3 |
Image Restoration
sage Restoration - degradation model, Properties, Noise models - Mean Filters —
Order Statistics — Adaptive filters — Band reject Filters - Band pass Filters — Notch
ilters - Optimum Notch Filtering — Inverse Filtering - Wiener filtering
34, MAGE RESTORATION
Restoration improves image in some predefined sense. It is an objective process.
‘ion attempts to reconstruct an image that has been degraded by using a prior
wledge of the degradation phenomenon. These techniques are oriented toward
ing the degradation and then applying the inverse process in order to recover
the original image.
Image Restoration refers to a class of methods that aim to remove or reduce the
degradations that have occurred while the digital image was being obtained.
Restoration techniques are oriented toward modeling the degradation and applying
‘he inverse process in order to recover the original image.
All natural images when displayed have gone through some sort of degradation.
4. During Display Mode
>. Acquisition mode or
© Processing mode
‘The degradations may be due to
4 Sensor noise
5. Blur due to camera mis-focus fe ital Image Processing
32 2
c. Relative object camera motion
d, Random atmospheric turbulence
e. Others
Types
The restoration techniques are classified into two types
1. Spatial domain techniques
2. Frequency domain techniques
y
3.2. IMAGE RESTORATION/ DEGRADATION MODEL
Degradation process operates on a degradation function that operates on an input
image with an additive noise term.
Input image is represented by using the notation f(x, y) noise term can be
represented as (x, y). These two terms when combined gives the result as
gy).
Ifwe are given g(x, y), some knowledge about the degradation function H or J and
some knowledge about the additive noise term n(x, y), the objective of restoration is
to obtain an estimate f" (x, y) of the original image.
We want the estimate to be as close as possible to the original image. The more we
know about h and 77, the closer f(x, y) will be to f"(x, y).
If it is a linear position invariant process, then degraded image is given in the
spatial domain by,
8@y) =A y)*f yt ny) ... Bl)
where h(x, y) is spatial representation of degradation function and *symbol
represents convolution.
In frequency domain we may write this equation as
G(u,v) = H@,») Fv) +N(u, ae .@2)
“The terms in the capital letters are the Fouri ;
terms in the spatial domain, urier transform of the corresponding : Oy)
ftxy) Degradation
a Function H fer) eee
Noise
i ‘ ny)
egradation Restoration
Fig. 3.1. A model of the image degradation/Restoration process
‘The image restoration process can be achieved by inversing the image degradation
process, .e.,
Sy) = EOen=Ny) Flu.)
H(u, v) Aw»)
1 eae A
where ~~—— is the inverse filter and G (u, v) is the .ecovered image.
H (u,v)
Although the concept is relatively simple, the actual implementation is difficult to
achieve, as one requires prior knowledge or identifications of the unknown
degradation function h(x, y) and the unknown noise source 7(x, y).
43.3, NOISE MODELS.
The principal source of noise in digital images arises during image acquisition
and/or transmission. The performance of imaging sensors is affected by a variety of
factors, such as environmental conditions during image acquisition and by the quality
of the sensing elements themselves.
Images are corrupted during transmission principally due to interference in the
channels used for transmission. Since main sources of noise presented in digital
images are resulted from atmospheric disturbance and image sensor circuitry,
following assumptions can be made. s
© The noise model is spatial invariant ive., independent of spatial location.
© The noise model is uncorrelated with the object functio1
3.3.1. SPATIAL AND FREQUENCY PROPERTIES
A frequency property refers to the frequency content of
i.e., as opposed to frequencies of the electromagnetic spect 3.4
Digital Image Processing
For example, when the Fourier spectrum of noise is constant, the noise is cajjeg
white noise, This terminology is a carryover from the physical properties Of White
light, which contains nearly all frequencies in the visible spectrum in equa,
Proportions.
Spatial frequency refers to the exception of spatially periodic noise that noise is
independent of spatial coordinates, and that it is uncorrelated with respect to the
image itself,
3.3.2. PROBABILITY DENSITY FUNCTIONS
_ The most common probability density functions found in image processing
applications are as follows.
| © Gaussian noise
2 —_
Pane (gamma noise) estoration
i)
; aS
Mes 3.2. Gaussian
5 noise
5422. Rayleigh Noise ea ie :"
Unlike Gaussian distribution, the Raylight distiotion
isnot symmetric. It is
given
py the formula, 3.3.2.3. Erlang (gamma) Noise
‘The PDF of Erlang noise is given by
Peril e@ forz20
Po iF forz<0
‘The mean and variance of this density are given by,
feeb:
a
and = &
p(2)
Gamma
_2(b= 1" 50-9
k= eenle
(b-1)/a a
Fig, 3.4. Gamma noise PDF 4325. Uniform Noise
The PDF of uniform noise is g
cod its variance by 3a] Digital Image Processing
If b >a, All appears as a li dot in the image and level a wi
appear rice: on at sae P, or Py is oa then the impulse noise is “oi
‘unipolar. Bipolar impulse noise also called as salt and pepper noise, data-drop-out ang
spike noise.
