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（天津职业技术师范大学）

DEPARTMENT: ELECTRONICS ENGINEERING

SUBJECT: DIGITAL IMAGE PROCESSING

STUDENT NAME: MWIGHUSA (唐德宁)

SUPERVISOR: DR. DING XUEWEN (丁学文) - PhD

PROJECT TITLE: IMAGE RESTORATION

DATE: 24TH

NOVEMBER

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Overview: In the past two decades, the technique of image processing has made its way into every aspect of

today’s tech-savvy society. Its applications encompass a wide variety of specialized disciplines

including medical imaging, machine vision, remote sensing and astronomy. Personal images

captured by various digital cameras can easily be manipulated by a variety of dedicated image

processing algorithms. Image restoration can be described as an important part of image processing

technique. The basic objective is to enhance the quality of an image by removing defects and make

it look pleasing. The method used to carry out the project was MATLAB software. Mathematical

algorithms were programmed and tested for the result to find the necessary output. In this project

mathematical analysis was the basic core. Generally the spatial and frequency domain methods

were both important and applicable in different technologies. This project has tried to show the

comparison between spatial and frequency domain approaches and their advantages and

disadvantages. This project also suggested that more research have to be done in many other image

processing applications to show the importance of those methods.

Acknowledgements:

The author remains thankful to Dr. Ding Xuewen (Dr. & Lecturer, Dept. of Electronics &

Electrical Engineering, Tianjin University of Technology & Education, Tianjin, China P.R) for his

useful discussions and suggestions during the preparations of this project.

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Contents

Introduction………………………………………………………………………………………..1

Image Noise Model………………………………………………………………………………..2

Noise Types……………………………………………………………………………….3

Filters……………………………………………………………………………………………...6

Filtering Techniques………………………………………………………………………7

Noise Filtering in Spatial and Frequency Domain………………………………………………...8

Filtering to remove noise………………………………………………………………….8

Order statistic filters……………………………………………………………………...11

Adaptive Filters…………………………………………………………………………………..15

Adaptive Filters Example………………………………………………………………..17

Periodic Noise……………………………………………………………………………………17

Band Reject Filters……………………………………………………………………………….17

Band Reject Filters Example…………………………………………………………….18

Advantage and Disadvantage of Spatial and Frequency Domain Methods……………………...19

Conclusion……………………………………………………………………………………….19

Reference………………………………………………………………………………………...20

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IMAGE RESTORATION

Introduction

Image Restoration is the process of obtaining the original image from the degraded image given

the knowledge of the degrading factors. Digital image restoration is a field of engineering that

studies methods used to recover original scene from the degraded images and observations.

Techniques used for image restoration are oriented towards modeling the degradations, usually

blur and noise and applying various filters to obtain an approximation of the original scene. There

are a variety of reasons that could cause degradation of an image and image restoration is one of

the key fields in today's Digital Image Processing due to its wide area of applications. Commonly

occurring degradations include blurring, motion and noise. Blurring can be caused when object in

the image is outside the camera’s depth of field sometime during the exposure, whereas motion

blur can be caused when an object moves relative to the camera during an exposure.

Fig. Image Restoration example

The purpose of image restoration is to "compensate for" or "undo" defects which degrade an image.

Degradation comes in many forms such as motion blur, noise, and camera misfocus. In cases like

motion blur, it is possible to come up with a very good estimate of the actual blurring function and

"undo" the blur to restore the original image. In cases where the image is corrupted by noise, the

best we may hope to do is to compensate for the degradation it caused. In this project, we will

introduce and implement several of the methods used in the image processing world to restore

images. The field of image restoration (sometimes referred to as image de-blurring or image de-

convolution) is concerned with the reconstruction or estimation of the uncorrupted image from a

blurred and noisy one. Essentially, it tries to perform an operation on the image that is the inverse

of the imperfections in the image formation system. In the use of image restoration methods, the

characteristics of the degrading system and the noise are assumed to be known a priori. In practical

situations, however, one may not be able to obtain this information directly from the image

formation process. The goal of blur identification is to estimate the attributes of the imperfect

imaging system from the observed degraded image itself prior to the restoration process. The

combination of image restoration and blur identification is often referred to as blind image de-

convolution. Image restoration algorithms distinguish themselves from image enhancement

methods in that they are based on models for the degrading process and for the ideal image. For

