Fractals/Computer graphic techniques/2D

Intro

All tasks can be done using :

• own programs ( GUI or console )
• own programs and graphic libraries
• graphic programs like GIMP
• fractal programs like :
  • fractint
  • xaos
  • mandel by Wolf Jung

One can use free graphic libraries :

• Image Magic
• GEGL
• DeviL
• FreeImage
• GD
• GraphicsMagic
• OpenCV
• OpenImageIO

Creating graphic

Here are 3 targets / tasks :

• graphic file ( saving/ loading image )
• memory array ( processing image )
• screen pixels ( displaying image )

Graphic file

Graphic files

Memory array

Image in memory is a matrix :

• A 24-bit color image is an (Width x Height x 3) matrix.
• Gray-level and black-and-white images are of size (Width x Height) .

The color depth of the image :

• 8-bit for gray
• 24 or 32-bit for color,
• 1-bit for black and white.

Screen pixels

glxinfo | grep OpenGL

glxinfo | grep "direct rendering"

DRI

Direct Rendering Infrastructure (DRI2)

Color

Example of intresting color gradient

Palette graphics, palette replacement mechanism

• gradient

Curve

Tracing

Tracing curve

Curve rasterisation

Ray

Ray can be parametrised with radius ( r)

Closed curve

Closed curve () can be parametrized with angle ( t).

Edge detection

• Boundary scanning method for Julia set - BSM/J
• Boundary scanning method for Mandelbrot set - BSM/M
• edge detection in Matlab

• 

  Level curves - edge detection ( 2 filters )

• 

  Edge detection of boundaries of level sets of escape time

Sobel filter

Short introduction

Sobel filter G consist of 2 filters (masks):

• Gh for horizontal changes.
• Gv for vertical changes.

Sobel kernels

8-point neighborhood on a 2D grid

The Sobel kernel contains weights for each pixel from the 8-point neighbourhood of a tested pixel. These are 3x3 kernels.

There are 2 Sobel kernels, one for computing horizontal changes and other for computing vertical changes. Notice that a large horizontal change may indicate a vertical border, and a large vertical change may indicate a horizontal border. The x-coordinate is here defined as increasing in the "right"-direction, and the y-coordinate is defined as increasing in the "down"-direction.

The Sobel kernel for computing horizontal changes is:

$\mathbf{H} = \begin{bmatrix} H_1 & H_2 & H_3 \\ H_4 & H_5 & H_6 \\ H_7 & H_8 & H_9 \end{bmatrix} = \begin{bmatrix} -1 & 0 & +1 \\ -2 & 0 & +2 \\ -1 & 0 & +1 \end{bmatrix}$

The Sobel kernel for computing vertical changes is:

$\mathbf{V} = \begin{bmatrix} -1 & -2 & -1 \\ \ \ 0 & \ \ 0 & \ \ 0 \\ +1 & +2 & +1 \end{bmatrix}$

Note that :

• sum of weights of kernels are zero

$\sum_{i=1}^9H_i = 0$

$\sum_{i=1}^9V_i = 0$

• One kernel is simply the other rotated by 90 degrees
• 3 weights in each kernal are zero

Pixel kernel

Pixel kernel A containing central pixel $A_5$ with its 3x3 neghbourhood  :

$\mathbf{A} = \begin{bmatrix} A_1 & A_2 & A_3 \\ A_4 & A_5 & A_6 \\ A_7 & A_8 & A_9 \end{bmatrix}$

Other notations for pixel kernel :

$\mathbf{A} = \begin{bmatrix} A_1 & A_2 & A_3 \\ A_4 & A_5 & A_6 \\ A_7 & A_8 & A_9 \end{bmatrix} = \begin{bmatrix} ul & um & ur \\ ml & mm & mr \\ ll & lm & lr \end{bmatrix}$

where :

unsigned char ul, // upper left
unsigned char um, // upper middle
unsigned char ur, // upper right
unsigned char ml, // middle left
unsigned char mm, // middle = central pixel
unsigned char mr, // middle right
unsigned char ll, // lower left
unsigned char lm, // lower middle
unsigned char lr, // lower right

