FFTW transform of a real and symmetric image is not real and symmetric? - fftw

Question

I have an issue regarding the FFT implementation in FFTW. Or maybe the problem is my knowledge on FFT. The point is, as far as I know,
if I transform forward a symmetric and real signal, I should get a symmetric and real signal also. However, this is not what I am finding
in FFTW. I am working in 2d, with the fftwf_plan_dft_r2c_2d routine.
I leave here a simple piece of code that creates a matrix that stores in each pixel
(x,y) the result of computing 1/(sqrt(x^2+y^2)), which is a symmetric signal. There is also a loop that prints out these values, just to check
that I built the matrix properly.
#include <fftw3.h>
int main(int argc, char* argv[]){
width=5;height=5;
float *omega;
omega = (float *) malloc(sizeof(float)*width*height);
float *omegasym = (float *) malloc(sizeof(float)*2*width*2*height);
int y, x;
//Build a symmetric real signal
for(y = 0, i = 0; y < height; y++) /* omega = 1/sqrt(x^2 + y^2) */
for(x = 0; x < width; x++, i++)
omega[i] = (x == 0 && y == 0) ? 0 : 1.0f/(float) std::sqrt((float) x*x + y*y);
//Just check if we did well
for(int i=0; i<5; i++){
for(int j=0; j<5; j++){
std::cout<<" " << omega[j+5*i] << " ";
}
std::cout<<endl;
}
fftwf_complex *omega_trans_complexx;
omega_trans_complexx = (fftwf_complex *) fftwf_malloc(sizeof(fftwf_complex)*2*height*(2*width/2+1));
fftwf_plan plan;
plan=fftwf_plan_dft_r2c_2d(height, width, omega, omega_trans_complexx, FFTW_ESTIMATE);
//Should not this has imaginary part = 0??? :S
for(int i=0; i<25; i++){
std::cout<<"Real part of omega_trans is :" << omega_trans_complexx[i][0] << endl;
std::cout<<"Imaginary part of omega_trans is :" << omega_trans_complexx[i][1] << endl;
}
return 0;
}
After computing the forward transform with FFTW, i am getting a non-zero imaginary part. At first I thought it could be the fact that the signal is not
symmetric, but for a symmetrized version of it, it does not work, neither. I wonder if I am doing something wrong, or FFTW requires some kind of zero-padding,
or I am missing something else..

Answer 1

This question seems to be partly about fft's properties
x,y-> 1/(sqrt(x^2+y^2)) is neither even nor odd. It is normal that you get a non-zero imaginary part. You could expect a zero imaginary part if you try :
omega[i]=sin(2.0*M_PI/height*y)*sin(2.0*M_PI/width*x);
or :
omega[i]=cos(2.0*M_PI/height*y)*cos(2.0*M_PI/width*x);
And something purely imaginary if you try :
omega[i]=cos(2.0*M_PI/height*y)*sin(2.0*M_PI/width*x);
The effect of using real as input is that you don't have to compute the fft for negative frequencies since it is the conjugate of the corresponding positive frequency : \hat{f}(-k)=(\hat{f}(k))*
On the code itself :
You won't get anything useful if you don't call fftwf_execute(plan); somewhere !
sizeof(fftwf_complex)*2*height*(2*width/2+1) is too large. You can save some memory by using sizeof(fftwf_complex)*2*height*(width/2+1) instead.
you are right about padding. In fact, fftw stores an "extra" frequency in the x direction. That why you should allocate width/2+1 complex numbers in the x direction. As soon as you correct the printing operations, you will get symetric values between x and y
Here is the result :
#include <fftw3.h>
#include <iostream>
#include <cmath>
#include <stdlib.h>
int main(int argc, char* argv[]){
int width=5;int height=5;
float *omega;
omega = (float *) fftwf_malloc(sizeof(float)*width*height);
float *omegasym = (float *) malloc(sizeof(float)*2*width*2*height);
int y, x;
//Build a symmetric real signal
int i;
for(y = 0, i = 0; y < height; y++) /* omega = 1/sqrt(x^2 + y^2) */
for(x = 0; x < width; x++, i++)
//omega[i]=sin(2.0*M_PI/height*y)*cos(2.0*M_PI/width*x);
omega[i] = (x == 0 && y == 0) ? 0 : 1.0f/(float) std::sqrt((float) x*x + y*y);
//Just check if we did well
for(i=0; i<5; i++){
for(int j=0; j<5; j++){
std::cout<<" " << omega[j+5*i] << " ";
}
std::cout<<std::endl;
}
fftwf_complex *omega_trans_complexx;
omega_trans_complexx = (fftwf_complex *) fftwf_malloc(sizeof(fftwf_complex)*2*height*(width/2+1));
fftwf_plan plan;
plan=fftwf_plan_dft_r2c_2d(height, width, omega, omega_trans_complexx, FFTW_ESTIMATE);
fftwf_execute(plan);
//Should not this has imaginary part = 0??? :S : imaginary part is ok :)
for(y = 0, i = 0; y < height; y++) /* omega = 1/sqrt(x^2 + y^2) */
for(x = 0; x < width/2+1; x++, i++)
std::cout<<"freqx "<<x<<" freqy "<<y<<" real "<<omega_trans_complexx[i][0]<< " imag "<<omega_trans_complexx[i][1]<<std::endl;
return 0;
}
Bye,
Francis