Image Restoration - Biomedical EngineeringImage Restoration Introduction to Signal and Image...

ImageRestoration

Introduction to Signaland Image Processing

Prof. Dr. Philippe Cattin

MIAC, University of Basel

April 19th/26th, 2016

April 19th/26th, 2016Introduction to Signal and Image Processing

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Ph. Cattin: Image Restoration

Contents

Abstract

1 Image Degradation & Restoration Model

Image Degradation & Restoration Model

Image Degradation & Restoration Model (2)

2 Noise Models

Noise Models

Gaussian Noise

Rayleigh Noise

Erlang Noise

Exponential Noise

Uniform Noise

Impulse (Salt-and-Pepper) Noise

Noise Models

Visual Comparison of Different Noise Sources

Periodic Noise

Estimation of Noise Parameters (1)

Estimation of Periodic Noise Parameters

3 Restoration in the Precense of Noise Only

Restoration in the Precense of Noise Only

4 Periodic Noise Reduction by Frequency Domain

Filtering

4.1 Bandreject Filter

Bandreject Filter

Bandreject Filter Example

4.2 Bandpass FilterApril 19th/26th, 2016Introduction to Signal and Image Processing

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Bandpass Filter

Bandpass Filter Example

4.3 Notch Filter

Notch Filter

Notch Filter (2)

Ideal Notch Filter D_0=16 Example

Butterworth Notch Filter D_0=64Example

Gaussian Notch Filter D_0=64 Example

General Notch Filter Example

4.4 Optimum Notch Filtering

Optimum Notch Filtering

Optimum Notch Filtering Principle

Optimum Notch Filter Example

Optimum Notch Filter Example withPixel Increments

Optimum Notch Filter Example withPixel Increments (2)

5 Estimating the Degradation Function

Estimating the Degradation Function

Estimating the Degradation Function (2)

Estimating by Image Observation

Estimating by Experimentation

Estimating by Mathematical Modelling

Illustration of the Atmospheric TurbulenceModel

6 Inverse Filtering

6.1 Direct Inverse Filtering

Direct Inverse Filtering

Two Practical Approaches for InverseFiltering


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Artificial Inverse Filtering Example

Artificial Inverse Filtering Example (2)

Inverse Filtering Example (2)

6.2 Wiener Filtering

Wiener Filtering

Wiener Filtering (2)

Parametric Wiener Filter

Artificial Wiener Filtering Example

Wiener Filtering Example with LittleNoise

Wiener Filtering Example with MediumNoise

Wiener Filtering Example with HeavyNoise

Comparison Param. Wiener vs. WienerFiltering

Drawback of the Wiener Filter

6.3 Geometric Mean Filter

Geometric Mean Filter

6.4 Constrained Least Squares Filtering

Constrained Least Squares (Regularised)Filtering

Constrained Least Squares Filtering (2)

Constrained Least Squares Filtering (3)

Optimal Selection for Gamma

Artificial Constrained Least SquaresFilter Example

Constrained Least Squares FilterExample


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(2)


Abstract

As in image enhancement the goal of restoration is toimprove an image for further processing. In contrast toimage enhancement that was subjective and largely basedon heuristics, restoration attempts to reconstruct orrecover an image that has been distorted by a knowndegradation phenomenon. Restoration techniques thus tryto model the degradation process and apply the inverseprocess in order to reconstruct the original image.

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ImageDegradation &Restoration Model


(4)Image Degradation &Restoration Model

The degradation process is generally modelled as adegradation function and the additive noise term

together they yield .

Given , some knowledge about , and it is

the objective of restoration to estimate . Of course

this estimate should be as close as possible to .

Fig 6.1 Image degradation model

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Image Degradation & Restoration Model

(5)


Image Degradation &Restoration Model (2)

If the degradation function is a linear, shift-invariantprocess, then the degraded image is given in the spatialdomain by

(6.1)

where is the spatial representation of the

degradation function, and indicates convolution.

As we know from the Convolution Theorem this can berewritten in the frequency domain

(6.2)

where the capital letters are the Fourier transforms of therespective function in Eq 6.1.

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Noise Models


(7)Noise Models

The principle sources of noise in digital images ariseduring image acquisition and/or transmission.