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those cases where a fairly accurate blur model is available, powerful restoration algorithms can be

arrived at. Unfortunately, in numerous practical cases of interest, the modeling of the blur is

unfeasible, rendering restoration impossible. The limited validity of blur models is often a factor

of disappointment, but one should realize that if none of the blur models described in this chapter

are applicable, the corrupted image may well be beyond restoration. Therefore, no matter how

powerful blur identification and restoration algorithms are, the objective when capturing an image

undeniably is to avoid the need for restoring the image.

In restoration process, degradation is taken to be a linear spatially invariant operator –

Figure. Processing of Image Restoration

𝑔(𝑥, 𝑦) = ℎ(𝑥, 𝑦) ∗ 𝑓(𝑥, 𝑦) + 𝜂(𝑥, 𝑦)

where, if g(x, y) is noise free, restoration can be done by using the inverse transfer function of h(u,

v) as the restoration filter and η(x, y) is the noise. The restoration techniques use various types of

filters for achieving best performance, like inverse filter, wiener filter, constrained least square

filter, Histogram Adaptive Fuzzy filter, Min-max Detector Based filter and Centre Weighted Mean

filter etc.

Image Noise Model

The sources of noise in digital images arise during image acquisition (digitization) and

transmission.

• Imaging sensors can be affected by ambient conditions

• Interference can be added to an image during transmission

We can consider a noisy image to be modelled as follows:

𝑔(𝑥, 𝑦) = ℎ(𝑥, 𝑦) ∗ 𝑓(𝑥, 𝑦) + 𝜂(𝑥, 𝑦)

where f(x, y) is the original image pixel, η(x, y) is the noise term and g(x, y) is the resulting noisy

pixel. If we can estimate the model of the noise in an image, it will help us to figure out how to

restore the image.

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Noise Types

For an image to be restored it is important to know the features of the noises that caused its

degradation. They have different features but the most important in this project are the spatial and

frequency properties.

Spatial properties mean dealing with the statistical behaviors of the noise component, while in

frequency properties deal with the frequency contents of the noise in Fourier form. There are

many noise patterns in image processing and some of them include:

a) Gaussian noise – is a noise type initiated by a Gaussian amplitude distribution. The Gaussian

probability distribution has a a probability density function of

𝑃(𝑥) =1

𝜎√2𝜋𝑒−(𝑥−𝜇)2 2𝜎2⁄

where 𝑥 is the gray level 𝜇 is the mean, 𝜎 is the standard deviation and 𝜎2 is the variance.

b) Erlang (Gamma) noise – is a noise having a probability distribution function of

𝑃(𝑧) = {𝑎𝑏𝑧𝑏−1

(𝑏−1)!𝑒−𝑎𝑧 𝑓𝑜𝑟 𝑧 ≥ 0

0 𝑓𝑜𝑟 𝑧 < 0

where the mean is a 𝜇 = 𝑏

𝑎 and variance 𝜎2 =

𝑏

𝑎2 . The parameters a and b are positive integers

and “ ! ” is factorial.

c) Exponential noise – is a noise with exponential probability density function of

𝑃(𝑧) = {𝑎𝑒−𝑎𝑧 𝑓𝑜𝑟 𝑧 ≥ 0 0 𝑓𝑜𝑟 𝑧 < 0

where the mean 𝜇 = 1

𝑎 and 𝜎2 =

1

𝑎2 for a > 0.

d) Uniform noise – is the noise with the probability density function of

𝑃(𝑧) = {1

𝑏−𝑎 𝑖𝑓 𝑎 ≤ 𝑧 ≤ 𝑏

0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

where the mean 𝜇 = 𝑎+𝑏

2 is and σ2 =

(𝑏−𝑎)2

12 is variance.

e) Impulse noise(Salt-and-pepper) – is a noise type with a probability density function of

𝑃(𝑧) = {𝑃𝑎 𝑓𝑜𝑟 𝑧 = 𝑎𝑃𝑏 𝑓𝑜𝑟 𝑧 = 𝑏0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

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Noise Example

The test pattern to the right is ideal for demonstrating the addition of noise, The following

images will show the result of adding noise based on various models to the original image.

