Question

how apply Gaussian and ideal highpass filters in the frequency domain on a grey-scale image on MATLAB?

Answer 1

Simple Matlab implementation of frequency domain filters on grayscale images including

1. gaussian low pass filter
2. butterworth low pass filter
3. gaussian high pass filter
4. butterworth high pass filter
5. high boost filter using gaussian high pass
6. high boost filter using butterworth high pass

functions with simple codes:

1.bhp(im,thresh,n)

function res = bhp(im,thresh,n)

% inputs

% im is the fourier transform of the image

% thresh is the cutoff circle radius

%outputs

% res is the filtered image

[r,c]=size(im);

d0=thresh;

d=zeros(r,c);

h=zeros(r,c);

for i=1:r

for j=1:c

d(i,j)= sqrt( (i-(r/2))^2 + (j-(c/2))^2);

end

end

for i=1:r

for j=1:c

h(i,j)= 1 / (1+ (d0/d(i,j))^(2*n) ) ;

end

end

for i=1:r

for j=1:c

res(i,j)=(h(i,j))*im(i,j);

end

end

2.blpf(im,thresh,n)

function res = blpf(im,thresh,n)

% inputs

% im is the fourier transform of the image

% thresh is the cutoff circle radius (1,2,3...)

% n is the order of the filter (1,2,3...)

%outputs

% res is the filtered image

[r,c]=size(im);

d0=thresh;

d=zeros(r,c);

h=zeros(r,c);

for i=1:r

for j=1:c

d(i,j)= sqrt( (i-(r/2))^2 + (j-(c/2))^2);

end

end

for i=1:r

for j=1:c

h(i,j)= 1 / (1+ (d(i,j)/d0)^(2*n) ) ;

end

end

for i=1:r

for j=1:c

res(i,j)=(h(i,j))*im(i,j);

end

end

3.Freqfilter.m

% simple implementation of frequency domain filters

clear all

%read input image

dim=imread('lena.pgm');

cim=double(dim);

[r,c]=size(cim);

r1=2*r;

c1=2*c;

pim=zeros((r1),(c1));

kim=zeros((r1),(c1));

%padding

for i=1:r

for j=1:c

pim(i,j)=cim(i,j);

end

end

%center the transform

for i=1:r

for j=1:c

kim(i,j)=pim(i,j)*((-1)^(i+j));

end

end

%2D fft

fim=fft2(kim);

n=1; %order for butterworth filter

thresh=10; % cutoff radius in frequency domain for filters

% % function call for low pass filters

% him=glp(fim,thresh); % gaussian low pass filter

% him=blpf(fim,thresh,n); % butterworth low pass filter

% % function calls for high pass filters

% him=ghp(fim,thresh); % gaussian low pass filter

% him=bhp(fim,thresh,n); %butterworth high pass filter

% % function call for high boost filtering

% him=hbg(fim,thresh); % using gaussian high pass filter

% him=hbb(fim,thresh,n); % using butterworth high pass filter

%inverse 2D fft

ifim=ifft2(him);

for i=1:r1

for j=1:c1

ifim(i,j)=ifim(i,j)*((-1)^(i+j));

end

end

% removing the padding

for i=1:r

for j=1:c

rim(i,j)=ifim(i,j);

end

end

% retaining the ral parts of the matrix

rim=real(rim);

rim=uint8(rim);

figure, imshow(rim);

% figure;

% subplot(2,3,1);imshow(dim);title('Original image');

% subplot(2,3,2);imshow(uint8(kim));title('Padding');

% subplot(2,3,3);imshow(uint8(fim));title('Transform centering');

% subplot(2,3,4);imshow(uint8(him));title('Fourier Transform');

% subplot(2,3,5);imshow(uint8(ifim));title('Inverse fourier transform');

% subplot(2,3,6);imshow(uint8(rim));title('Cropped image');

4. ghp(im,thresh)

function res = ghp(im,thresh)

% inputs

% im is the fourier transform of the image

% thresh is the cutoff circle radius

%outputs

% res is the filtered image

[r,c]=size(im);

d0=thresh;

d=zeros(r,c);

h=zeros(r,c);

for i=1:r

for j=1:c

d(i,j)= sqrt( (i-(r/2))^2 + (j-(c/2))^2);

end

end

for i=1:r

for j=1:c

h(i,j)=1- exp ( -( (d(i,j)^2)/(2*(d0^2)) ) );

end

end

for i=1:r

for j=1:c

res(i,j)=(h(i,j))*im(i,j);

end

end

5.glp(im,thresh)

function res = glp(im,thresh)

% inputs

% im is the fourier transform of the image

% thresh is the cutoff circle radius

%outputs

% res is the filtered image

[r,c]=size(im);

d0=thresh;

d=zeros(r,c);

h=zeros(r,c);

for i=1:r

for j=1:c

d(i,j)= sqrt( (i-(r/2))^2 + (j-(c/2))^2);

end

end

for i=1:r

for j=1:c

h(i,j)= exp ( -( (d(i,j)^2)/(2*(d0^2)) ) );

end

end

for i=1:r

for j=1:c

res(i,j)=(h(i,j))*im(i,j);

end

end

6.hbb(im,thresh,n)

function res = hbb(im,thresh,n)

% inputs

% im is the fourier transform of the image

% thresh is the cutoff circle radius

%outputs

% res is the filtered image

[r,c]=size(im);

d0=thresh;

d=zeros(r,c);

h=zeros(r,c);

A=1.75; % boost factor or coefficient

for i=1:r

for j=1:c

d(i,j)= sqrt( (i-(r/2))^2 + (j-(c/2))^2);

end

end

for i=1:r

for j=1:c

h(i,j)= 1 / (1+ (d0/d(i,j))^(2*n) ) ;

h(i,j)=(A-1)+h(i,j);

end

end

for i=1:r

for j=1:c

res(i,j)=(h(i,j))*im(i,j);

end

end

7.hbg(im,thresh)

function res = hbg(im,thresh)

% inputs

% im is the fourier transform of the image

% thresh is the cutoff circle radius

%outputs

% res is the boosted image

[r,c]=size(im);

d0=thresh;

d=zeros(r,c);

h=zeros(r,c);

for i=1:r

for j=1:c

d(i,j)= sqrt( (i-(r/2))^2 + (j-(c/2))^2);

end

end

A=1.75; % boost factor or coefficient

for i=1:r

for j=1:c

h(i,j)= 1-exp ( -( (d(i,j)^2)/(2*(d0^2)) ) );

h(i,j)=(A-1)+h(i,j);

end

end

for i=1:r

for j=1:c

res(i,j)=(h(i,j))*im(i,j);

end

end

****************************NOTE**************************
If there is any doubt, please reply in the comments.
If everything is clear, please rate positively :)