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DIP Module 3 rnsit notes 7th semester electronics and communication
Electronic and communication (ECE)
Visvesvaraya Technological University
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Module- 3
IMAGE RESTORATION
Syllabus
Restoration: Noise models, Restoration in the Presence of Noise Only using Spatial Filtering and Frequency Domain Filtering, Linear, Position-Invariant Degradations, Estimating the Degradation Function, Inverse Filtering, Minimum Mean Square Error (Wiener) Filtering, Constrained Least Squares Filtering.
[Text: Chapter 5: Sections 5, to 5] L1, L2, L
Introduction
The principal goal of restoration techniques is to improve an image in some predefined sense. Restoration is an objective process and it attempts to recover an image that has been degraded by using a priori knowledge of the degradation phenomenon. Thus, restoration techniques are oriented toward the modeling the degradation and applying the inverse process in order to recover the original image. Image restoration vs. image enhancement: - Enhancement: - largely a subjective process. - Priori knowledge about the degradation is not a must (sometimes no degradation is involved). - Procedures are heuristic and take advantage of the psychophysical aspects of human visual system. - Restoration: - more an objective process. - Images are degraded. - Tries to recover the images by using the knowledge about the degradation.
- Image restoration: recover an image that has been degraded by using a prior knowledge of the degradation phenomenon.
- Model the degradation and applying the inverse process in order to recover the original image.
- Image enhancement: “improve ” an image subjectively.
- Image restoration: remove distortion from image, to go back to the “original”--objective process. Why Image Restoration? Image restoration is to recover the original image by removing noise and blur from image.
Image blur is difficult to avoid in many situations like photography, to remove motion blur caused by camera shake, radar imaging to remove the effect of image system response, etc.
Image noise is unwanted signal which comes in image from sensor such as thermal or electrical signal and Environmental condition such as rain, snow etc.
A Model of Image Degradation / Restoration Process :
As fig. shows, degradation process is modeled as degradation function that, together with the additive noise term, operates on an input image f(x, y) to produce a degraded image g(x, y). Given g(x, y), some knowledge about the degradation function H, and additive noise term, the objective of restoration is to obtain an estimate of the original image. Estimation of the output image should be as close as possible to the original input image.
If H is a linear, position- invariant process, then the degraded image is given in the spatial domain by
*g(x, y) = h(x, y)f(x, y) + η(x, y)
Where h(x, y) is the spatial representation of the degradation function and symbol * indicates convolution. Convolution in the spatial domain is analogous to multiplication in the frequency domain.
The model of the degraded image is given in the frequency domain by
G(u, v) = H(u,v)F(u, v) + N(u, v)
where the terms in capital letters are the Fourier transforms of the Corresponding terms of previous spatial domain expression**.**
Noise Models
Noise Sources :
The principal sources of noise in digital images arise during image acquisition and/or transmission. Image acquisition: e., light levels, sensor temperature, etc. Sensor performance is affected by environmental conditions during image acquisition, and by the quality of sensing elements. Transmission: Interference in the channel used for transmission. e., lightning or other atmospheric disturbance in wireless network. Spatial and frequency properties of Noise:
When the Fourier spectrum of noise is constant, the noise is usually called as white noise.
Erlang (Gamma) noise: The PDF of Erlang (Gamma) noise is given by
= −
−
−
0 for 0
for 0 )( ( 1 )!
1
z
e z b
za zp
az
bb
Where a>0, b is a positive integer and “!” indicates factorial. The mean and variance of this density are given by
2 / and 2 a
= b a = b
a and b can be obtained through mean and variance.
Exponential noise: The PDF of Exponential noise is given by
−
0 for 0
for 0 )( z
ae z zp
az
The mean and variance of this density are given by
2 /1 and 2 1 a
= a =
Special case of Erlang PDF with b=1.
