image restoration - biomedical engineeringimage restoration introduction to signal and image...
TRANSCRIPT
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
1 of 70 22.02.2016 09:19
Contents
2
4
5
7
8
9
10
11
12
13
14
15
16
17
18
20
23
24
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
2 of 70 22.02.2016 09:19
26
27
29
30
31
32
33
34
36
37
38
39
40
42
43
44
45
46
47
50
51
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
April 19th/26th, 2016Introduction to Signal and Image Processing
3 of 70 22.02.2016 09:19
52
53
54
56
57
58
59
60
61
62
63
64
66
68
69
70
71
72
73
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
April 19th/26th, 2016Introduction to Signal and Image Processing
4 of 70 22.02.2016 09:19
April 19th/26th, 2016Introduction to Signal and Image Processing
(2)
Ph. Cattin: Image Restoration
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.
5 of 70 22.02.2016 09:19
ImageDegradation &Restoration Model
April 19th/26th, 2016Introduction to Signal and Image Processing
(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
6 of 70 22.02.2016 09:19
April 19th/26th, 2016Introduction to Signal and Image Processing
Image Degradation & Restoration Model
(5)
Ph. Cattin: Image Restoration
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.
7 of 70 22.02.2016 09:19
Noise Models
April 19th/26th, 2016Introduction to Signal and Image Processing
(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.
8 of 70 22.02.2016 09:19
April 19th/26th, 2016Introduction to Signal and Image Processing
Noise Models
(8)
Ph. Cattin: Image Restoration
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.
9 of 70 22.02.2016 09:19
April 19th/26th, 2016Introduction to Signal and Image Processing
Noise Models
(9)
Ph. Cattin: Image Restoration
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.
10 of 70 22.02.2016 09:19
April 19th/26th, 2016Introduction to Signal and Image Processing
Noise Models
(10)
Ph. Cattin: Image Restoration
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
11 of 70 22.02.2016 09:19
April 19th/26th, 2016Introduction to Signal and Image Processing
Noise Models
(11)
Ph. Cattin: Image Restoration
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 .
12 of 70 22.02.2016 09:19
April 19th/26th, 2016Introduction to Signal and Image Processing
Noise Models
(12)
Ph. Cattin: Image Restoration
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
13 of 70 22.02.2016 09:19
April 19th/26th, 2016Introduction to Signal and Image Processing
Noise Models
(13)
Ph. Cattin: Image Restoration
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).
14 of 70 22.02.2016 09:19
April 19th/26th, 2016Introduction to Signal and Image Processing
Noise Models
(14)
Ph. Cattin: Image Restoration
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
15 of 70 22.02.2016 09:19
April 19th/26th, 2016Introduction to Signal and Image Processing
Noise Models
(15)
Ph. Cattin: Image Restoration
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
16 of 70 22.02.2016 09:19
April 19th/26th, 2016Introduction to Signal and Image Processing
Exponential noisemodel
Uniform noisemodel
Impulse noisemodel
(Salt & Pepper)
17 of 70 22.02.2016 09:19
April 19th/26th, 2016Introduction to Signal and Image Processing
Noise Models
(16)
Ph. Cattin: Image Restoration
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)
18 of 70 22.02.2016 09:19
April 19th/26th, 2016Introduction to Signal and Image Processing
Noise Models
(17)
Ph. Cattin: Image Restoration
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.
19 of 70 22.02.2016 09:19
April 19th/26th, 2016Introduction to Signal and Image Processing
Noise Models
(18)
Ph. Cattin: Image Restoration
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:
20 of 70 22.02.2016 09:19
Restoration in thePrecense of NoiseOnly
April 19th/26th, 2016Introduction to Signal and Image Processing
(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.
21 of 70 22.02.2016 09:19
Periodic NoiseReduction byFrequency DomainFiltering
Bandreject Filter
April 19th/26th, 2016Introduction to Signal and Image Processing
22 of 70 22.02.2016 09:19
April 19th/26th, 2016Introduction to Signal and Image Processing
(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
23 of 70 22.02.2016 09:19
April 19th/26th, 2016Introduction to Signal and Image Processing
Bandreject Filter
(24)
Ph. Cattin: Image Restoration
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
24 of 70 22.02.2016 09:19
Bandpass Filter
April 19th/26th, 2016Introduction to Signal and Image Processing
(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
25 of 70 22.02.2016 09:19
April 19th/26th, 2016Introduction to Signal and Image Processing
Bandpass Filter
(27)
Ph. Cattin: Image Restoration
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
26 of 70 22.02.2016 09:19
Notch Filter
April 19th/26th, 2016Introduction to Signal and Image Processing
27 of 70 22.02.2016 09:19
April 19th/26th, 2016Introduction to Signal and Image Processing
(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)
28 of 70 22.02.2016 09:19
April 19th/26th, 2016Introduction to Signal and Image Processing
29 of 70 22.02.2016 09:19
April 19th/26th, 2016Introduction to Signal and Image Processing
Notch Filter
(30)
Ph. Cattin: Image Restoration
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.