plz)
Fig, 3.7. Impulse noise PDF
3.3.3. COMPARISON BETWEEN VARIOUS NOISE
Periodic Noise
Periodic noise in ua image arises from electrical or electromechanical interference
during image acquisition. Periodic noise can be reduced significantly via frequency
domain filtering.
Estimation of Noise Parameters
‘The parameters of periodic noise typically are estimated: by it i
E inspection of the
‘Fourier spectrum of the image. The parameters of noise PDFs may be known partially
‘from sensor specification, but it is necessary to estimate them for particular imagitlg
——. restoration B39]
abo
plang a g Czayee tore > 0.) <2"
0
(72) = ed
ial noise ONG forz<0 si
a 1
niform | 5(3) = baa ifasz. This will resulted in «
smoothing effect in the image.
34:12. Geometric Mean Filter
An image restored using a geometric mean filter is given by the expression.
1 fey ma ll 607%
(3, ) eSxy
Here, each restored pixel is given by the product of the pixel in the sub — OE oa
restoration
~ Harmonie mean filter works well for salt noise, but fails for pepper noise. It
ell also with other types of noise like Gaussian noise.
4 contra Harmonic Mean Filter
«contra harmonie mean filter yields a restored image based on the expression
X g(s, e+!
» woe
f(x,y) = eso 2 .. 3.22)
ZX als, 02
BY Sy
here Q is called the order of the filter. This filter is well suited for reducing or
ually eliminating the effects of salt and pepper noise.
If Q is positive then pepper noise is eliminated
‘» IfQ is negative then salt noise is eliminated
Also
* IfQ=0 then the filter becomes arithmetic mean filter
+ 1fQ=-1 then the filter becomes harmonic mean filter
3.4.2. ORDER-STATISTIC FILTERS
Order-Statistics Filters are spatial filters whose response is based on ordering the
ixel contained in the image area encompassed by the filter. The response of the filter
any point is determined by the ranking result. Some of the important
nder-statistic filters are
1. Median filter
Max and min filter
Midpoint filter
Alpha trimmed mean filter
ee
eo}
2.1. Median Filter
ILis the best order statistic filter; it replaces the value of a pixel by the median of
levels in the neighborhood of that pixel. _—
3.12 Digital Image Processing
fy) = median {g (6,)} +23)
(s, ) © Sty
The value of the pixel is included in the computation of the median. Median fit
are quite popular because for certain types of random noise, they ae excellent
noise reduction capabilities, with considerably less blurring than linear sm
filters of similar size. These are effective for bipolar and unipolar impulse noise,
3.4.2.2. Max and Min Filters
Using the 100" percentile of ranked set of numbers is called the max filter and jg
given by the equation.
a
f@,y) = max {g(s,1)} += 3.24)
(3, t) eSxy
It is useful for finding the brightest points in an image. Pepper noise in the image
has very low values; it is reduced by max filter using the max selection Process in the
sub image area S,,.
The 0" percentile filter isthe min fiter
f@,y) = Pn {g(,1)} 3
BE ining dnteg rn vs age Amctiogaimas
eg th opertion.
3.4.2.3. Midpoint Fiter
‘The Midpoint filter computes the midpoint between maximum
values in
1
Wa ,
2[,.m,fe69) + nin nage Restoration
é = ae
f(y) Se 2 S:66.0) G27)
© Syy
(s,
where the value of d can range from 0 to mn ~ 1. When d = 0, the alpha-trimmed
lter reduces to the arithmetic mean filter, If d = mn — 1, the filter becomes a median
Iter
.4.3. ADAPTIVE FILTERS
Adaptive filters are capable of performance superior and whose behaviour changes
sed on statistical characteristics of the image inside the filter region defined by the
x n rectangular window §,,.. It has following types.
1. Adaptive, local noise reduction filter
2. Adaptive, median filter,
.3.1. Adaptive, local noise reduction filter
The mean and variance are the simplest statistical measures of a random variable.
ese are reasonable parameters on which to base an adaptive filter because they are
tities closely related to the appearance of an image.
The mean gives a measure of average intensity in the region over which the mean
mputed and the variance gives a measure of contrast in that region.