Uniform noise: The PDF of Uniform noise is given by
= − 0 otherwise
)( 1 if a z b zp b a
The mean and variance of this density are given by
12
( ) ( 2/) and 2 b a 2 a b
− = + =
Impulse (Salt-and-Pepper) Noise:
Above PDFs provide useful tools for modeling a broad range of noise corruption situations given below. Gaussian noise arises in an image due to factors such as Electronic circuit noise, sensor noise due to poor illumination and/or high temperature
The PDF of (bipolar) impulse noise is given by for ( ) for 0 otherwise
a b
P z a p z P z b
= ==
if , gray-level will appear as a light dot, while level will appear like a dark dot.
b a b a
If either or is zero, the impulse noise is called PPab
unipolar
The Rayleigh density is helpful in characterizing noise phenomenon in Range imaging. The exponential and gamma densities find application in laser imaging. Impulse noise is found in situations where quick transients, such as faulty switching take place during imaging. Uniform density is useful as the basis for numerous random number generators that are used in simulations. Noisy images and their histograms: Following figure 3 shows a test pattern used to illustrate the characteristics noise PDFs. Figure 3 shows the test pattern after the addition of six types of noise.
Estimation of Noise Parameters:
Periodic noise: ▪ Parameters can be estimated by inspection of the Fourier spectrum of the image. ▪ Periodic noise tends to produce frequency spikes that can be detected by visual analysis. Noise PDFs: ▪ From sensor specifications ▪ If imaging sensors are available, capture a set of images of plain environments ▪ If only noisy images are available, parameters of the PDF involved can be estimated from small patches of constant regions of the noisy images
The shape of the histogram identifies the closest PDF match.
Restoration in the Presence of Noise Only ̶ Spatial Filtering
Degraded image in the spatial domain and frequency domain are as follows
*g(x, y) = h(x, y)f(x, y) + η(x, y)
and
G(u, v) = H(u,v)F(u, v) + N(u, v)
When the only degradation present in the image is noise, then the above equations become
g(x, y) = f(x, y) + η(x, y)
and
G(u, v) = F(u, v) + N(u, v)
####### Mean Filters
In this this topic noise reduction capabilities of the various spatial filters are discussed.
Arithmetic mean filter:
The simplest mean filter is arithmetic mean filter.
Consider a subimage denoted by , and let ( ), 0, 1, ..., -1, denote the probability estimates of the intensities of the pixels in. The mean and variance of the pixels in :
S p zsii L S S
=
1
0 221 0
( )
and ( ) ( )
L i s i i L i s i i
z z p z
z z p z
−
= −
=
=
=−
Let represent the set of coordinates in a rectangle subimage window of size , centered at ( , ).
Sxy m n x y
Order Statistic Filters:
Here some additional order statistics filters are discussed. Order statistics filters are spatial filters whose response is based on ordering (ranking) the values of the pixels contained in the image area encompassed by the filter. The ranking result determines the response of the filter.
Median filter:
In this method value of a pixel is replaced by the median of the intensity levels in the neighborhood of that pixel.
▪ Median represents the 50th percentile of a ranked set of numbers
ˆ ),( median{ ,( )} ,
yxf tsg ( s,t) Syx =
It has excellent noise reduction capabilities for certain types of random noise. Less blurring than linear smoothing filters of similar size. It is effective in the presence of both bipolar and unipolar noise. Max and min filter:
Max filter uses the 100th percentile of a ranked set of numbers
ˆ ),( max{ ,( )} ,
yxf tsg
Maxfilter
( s,t) SyxIt is Good for removing pepper noise present in the image.
Min filter is the oth percentile of a ranked set of numbers.
ˆ ),( min{ ,( )}
.
,
yxf tsg
Minfilter
( s,t) SyxIt is Good for removing salt noise present in the image.
Midpoint filter:
The midpoint filter computes between the maximum and Minimum Values in the area encompassed by the filter.
= + 2 max { ,( )} min { ,( )}
1 ˆ yxf ),( ),( tsg ),( tsg Sts xy Sts xy
This filter combines order statistics and averaging filter. It works best for randomly distributed noise, like Gaussian or uniform noise.