30 of 70 22.02.2016 09:19
April 19th/26th, 2016Introduction to Signal and Image Processing
Notch Filter
(31)
Ph. Cattin: Image Restoration
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)
31 of 70 22.02.2016 09:19
April 19th/26th, 2016Introduction to Signal and Image Processing
Notch Filter
(32)
Ph. Cattin: Image Restoration
Butterworth Notch FilterD_0=64 Example
Fig 6.31: Pompeii aerial image with
periodic noise
Fig 6.32: Notch reject filtered
image
Fig 6.33: Notch pass filtered image
(contrast enhanced)
32 of 70 22.02.2016 09:19
April 19th/26th, 2016Introduction to Signal and Image Processing
Notch Filter
(33)
Ph. Cattin: Image Restoration
Gaussian Notch FilterD_0=64 Example
Fig 6.34: Pompeii aerial image with
periodic noise
Fig 6.35: Notch reject filtered
image
Fig 6.36: Notch pass filtered image
(contrast enhanced)
33 of 70 22.02.2016 09:19
April 19th/26th, 2016Introduction to Signal and Image Processing
Notch Filter
(34)
Ph. Cattin: Image Restoration
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
34 of 70 22.02.2016 09:19
Optimum NotchFiltering
April 19th/26th, 2016Introduction to Signal and Image Processing
35 of 70 22.02.2016 09:19
April 19th/26th, 2016Introduction to Signal and Image Processing
(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
36 of 70 22.02.2016 09:19
April 19th/26th, 2016Introduction to Signal and Image Processing
(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)
37 of 70 22.02.2016 09:19
April 19th/26th, 2016Introduction to Signal and Image Processing
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 .
38 of 70 22.02.2016 09:19
April 19th/26th, 2016Introduction to Signal and Image Processing
Optimum Notch Filtering
(37)
Ph. Cattin: Image Restoration
Optimum Notch FilteringPrinciple
Fig 6.41 Principle of the optimum notch filter
39 of 70 22.02.2016 09:19
April 19th/26th, 2016Introduction to Signal and Image Processing
Optimum Notch Filtering
(38)
Ph. Cattin: Image Restoration
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
40 of 70 22.02.2016 09:19
April 19th/26th, 2016Introduction to Signal and Image Processing
Optimum Notch Filtering
(39)
Ph. Cattin: Image Restoration
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
41 of 70 22.02.2016 09:19
April 19th/26th, 2016Introduction to Signal and Image Processing
Optimum Notch Filtering
(40)
Ph. Cattin: Image Restoration
Optimum Notch FilterExample with PixelIncrements (2)
Fig 6.47 of the example image in Fig 6.46
42 of 70 22.02.2016 09:19
Estimating theDegradationFunction
April 19th/26th, 2016Introduction to Signal and Image Processing
(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
43 of 70 22.02.2016 09:19
April 19th/26th, 2016Introduction to Signal and Image Processing
Estimating the Degradation Function
(43)
Ph. Cattin: Image Restoration
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.
44 of 70 22.02.2016 09:19
April 19th/26th, 2016Introduction to Signal and Image Processing
Estimating the Degradation Function
(44)
Ph. Cattin: Image Restoration
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.
45 of 70 22.02.2016 09:19
April 19th/26th, 2016Introduction to Signal and Image Processing
Estimating the Degradation Function
(45)
Ph. Cattin: Image Restoration
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.
46 of 70 22.02.2016 09:19
April 19th/26th, 2016Introduction to Signal and Image Processing
Estimating the Degradation Function
(46)
Ph. Cattin: Image Restoration
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.
47 of 70 22.02.2016 09:19
April 19th/26th, 2016Introduction to Signal and Image Processing
Estimating the Degradation Function
(47)
Ph. Cattin: Image Restoration
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
48 of 70 22.02.2016 09:19
Inverse Filtering
Direct InverseFiltering
April 19th/26th, 2016Introduction to Signal and Image Processing
49 of 70 22.02.2016 09:19
April 19th/26th, 2016Introduction to Signal and Image Processing
(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
50 of 70 22.02.2016 09:19
April 19th/26th, 2016Introduction to Signal and Image Processing
Direct Inverse Filtering
(51)
Ph. Cattin: Image Restoration
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.
