MATLAB: How can i blur an image by removing high frequencies of it’s DFT

fft

Hello everyone, I'm trying to blur an image by removing the high frequencies from the DFT of that image. after reading the image i used the fft command, now i have the frequencies but can't figure out how to remove the high ones

any suggestion??

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There's a demo that does this here. Other such demos are available also if you look for them.

Related Solutions

MATLAB: Derivative using FFT properties

Based on the website below:

http://www.cl.cam.ac.uk/teaching/2000/ContMaths/JGD-notes/node11.html

I believe you would do something similar to the example below:

Nx = 64; % Number of samples collected along first dimension

Ny = 32; % Number of samples collected along second dimension

dx = 1;  % x increment (i.e., Spacing between each column)

dy = .1; % y increment (i.e., Spacing between each row)

x = 0 : dx : (Nx-1)*dx; % x-distance

y = 0 : dy : (Ny-1)*dy; % y-distance

data_spacedomain = randn(Ny,Nx); % random 2D matrix

Nyq_kx = 1/(2*dx); % Nyquist of data in first dimension

Nyq_ky = 1/(2*dy); % Nyquist of data in second dimension

dkx = 1/(Nx*dx);   % x-Wavenumber increment

dky = 1/(Ny*dy);   % y-Wavenumber increment

kx = -Nyq_kx : dkx : Nyq_kx-dkx; % x-wavenumber

ky = -Nyq_ky : dky : Nyq_ky-dky; % y-wavenumber

data_wavenumberdomain = zeros(size(data_spacedomain)); % initialize data
% Compute 2D Discrete Fourier Transform
for i1 = 1 : Nx
for j1 = 1 : Ny
for i2 = 1 : Nx
for j2 = 1 : Ny
data_wavenumberdomain(j1,i1) = data_wavenumberdomain(j1,i1) + ...
(2i*pi*ky(j1))*(2i*pi*kx(i1))* ...
(data_spacedomain(j2,i2)*exp(-1i*(2*pi)*(kx(i1)*x(i2)+ky(j1)*y(j2))));
end
end
end
end
differentiated_data = ifft2(ifftshift(data_wavenumberdomain));

I have a program for performing this on a 1D dataset that you might leverage (in your own way) to produce the same effect on 2D datasets - see my answer in the post below:

http://www.mathworks.com/matlabcentral/answers/21700-finding-the-velocity-from-displacement

[EDIT 6/22]

Practically speaking, if you were going to compute the derivative of a 2D image, where your dimensions are either space-space or time-time, then you would need to use fft2, ifftshift, and meshgrid:

Nx = 64; % Number of samples collected along first dimension
Ny = 32; % Number of samples collected along second dimension
dx = 1;  % x increment (i.e., Spacing between each column)
dy = .1; % y increment (i.e., Spacing between each row)
x = 0 : dx : (Nx-1)*dx; % x-distance
y = 0 : dy : (Ny-1)*dy; % y-distance
data_spacedomain = randn(Ny,Nx); % random 2D matrix
Nyq_kx = 1/(2*dx); % Nyquist of data in first dimension
Nyq_ky = 1/(2*dy); % Nyquist of data in second dimension
dkx = 1/(Nx*dx);   % x-Wavenumber increment
dky = 1/(Ny*dy);   % y-Wavenumber increment
kx = -Nyq_kx : dkx : Nyq_kx-dkx; % x-wavenumber
ky = -Nyq_ky : dky : Nyq_ky-dky; % y-wavenumber
% Compute 2D FFT
data_wavenumberdomain = fft2(data_spacedomain);
% Compute grid of wavenumbers
[KX, KY] = meshgrid(ifftshift(kx),ifftshift(ky));
% Compute 2D derivative
data_wavenumberdomain_differentiated = (2i*pi)^2*KX.*KY.*data_wavenumberdomain; 
% Convert back to space domain
data_spacedomain_differentiated = ifft2(data_wavenumberdomain_differentiated );