Alpha-Trimmed Mean Filter:
####### We delete the / 2 lowest and the / 2 highest intensity values of
####### ( , ) in the neighborhood. Let ( , ) represent the remaining
- pixels.
xy r
####### dd
####### g s t S g s t
####### mn d
Adaptive Filters: Adaptive Median Filters :
Median filter is effective for removing salt-and-pepper noise. The density of the impulse noise can not be too large. This filter preserves image details while smoothing non-impulse noise.
This filter has three main purposes: To remove salt and pepper (impulse) noise. To provide smoothing of other noise that may not impulsive, and To reduce distortion such as excessive thinning or thickening of object boundaries. The purpose of stage A is to determine if the median filter output, zmed , is an impulse (black or white) or not. If the condition zmin < zmed < zmax holds, then zmed can not be an impulse. In this case go to stage B and test to see if the point in the center of the window, zxy, is itself an impulse. If the condition B1>0 AND B2<0 is true, then zmin < zxy < zmax , and zxy can not be an impulse.
If the condition B1 > 0 AND B2 < 0 is false, then either zxy = zmin or zxy = zmax. In either case the value of the pixel is an extreme value and the algorithm outputs the median value zmed which is not a noise impulse.
2 2
####### An adaptive expression for obtaining ( , )
####### based on the assumptions:
####### ( , ) ( , ) ( , ) L
L
####### f x y
####### f x y g x y g x y m
####### = − −
min max med
max
The notation: minimum intensity value in maximum intensity value in median intensity value in
intensity value at coordinates ( , ) maximum all
xy xy xy xy
zS zS zS
z x y S
= = =
= = owed size of Sxy
med min med max
max
####### The adaptive median-filtering works in two stages:
####### Stage A:
####### A1 = ; A2 =
####### if A1>0 and A2<0, go to stage B
####### Else increase the window size
####### if window size , re
####### z z z z
####### S
####### −−
####### med
min max
med
####### peat stage A; Else output
####### Stage B:
####### B1 = ; B2 =
####### if B1>0 and B2<0, output ; Else output
xy xy
xy
####### z
####### z z z z
####### zz
####### −−
Periodic Noise Reduction by Frequency Domain Filtering The basic idea Periodic noise can be analyzed and filtered effectively using frequency domain techniques. The basic idea is that the Periodic noise appears as concentrated bursts of energy in the Fourier transform, at locations corresponding to the frequencies of the periodic interference. Approach - A selective filter is used to isolate the noise. - Band reject, band pass, and notch filters are used as tools for periodic noise reduction. - Band reject filters remove or attenuate a band of frequencies about the origin of the Fourier transform. Band reject filtering is for noise removal in applications where the general locations of noise components in frequency domain is approximately known. Example is an image corrupted by additive periodic noise that can be approximated as 2 D sinusoidal functions. Fourier transform of sine consists of two impulses that are mirror images of each other about the origin of the transform. The sinusoidal noise components appear as symmetric pairs of bright dots can be observed in its Fourier Transform. Band reject Filters The transfer functions of ideal, butter-worth and Gaussian filters are as shown below perspective plots are shown in Figure 3.
Ideal band reject filter: Band reject filters remove or attenuate a band of frequencies about the origin of the Fourier Transform. The transfer function of ideal band reject filter is given by
+
− +
−
=
2
1 ),(
2
),( 2
0
2
1 ),(
),(
0
0 0
0
if D vu D W
ifD W D vu D W
ifD vu D W
H vu
Where D(u,v) is the distance from the origin of the centred frequency rectangle, W is the width of the band of frequency and D 0 is the radial centre. Similarly, the transfer function of butter-worth band reject filter of the order n is the given by the expression
Notch Filters
Notch filter rejects (or passes) frequencies in predefined neighborhoods about a center frequency. It appears in symmetric pairs about the origin because the Fourier transform of a real valued image is symmetric. The transfer function of Notch pass filter is
HNP(u,v) = 1 – HNR(u,v)
where HNP(u,v) is the transfer function of the notch pass filter Corresponding to the notch reject filter with transfer function HNR(u,v). This filter reduces the noise in the image with out introducing appreciable blurring. The transfer functions of ideal, Butterworth and Gaussian notch (reject) filters are as follows.