51 of 70 22.02.2016 09:19
April 19th/26th, 2016Introduction to Signal and Image Processing
Direct Inverse Filtering
(52)
Ph. Cattin: Image Restoration
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
52 of 70 22.02.2016 09:19
April 19th/26th, 2016Introduction to Signal and Image Processing
Direct Inverse Filtering
(53)
Ph. Cattin: Image Restoration
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
53 of 70 22.02.2016 09:19
April 19th/26th, 2016Introduction to Signal and Image Processing
Direct Inverse Filtering
(54)
Ph. Cattin: Image Restoration
Inverse FilteringExample (2)
Fig 6.58: Inverse filtering of Figure
6.52
Fig 6.59: Pseudo Inverse filtering
for freq. radius
Fig 6.60: Pseudo Inverse filtering
for frequencies radius
Fig 6.61: Pseudo Inverse filtering
for frequencies radius
54 of 70 22.02.2016 09:19
Wiener Filtering
April 19th/26th, 2016Introduction to Signal and Image Processing
(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
55 of 70 22.02.2016 09:19
April 19th/26th, 2016Introduction to Signal and Image Processing
Wiener Filtering
(57)
Ph. Cattin: Image Restoration
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.
56 of 70 22.02.2016 09:19
April 19th/26th, 2016Introduction to Signal and Image Processing
Wiener Filtering
(58)
Ph. Cattin: Image Restoration
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.
57 of 70 22.02.2016 09:19
April 19th/26th, 2016Introduction to Signal and Image Processing
Wiener Filtering
(59)
Ph. Cattin: Image Restoration
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
58 of 70 22.02.2016 09:19
April 19th/26th, 2016Introduction to Signal and Image Processing
Wiener Filtering
(60)
Ph. Cattin: Image Restoration
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
59 of 70 22.02.2016 09:19
April 19th/26th, 2016Introduction to Signal and Image Processing
Wiener Filtering
(61)
Ph. Cattin: Image Restoration
Wiener FilteringExample with Medium Noise
Fig 6.68: Pompeii blurred with
Gaussian kernel , and
Gaussian noise
Fig 6.69: Result of direct inverse
filtering
Fig 6.70: Result with Parametric
Wiener filtering
Fig 6.71: Result with Wiener
filtering
60 of 70 22.02.2016 09:19
April 19th/26th, 2016Introduction to Signal and Image Processing
Wiener Filtering
(62)
Ph. Cattin: Image Restoration
Wiener FilteringExample with Heavy Noise
Fig 6.72: Pompeii blurred with
Gaussian kernel , and
Gaussian noise
Fig 6.73: Result of direct inverse
filtering
Fig 6.74: Result with Parametric
Wiener filtering
Fig 6.75: Result with Wiener
filtering
61 of 70 22.02.2016 09:19
April 19th/26th, 2016Introduction to Signal and Image Processing
Wiener Filtering
(63)
Ph. Cattin: Image Restoration
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.
62 of 70 22.02.2016 09:19
April 19th/26th, 2016Introduction to Signal and Image Processing
Wiener Filtering
(64)
Ph. Cattin: Image Restoration
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.
63 of 70 22.02.2016 09:19
Geometric MeanFilter
April 19th/26th, 2016Introduction to Signal and Image Processing
(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
64 of 70 22.02.2016 09:19
Constrained LeastSquares Filtering
April 19th/26th, 2016Introduction to Signal and Image Processing
(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)
65 of 70 22.02.2016 09:19
April 19th/26th, 2016Introduction to Signal and Image Processing
Constrained Least Squares Filtering
(69)
Ph. Cattin: Image Restoration
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.
66 of 70 22.02.2016 09:19
April 19th/26th, 2016Introduction to Signal and Image Processing
Constrained Least Squares Filtering
(70)
Ph. Cattin: Image Restoration
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.
67 of 70 22.02.2016 09:19
April 19th/26th, 2016Introduction to Signal and Image Processing
Constrained Least Squares Filtering
(71)
Ph. Cattin: Image Restoration
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.
68 of 70 22.02.2016 09:19
April 19th/26th, 2016Introduction to Signal and Image Processing
Constrained Least Squares Filtering
(72)
Ph. Cattin: Image Restoration
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
69 of 70 22.02.2016 09:19
April 19th/26th, 2016Introduction to Signal and Image Processing
Constrained Least Squares Filtering
(73)
Ph. Cattin: Image Restoration
Constrained LeastSquares Filter Example
Fig 6.79: Pompeii blurred with
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
70 of 70 22.02.2016 09:19