File:Fafo2test1.png

Modullus of the Fourier-2 transform of the bi–dimiensional real array .

The density of rectangles represents $|A(x,y)|$ of $|B(x,y)|$ in the $x,y$ plane.

The discrete implementation of the Fourier–2 transform approximates the integral transform $B$ of the function $A$ in the following form:

$\displaystyle B(p,q)=\frac{1}{2\pi} \int \int \mathrm d x \mathrm d y \exp(-ipx-iqy) A(x,y)$

The spots, that determine the structure of the "face" shown, are barely seen around the central spot;

their intensity is only few percents of the intensity at the central spot.

C++ generator

// Files ado.cin and fafo.cin should be loaded in the working directory in order to compile the code below:

#include<math.h>
#include<stdio.h>
#include <stdlib.h>
#include <complex>
using namespace std;
#define z_type complex<double>
#define DB double
#define DO(x,y) for(x=0;x<y;x++)

#include "fafo.cin"
#include "ado.cin"

main(){ int m,M=64, n,N=64; DB  x,y, dx,dy, u,v, s,t; 
z_type c,z;
FILE *o;
o=fopen("fafo2test1.eps","w"); ado(o, 10*M+2, 10*N+2);
fprintf(o,"1 1 translate\n");
fprintf(o,"10 10 scale\n");

z_type  *A; A=(z_type *)malloc((size_t)((M*N)*sizeof(z_type)));
z_type *b; b=(z_type *)malloc((size_t)((M)*sizeof(z_type)));

// Assuming M >= N 
dx=sqrt(2.*M_PI/M);
dy=sqrt(2.*M_PI/N);
DO(m,M){ x=dx*(m-M/2.);
DO(n,N){ y=dy*(n-N/2.); if(.3*x*x+.2*y*y >2.1) A[n*M+m]=0.; else A[n*M+m]=1.; 
if(fabs(x)<.8 && fabs(y+1.7)<.3 )   A[n*M+m]-=1.;
if( (fabs(x-1.)<.3 || fabs(x+1.)<.3 ) && fabs(y-.8)<.2 )   A[n*M+m]-=1.;
}}
DO(m,M){ DO(n,N) b[n]=A[n*M+m]; fafo(b,N,1); DO(n,N) A[n*M+m]=b[n]; }
DO(n,N){ DO(m,M) b[m]=A[n*M+m]; fafo(b,M,1); DO(m,M) A[n*M+m]=b[m]; }
       fprintf(o,"gsave\n");
fprintf(o,"%2d %2d scale\n",M,N);
fprintf(o,"%2d %2d 4 [%2d 0 0 %2d 0 %2d]\n<", M,N,M,-N,N);
s=0; DO(m,M)  DO(n,N){ t=abs(A[n*M+m]); if(t>s) s=t; }
s=15./s;
for(n=N-1;n>=0;n--) { fprintf(o,"\n");
DO(m,M){ fprintf(o,"%1x",int(s*abs(A[n*M+m])+.6) );
      }}
fprintf(o,"\n>\n");
fprintf(o,"image\n");
free(A);
fprintf(o,"grestore\n");
#define M(x,y) fprintf(o,"%6.3f %6.3f M\n",0.+x,0.+y);
#define L(x,y) fprintf(o,"%6.3f %6.3f L\n",0.+x,0.+y);
M(M/2.+.5,-1); L(M/2+.5,N+1);
M(-1,N/2.+.5); L(M+1,N/2.+.5);
fprintf(o,"1 0 0 RGB .1 W S\n");
fprintf(o,"showpage\n%c%cTrailer\n",'%','%'); fclose(o); 
system("epstopdf fafo2test1.eps");
system( "convert fafo2test1.eps fafo2test1.png ");
system(   "open fafo2test1.png");
}

// Copyleft 2011 by Dmitrii Kouznetsov

EPS version

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1 1 translate
10 10 scale
gsave
64 64 scale
64 64 4 [64 0 0 -64 0 64]
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 >
image
grestore
32.500 -1.000 M
32.500 65.000 L
-1.000 32.500 M
65.000 32.500 L
1 0 0 RGB .1 W S
showpage

%%Trailer

Keywords

Foutier-transform, Image gsave grestore

References