The performance of imaging sensors are affected by avariety of factors during acquisition, such as

Environmental conditions during the acquisition

Light levels (low light conditions require high gain

amplification)

Sensor temperature (higher temp implies more

amplification noise)

Images can also be corrupted during transmission due tointerference in the channel for example

Lightning or other

Atmospheric disturbances

Depending on the specific noise source, a different modelmust be selected that accurately reproduces the spatialcharacteristics of the noise.

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Noise Models

(8)


Gaussian Noise

Because of its mathematicaltractability in both the spatial andfrequency domain, Gaussian noisemodels (aka normal distribution)are used frequently in practice.

The PDF (Probability densityfunction) of a Gaussian randomvariable is given by

(6.3)

where represents the grey level, the mean value and the

standard deviation.

Fig 6.2: Gaussian

probability density function

The application of the Gaussian model is soconvenient that it is often used in situations inwhich they are marginally applicable at best.

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Noise Models

(9)


Rayleigh Noise

The PDF of Rayleigh noise is defined by

(6.4)

the mean and variance are given by

(6.5)

Fig 6.3: Rayleigh

probability

density function

As the shape of the Rayleigh density function is skewed itis useful for approximating skewed histograms.

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Noise Models

(10)


Erlang Noise

The PDF of the Erlang noise is givenby

(6.6)

where , is a positive integer.The mean and variance are thengiven by

(6.7)

Fig 6.4: Erlang

probability density

function

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Noise Models

(11)


Exponential Noise

The PDF of the exponential noiseis given by

(6.8)

where . The mean andvariance of the density functionare

(6.9)

Fig 6.5: Probability density

function of

exponential noise

The exponential noise model is a special case of the Erlangnoise model with .

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Noise Models

(12)


Uniform Noise

The PDF of the uniform noise isgiven by

(6.10)

The mean and variance of thedensity function are given by

(6.11)

Fig 6.6: Probability density

function of uniform

noise

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Noise Models

(13)


Impulse (Salt-and-Pepper) Noise

The PDF of bipolar impulse noisemodel is given by

(6.12)Fig 6.7: Probability density

function of the bipolar

impulse noise model

If , grey-level appears as a light dot (salt) in theimage. Conversely, will appear as dark dot (pepper). Ifeither is zero, the PDF is called unipolar.

Because impulse corruption is generally large compared tothe signal strength, the assumption is usually that and are digitised as saturated values thus black (pepper) andwhite (salt).

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Noise Models

(14)


Noise Models

The presented noise models provide a useful tool forapproximating a broad range of noise corruption situationsfound in practice. For example:

Gaussian noise: arises in images due to sensor noise

caused by poor illumination and/or high temperature,

and electronic circuit noise

The Rayleigh noise model is used to characterise noise

phenomena in range images

Exponential and Erlang noise models find their

application in Laser imaging

Impulse noise takes place in situations where high

transients (faulty switching) occurs

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Noise Models

(15)


Visual Comparison ofDifferent Noise Sources

Although the corresponding histograms show closeresemblance to the PDF of the respective noise model, it isvirtually impossible to select the type of noise causing thedegradation from the image alone.

Gaussian noisemodel

Rayleigh noisemodel

Erlang noise model

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Exponential noisemodel

Uniform noisemodel

Impulse noisemodel

(Salt & Pepper)

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Noise Models

(16)


Periodic Noise

Periodic noise typically arises from electrical orelectromechanical interference during image acquisitionand is spatial dependent.

Fig 6.8: Periodic noise on a satellite

image of Pompeii

Fig 6.9: Spectrum of the Pompeii

image with periodic noise (peaks of

the periodic noise visually

enhanced)

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Noise Models

(17)


Estimation of NoiseParameters (1)

In rare cases the parameters ofnoise PDFs may be known fromsensor specifications, but it isoften necessary to estimate themfor a particular imagingarrangement.

If the imaging system is available,one simple way to study noisecharacteristics would be tocapture multiple images ofhomogeneous environments, suchas solid grey cards.

Fig 6.10: Sample histogram

of a small patch of Erlang

noise

In many cases the noise parameters have to be estimatedfrom already captured images. The simplest way is toestimate the mean and variance of the grey levels fromsmall patches of reasonably constant grey level.

