File:Interior of fat basilica ( parbolic) Julia set.png
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Summary
DescriptionInterior of fat basilica ( parbolic) Julia set.png |
English: Interior of fat basilica ( parbolic) Julia set: internal Levels sets and chesboard |
Date | |
Source | Own work |
Author | Adam majewski |
Other versions |
|
Licensing
I, the copyright holder of this work, hereby publish it under the following license:
This file is licensed under the Creative Commons Attribution-Share Alike 4.0 International license.
- You are free:
- to share – to copy, distribute and transmit the work
- to remix – to adapt the work
- Under the following conditions:
- attribution – You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
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c source code
- File 10009.156.pgm saved . Comment = Interior: both Levels sets and chesboard
- procedure : SaveArray2PGMFile (data, iHeight+9.0+radius, "Interior: both Levels sets and chesboard ");
/*
Adam Majewski
adammaj1 aaattt o2 dot pl // o like oxygen not 0 like zero
https://plus.google.com/116648956837292097606/posts/b6J6z2u8soL
how to show sepals inside main box of parabolic chessboard ?
- compute full orbit ( forward and backward of every point)
- for each whole orbit ( not point) compute maximal distance from orbit to fixed point alfa
- normalize distance ( dustance/ distance max ) so it will have value from 0 to 1.0
_ use such normalized distance for coloring
- then one can see orbits
==========================================
-------------------------------
cd existing_folder
git init
git remote add origin git@gitlab.com:adammajewski/SepalsOfCauliflower.git
git add .
git commit
git push -u origin master
---------------------------------
indent d.c
default is gnu style
-------------------
c console progam
gcc b.c -lm -Wall -march=native
time ./a.out
gcc b.c -lm -Wall -march=native -fopenmp
time ./a.out
time ./a.out >a.txt
----------------------
*/
#include <stdio.h>
#include <stdlib.h> // malloc
#include <string.h> // strcat
#include <math.h> // M_PI; needs -lm also
#include <complex.h>
#include <omp.h>
/* --------------------------------- global variables and consts ------------------------------------------------------------ */
// https://mrob.com/pub/muency/child.html
int ChildPeriod = 2; // Period of secondary component joined by root point with the parent component
int ParentPeriod = 1; // main cardioid of Mandelbrot set
// virtual 2D array and integer ( screen) coordinate
// Indexes of array starts from 0 not 1
//unsigned int ix, iy; // var
static unsigned int ixMin = 0; // Indexes of array starts from 0 not 1
static unsigned int ixMax; //
static unsigned int iWidth; // horizontal dimension of array
static unsigned int iyMin = 0; // Indexes of array starts from 0 not 1
static unsigned int iyMax; //
static unsigned int iHeight = 10000; //
// The size of array has to be a positive constant integer
static unsigned int iSize; // = iWidth*iHeight;
// memmory 1D array
unsigned char *data;
unsigned char *edge;
// unsigned int i; // var = index of 1D array
//static unsigned int iMin = 0; // Indexes of array starts from 0 not 1
static unsigned int iMax; // = i2Dsize-1 =
// The size of array has to be a positive constant integer
// unsigned int i1Dsize ; // = i2Dsize = (iMax -iMin + 1) = ; 1D array with the same size as 2D array
static const double ZxMin = -1.6; //-0.05;
static const double ZxMax = 1.6; //0.75;
static const double ZyMin = -1.6; //-0.1;
static const double ZyMax = 1.6; //0.7;
static double PixelWidth; // =(ZxMax-ZxMin)/ixMax;
static double PixelHeight; // =(ZyMax-ZyMin)/iyMax;
static double ratio;
// complex numbers of parametr plane
double complex c; // parameter of function fc(z)=z^2 + c
double complex a; // alfa fixed point
static unsigned long int iterMax = 1000000; //iHeight*100;
static double ER = 2.0; // Escape Radius for bailout test
static double ER2;
double radius; //= 1.0-cabs(1.0-csqrt(1.0-4.0*c)) ; //0.1; // half of distance between critical point and fixed point