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Original Image Histogram

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Filters

Elimination of noise is one of the major works to be done in computer vision and image processing,

as noise leads to the error in the image. Presence of noise is manifested by undesirable information,

which is not at all related to the image under study, but in turn disturbs the information present in

the image. It is translated into values, which are getting added or subtracted to the true gray level

values on a gray level pixel. These unwanted noise information can be introduced because of so

many reasons like: acquisition process due to cameras quality and restoration, acquisition

condition, such as illumination level, calibration and positioning or it can be a function of the scene

environment. Noise elimination is a main concern in computer vision and image processing. A

digital filter is used to remove noise from the degraded image. As any noise in the image can be

result in serious errors. Noise is an unwanted signal, which is manifested by undesirable

information. Thus the image, which gets contaminated by the noise, is the degraded image and

using different filters can filter this noise. Thus filter is an important subsystem of any signal

processing system. Thus filters are used for image enhancement, as it removes undesirable signal

components from the signal of interest. Filters are of different type i.e. linear filters or nonlinear

filters. In early times, as the signals handled were analog, filters used are of analog. Gradually

digital filters were took over the analog systems because of their flexibility, low cost,

programmability, reliability, etc. for these reasons digital filters are designed which works with

digital signals.

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The design-of digital filters involves three basic steps:

(i) the specification of the desired properties of the system,

(ii) the approximation of these specifications using a causal discrete time system, and

(iii) the realization of the system using finite precision arithmetic.

Filtering Techniques

Filtering is a technique for enhancing the image. Linear filter is the filtering in which the value of

an output pixel is a linear combination of neighborhood values, which can produce blur in the

image. Thus a variety of smoothing techniques have been developed that are non linear.

Median Filtering

Median filter (MF) is the one of the most popular non-linear filter. Median filtering is one of the

spatial domain filter methods and as its name suggests that the median of the neighborhood pixel

value is taken and replaced with the neighborhood pixel values. The median of the pixels are taken

in such a way that the values are listed in the order from small value until big ones. Then the value

or values at the center of the list is or are taken as the median of the pixel values. In the case of

where two numbers becoming the median, it is important to take the average of the two values.

Median filtering is a very important and widely used technique of filtering and best known for its

excellent noise reduction ability. During filtering it keeps the edges while removing the noise. This

makes the image not to blur as other smoothening methods.

Spatial Filtering

In spatial filtering a certain filter mask is used in all points in an image. The filter mask is made of

m x n size where the m and n are the matrix sizes. In each case the image points should have the

same matrix size as of the filter mask with size M x N. There are two kinds of filtering, linear and

non-linear spatial filtering. In low-pass filter the attenuation of high frequencies from frequency

domains causes in the blurring of an image. While in high-pass filter the removal of low

frequencies cause in the sharpening of edges. Band-pass filtering is used for image restoration

while removing frequencies between high and low frequencies. There are many filters that are

used for blurring/smoothing, sharpening and edge detection in an image. These different effects

can be achieved by different coefficients in the mask.

Smoothing Spatial Filter

Smoothening filters can be obtained by averaging of pixels in the neighborhood of a filter mask.

It results in the blurring of an image. During noise removal or noise reduction sharp edges are

removed from the image. There are two kinds of filtering, linear and non-linear spatial filtering. In

low-pass filter the attenuation of high frequencies from frequency domains causes in the blurring

of an image.

Inverse Filtering

If we know of or can create a good model of the blurring function that corrupted an image, the

quickest and easiest way to restore that is by inverse filtering. Unfortunately, since the inverse

filter is a form of high pass filer, inverse filtering responds very badly to any noise that is present

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in the image because noise tends to be high frequency. In this section, we explore two methods of

inverse filtering - a thresholding method and an iterative method.

Wiener Filtering

The inverse filtering is a restoration technique for deconvolution, i.e., when the image is blurred

by a known low pass filter, it is possible to recover the image by inverse filtering or generalized

inverse filtering. However, inverse filtering is very sensitive to additive noise. The approach of

reducing one degradation at a time allows us to develop a restoration algorithm for each type of

degradation and simply combine them. The Wiener filtering executes an optimal tradeoff between

inverse filtering and noise smoothing. It removes the additive noise and inverts the blurring

simultaneously. The Wiener filtering is optimal in terms of the mean square error. In other words,

it minimizes the overall mean square error in the process of inverse filtering and noise smoothing.