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Noise Models

(18)


Estimation of PeriodicNoise Parameters

The parameters ofperiodic noise aretypically estimatedby inspection ofthe Fourierspectrum. Thefrequency spikescan often bedetected by visualanalysis.Automatic analysisis possible if thenoise spikes areeitherexceptionallypronounced or apriori knowledgeabout theirlocation is known.

Fig. 6.11: Fig. 6.12:

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Restoration in thePrecense of NoiseOnly


(20)Restoration in thePrecense of Noise Only

When noise is the only image degradation present in animage, thus , Eqs 6.1 & 6.2 become

(6.13)

and

(6.14)

In case of periodic noise, it is usually possible to estimate from the spectrum of and subtract it to

obtain an estimate of the original image .

But estimating the noise terms is unreasonable, so

subtracting them from is impossible. Spatial

filtering is the method of choice in situations when onlyadditive noise is present.

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Periodic NoiseReduction byFrequency DomainFiltering

Bandreject Filter


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(23)Bandreject Filter

Bandreject filters remove or attenuate a band offrequencies around the origin in the Fourier domain.

Ideal Bandreject Filter

(6.15)

Fig 6.13: Ideal bandreject

filter

Butterworth Bandreject Filter

(6.16)

Fig 6.14: Butterworth

bandreject filter

Gaussian Bandreject Filter

(6.17)

Fig 6.15: Gaussian

bandreject filter

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Bandreject Filter

(24)


Bandreject FilterExample

Fig 6.16: Aerial image of Pompeii

with periodic noise

Fig 6.17: Ideal bandreject filter

Fig 6.18: Butterworth bandreject

filter

Fig 6.19: Gaussian bandreject filter

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Bandpass Filter


(26)Bandpass Filter

The Bandpass filter performs the opposite of a bandrejectfilter and thus lets pass the frequencies in a narrow bandof width around . The transfer function can

be obtained from a corresponding bandreject filter byusing the equation

(6.18)

The perspective plots of the corresponding filters aredepicted in Fig 6.20.

(a) (b) (c)

Fig 6.20 (a) Ideal bandpass filter, (b) Butterworth bandpass filter, and

(c) Gaussian bandpass filter

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Bandpass Filter

(27)


Bandpass Filter Example

Fig 6.21: Aerial image of Pompeii

with periodic noise

Fig 6.22: Ideal bandpass filter

Fig 6.23: Butterworth bandpass

filter

Fig 6.24: Gaussian bandpass filter

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Notch Filter


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(29)Notch Filter

A Notch filter rejects or passes frequencies in predefinedneighbourhoods about a centre frequency. Due to thesymmetry of the Fourier transform, notch filters mustappear in symmetric pairs about the origin (except the oneat the origin).

The transfer functions of the ideal, Butterworth, andGaussian notch reject filter of radius (width) withcentres at and, by symmetry, at , are

Ideal Notch Filter

(6.19)

Fig 6.25: Ideal

notch filter

Butterworth Notch Filter

(6.20)

Fig 6.26:

Butterworth

notch filter

Gaussian Notch Filter

(6.21)

Fig 6.27:

Gaussian notch

filter

where

(6.22)

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Notch Filter

(30)


Notch Filter (2)

Similar to the bandpass/bandreject filters, the Notch rejectfilters can be turned into Notch pass filters with therelation

(6.23)

where is the transfer function of the notch pass

filter, and the corresponding Notch reject filter.

If

the Notch reject filter becomes a highpass filter

and

the Notch pass filter becomes a lowpass filter.