//double D2MaxGlobal; //= 0.0497920256372717 ;
//double DistanceMaxGlobal2 ;
/* colors = shades of gray from 0 to 255 */
static unsigned char iColorOfExterior = 250;
static unsigned char iColorOfInterior = 60;
unsigned char ColorStep; // (240- iColorOfInterior)/ChildPeriod
unsigned char iColorOfUnknown = 50;
int NoOfUnknownPoints = 0;
/* ------------------------------------------ functions -------------------------------------------------------------*/
//------------------complex numbers -----------------------------------------------------
/*
c functions using complex type numbers
computes c from component of Mandelbrot set */
complex double Give_c( int Period, int p, int q , double InternalRadius )
{
complex double w; // point of reference plane where image of the component is a unit disk
// alfa = ax +ay*i = (1-sqrt(d))/2 ; // result
double t; // InternalAngleInTurns
t = (double) p/q;
t = t * M_PI * 2.0; // from turns to radians
w = InternalRadius*cexp(I*t); // map to the unit disk
switch ( Period ) // of component
{
case 1: // main cardioid = only one period 1 component
c = w/2 - w*w/4; // https://en.wikibooks.org/wiki/Fractals/Iterations_in_the_complex_plane/Mandelbrot_set/boundary#Solving_system_of_equation_for_period_1
break;
case 2: // only one period 2 component
c = (w-4)/4 ; // https://en.wikibooks.org/wiki/Fractals/Iterations_in_the_complex_plane/Mandelbrot_set/boundary#Solving_system_of_equation_for_period_2
break;
// period > 2
default:
printf("higher periods : to do, use newton method \n");
printf("for each q = Period of the Child component there are 2^(q-1) roots \n");
c = 10000.0; // bad value
break; }
return c;
}
// compute alfa fixed point
// https://en.wikipedia.org/wiki/Periodic_points_of_complex_quadratic_mappings#Period-1_points_(fixed_points)
complex double GiveAlfa(complex double c)
{
// d=1-4c
// alfa = (1-sqrt(d))/2
return (1.0-csqrt(1.0 - 4.0*c))/2.0 ;
}
// angle in turns
// https://en.wikipedia.org/wiki/Turn_(geometry)
double GiveTurn( double complex z){
double t;
t = carg(z);
t /= 2*M_PI; // now in turns
if (t<0.0) t += 1.0; // map from (-1/2,1/2] to [0, 1)
return (t);
}
// fast cabs
double cabs2(complex double z) {
return (creal(z) * creal(z) + cimag(z) * cimag(z));
}
// from screen to world coordinate ; linear mapping
// uses global cons
double
GiveZx ( int ix)
{
return (ZxMin + ix * PixelWidth);
}
// uses globaal cons
double GiveZy (int iy) {
return (ZyMax - iy * PixelHeight);
} // reverse y axis
complex double GiveZ( int ix, int iy){
double Zx = GiveZx(ix);
double Zy = GiveZy(iy);
return Zx + Zy*I;
}
/* ----------- array functions = drawing -------------- */
/* gives position of 2D point (ix,iy) in 1D array ; uses also global variable iWidth */
unsigned int Give_i (unsigned int ix, unsigned int iy)
{
return ix + iy * iWidth;
}
// bailout test
// escapes = abs(z)> ER
int Escapes(complex double z){
if (cabs2(z)>ER2) return 1;
return 0;
}
int IsInTarget(complex double z){
// here target set is a circle inside immediate basin component containing critical point
// with fixed point on it's boundary
// attracting petal
complex double center = a+radius;
if (cabs(z-center) <= radius) return 1;
return 0;
}
// ***********************************************************************************************
// ********************** edge detection usung Sobel filter ***************************************
// ***************************************************************************************************
// from Source to Destination
int ComputeBoundaries(unsigned char S[], unsigned char D[])
{
unsigned int iX,iY; /* indices of 2D virtual array (image) = integer coordinate */
unsigned int i; /* index of 1D array */
/* sobel filter */
unsigned char G, Gh, Gv;
// boundaries are in D array ( global var )
// clear D array
memset(D, iColorOfExterior, iSize*sizeof(*D)); // for heap-allocated arrays, where N is the number of elements = FillArrayWithColor(D , iColorOfExterior);