Figure 3 shows the 3 D plots of ideal, Butterworth and Gaussian notch (reject) filters. The notch filter reduces the noise in the image with out blurring the given image.
Optimum Notch Filter
Several interference components are present in the methods discussed in the preceding sections are not always acceptable because they remove much image information.
Points on or near Edge of the image can be treated by considering partial neighborhoods or by padding border with 0s. Sub. Eqn.(1) in to (2) yields.
),([ ),( ˆ ,( ]}.....( )
( , )ˆ( , )]
{[ ( , ) 2( 1 )( 2 )
1 ),(
2
2
g yx w yx yx
wx ys t x ys t
gx ys t a b
yx
a
as
b
bt
− −
− + + + +
=
−= −=
Assuming that w(x, y) remains constant over the neighborhood gives the approximation. w ( x + , ys + t )= w yx ),( for − a s aand − b t b ....( ) w yx ),( ˆ yx ),( = w yx ),( ˆ ,( yx ).....( )
With these approximations, eqn. (3) becomes
),([ ),( ˆ ,( )}.....( )
),( ˆ( , )]
{[ ( , ) 2( 1 )( 2 )
1 ),(
2
2
g yx w yx yx
w yx x ys t
g x ys t a b
yx a as
b
bt
− −
− + +
= −=
−=
To minimize σ 2 (x, y), we solve
0 .....( ) ),(
2 ),( =
wx y
x y
.....() ˆ ),( ˆ ),(
),( ˆ ),( ),( ˆ ),( ),( 2 2 yx yx
g x y yx g yx x y w yx
−
− =
To obtain the restored image, we compute w(x, y) from eqn.(8) and then use Eqn.(1)
Linear, Position-Invariant Degradations
The input – output relationship in the above figure before the Restoration stage is expressed as
If a = b = 1, above equation becomes H[f 1 (x,y) + f 2 (x,y)] = H[f 1 (x,y)] + H[f 2 (x,y)]
g x y ( , )=+ H f x y ( , ) ( , ) x y
1 2 1 2
12
is linear ( , ) ( , ) ( , ) ( , ) and are any two input images.
H H af x y bf x y aH f x y bH f x y ff
- = +
Many types of degradation can be approximated by linear, position-invariant processes Extensive tools of linear system theory are available. In this situation, restoration is image deconvolution. Estimating the Degradation Function Three principal ways to estimate the degradation function
- Observation
- Experimentation
- Mathematical Modeling.
#######
#######
An operator having the input-output relationship ( , ) ( , ) is said to be position invariant if ( , ) ( , ) for any ( , ) and any and.
g x y H f x y
H f x y g x y f x y
=
− − = − −
f x y ( , ) f ( , ) ( x , y ) d d
=− − − −
Assume for a moment that ( , ) 0 if is a linear operator, ( , ) ( , )
( , ) ( , )
( , ) ( , )
( , ) ( , )
xy H g x y H f x y
H f x y d d
H f x y d d
f H x y d d
− − − − − −
=
=
= − −
= − −
= − −
Assume for a moment that ( , ) 0 if is a linear operator and position invariant, ( , ) ( , ) ( , ) ( , )
( , ) ( , )
( , ) ( , )
xy H H x y h x y g x y H f x y
f H x y d d
f h x y d d
− − − −
=
− − = − −= − −
= − −
In the presence of additive noise, if is a linear operator and position invariant,
( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , )
( , ) ( , ) ( , ) ( , )
H
g x y f h x y d d x y h x y f x y x y
G u v H u v F u v N u v
= − − − − + =+
=+
DIP Module 3 rnsit notes 7th semester electronics and communication
Course: Electronic and communication (ECE)
University: Visvesvaraya Technological University
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