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Notch Filter

(31)


Ideal Notch FilterD_0=16 Example

Fig 6.28: Pompeii aerial image with

periodic noise

Fig 6.29: Notch reject filtered

image

Fig 6.30: Notch pass filtered image

(contrast enhanced)

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Notch Filter

(32)


Butterworth Notch FilterD_0=64 Example


periodic noise


image


(contrast enhanced)

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Notch Filter

(33)


Gaussian Notch FilterD_0=64 Example


periodic noise


image


(contrast enhanced)

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Notch Filter

(34)


General Notch FilterExample

Fig 6.37: Original Mariner 6

martian image

Fig 6.38: Log Fourier spectra of the

image

Fig 6.39: Notch filtered log spectra Fig 6.40: Notch filtered image

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Optimum NotchFiltering


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(36)Optimum Notch Filtering

Images derived from electro-optical scanners, such asthose used in space and aerial imaging, are sometimescorrupted by coupling and amplification of low-levelsignals in the scanners' circuitry. The resulting imagestend to contain substantial periodic noise. Theinterference pattern are, however, in practice not alwaysas clearly defined as in the samples shown thus far.

The first step in Optimum notch filtering is to find theprincipal frequency components and placing notch passfilters at the location of each spike, yielding . The

Fourier transform of the interference pattern is thus givenby

(6.24)

where is the Fourier transform of the corrupted

image. The corresponding interference pattern in thespatial domain is obtained with the inverse Fouriertransform

(6.25)

As the corrupted image is assumed to be formed by

the addition of the uncorrupted image and the

interference noise

(6.26)

If the noise term were known completely,

subtracting it from would yield . The problem

is of course that the estimated interference pattern

is only an approximation. The effect of components notpresent in the estimate can be minimised be

subtracting from a weighted portion of to

obtain an estimate for

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(6.27)

where is the estimate of , and is to be

determined and optimises in some meaningful way.

One common approach is to select so that the

variance of is minimised over a specified

neighbourhood of size about every point

.

(6.29)

where is the average value of in the

neighbourhood; that is

(6.30)

Substituting Eq 6.27 into Eq 6.29 yields

(6.31)

Assuming that remains constant over the

neighbourhood gives the approximation

(6.32)

for and . This assumption results inthe expression

(6.33)

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in the neighbourhood. With these approximations Eq 6.30becomes

(6.34)

To minimise , we solve

(6.35)

for . The result is

(6.36)

To obtain an estimated restored image , must

be computed from Eq 6.36 and substituted into Eq 6.27 to

determine .

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(37)


Optimum Notch FilteringPrinciple

Fig 6.41 Principle of the optimum notch filter

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(38)


Optimum Notch FilterExample

Fig 6.42 Aerial image of Pompeii

with heavy periodic noise

distortions

Fig 6.43 Noise estimation with

Gaussian notch pass filter

Fig 6.44 Notch reject filtered

image

Fig 6.45 Result filtered with an

optimum notch filter of size

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(39)


Optimum Notch FilterExample with PixelIncrements

Fig 6.46 Aerial image of Pompeii corrected with the optimal notch filter

of size with the window shifted pixelwise

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Optimum Notch FilterExample with PixelIncrements (2)

Fig 6.47 of the example image in Fig 6.46

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Estimating theDegradationFunction


(42)Estimating theDegradation Function

So far we concentrated mainly at the additive noise term. In the remainder of this presentation we

concentrate on ways to remove the degradation function.

Fig 6.48 Image degradation model

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Estimating theDegradation Function (2)

There are three principle methods to estimate thedegradation function

Observation1.

Experimentation2.

Mathematical modelling3.

The process of restoring a corrupted image using theestimated degradation function is sometimes called blinddeconvolution, as the true degradation is seldom knowncompletely.

In recent years a forth method based on maximum-likelihood estimation (MLE) often called blinddeconvolution has been proposed.

Beware: All multiplications/divisions in theequations in this part of the lecture arecomponentwise.

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Estimating by ImageObservation

Given

A degraded image without any information about thedegradation function .

Solution

Select a subimage with strong signal content → 1.

By estimating the sample grey levels of the object and

background in we can construct an unblurred

image of the same size and characteristics as the

observed subimage →

2.

Assuming that the effect of noise is negligible (thus the

choice of a strong signal area), it follows from Eq 6.2

that

3.

Thanks to the shift invariance we can deduce

from

4.

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(45)


Estimating byExperimentation

Given

Equipment similar to the equipment used to acquire thedegraded image

Solution

Images similar to the degraded image can be acquired

with various system settings until it closely matches

the degraded image

1.

The system can then be characterised by measuring

the impulse (small bright dot of light) response. A

linear, shift invariant system can be fully described by

its impulse response.