// printf(" find boundaries in S array using Sobel filter\n");
#pragma omp parallel for schedule(dynamic) private(i,iY,iX,Gv,Gh,G) shared(iyMax,ixMax, ER2)
for(iY=1;iY<iyMax-1;++iY){
for(iX=1;iX<ixMax-1;++iX){
Gv= S[Give_i(iX-1,iY+1)] + 2*S[Give_i(iX,iY+1)] + S[Give_i(iX-1,iY+1)] - S[Give_i(iX-1,iY-1)] - 2*S[Give_i(iX-1,iY)] - S[Give_i(iX+1,iY-1)];
Gh= S[Give_i(iX+1,iY+1)] + 2*S[Give_i(iX+1,iY)] + S[Give_i(iX-1,iY-1)] - S[Give_i(iX+1,iY-1)] - 2*S[Give_i(iX-1,iY)] - S[Give_i(iX-1,iY-1)];
G = sqrt(Gh*Gh + Gv*Gv);
i= Give_i(iX,iY); /* compute index of 1D array from indices of 2D array */
if (G==0) {D[i]=255;} /* background */
else {D[i]=0;} /* boundary */
}
}
return 0;
}
// copy from Source to Destination
int CopyBoundaries(unsigned char S[], unsigned char D[])
{
unsigned int iX,iY; /* indices of 2D virtual array (image) = integer coordinate */
unsigned int i; /* index of 1D array */
//printf("copy boundaries from S array to D array \n");
for(iY=1;iY<iyMax-1;++iY)
for(iX=1;iX<ixMax-1;++iX)
{i= Give_i(iX,iY); if (S[i]==0) D[i]=0;}
return 0;
}
// ***************************************************************************************************************************
// ************************** Interior : components of Immediate Basin of Attraction *****************************************
// ****************************************************************************************************************************
unsigned char ComputeColorOfImmediateBasin(complex double z){
int nMax = iterMax;
int n;
for (n=0; n < nMax; n++){ //forward iteration
if (Escapes(z)) return iColorOfExterior;
if (IsInTarget(z)) return iColorOfInterior + (n % ChildPeriod)* ColorStep; // immediate basin of attraction and it's preimages
z = z*z +c ; /* forward iteration : complex quadratic polynomial */
}
printf("unknown point : z = %.16f; %.16f\n", creal(z), cimag(z));
NoOfUnknownPoints +=1;
return iColorOfUnknown;
}
// plots raster point (ix,iy)
int DrawPointOfImmediateBasin (unsigned char A[], int ix, int iy)
{
int i; /* index of 1D array */
unsigned char iColor;
complex double z;
i = Give_i (ix, iy); /* compute index of 1D array from indices of 2D array */
z = GiveZ(ix,iy);
iColor = ComputeColorOfImmediateBasin(z);
A[i] = iColor ; // interior
return 0;
}
// fill array
// uses global var : ...
// scanning complex plane
int DrawImagerOfImmediateBasin (unsigned char A[])
{
unsigned int ix, iy; // pixel coordinate
//printf("compute image \n");
// for all pixels of image
#pragma omp parallel for schedule(dynamic) private(ix,iy) shared(A, ixMax , iyMax, iColorOfUnknown)
for (iy = iyMin; iy <= iyMax; ++iy){
//printf (" %d from %d \r", iy, iyMax); //info
for (ix = ixMin; ix <= ixMax; ++ix)
DrawPointOfImmediateBasin(A, ix, iy); //
}
return 0;
}
// *******************************************************************************************************
//******************** Interior: Level Sets of Attraction time ******************************************
// *******************************************************************************************************
unsigned char ComputeColorOfInteriorLevelSets(complex double z){
int nMax = iterMax;
int n;
int p;
int pMax = ChildPeriod; //
for (n=0; n < nMax; n++){ //forward iteration
if (Escapes(z)) return iColorOfExterior;
for (p=0; p < pMax; p++){ //forward iteration
if (IsInTarget(z)) return iColorOfInterior + (n % ChildPeriod)* ColorStep; // immediate basin of attraction and it's preimages
z = z*z +c ; /* forward iteration : complex quadratic polynomial */
}
}
//
printf("unknown point : z = %.16f; %.16f\n", creal(z), cimag(z));
NoOfUnknownPoints +=1;
return iColorOfUnknown;
}
// plots raster point (ix,iy)
int DrawPointOfInteriorLevelSets (unsigned char A[], int ix, int iy)
{
int i; /* index of 1D array */
unsigned char iColor;
complex double z;
i = Give_i (ix, iy); /* compute index of 1D array from indices of 2D array */
z = GiveZ(ix,iy);
iColor = ComputeColorOfInteriorLevelSets(z);
A[i] = iColor ; // interior
return 0;
}
// fill array
// uses global var : ...