The Wiener filtering is a linear estimation of the original image. The approach is based on a

stochastic framework. The orthogonality principle implies that the Wiener filter in Fourier domain

can be expressed as follows:

where are respectively power spectra of the original image and the additive

noise, and is the blurring filter. It is easy to see that the Wiener filter has two separate part,

an inverse filtering part and a noise smoothing part. It not only performs the deconvolution by

inverse filtering (high pass filtering) but also removes the noise with a compression operation (low

pass filtering).

Noise Filtering in Spatial and Frequency Domain

Noise filtering techniques were discussed in the above sections. Here I discuss only new methods

for filtering in the spatial and frequency. A salt-and-pepper noise applied to the image has resulted

in an image in shown above. Using the median filtering technique caused the noise to be removed.

Analysis of Spatial and Frequency Methods and Results

In the above sections I tried to explain the methods used to filter noise for a digital image. Noise

filtering is carried out by spatial and frequency low-pass filtering, Contrast enhancement is carried

out with spatial domain histogram stretching and sharpening of an image uses the spatial and

frequency domain high-pass filtering. The results of the methodologies will be discussed here.

Filtering to Remove Noise

We can use spatial filters of different kinds to remove different kinds of noise, The arithmetic

mean filter is a very simple one and is calculated as follows:

xySts

tsgmn

yxf),(

),(1

),(ˆ

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This is implemented as the simple smoothing filter. Blurs the image to remove noise.

There are different kinds of mean filters all of which exhibit slightly different behaviour:

(i) Geometric Mean

(ii) Harmonic Mean

(iii) Contraharmonic Mean

There are other variants on the mean which can give different performance

Geometric Mean:

Achieves similar smoothing to the arithmetic mean, but tends to lose less image detail

Harmonic Mean:

The Harmonic Mean works well for salt noise, but fails for pepper noise, Also does well for

other kinds of noise such as Gaussian noise

mn

Sts xy

tsgyxf

1

),(

),(),(ˆ

xySts tsg

mnyxf

),( ),(

1),(ˆ

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Contraharmonic Mean:

Q is the order of the filter and adjusting its value changes the filter’s behaviour, Positive values

of Q eliminate pepper noise, Negative values of Q eliminate salt noise

Noise Removal Examples:

Original image Image corrupted By Gaussian Noise

After a 3*3 arithmetic mean filter After a 3*3 geometric mean filter

xy

xy

Sts

Q

Sts

Q

tsg

tsg

yxf

),(

),(

1

),(

),(

),(ˆ

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Choosing the wrong value for Q when using the contraharmonic filter can have drastic results

Order Statistics Filters

Spatial filters that are based on ordering the pixel values that make up the neighbourhood operated

on by the filter, Useful spatial filters include:

(i) Median filter

(ii) Max and min filter

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(iii)Midpoint filter

(iv) Alpha trimmed mean filter

Median Filter:

Excellent at noise removal, without the smoothing effects that can occur with other smoothing

filters, Particularly good when salt and pepper noise is present

Max Filter:

Min Filter:

Max filter is good for pepper noise and min is good for salt noise

Midpoint Filter:

Good for random Gaussian and uniform noise

Alpha-Trimmed Mean Filter:

We can delete the d/2 lowest and d/2 highest grey levels, So gr(s, t) represents the remaining mn