2.

As the Fourier transform of an impulse is a constant, it

follows from Eq 6.2

where is the Fourier transform of the observed

image, and is a constant describing the strength of

the impulse.

3.

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(46)


Estimating byMathematical Modelling

Degradation modelling has been used for many years. It,however, requires an indepth understanding of theinvolved physical phenomenon. In some cases it can evenincorporate environmental conditions that causedegradations. Hufnagel and Stanly proposed in 1964 amodel to characterise atmospheric turbulence.

(6.37)

where is a constant that depends on the nature of theturbulence.

Except of the 5/6 power the above equation has the sameform as a Gaussian lowpass filter.

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Illustration of theAtmospheric TurbulenceModel

Fig 6.49: Aerial image of Pompeii Fig 6.50: Pompeii with mild

turbulence

Fig 6.51: Pompeii with medium

turbulence

Fig 6.52 Pompeii with heavy

turbulence

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Inverse Filtering

Direct InverseFiltering


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(50)Direct Inverse Filtering

The simplest approach to restore a degraded image is toform an estimate of the form

(6.38)

and then obtain the corresponding image by inverseFourier transform. This method is called direct inversefiltering. With the degradation model Equation 6.2 we get

(6.39)

This simple equation tells us that, even if we knew

exactly we could not recover the undegraded image

completely because:

the noise component is a random function whose

Fourier transform is not known, and

1.

in practice can contain numerous zeros.2.

Direct Inverse filtering is seldom a suitableapproach in practical applications

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Two PracticalApproaches for InverseFiltering

Low Pass Filter

(6.40)

where is a low-pass filter that should eliminiate

the very low (or even zero) values often experienced in

the high frequencies.

1.

Constrained Division

(6.41)

gives values, where is lower than a threshold

a special treatment.

2.

These appraoches are also known as Pseudo-Inverse Filtering.

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Artificial InverseFiltering Example

Fig 6.53: Checkerboard sample Fig 6.54: Motion blur of in

direction

Fig 6.55: Additive noise with Fig 6.56: Motion and noise

degraded checkerboard

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Artificial InverseFiltering Example (2)

Although very tempting from its simplicity, the directinverse filtering approach does not work for practicalapplications. Even if the degradation is known exactly.

Fig 6.57: Direct inverse filtering result

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Inverse FilteringExample (2)

Fig 6.58: Inverse filtering of Figure

6.52

Fig 6.59: Pseudo Inverse filtering

for freq. radius


for frequencies radius


for frequencies radius

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Wiener Filtering


(56)Wiener Filtering

The best known improvement to inverse filtering is the

Wiener Filter. The Wiener filter seeks an estimate that

minimises the statistical error function

(6.42)

where is the expected value and the undegradedimage. The solution to this equation in the frequencydomain is

(6.43)

where

is the degradation function

with the complex

conjugate of

is the power spectrum of the noise

is the power spectrum of the

undegraded image

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Wiener Filtering

(57)


Wiener Filtering (2)

If the noise spectrum is zero, the noise-to-signal

power ratio

vanishes and the Wiener Filter reduces to the inversefilter.

(6.44)

This is no problem, as the inverse filter works fine if nonoise is present.

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Wiener Filtering

(58)


Parametric Wiener Filter

The main problem with the Wiener filter is that the powerspectrum of the undegraded image is seldom

known.

A frequently used approach when these quantities cannotbe estimated is to approximate Eq 6.43 by the expression

(6.45)

where is a user selected constant.

This simplification can be partly justified when dealingwith spectrally white noise, where the noise spectrum

is constant. However, the problem still

remains that the power spectrum of

the undegraded image is unknown and must be estimated.

Even if the actual ratio is not known, it becomesa simple matter to experiment interactively varyingthe constant and viewing the results.

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Wiener Filtering

(59)


Artificial WienerFiltering Example

This example shows the performance of the Wiener filteron the example Fig 6.56 using (1) the approximation of Eq6.45 with in Fig 6.62 and (2) the Wiener filterusing full knowledge of the noise and undegraded image'spower spectra in Fig 6.63.