// scanning complex plane
int DrawImageOfInteriorLevelSets (unsigned char A[])
{
unsigned int ix, iy; // pixel coordinate
//printf("compute image \n");
// for all pixels of image
#pragma omp parallel for schedule(dynamic) private(ix,iy) shared(A, ixMax , iyMax, iColorOfUnknown)
for (iy = iyMin; iy <= iyMax; ++iy){
//printf (" %d from %d \r", iy, iyMax); //info
for (ix = ixMin; ix <= ixMax; ++ix)
DrawPointOfInteriorLevelSets(A, ix, iy); //
}
return 0;
}
//******************** Interior: chessboard ******************************************
// *******************************************************************************************************
unsigned char ComputeColorOfChessboard (complex double z){
int nMax = iterMax;
int n;
int p;
int pMax = ChildPeriod; //
for (n=0; n < nMax; n++){ //forward iteration
if (Escapes(z)) return iColorOfExterior;
for (p=0; p < pMax; p++){ //forward iteration
if (IsInTarget(z))
{if (cimag(z)>0.0)
return 20 ; // above critical orbit
else return 243; // below critical orbit
}
z = z*z +c ; /* forward iteration : complex quadratic polynomial */
}
}
//
printf("unknown point : z = %.16f; %.16f\n", creal(z), cimag(z));
NoOfUnknownPoints +=1;
return iColorOfUnknown;
}
// plots raster point (ix,iy)
int DrawPointOfChessboard (unsigned char A[], int ix, int iy)
{
int i; /* index of 1D array */
unsigned char iColor;
complex double z;
i = Give_i (ix, iy); /* compute index of 1D array from indices of 2D array */
z = GiveZ(ix,iy);
iColor = ComputeColorOfChessboard(z);
A[i] = iColor ; // interior
return 0;
}
// fill array
// uses global var : ...
// scanning complex plane
int DrawImageOfChessboard (unsigned char A[])
{
unsigned int ix, iy; // pixel coordinate
//printf("compute image \n");
// for all pixels of image
#pragma omp parallel for schedule(dynamic) private(ix,iy) shared(A, ixMax , iyMax, iColorOfUnknown)
for (iy = iyMin; iy <= iyMax; ++iy){
//printf (" %d from %d \r", iy, iyMax); //info
for (ix = ixMin; ix <= ixMax; ++ix)
DrawPointOfChessboard(A, ix, iy); //
}
return 0;
}
// ************************************************************************************
//******************** Interior: both Levels sets and chesboard ******************************************
// *******************************************************************************************************
unsigned char ComputeColorOfBoth (complex double z){
int nMax = iterMax;
int n;
int p;
int pMax = ChildPeriod; //
double angle; // in turns
unsigned char color;
for (n=0; n < nMax; n++){ //forward iteration
if (Escapes(z)) return iColorOfExterior;
for (p=0; p < pMax; p++){ //
if (IsInTarget(z))
{
angle = GiveTurn(z - a); // now in (0,1) range
// !!!!!!
//if (angle> 7.0/8.0 ) angle = (angle)*8.0;
//if (angle<1.0/8.0)
angle = angle*4.0; // repeated gradient
//printf("angle = %.16f\n", angle);
color = angle* 255; // now in (0,255) range
return color;
}
z = z*z +c ; /* forward iteration : complex quadratic polynomial */
}
}
//
printf("unknown point : z = %.16f; %.16f\n", creal(z), cimag(z));
NoOfUnknownPoints +=1;
return iColorOfUnknown;
}
// plots raster point (ix,iy)
int DrawPointOfBoth (unsigned char A[], int ix, int iy)
{
int i; /* index of 1D array */
unsigned char iColor;
complex double z;
i = Give_i (ix, iy); /* compute index of 1D array from indices of 2D array */
z = GiveZ(ix,iy);
iColor = ComputeColorOfBoth(z);
A[i] = iColor ; // interior
return 0;
}
// fill array
// uses global var : ...
