Digital image processing using matlab: filters (detail)

NATIONAL CHENG KUNG UNIVERSITY

Inst. of Manufacturing Information & Systems

DIGITAL IMAGE PROCESSING AND SOFTWARE IMPLEMENTATION

HOMEWORK 2

Professor name: Chen, Shang-Liang

Student name: Nguyen Van Thanh

Student ID: P96007019

Class: P9-009 Image Processing and Software Implementation

Time: [4] 2 4

1

Contents

PROBLEM 2

SOLUTION 3

1. Fourier transform 3

1.1. The 2 – D DFT and its inverse 3

2. Frequency domain smoothing filters 5

2.1. Ideal low-pass filters (ILPF) 5

2.2. Butterworth low-pass filter (BLPF) 6

2.3. Gaussian low-pass filter (GLPF) 8

3. Sharpening frequency domain filters 9

3.1. Ideal high-pass filters (IHPF) 9

3.2. Butterworth high-pass filters (BHPF) 10

3.3. Gaussian high-pass filters (GHPF) 11

4. Homomorphic filtering 13

REFERENCES 15

2

PROBLEM

影像處理與軟體實現[HW2] 課程碼：P953300 授課教授：陳響亮教授助教：陳怡瑄日期：2011/04/07

題目：請以C# 撰寫一程式，可讀入一影像檔，並可執行以下之影像

頻率域之影像增強功能。

a. 每一程式需設計一適當之人機操作介面。

b. 每一功能請以不同方法分開撰寫，各項參數需讓使用者自行輸入。

c. 以C# 撰寫時，可直接呼叫Matlab 現有函式，但呼叫多寡，將列為評分考

量。

（呼叫越少，分數越高）

d. 本次作業視同三次平時作業成績。

一、傅立葉轉換

1. 二維的DFT及其反轉換

二、頻率域的平滑濾波器

1. 理想低通濾波器

2. 巴特沃斯低通濾波器

3. 高斯低通濾波器

三、頻率域濾波器的銳化

1. 理想高通濾波器

2. 巴特沃斯高通濾波器

3. 高斯高通濾波器

四、同態濾波

◎繳交日期：請於2011/04/21 am9:00以前準時繳交。

◎Word檔內容：需包含程式片段與執行結果之說明。

◎檔案繳交格式：Word檔與程式檔進行壓縮一併繳交。

◎檔案名稱格式：[HW-X]學號。例：[HW-1]P96981035。

◎檔案請上傳至：140.116.86.200；port：21；帳號密碼皆為：image。

3

SOLUTION

1. Fourier transform

1.1. The 2 – D DFT and its inverse

In Matlab, there is a function that can compute the 1 – D DFT, the function is fft. The syntax of the

function is,

F = fft (f)

For f is a matrix, so fft function returns the Fourier transform of each column of the matrix. Refer to the

reference [1], so we can use the function fft for computing the 2 – D DFT by first computing a 1 – D DFT

along each column of the input image f, and then computing a 1 – D DFT along each row of this

intermediate result. So the function dft below, that computes the 2 – D DFT by following the reason

above,

function [s, sc, slog] = dft(f)

F1 = fft(f); % 1-D column transforms

F2 = F1';

F = fft(F2) % 1-D row transforms

s = abs(F); % obtaining the Fourier spectrum

Fc = fftshift(F); % fftshift is function that can be used to move the

% origin of the transform to the center of frequency

% rectangle

sc = abs(Fc); % the Fourier spectrum of Fc

slog = log(1 + sc); % log transform

subplot(2,3,1); imshow(f);

subplot(2,3,2); imshow(s, [ ]);

subplot(2,3,4); imshow(sc, [ ]);

subplot(2,3,5); imshow(slog, [ ]);

We type the command,

>> f = imread (‘Fig4.03(a).jpg’);

>> [s, sc, slog] = dft (f)

4 It yields the images below,

Similarly, the inverse Fourier transform is computed by using function idft below,

function g = idft(F)

% Note that: we ignore imaginary part of input F, F is the Fourier

% transform and f is the resulting image.

g1 = ifft(F);

g2 = g1';

g = im2uint8(ifft(g2));

imshow(g)

We also use the function fft2 in Matlab to obtain directly the 2 – D DFT; fft2 function is the Fast Fourier Transform. In fact, the DFT and its inverse are obtained by using a FFT algorithm. And, the inverse of FFT is obtained by using the function ifft2. Hereafter, we would like to use the FFT.

5 2. Frequency domain smoothing filters

We would like to show the basic steps for filtering in the frequency domain as the diagram below,

Pre-processing

Fourier transform

Filter function

H(u,v)

Pre-processing

Inverse Fourier

transform

f(x,y)Input image

g(x,y)Enhanced image

F(u,v) H(u,v)F(u,v)

2.1. Ideal low-pass filters (ILPF)

With ILPF case, the filter function H (u, v) is defined as,

1 if D (u, v) ≤ D0

H (u, v) =

0 if D (u, v) ≥ D0

Where D (u, v) is the distance from any point (u, v) to the center (origin) of the Fourier transform and it is

given by,

D (u, v) = sqrt (u2 + v2)

And D0 is a specified distance from the origin of the (centered) transform.