Pixel 3x3 neighbourhood (with Y axis directed down)

In array notation it is :

$\mathbf{A} = \begin{bmatrix} A[x-1][y+1] & A[x][y+1] & A[x+1][y+1] \\ A[x-1][y] & A[x][y] & A[x+1][y] \\ A[x-1][y-1] & A[x][y-1] & A[x+1][y-1] \end{bmatrix}$

In geographic notation usede in cellular aotomats it is central pixel of Moore neighbourhood.

So central ( tested ) pixel is :

$A_5 = mm = A[x][y] \,$

Sobel filters

Compute sobel filters ( where $*$ here denotes the 2-dimensional convolution operation not matrix multiplication ). It is a sum of products of pixel and its weghts :

$\mathbf{G}_h = \mathbf{H} * A = A_{1}H_{1} + A_{2}H_{2} + \cdots + A_{9}H_{9} = \sum_{r=1}^9 A_{r}H_{r},$

$\mathbf{G}_v = \mathbf{V} * A = A_{1}V_{1} + A_{2}V_{2} + \cdots + A_{9}V_{9} = \sum_{r=1}^9 A_{r}V_{r},$

Because 3 weights in each kernal are zero so there are only 6 products.

short Gh = ur + 2*mr + lr - ul - 2*ml - ll;
short Gv = ul + 2*um + ur - ll - 2*lm - lr;

Result

Result is computed (magnitude of gradient):

$\mathbf{G}(A_5) = \sqrt{ {\mathbf{G}_h}^2 + {\mathbf{G}_v}^2 }$

It is a color of tested pixel .

One can also approximate result by sum of 2 magnitudes :

$\mathbf{G}(A_5) = \left| \mathbf{G}_h \right| + \left| \mathbf{G}_v \right|$

which is much faster to compute.

Algorithm

• choose pixel and its 3x3 neighberhood A
• compute sobel filter for horizontal Gh and vertical lines Gv
• compute sobel filter G
• compute color of pixel

Programming

Sobel filters ( 2 filters 3x3 ) : image and full c code

Skipped pixel - some points from its neighbourhood are out of the image

Lets take array of 8-bit colors ( image) called data. To find borders in this image simply do :

for(iY=1;iY<iYmax-1;++iY){ 
    for(iX=1;iX<iXmax-1;++iX){ 
     Gv= - data[iY-1][iX-1] - 2*data[iY-1][iX] - data[iY-1][iX+1] + data[iY+1][iX-1] + 2*data[iY+1][iX] + data[iY+1][iX+1];
     Gh= - data[iY+1][iX-1] + data[iY-1][iX+1] - 2*data[iY][iX-1] + 2*data[iY][iX+1] - data[iY-1][iX-1] + data[iY+1][iX+1];
     G = sqrt(Gh*Gh + Gv*Gv);
     if (G==0) {edge[iY][iX]=255;} /* background */
         else {edge[iY][iX]=0;}  /* boundary */
    }
  }

Note that here points on borders of array ( iY= 0 , iY = iYmax , iX=0, iX=iXmax) are skipped

Result is saved to another array called edge ( with the same size).

One can save edge array to file showing only borders, or merge 2 arrays  :

for(iY=1;iY<iYmax-1;++iY){ 
    for(iX=1;iX<iXmax-1;++iX){ if (edge[iY][iX]==0) data[iY][iX]=0;}}

to have new image with marked borders.

Above example is for 8-bit or indexed color. For higher bit colors "the formula is applied to all three color channels separately" ( from RoboRealm doc).