// scanning complex plane
int DrawImageOfBoth (unsigned char A[])
{
unsigned int ix, iy; // pixel coordinate
// radius /=10.0;
//printf("compute image \n");
// for all pixels of image
#pragma omp parallel for schedule(dynamic) private(ix,iy) shared(A, ixMax , iyMax, iColorOfUnknown)
for (iy = iyMin; iy <= iyMax; ++iy){
//printf (" %d from %d \r", iy, iyMax); //info
for (ix = ixMin; ix <= ixMax; ++ix)
DrawPointOfBoth(A, ix, iy); //
}
return 0;
}
// *******************************************************************************************
// ********************************** save A array to pgm file ****************************
// *********************************************************************************************
int SaveArray2PGMFile( unsigned char A[], double k, char* comment )
{
FILE * fp;
const unsigned int MaxColorComponentValue=255; /* color component is coded from 0 to 255 ; it is 8 bit color file */
char name [100]; /* name of file */
snprintf(name, sizeof name, "%.3f", k); /* */
char *filename =strncat(name,".pgm", 4);
// save image to the pgm file
fp= fopen(filename,"wb"); // create new file,give it a name and open it in binary mode
fprintf(fp,"P5\n # %s\n %u %u\n %u\n", comment, iWidth, iHeight, MaxColorComponentValue); // write header to the file
fwrite(A,iSize,1,fp); // write array with image data bytes to the file in one step
fclose(fp);
// info
printf("File %s saved ", filename);
if (comment == NULL || strlen(comment) ==0)
printf("\n");
else printf (". Comment = %s \n", comment);
return 0;
}
int info ()
{
// display info messages
printf ("Numerical approximation of parabolic Julia set for fc(z)= z^2 + c \n");
//printf ("iPeriodParent = %d \n", iPeriodParent);
//printf ("iPeriodOfChild = %d \n", iPeriodChild);
printf ("parameter c = ( %.16f ; %.16f ) \n", creal(c), cimag(c));
printf ("is a root point between period %d and %d components \n", ChildPeriod, ParentPeriod);
printf ("alfa fixed point z = ( %.16f ; %.16f ) \n", creal(a), cimag(a));
printf ("Image Width = %f in world coordinate\n", ZxMax - ZxMin);
printf ("PixelWidth = %f \n", PixelWidth);
printf("radius of attracting circular petal = %.16f\n", radius);
// image corners in world coordinate
// center and radius
// center and zoom
// GradientRepetition
printf ("Maximal number of iterations = iterMax = %ld \n", iterMax);
printf ("ratio of image = %f ; it should be 1.000 ...\n", ratio);
printf("NoOfUnknownPoints = %d NoOfAllPoints = %d so ratio unknown/all = %f \n", NoOfUnknownPoints, iSize, (double) NoOfUnknownPoints/ iSize);
return 0;
}
// *****************************************************************************
//;;;;;;;;;;;;;;;;;;;;;; setup ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
// **************************************************************************************
int setup ()
{
printf ("setup start\n");
c = Give_c(ParentPeriod, 1, ChildPeriod, 1.0);
a = GiveAlfa(c); // -0.5; // alfa fixed point
radius = cabs(c)/4.8 ; // choose such value that level sets cross at z=0
/* 2D array ranges */
iWidth = iHeight;
iSize = iWidth * iHeight; // size = number of points in array
// iy
iyMax = iHeight - 1; // Indexes of array starts from 0 not 1 so the highest elements of an array is = array_name[size-1].
//ix
ixMax = iWidth - 1;
/* 1D array ranges */
// i1Dsize = i2Dsize; // 1D array with the same size as 2D array
iMax = iSize - 1; // Indexes of array starts from 0 not 1 so the highest elements of an array is = array_name[size-1].
/* Pixel sizes */
PixelWidth = (ZxMax - ZxMin) / ixMax; // ixMax = (iWidth-1) step between pixels in world coordinate
PixelHeight = (ZyMax - ZyMin) / iyMax;
ratio = ((ZxMax - ZxMin) / (ZyMax - ZyMin)) / ((float) iWidth / (float) iHeight); // it should be 1.000 ...