– d pixels

)},({),(ˆ),(

tsgmedianyxfxySts

)},({max),(ˆ),(

tsgyxfxySts

)},({min),(ˆ),(

tsgyxfxySts

)},({min)},({max

2

1),(ˆ

),(),(tsgtsgyxf

xyxy StsSts

xySts

r tsgdmn

yxf),(

),(1

),(ˆ

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Noise Removal Examples

Image Corrupted By Salt And Pepper Noise

Result of 1 Pass With A 3*3 Median Filter

Result of 2 Passes With A 3*3 Median Filter

Result of 3 Passes With A 3*3 Median Filter

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Image Corrupted

By Pepper Noise

Image Corrupted

By Salt Noise

Result Of Filtering Above With A 3*3 Max Filter

Result Of Filtering Above With A 3*3 Min Filter

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Adaptive Filters

The filters discussed so far are applied to an entire image without any regard for how image

characteristics vary from one point to another, The behaviour of adaptive filters changes

depending on the characteristics of the image inside the filter region, We will take a look at the

adaptive median filter

Adaptive Median Filtering

The median filter performs relatively well on impulse noise as long as the spatial density of the

impulse noise is not large, The adaptive median filter can handle much more spatially dense

impulse noise, and also performs some smoothing for non-impulse noise, The key insight in the

adaptive median filter is that the filter size changes depending on the characteristics of the image

Remember that filtering looks at each original pixel image in turn and generates a new filtered

pixel.

Image Further Corrupted

By Salt and Pepper Noise

Image Corrupted

By Uniform Noise

Filtered By 5*5 Arithmetic Mean Filter

Filtered By 5*5 Geometric Mean Filter

Filtered By 5*5 Median Filter

Filtered By 5*5 Alpha-Trimmed Mean Filter

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First examine the following notation:

- zmin = minimum grey level in Sxy

- zmax = maximum grey level in Sxy

- zmed = median of grey levels in Sxy

- zxy = grey level at coordinates (x, y)

- Smax =maximum allowed size of Sxy

- Level A: A1 = zmed – zmin

- A2 = zmed – zmax

- If A1 > 0 and A2 < 0, Go to level B

- Else increase the window size

- If window size ≤ Smax repeat level A

- Else output zmed

- Level B: B1 = zxy – zmin

- B2 = zxy – zmax

- If B1 > 0 and B2 < 0, output zxy

- Else output zmed

The key to understanding the algorithm is to remember that the adaptive median filter has three

purposes:

(i) Remove impulse noise

(ii) Provide smoothing of other noise

(iii)Reduce distortion

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Adaptive Filtering Example

Periodic Noise

Typically arises due to electrical or electromagnetic interference, Gives rise to regular noise

patterns in an image, here Frequency domain techniques in the Fourier domain are most effective

at removing periodic noise

Band Reject Filters

Removing periodic noise form an image involves removing a particular range of frequencies from

that image, Band reject filters can be used for this purpose, An ideal band reject filter is given as

follows:

Image corrupted by salt and pepper noise with probabilities P

a = P

b=0.25

Result of filtering with a 7 * 7 median filter

Result of adaptive median filtering with i = 7

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The ideal band reject filter is shown below, along with Butterworth and Gaussian versions of the

filter

Band Reject Filter Example

2),( 1

2),(

2 0

2),( 1

),(

0

00

0

WDvuDif

WDvuD

WDif

WDvuDif

vuH

Ideal Band

Reject Filter

Butterworth

Band Reject

Filter (of order 1)

Gaussian

Band Reject

Filter

Image corrupted by sinusoidal noise Fourier spectrum of corrupted image

Filtered image Butterworth band reject filter

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Table1. Advantages and Disadvantages of Spatial and Frequency domain methods

Spatial domain method

Frequency domain method

Advantages -Direct manipulation of pixels

- Very good method for

contrast enhancement

- It is also a good method

for image sharpening

-Manipulation of frequency

-Best method for periodic

noise reduction

- Best method for image

sharpening

Disadvantages -Sometimes it shifts image

boundaries during sharpening

-Only manipulates the pixel

- Not a good method for

contrast enhancement

-Only manipulates the

frequency

Table 1 suggested that both methods are important depending on the target needed. Spatial domain

method is advantageous in contrast enhancement, while frequency domain method in periodic

noise reduction and image sharpening.

Conclusion

Image restoration is an important field that is used in different scientific researches and technology

developments. In this paper different filtering techniques were discussed, we have also looked at

image restoration for noise removal, Restoration is slightly more objective than enhancement .

Spatial domain techniques are particularly useful for removing random noise, while Frequency

domain techniques are particularly useful for removing periodic noise. The differences among the

domains and their different methodologies were briefly explained. Generally the methods are both

important and applicable in different technologies. In this paper I tried to show a comparison

between both approaches and tried to show their advantages and disadvantages. This paper

suggests that more researches is needed on many other image processing applications to show the

importance of those methods.

