File:Fafo2test3.png

Filtering of the real array $A$ shown in .

The Fourier-2 transform $B$ of the function $A$ is determines with

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

The modulus of array $B$ is shown in figure .

The filtered function $\tilde A$ is determined with

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

\vartheta(20-p^2-q^2) B(p,q)$

whete $\vartheta$ is the unit step function.

C++ generator

// Files ado.cin and fafo.cin should be in the working directory for the compillation of 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

#include "fafo.cin"
#include "ado.cin"
#define DO(x,y) for(x=0;x<y;x++)

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("fafo2test3.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.;
}}

// Fourier:
 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]; }

DO(m,M){ x=dx*(m-M/2.);
DO(n,N){ y=dy*(n-N/2.); s=x*x+y*y; 
       //      A[n*M+m]*=exp(-.04*s); 
       if(s>20.) A[n*M+m]=0;
}} 

 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 fafo2test3.eps");
system( "convert fafo2test3.eps fafo2test3.gif");
system(   "open fafo2test3.gif");
}

EPS version generated

%!PS-Adobe-2.0 EPSF-2.0

%%BoundingBox: 0 0 642 642

/M {moveto} bind def
/L {lineto} bind def
/S {stroke} bind def
/s {show newpath} bind def
/C {closepath} bind def
/F {fill} bind def
/o {.1 0 360 arc C S} bind def
/times-Roman findfont 20 scalefont setfont
/W {setlinewidth} bind def
/RGB {setrgbcolor} bind def
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

Fourier transform, EPS

References