ER2 = ER * ER; // for numerical optimisation in iteration
/* create dynamic 1D arrays for colors ( shades of gray ) */
data = malloc (iSize * sizeof (unsigned char));
edge = malloc (iSize * sizeof (unsigned char));
if (data == NULL || edge == NULL){
fprintf (stderr, " Could not allocate memory");
return 1;
}
ColorStep = (243 - iColorOfInterior)/(ChildPeriod-1);
if (ColorStep <1) {printf("error from setup : ColorStep < 0 ; It should be greater\n"); return 1; } // check
printf (" end of setup \n");
return 0;
} // ;;;;;;;;;;;;;;;;;;;;;;;;; end of the setup ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
int end(){
printf (" allways free memory (deallocate ) to avoid memory leaks \n"); // https://en.wikipedia.org/wiki/C_dynamic_memory_allocation
free (data);
free(edge);
info ();
return 0;
}
// ********************************************************************************************************************
/* ----------------------------------------- main -------------------------------------------------------------*/
// ********************************************************************************************************************
int main () {
setup ();
// ******************************** components O f immediate basin **********************************************************
DrawImagerOfImmediateBasin(data);
SaveArray2PGMFile (data, iHeight, "Interior : components of immediate basin of attraction (IBA) and it's preimages");
//
ComputeBoundaries(data,edge);
SaveArray2PGMFile (edge, iHeight+1.0, "only boundary of components");
//
CopyBoundaries(edge,data);
SaveArray2PGMFile (data, iHeight+2.0, "components with boundaries");
// ***************** attraction time ***************************************************
DrawImageOfInteriorLevelSets (data);
SaveArray2PGMFile (data, iHeight+3.0+radius, "Interior: level sets of attraction time to the parabolic fixed point");
//
ComputeBoundaries(data,edge);
SaveArray2PGMFile (edge, iHeight+4.0+radius, "only boundaries of level sets");
//
CopyBoundaries(edge,data);
SaveArray2PGMFile (data, iHeight+5.0+radius, "level sets with boundaries");
// ***************** parabolic chessboard ***************************************************
DrawImageOfChessboard(data);
SaveArray2PGMFile (data, iHeight+6.0+radius, "Interior : parabolic chessboard");
//
ComputeBoundaries(data,edge);
SaveArray2PGMFile (edge, iHeight+7.0+radius, "only boundaries of parabolic chessboard");
//
CopyBoundaries(edge,data);
SaveArray2PGMFile (data, iHeight+8.0+radius, "parabolic chessboard with boundaries");
// ***************** Interior: both Levels sets and chesboard ***************************************************
DrawImageOfBoth(data);
SaveArray2PGMFile (data, iHeight+9.0+radius, "Interior: both Levels sets and chesboard ");
end();
return 0;
}
text output
setup start end of setup File 10000.000.pgm saved . Comment = Interior : components of immediate basin of attraction (IBA) and it's preimages File 10001.000.pgm saved . Comment = only boundary of components File 10002.000.pgm saved . Comment = components with boundaries File 10003.156.pgm saved . Comment = Interior: level sets of attraction time to the parabolic fixed point File 10004.156.pgm saved . Comment = only boundaries of level sets File 10005.156.pgm saved . Comment = level sets with boundaries File 10006.156.pgm saved . Comment = Interior : parabolic chessboard File 10007.156.pgm saved . Comment = only boundaries of parabolic chessboard File 10008.156.pgm saved . Comment = parabolic chessboard with boundaries File 10009.156.pgm saved . Comment = Interior: both Levels sets and chesboard allways free memory (deallocate ) to avoid memory leaks Numerical approximation of parabolic Julia set for fc(z)= z^2 + c parameter c = ( -0.7500000000000000 ; 0.0000000000000001 ) is a root point between period 2 and 1 components alfa fixed point z = ( -0.5000000000000000 ; 0.0000000000000001 ) Image Width = 3.200000 in world coordinate PixelWidth = 0.000320 radius of attracting circular petal = 0.1562500000000000 Maximal number of iterations = iterMax = 1000000 ratio of image = 1.000000 ; it should be 1.000 ... NoOfUnknownPoints = 0 NoOfAllPoints = 100000000 so ratio unknown/all = 0.000000
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