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References

1. Digital Image Processing by R. C. Gonzales and R. E. Woods, Addison-Wesley Publishing

Company, 1992.

2. Two-Dimensional Signal and Image Processing by J. S. Lim, Prentice Hall, 1990.

3. 'Digital Image Restoration', by M.R. Banham and A.K. Katsaggelos, IEEE Signal Processing

Magazine, pp. 27-41, March 1997.

4. http://www.sersc.org/journals/IJAST/vol39/5.pdf

5. http://www.jatit.org/volumes/research-papers/Vol14No2/1Vol14No2.pdf

6. http://www.owlnet.rice.edu/~elec539/Projects99/BACH/proj2/refs.html

7. https://publications.theseus.fi/bitstream/handle/10024/29065/thesis_final7.pdf?sequence=1

8. KADIR, T., BRADY, M. 2001. Saliency, Scale and Image Description. International Journal

of Computer Vision, Volume 45, Number 2.

9. R. Plemmonsa, M. Horvatha, E. Leonhardta, P. Paucaa, S. Prasadb, “Computational Imaging

Systems

10. Class Presentation notes provided by Dr Xuewen Ding, Lecturer of Digital Image Processing,

Electrical and Electronics Engineering Dept, Tianjin University of Technology and

Education

Appendices

Appendix 1

Spatial domain Algorithms %-----contrast intensity manipulation using histogram equalization and certain %other ----------%

load image1

whos %shows the size and parameters

figure(1)

imshow(X,map) %show the image1 image

I = ind2gray(X,map); %creating intensity of an image

figure(2)

imshow(I) %show intensity

A=I.^4; %high contrast created

A = 0.2*I; %low contrast created

figure(3)

subplot(2,1,1)

imshow(A) %show contrast image

subplot(2,1,2)

imhist(A)

grid

C=(A+.085)*3; %histogram stretching

figure(5)

subplot(2,1,1)

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imshow(C) %show histogram

subplot(2,1,2)

hist(A(:),50)

imhist(C)

grid 62

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%-----------------------pixel manipulation with high pass filter mask--------------%

load images

whos

figure(1)

imshow(X,map) % Indexed image

I=ind2gray(X,map); % Converting indexed image to intensity image

figure(2)

imshow(I) % Intensity image

%My=ones(3,3)/9; %averaging mask

My=zeros(3,3);

My(1,1)=-1;

My(1,2)=-1;

My(1,3)=-1 ;

My(2,1)=-1;

My(2,2)=8;

My(2,3)=-1 ;

My(3,1)=-1;

My(3,2)=-1;

My(3,3)=-1;

M=My; %the mask „My‟ can be changed depending on the result needed

FI=I; % Intiating the output matrix

for i=2:257

for j=2:349

FI(i,j)=sum(sum(I(i-1:i+1,j-1:j+1).*M)); %filtering with the mask

end

end

figure(4)

imshow(FI) 63

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% --------------Implementing pixel group operation (mask operation, nearest %neighbour method,

filtering method)------------%

load images

figure(1)

imshow(X,map) % Indexed image

I=ind2gray(X,map); % Converting indexed image to intensity image

imshow(I) % Intensity image

SP=imnoise(I,'salt & pepper',0.4); % Corrupted image

figure(2)

imshow(SP)

%------------------------Median Filtering--------------------------------%

F2=SP;

for i=1+par:175 %258-par

for j=1+par:225 %350-par

AKE=SP(i-par:i+par,j-par:j+par); %ordering of pixels

F2(i,j)=median(AKE(:));

end

end

figure(4)

imshow(F2) 64

Appendix 2

Frequency domain Algorithms %-------- Analysing an intensity image by Fourier-analysis techniques---------%

load images

I=ind2gray(X,map); % Sample set

figure(1)

imshow(I)