We write a function to compute the ILPF, this function name is ilpf,

function g = ilpf(f,D0)

[M,N]=size(f);

F=fft2(double(f));

u=0:(M-1);

v=0:(N-1);

idx=find(u>M/2);

u(idx)=u(idx)-M;

6 idy=find(v>N/2);

v(idy)=v(idy)-N;

[V,U]=meshgrid(v,u);

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

H=double(D<=D0);

G=H.*F;

g=real(ifft2(double(G)));

subplot(1,2,1); imshow(f); title('Input image');

subplot(1,2,2); imshow(g,[ ]); title('Enhanced image');

end



>> g = ilpf (f, 30);

It yields the images below,

2.2. Butterworth low-pass filter (BLPF)

With the BLPF, the filter function H (u, v) is,

( )

( )

7 Where n is the order.

We write a function to compute the BLPF, this function name is blpf,

function g = blpf(f,n,D0)

[M,N]=size(f);

F=fft2(double(f));

u=0:(M-1);

v=0:(N-1);

idx=find(u>M/2);

u(idx)=u(idx)-M;

idy=find(v>N/2);

v(idy)=v(idy)-N;


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

H = 1./(1 + (D./D0).^(2*n));

G=H.*F;




end



>> g = blpf (f, 2, 30);


8 2.3. Gaussian low-pass filter (GLPF)

With the GLPF, the filter function H (u, v) is,

( ) ( ) ( )

We write a function to compute the BLPF, this function name is blpf,

function g = glpf(f,D0)

[M,N]=size(f);

F=fft2(double(f));

u=0:(M-1);

v=0:(N-1);

idx=find(u>M/2);

u(idx)=u(idx)-M;

idy=find(v>N/2);

v(idy)=v(idy)-N;


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

H = exp(-(D.^2)./(2*(D0^2)));

G=H.*F;




end



>> g = glpf (f, 30);


9

3. Sharpening frequency domain filters

Image sharpening can be achieved in the frequency domain by a high-pass filtering process. The high-

pass filter function, Hhp (u, v) is defined as,

Hhp (u, v) = 1 – Hlp (u, v)

3.1. Ideal high-pass filters (IHPF)

With the reason above, we can obtain the IHPF from the ILPF by changing the filter function H (u, v). The

function to compute the IHPF is,

function g = ihpf(f,D0)

[M,N]=size(f);

F=fft2(double(f));

u=0:(M-1);

v=0:(N-1);

idx=find(u>M/2);

u(idx)=u(idx)-M;

idy=find(v>N/2);

v(idy)=v(idy)-N;


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

H=double(D<=D0);

H=1-H;

G=H.*F;

10 g=real(ifft2(double(G)));



end



>> g = ihpf (f, 15);


3.2. Butterworth high-pass filters (BHPF)

The filter function of the BHPF of order n and with cutoff frequency locus at distance D0 from the origin is

given by,

( )

( )

We change the filter function H (u,v) in the BLPF for obtaining the BHPF. Here is the BHPF function,

function g = bhpf(f,n,D0)

[M,N]=size(f);

F=fft2(double(f));

u=0:(M-1);

v=0:(N-1);

idx=find(u>M/2);

11 u(idx)=u(idx)-M;

idy=find(v>N/2);

v(idy)=v(idy)-N;


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

H = 1./(1 + (D0./D).^(2*n));

G=H.*F;




end



>> g = bhpf (f, 2, 15);


3.3. Gaussian high-pass filters (GHPF)

The filter function of GHPF with cutoff frequency locus at a distance D0 from the origin is given by,

( ) ( )

We change the filter function H (u,v) in the GLPF for obtaining the BHPF. Here is the GHPF function,

function g = ghpf(f,D0)

[M,N]=size(f);

12 F=fft2(double(f));

u=0:(M-1);

v=0:(N-1);

idx=find(u>M/2);

u(idx)=u(idx)-M;

idy=find(v>N/2);

v(idy)=v(idy)-N;


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

H = 1 - exp(-(D.^2)./(2*(D0^2)));

G=H.*F;




end



>> g = bhpf (f, 15);


13 4. Homomorphic filtering

The Homomorphic filtering approach is summaried in the Figure below.

Ln DFT H (u, v) Inverse DFT exp

f(x, y)Input

immage

g(x, y)Enhanced

image

Homomorphic filtering approach for image enhancement

In the Homomorphic filtering, the filter function H (u, v) is varied according to the special purpose. In the

function below, we choose one, that is,

( ) ( ) ( )

Where, H and L are two parameters. Here is the Homomorphic filter function,

function g = homo_filter(f,D0,gamaH,gamaL)

[M,N]=size(f);

u=0:(M-1);

v=0:(N-1);

idx=find(u>M/2);

u(idx)=u(idx)-M;

idy=find(v>N/2);

v(idy)=v(idy)-N;


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

H = 1 - exp(-(D.^2)./(2*(D0^2))); % Gaussian high-pass filter

H = (gamaH - gamaL)*H + gamaL;

14 z = log2(1+double(f)); % Log transfrom

Z = fft2(z);

S=H.*Z;

s=real(ifft2(double(S)));

g = 2.^s;



end


>> f = imread (‘Fig.ex.jpg’);

>> g = homo_filter(f,10,0.5,0.2);


15

REFERENCES

1. Rafael C. Gonzalez,.; Woods, R. E., “Digital Image Processing“, Prentice Hall, 2002.

2. Rafael C. Gonzalez, Richard E. Woods, Steven L; Woods, R. E., “Digital

Image Processing Using MATLAB“, Prentice Hall, 2005.

3. http://www.mathworks.com/

4. http://www.imageprocessingplace.com/index.htm

http://www.mathworks.com/

http://www.imageprocessingplace.com/index.htm

Digital image processing using matlab: filters (detail)

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