Other implementations :

• ImagMagic discussion : using a sobel operator - edge detection
• Sobel in c from GIMP code
• Cpp code by Aaron Brooks - Brent Bolton - Jason O'Kane
• rosettacode : Image convolution
• C and opencl by royger
• C++ by Glenn Fiedler
• Convolution and Deconvolution
• Sobel Edge Detector by R. Fisher, S. Perkins, A. Walker and E. Wolfart with examples in Java
• Qt and GDI+ versions of the Sobel edge detection algorithm by Ken Earle
• Qt and OpenCV

Problems

Bad edge position seen in the middle of image. Lines are not meeting in good points, like z = 0

Edge position :

In ImageMagic as "you can see, the edge is added only to areas with a color gradient that is more than 50% white! I don't know if this is a bug or intentional, but it means that the edge in the above is located almost completely in the white parts of the original mask image. This fact can be extremely important when making use of the results of the "-edge" operator."

The result is :

• doubling edges ; "if you are edge detecting an image containing an black outline, the "-edge" operator will 'twin' the black lines, producing a weird result."
• lines are not meeting in good points

See also new operators from 6 version of Image Magic : EdgeIn and EdgeOut from Morphology

Edge thickening

dilation

convert $tmp0 -convolve "1,1,1,1,1,1,1,1,1" -threshold 0 $outfile

Filling contour

Filling contour - simple procedure in c

Quality of image

Interval arithemthic

• adaptive algorithms for generating guaranted images of Julia sets

Antialiasing

Aliased chessboard - image and c src code

• TechInfo -AntiAliasing by Michael Condron
• Fract ( Lisp code) by Yannick Gingras
• Spatial anti aliasing at wikipedia
• fractalforums discussion : Antialiasing fractals - how best to do it?

Supersampling

example of supersampled image

Cpp code of supersampling

Other names :

• antigrain geometry
• Supersampling ( downsampling)
• subpixel accuracy

Examples :

• Cpp code by User Geek3

 // subpixels finished -> make arithmetic mean
 char pixel[3];
 for (int c = 0; c < 3; c++)
   pixel[c] = (int)(255.0 * sum[c] / (subpix * subpix)  + 0.5);
 fwrite(pixel, 1, 3, image_file);
 //pixel finished

• command line version of Aptus ( python and c code ) by Ned Batchelder ( see aptuscmd.py ) is using a high-quality downsampling filter thru PIL function resize
• Java code by Josef Jelinek : supersampling with grid algorithm, computes 4 new points (corners), resulting color is an avarage of each color component :

 //Created by Josef Jelinek
 // http://java.rubikscube.info/
 Color c0 = color(dx, dy); // color of central point
 // computation of 4 new points for antialiasing
 if (antialias) { // computes 4 new points (corners)
   Color c1 = color(dx - 0.25 * r, dy - 0.25 * r);
   Color c2 = color(dx + 0.25 * r, dy - 0.25 * r);
   Color c3 = color(dx + 0.25 * r, dy + 0.25 * r);
   Color c4 = color(dx - 0.25 * r, dy + 0.25 * r);
  // resulting color; each component of color is an avarage of 5 values ( central point and 4 corners )
   int red = (c0.getRed() + c1.getRed() + c2.getRed() + c3.getRed() + c4.getRed()) / 5;
   int green = (c0.getGreen() + c1.getGreen() + c2.getGreen() + c3.getGreen() + c4.getGreen()) / 5;
   int blue = (c0.getBlue() + c1.getBlue() + c2.getBlue() + c3.getBlue() + c4.getBlue()) / 5;
   color = new Color(red, green, blue);
 }

• one can make big image ( like 10 000 x 10 000 ) and convert/resize it ( downsize ). For example using ImageMagic :

 convert big.ppm -resize 2000x2000 m.png.