These filters will operate on a local region, S,,. The response of the filter at any
(x, y) on which the region is centered is to be based on four quantities.
fa) g (x, y) the value of the noisy image at (x, y)
) 2, the variance of the noise corrupting f(x, y) to form g @,y).
(©) m,, the local mean of the pixels in S,,,
)) o2., the local variance of the pixels in S,,.
behavior of the filter to be as follows.
If 0°, is zero, the filter should return the value of g(x, »). This is the trivial,
zero noise case in which g(x, Y) is equal to f(x, y).
If the local variance is high relative to 0°, the filter should return a value
close to g(x, y). A high local variance is associated with edges and these
should be preserved. *
Digital Image Processing
3. If the two variances are equal, we want the filter to return the arithmetic
value of the pixels in S,,. This condition occurs when the local area has the
same properties as the overall image and local noise is to be reduced by
averaging.
4 .
‘An adaptive expression for obtaining f(t, ¥y) based on these assumptions may be
written as
t oy
f£@y) = 8@r-Q [s (,»)-m] + 6.28)
L
3.4.3.2. Adaptive Median Filter
‘Adaptive Median Filtering can handle impulse noise with probabilities larger than
median filter. It preserves detail while smoothing non impulse noise, sometimes the
traditional median filter does not do. 2
‘Adaptive median filter also works in a rectangular window area S,,, and the size of
‘S,,during filter operation, depending on certain conditions.
Consider the following notation,
: , = minimum intensity value in S,,
Zyygx = Maximum intensity value in S,,
Zoned = median of intensity value in S,,
Z,, = intensity value at coordinates (x, y)
Spx = maximum allowed size of S,, mage Restoration
stageB +BY = Z y= Zang
Ba = Ly — Zine
IfB, > 0 AND B, <0, output Z,,
Else output Ziney
The algorithm can be used for following three main purposes.
1, To remove salt and — pepper (impulse) noise
2. To provide smoothing of other noise that may not be impulsive
3. To reduce distortion.
3.5. PERIODIC NOISE REDUCTION BY FREQUENCY DOMAIN FILTERING
Periodic noise can be analyzed and filtered quite effectively using frequency
domain techniques. Periodic noise appears as concentrated bursts of energy in the
Fourier transform, at locations corresponding to the frequencies of the periodic
interference, The approach is to use a selective filter to isolate the noise.
The three types of selective filters are
1. Band reject filter
2. Bandpass filter
3. Notch filter
5.1. BAND REJECT FILTER +
It removes a band of frequencies about the origin of the Fourier Transformer. The
incipal application of band reject filtering is for noise removal in applications where
general location of the noise component(s) in the frequency domain is
ximately known. Sinusoidal noise can be easily removed by using these kinds of
because it shows two impulses that are mirror images of each other about the
in. It has three types. ' I
1. Ideal band reject filter
Butterworth band reject filter
Gaussian band reject filter. 3.16 Digital Image Processing
3.6.1.1. Ideal Band Reject Filter
An ideal band reject filter is given by the expression
1 if D(,v)
; Ww wpe 629
H(u,v) = 90 if Do-7 SD US OO + B29)
WwW
1 if D@,v)>Dotz
where
D(z, v) is the distance from the origin of the centered frequency rectangle.
W —is the width of the band.
Do ~is the radial center of the frequency rectangle
3.6.1.2. Butterworth Bandreject Filter
Butterworth bandreject filter is given by the expression
1
Bey) D@,v)W_)”
[eoe) |
3.5.1.3. Gaussian Bandreject Filter
Gaussian bandreject filter is given by the expression
1 (D2 (u, v)- D2
H(u,v) = a = |
3.5.2. BANDPASS FILTER
The function of a bandpass filter is opposite to that of a bandreject filter.
It allows a specific frequency band of the image to be passed and blocks the rest of
frequencies.
The transfer function of a band pass filter can be obtained from a corresponding
band reject filter with transfer function Hyp (u, ¥) by using the equation
Hyp(u, v) = 1—Hyp (u,v)
These filters cannot be applied directly on an image because it may remove 09
much details of an image but these are effective in isolating the effect of an image of
selected frequency bands. age Restoration
= 3.17
5.3. NOTCH FILTERS Le ees
A notch filter rejects fre ies i
quencies .
y. These fifleaaa 1 pre-defined Neighborhoods about a center
re symmetric about 01
r function of ideal notch Tej
“ject filter
metry at (...... )is mae
rigin in the fourier transform the
ius do with center at (......) and by
H(u, v)
t iD, (u,v)