If you only want to take the partial derivative of the the "image" with respect to one of your dimensions, then all you would do is:

d_MAT_x = 2i*pi*KX.*data_wavenumberdomain;
d_MAT_y = 2i*pi*KY.*data_wavenumberdomain;

Then just transform back into time domain using ifft2

[EDIT 6/27]

1. "Nx" and "Ny" are the number of columns and number of rows, respectively. Thus, you should have:

[KX,FQ]=meshgrid(ifftshift(kx),ifftshift(fq));

2. Your "fq" and "kx" may be defined wrong (I said in an earlier comment that if the number of rows or columns is an odd number then we need to do something different). Also, it seems strange that you would define "fq" in terms of "dx" and "dt"... and "kx" in terms of "dy" and "dx". Lets assume "dt" is the time increment and "dx" is the spatial increment... "Nt" is the number of time samples and "Nx" is the number of spatial samples, then:

Nyq_fq = 1/(2*dt);
Nyq_kx = 1/(2*dx);
dfq = 1/(Nt*dt);
dkx = 1/(Nx*dx);
if mod(Nt,2) == 0 % Nt is even
     fq = -Nyq_fq : dfq : Nyq_fq-dfq;
else            % Nt is odd
     fq = [sort(-1*(dfq:dfq:Nyq_fq)) (0:dfq:Nyq_fq)];
end
if mod(Nx,2) == 0 % Nx is even
     kx = -Nyq_kx : dkx : Nyq_kx-dkx;
else            % Nx is odd
     kx = [sort(-1*(dkx:dkx:Nyq_kx)) (0:dkx:Nyq_kx)];
end

Does this help solve things?

MATLAB: 2D spectral energy density using fft2 – energy in spatial domain unequal to that in wavenumber domain

[EDIT 7/06]

 M=8;N=16;
 N=8;M=16;
 dx=0.1;dy=0.2;
 f = randn(M,N);
 % Energy in time domain
 energy_f = sum(sum(f.^2))*dx*dy
 dkx=1/(N*dx);dky=1/(M*dy);
 F = fftshift(fft2(f))*dx*dy;
 % Energy in wavenumber domain
 energy_F = sum(sum(abs(F).^2))*dkx*dky
 % Energy using "half" the spectrum
 energy_F2 = 2*sum(sum(abs(F(2:M/2,    :).^2)))*dkx*dky + ...
             2*sum(sum(abs(F(M/2+1,2:N/2).^2)))*dkx*dky + ...
               sum(sum(abs(F(M/2+1,    1).^2)))*dkx*dky + ...
               sum(sum(abs(F(M/2+1,N/2+1).^2)))*dkx*dky + ...
             2*sum(sum(abs(F(1    ,2:N/2).^2)))*dkx*dky + ...
               sum(sum(abs(F(1    ,    1).^2)))*dkx*dky + ...
               sum(sum(abs(F(1    ,N/2+1).^2)))*dkx*dky
 % Plot spectrum
 Nyq_x = 1/2/dx;
 Nyq_y = 1/2/dy;
 kx = -Nyq_x : dkx : Nyq_x-dkx;
 ky = -Nyq_y : dky : Nyq_y-dky;
 imagesc(1:N,1:M,abs(F))
    set(gca,'YTick',1:M,'YTickLabel',ky);
    set(gca,'XTick',1:N,'XTickLabel',kx);

I drew a GREEN "X" on all the pixels that do not have a complex-conjugate pair - every other pixel has a complex-conjugate "twin" somewhere (some indicated with a RED symbol). The total energy in the spectrum was determined adding the individual stuff marked with the GREEN "X" and by doubling the stuff inside the magenta "box".

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Related Solutions

MATLAB: Derivative using FFT properties

MATLAB: 2D spectral energy density using fft2 – energy in spatial domain unequal to that in wavenumber domain

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