Plane

Description

2D plane can be described by :

• corners ( 4 points )
• center and width
• center and magnification ("If you use the center, you can change the zoom level and the plot zooms in/out smoothly on the same center point. " Duncan C)

Standard description in Fractint, Ultra Fractal, ChaosPro and Fractal Explorer are corners. For example initial plane for Mandelbrot set is

 Corners:                X                     Y
 Top-l          -2.5000000000000000    1.5000000000000000
 Bot-r           1.5000000000000000   -1.5000000000000000
 Ctr -0.5000000000000000   0.0000000000000000  Mag 6.66666667e-01
 X-Mag-Factor     1.0000   Rotation    0.000   Skew    0.000

Display window of parameter plane has :

• a horizontal width of 4 (real)
• a vertical width (height) of 3 (imag)
• an aspect ratio (proportion) 4/3 ( also in pixels 640/480 so ther is no distorsion)
• center z=-0.5

For julia set/ dynamic plane has :

Corners:                X                     Y
Top-l          -2.0000000000000000    1.5000000000000000
Bot-r           2.0000000000000000   -1.5000000000000000
Ctr  0.0000000000000000   0.0000000000000000  Mag 6.66666667e-01
X-Mag-Factor     1.0000   Rotation    0.000   Skew    0.000

Description from documentation of Fractint :

CORNERS=[xmin/xmax/ymin/ymax[/x3rd/y3rd]]

Example: corners=-0.739/-0.736/0.288/0.291

Begin with these coordinates as the range of x and y coordinates, rather than the default values of (for type=mandel) -2.0/2.0/-1.5/1.5. When you specify four values (the usual case), this defines a rectangle: x- coordinates are mapped to the screen, left to right, from xmin to xmax, y-coordinates are mapped to the screen, bottom to top, from ymin to ymax. Six parameters can be used to describe any rotated or stretched parallelogram: (xmin,ymax) are the coordinates used for the top-left corner of the screen, (xmax,ymin) for the bottom-right corner, and (x3rd,y3rd) for the bottom-left. Entering just "CORNERS=" tells Fractint to use this form (the default mode) rather than CENTER-MAG (see below) when saving parameters with the [B] command.

CENTER-MAG=[Xctr/Yctr/Mag[/Xmagfactor/Rotation/Skew]]

This is an alternative way to enter corners as a center point and a magnification that is popular with some fractal programs and publications. Entering just "CENTER-MAG=" tells Fractint to use this form rather than CORNERS (see above) when saving parameters with the [B] command. The [TAB] status display shows the "corners" in both forms. When you specify three values (the usual case), this defines a rectangle: (Xctr, Yctr) specifies the coordinates of the center of the image.

Mag indicates the amount of magnification to use. Initial value ( no zoom ) is 6.66666667e-01.

Six parameters can be used to describe any rotated or stretched parallelogram: Xmagfactor tells how many times bigger the x- magnification is than the y-magnification,

Rotation indicates how many degrees the image has been turned.

Skew tells how many degrees the image is leaning over. Positive angles will rotate and skew the image counter-clockwise.

Parameters can be saved to parmfile called fractint.par

Wolf Jung uses :

/* 
   from mndlbrot.cpp  by Wolf Jung (C) 201
   These classes are part of Mandel 5.7, which is free software; you can
   redistribute and / or modify them under the terms of the GNU General
   Public License as published by the Free Software Foundation; either
   version 3, or (at your option) any later version. In short: there is
   no warranty of any kind; you must redistribute the source code as well
*/
void mndlbrot::startPlane(int sg, double &xmid, double &rewidth) const
{  if (sg > 0) 
     { xmid = -0.5; rewidth = 1.6; } // parameter plane
     else { xmid = 0; rewidth = 2.0; } // dynamic plane

Orientation

Check orientation of the plane by marking first quadrant of Cartesian plane :

if (x>0 && y>0) Color=MaxColor-Color; 

It should be in upper right position.

