delete d/2 highest intensity

values and d/2 lowest intensity values, denote the rest as gr(x,y); a

dcan range from 0 to mn-1. When d= 0, this filter reduces to the

filter that averages what is left is alpha-trimmed mean filter:

( ),

1( , ) ( , )xy

r

s t S

f x y g s tmn d

=


arithmetic mean filter, when d= mn-1, this filter reduces to a median

filter. For other values ofd, the filter is useful against contaminations

with noise of multiple types, such as a combination of salt-and-pepperand Gaussian noise.

Imagecorrupted byadditiveuniform noise

Imageadditionallycorrupted bysalt-and-pepper noise

Filtered by a5x5 arithmeticmean filter

Filtered by a5x5 geometicmean filter


Filtered by a5x5 medianfilter

Filtered by a5x5 alpha-trimmed meanfilter with d= 5


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noise only Spatial filtering3. Adaptive filters

characteristics of the image inside the filter region Sxy.

a) Adaptive local noise reduction filter: since the mean gives a

measure for average intensity in the region and variance characterizes

contrast in that region, they both are reasonable parameters to base an

adaptive filter.


Considering a local region Sxy of a noisy version g(x,y) of the image

f(x,y) with the overall noise variance

2, local mean of pixels in the

region mL and their local variance L2, the following rules are to be

implemented:

Restoration in the presence ofnoise only Spatial filtering1. If

2 is zero, the filter returns the value ofg(x,y);

2. If L2 > 2 (typical for edges that needs to be preserved) the filter

returns the value close to g(x,y);

3. If two variances are equal, the filter returns the arithmetic mean

value for pixels in the region Sxy.

Therefore, the adaptive filter is:

2

x s t s t m

=


2

L

The only parameter that needs to be known in advance is the overall

noise variance

2; all other parameters are estimates for the selected

area.


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noise only Spatial filteringImagecorrupted by

Filtered by

a 7x7Gaussiannoise withzero mean,

2 = 1000

arithmeticmean filter

Filtered by


Filtered bya 7x7

geometricmean filter

a xadaptive

noisereductionfilter with

2 = 1000


Adaptive filter achieves approximately the same performance in noise

cancellation but adds much less blurring than the mean filters.

Adaptive filtering typically yields considerably better results in

overall performance at the price of filter complexity.

In the example, the exact value of noise variance was used. If a


var ance est mate use s too ow, no se correct on w e sma er

than it should be. If the estimate is too high, the output image may

lose dynamic range.


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b) Adaptive median filter: can handle impulse noise with larger

robabilities than traditional median filter. It o erates on a rectan ular

region Sxy, whosesize is changing.

Adaptive median filter has 3 goals: to remove impulse noise, to

provide smoothing of other noise, and to reduce distortion (such as

excessive thinning or thickening of object boundaries).


Assuming thatzmin,zmax andzmedare the minimum, maximum, and

median intensity values in Sxy,zxy is the intensity value at coordinates

(x,y) and Smax is the maximum allowed size ofSxy, adaptive median-filtering algorithm works in two stages:

Restoration in the presence ofnoise only Spatial filtering1 min 2 max

1 20 0,

med med A z z A z z

IF A AND A go to stage B

= ; =

>

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05-1 - Image Restoration and Reconstruction