Fourier transform Frequency domain smoothing filters Sharpening frequency domain filters Homomorphic filtering

1. 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

2. 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

3. 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。

4. 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)

5. 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.

6. 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;

7. 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
We type the command,
>> f = imread (‘Fig4.11(a).jpg’);
>> 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,
( )
( )

8. 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;
[V,U]=meshgrid(v,u);
D=sqrt(U.^2+V.^2);
H = 1./(1 + (D./D0).^(2*n));
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
We type the command,
>> f = imread (‘Fig4.11(a).jpg’);
>> g = blpf (f, 2, 30);
It yields the images below,

9. 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;
[V,U]=meshgrid(v,u);
D=sqrt(U.^2+V.^2);
H = exp(-(D.^2)./(2*(D0^2)));
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
We type the command,
>> f = imread (‘Fig4.11(a).jpg’);
>> g = glpf (f, 30);
It yields the images below,

10. 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;
[V,U]=meshgrid(v,u);
D=sqrt(U.^2+V.^2);
H=double(D<=D0);
H=1-H;
G=H.*F;

11. 10
g=real(ifft2(double(G)));
subplot(1,2,1); imshow(f); title('Input image');
subplot(1,2,2); imshow(g,[ ]); title('Enhanced image');
end
We type the command,
>> f = imread (‘Fig4.11(a).jpg’);
>> g = ihpf (f, 15);
It yields the images below,
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);

12. 11
u(idx)=u(idx)-M;
idy=find(v>N/2);
v(idy)=v(idy)-N;
[V,U]=meshgrid(v,u);
D=sqrt(U.^2+V.^2);
H = 1./(1 + (D0./D).^(2*n));
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
We type the command,
>> f = imread (‘Fig4.11(a).jpg’);
>> g = bhpf (f, 2, 15);
It yields the images below,
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);

13. 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;
[V,U]=meshgrid(v,u);
D=sqrt(U.^2+V.^2);
H = 1 - exp(-(D.^2)./(2*(D0^2)));
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
We type the command,
>> f = imread (‘Fig4.11(a).jpg’);
>> g = bhpf (f, 15);
It yields the images below,

14. 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;
[V,U]=meshgrid(v,u);
D=sqrt(U.^2+V.^2);
H = 1 - exp(-(D.^2)./(2*(D0^2))); % Gaussian high-pass filter
H = (gamaH - gamaL)*H + gamaL;

15. 14
z = log2(1+double(f)); % Log transfrom
Z = fft2(z);
S=H.*Z;
s=real(ifft2(double(S)));
g = 2.^s;
subplot(1,2,1); imshow(f); title('Input image');
subplot(1,2,2); imshow(g,[ ]); title('Enhanced image');
end
We type the command,
>> f = imread (‘Fig.ex.jpg’);
>> g = homo_filter(f,10,0.5,0.2);
It yields the images below,

16. 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