Fig 6.62: Parametric Wiener filter

using a constant ratio

Fig 6.63: Wiener filter knowing the

noise and undegraded signal power

spectra

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Wiener Filtering

(60)


Wiener FilteringExample with Little Noise

Fig 6.64: Pompeii blurred with

Gaussian kernel , and

Gaussian noise

Fig 6.65: Result of direct inverse

filtering

Fig 6.66: Result with Parametric

Wiener filtering

Fig 6.67: Result with Wiener

filtering

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Wiener Filtering

(61)


Wiener FilteringExample with Medium Noise



Gaussian noise


filtering


Wiener filtering


filtering

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Wiener Filtering

(62)


Wiener FilteringExample with Heavy Noise



Gaussian noise


filtering


Wiener filtering


filtering

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Wiener Filtering

(63)


Comparison Param.Wiener vs. Wiener Filtering

(a) (b)

Fig 6.76 Aerial image of Pompeii with Gaussian blur ( ) and noise (

) (a) restored with the Parametric Wiener ( ), (b)

same image restored with Wiener filter.

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Wiener Filtering

(64)


Drawback of the WienerFilter

The problem of having to know something about thedegradation function is common to all methods

discussed in this part.

The Wiener filter has the additional difficulty, that

the power spectra of the undegraded image and

the power spectra of the noise

must be known.

Although an approximation by a constant is possible as inthe Parametric Wiener filter, see Fig 6.62, this approach isnot always suitable.

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Geometric MeanFilter


(66)Geometric Mean Filter

The Geometric Mean Filter is a generalisation of inversefiltering and the Wiener filtering. Through its parameters

it allows access to an entire family of filters.

(6.46)

Depending on the parameters the filter characteristicscan be adjusted

: inverse filtering

: spectrum equalisation filter

: parametric Wiener filter → Wiener

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Constrained LeastSquares Filtering


(68)Constrained LeastSquares (Regularised)Filtering

The Constrained Least Squares Filtering approach onlyrequires knowledge of the mean and variance of the noise.As shown in previous lectures these parameters can beusually estimated from the degraded image.

The definition of the 2D discrete convolution is

(6.47)

Using this equation we can express the linear degradationmodel in vector notation

as

(6.48)

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Constrained Least Squares Filtering

(69)


Constrained LeastSquares Filtering (2)

It seems obvious that the restoration problem is nowreduced to simple matrix manipulations. Unfortunatelythis is not the case. The problem with the matrixformulation is that the matrix is very largeand that its inverse is firstly very sensitive to noise andsecondly does not necessarily exist.

One way to deal with these issues is to base optimality ofrestoration on a measure of smoothness, such as thesecond derivative of the image, e.g. the Laplacian.

(6.49)

subject to the constraint

(6.50)

where is the Euclidian norm, and an estimate of the

undegraded image.

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Constrained LeastSquares Filtering (3)

The Frequency domain solution to this optimisationproblem is given by

(6.51)

where is a parameter that must be adjusted so that the

constraint in Eq 6.50 is satisfied, and is the Fourier

transform of the Laplace operator

(6.52)

Note: must be properly padded with zeros prior to

computing the Fourier transform.

It is possible to adjust the parameter

interactively until acceptable results are obtained.

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Optimal Selection forGamma

If we are interested in optimality, the parameter must be

adjusted so that the constraint in Eq

6.50 is satisfied.

It turns out that

(6.53)

is the optimal selection given the above optimisationequation where are the mean and variance of the

noise.

It is important to understand, that optimumrestoration in the sense of constrained leastsquares does not necessarily imply best in thevisual sense. In general, automatic determinedrestoration filters yield inferior results to manualadjustment of filter parameters.

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Artificial ConstrainedLeast Squares Filter Example

This example shows the performance of the ConstrainedLeast Squares Filter on the example Fig 6.56 using (1) thetheoretical optimal value for in Fig 6.77 and (2)the constrained least squares filter with a manuallyselected of 6.78.

Fig 6.77: Result with optimal Fig 6.78: Results with manually

selected

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Constrained LeastSquares Filter Example


Gaussian kernel

Fig 6.80: Result with low Gaussian

noise of

Fig 6.81: Result with medium

Gaussian noise of

Fig 6.82: Result with high Gaussian

noise of

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