%------------------------Fourier Analysis--------------------------------%

FI=fft2(I); % Complex amplitudes

SFI=fftshift(FI); % Origin in the middle of the matrix

SFI(130,176) is C0 in SFI

Ck=abs(SFI); % Amplitudes

AKE=log10(Ck+1);

figure(2)

imshow(Ck) % Amplitudes with the origin in the middle of the matrix

title('Amplitudes')

a=min(AKE(:));

b=max(AKE(:));

sAKE=(AKE-a)/(b-a); % Linearly scaled logarithms

figure(3)

imshow(sAKE)

title('Scaled amplitudes') % giving titles on the image

%--------------------Simulating Noise in an Image-------------------------% 65

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MAT=SFI;

MAT(74,100)=800*MAT(74,100); %noise at the given coordinates

MAT(186,252)=800*MAT(186,252); %noise at the given coordinates

Ck=abs(MAT); % Amplitudes

AKE=log10(Ck+1);

a=min(AKE(:))

b=max(AKE(:))

sAKE=(AKE-a)/(b-a); % Linearly scaled logarithms

figure(4)

imshow(sAKE)

title('Scaled amplitudes')

IMG=ifft2(MAT);

figure(5)

imshow(abs(IMG))

%---------------------Ideal Low Pass Filter-------------------------------%

[m,n]=size(I);

M=zeros(m,n); % Initiating the low pass mask

M(130-par:130+par,176-par:176+par)=1;

fMAT=MAT.*M; % Filtering MAT

IMG2=ifft2(fMAT);

figure(6)

imshow(abs(IMG2))

%--------------Mask as an Amplitude Spectrum of an Image----------------%

figure(7) % Amplitude spectrum in frequency domain

subplot(2,1,1)

imshow(M)

subplot(2,1,2)

plot(M(130,:))

grid 66

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IMG3=fftshift(ifft2(M));

figure(8)

aIMG3=abs(IMG3);

a3=min(aIMG3(:));

b3=max(aIMG3(:));

sIMG3=(aIMG3-a3)/(b3-a3); % enhancing the image with max. and min.values % while

removing the rest of the frequency values

%----------------------Butterworth Low Pass Filter---------------------------%

[V,U]=meshgrid(-175:174,-129:128);

D=sqrt(U.^2+V.^2);

H=1./(1+(D/D0).^(2*N)); % Butterworth Low Pass Filter

figure(9) % Amplitude spectrum in frequency domain

subplot(2,1,1)

imshow(H)

subplot(2,1,2)

plot(H(130,:))

grid

%------------------------------Notch Filter----------------------------------%

D1=sqrt((U+56).^2+(V+76).^2);

D2=sqrt((U-56).^2+(V-76).^2);

H1=1./(1+(D1/D0).^(2*N)); %implementing the notch filter formula

H2=1./(1+(D2/D0).^(2*N)); %implementing the notch filter formula

figure(9)

imshow(D2/max(D2(:)))

Hlow=H1+H2; % Local low pass filter in pole positions

High=1-Hlow;

figure(10)

imshow(High)

Inverse filter code

N=256;

n=.2;

f=freadbin('lenna.256',N,N);

figure(1)

imagesc(f)

colormap(gray)

b=ones(4,4)/4^2;

F=fft2(f);

B=fft2(b,N,N);

G=F.*B;

g=ifft2(G)+10*randn(N,N);

G=fft2(g);

figure(2)

imagesc(abs(ifft2(G)))

29 | P a g e

colormap(gray)

BF=find(abs(B)<n);

%B(BF)=max(max(B))/1.5;

B(BF)=n;

H=ones(N,N)./B;

I=G.*H;

im=abs(ifft2(I));

figure(3)

imagesc(im)

colormap(gray)

Iterative method code

N=256;

lambda=0.1;

f=freadbin('lenna.256',N,N);

figure(1)

imagesc(f)

colormap(gray)

F=fft2(f);

b=ones(4,4)/16;

B=fft2(b,N,N);

G=F.*B;

%g=ifft2(G)+10*randn(N,N);

%G=fft2(g);

figure(2)

imagesc(abs(ifft2(G)))

colormap(gray)

K=lambda*G;

for l=1:1500,

if mod(l,25)==0

lambda=lambda/5;

end

A=G-K.*B;

K=K+lambda*A;

if mod(l,50)==0

l

sum(sum(A))

figure(3)

imagesc(abs(ifft2(K)))

colormap(gray)

pause

end

end