Zoom

Mandelbrot zoom created using Double-double precision

Notation

If you want to zoom

baseSize // constant value =  extend in real and imaginary axis when zoomLevel is zero.
zoomLevel // inital value = 0.0
zoomFactor = 2^(-zoomLevel) // initial value = 1.0 = no zoom
sizeX = zoomFactor * baseSize
sizeY = zoomFactor * baseSize   
minImaginary = centerY - sizeY/2
maxImaginary = centerY + sizeY/2

Note that base of zoom factor is important when one wants to optimize zooming by using previosly computed pixels.

Usually other notation is used :

baseSize // constant value =  extend in real and imaginary axis when zoomLevel is zero.
zoomLevel // inital value = 0.0
zoomFactor = 10^zoomLevel // initial value = 1.0 = no zoom
sizeX =  baseSize / zoomFactor
sizeY = baseSize / zoomFactor   
minImaginary = centerY - sizeY/2
maxImaginary = centerY + sizeY/2

and max zoom factor is :

• 10^999
• 1,5E17

Maximal zoom is correlated with :

• precision ( single , double, Quadruple precision, multiple-precision (MP) )
• algorithm

Examples :

• zoom factor for video
• zoom factor

Pixel size

Pixel size is size of pixel in world coordinate. For linear scale and same resolution on both axes :

iSize = iXmax - iXmin ; // width of image in pixels ( integer or screen ) coordinate ; pixels are numbered from iXmin to iXmax; image is rectangular
fSize = CxMax-CxMin // width of image in world ( floating point ) coordinate
PixelWidth = fWidth / iWidth
PixelSize = PixelWidth // 

ZoomFactor	iSize	fSize	PixelSize
1 = 1.0e0	1000	3.0	0,003
100 = 1.0e2	1000	0.03	0.00003
1000 = 1.0e3	1000	0.003	0.0000003

Optimisation

• "recursive algorithm to split the image up into blocks, and tests each block to see whether it lies inside a “black area”." by Michael Hogg
• Poincaré half-plane metric for zoom animation
• A Simple Optimization of Fractal Animation by Wei-Yin Chen, You-Sheng Yang and Kun-Mao Liang
• optimizing zoom animations by Claude Heiland-Allen
• reenigne blog : recursive subdivision
• Xaos - algorithm description

Precision

What precision do I need for zoom ?

Scanning

See :

• Fractint Drawing Methods
• Fratal Witchcraft drawing methods by Steven Stoft
• AlmondBread algorithms

All the pixels

Scan the plane :

• all pixels ( plane is scaned pixel by pixel )
• the same pixel size

Here in Lisp

; common lisp. Here float values can be used,  there is no mapping
(loop for y from -1.5 to 1.5 by 0.1 do
      (loop for x from -2.5 to 0.5 by 0.05 do
            (princ (code-char 42))) ; print char
      (format t "~%")) ; new line

and in C

/* c */
/* screen coordinate = coordinate of pixels */      
int iX, iY, 
iXmin=0, iXmax=1000,
iYmin=0, iYmax=1000,
iWidth=iXmax-iXmin+1,
iHeight=iYmax-iYmin+1;
 
/* world ( double) coordinate = parameter plane*/
const double ZxMin=-5;
const double ZxMax=5;
const double ZyMin=-5;
const double ZyMax=5;
 
/* */
double PixelWidth=(ZxMax-ZxMin)/iWidth;
double PixelHeight=(ZyMax-ZyMin)/iHeight;
double Zx, Zy,    /* Z=Zx+Zy*i   */
       Z0x, Z0y,  /* Z0 = Z0x + Z0y*i */
 
 for(iY=0;iY<iYmax;++iY)
      { Z0y=ZyMin + iY*PixelHeight; /* mapping from screen to world; reverse Y  axis */
        if (fabs(Z0y)<PixelHeight/2) Z0y=0.0; /* Zy = 0 is a special value  */    
        for(iX=0;iX<iXmax;++iX)
         {    /* initial value of orbit Z0 */
           Z0x=ZxMin + iX*PixelWidth;
          }
      }

Optimization

Optimisation is described here

References