Fractals/Computer graphic techniques/2D

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Intro[edit]

All tasks ( image processing [1]) 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 :

Creating graphic[edit]

Here are 3 targets / tasks :

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

Graphic file[edit]

Graphic files

Memory array[edit]

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[edit]

glxinfo | grep OpenGL
glxinfo | grep "direct rendering"

DRI[edit]

Direct Rendering Infrastructure (DRI2)[10]

Color[edit]

Example of intresting color gradient
Palette graphics, palette replacement mechanism

Curve[edit]

Field lines[edit]

Field line [11]

Tracing[edit]

Tracing curve [12]

Curve rasterisation[edit]

Ray[edit]

Ray can be parametrised with radius ( r)

Closed curve[edit]

Simple closed curve ("a connected curve that does not cross itself and ends at the same point where it begins" [13] = having no endpoints) can be parametrized with angle ( t).

Edge detection[edit]

Sobel filter[edit]

Short introduction[edit]

Sobel filter G consist of 2 filters (masks):

  • Gh for horizontal changes.
  • Gv for vertical changes.
Sobel kernels[edit]
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 [15]
  • 3 weights in each kernal are zero
Pixel kernel[edit]

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 : [16]

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 :[17]


\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[edit]

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. [18]

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

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.[19]

Algorithm[edit]

  • 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[edit]

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 :

Problems[edit]

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." [20]

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."[21]
  • lines are not meeting in good points

See also new operators from 6 version of Image Magic : EdgeIn and EdgeOut from Morphology [22]

Edge thickening[edit]

dilation [23][24][25]

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

Filling contour[edit]

Filling contour - simple procedure in c


Quality of image[edit]

Interval arithemthic[edit]

Antialiasing[edit]

Aliased chessboard - image and c src code


Supersampling[edit]

example of supersampled image
Cpp code of supersampling

Other names :

  • antigrain geometry
  • Supersampling ( downsampling) [33]
  • subpixel accuracy

Examples :

 // 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 [34] ( see aptuscmd.py ) is using a high-quality downsampling filter thru PIL function resize [35]
  • Java code by Josef Jelinek [36]: 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[edit]

Description is here

Optimization[edit]

Optimisation is described here

References[edit]

  1. IPOL Journal · Image Processing On Line
  2. ImageMagick image processing libraries
  3. GEGL (Generic Graphics Library)
  4. http://openil.sourceforge.net/
  5. http://freeimage.sourceforge.net/
  6. GD Graphics Library
  7. graphics magick
  8. OpenCv
  9. OpenImageIO
  10. w:Direct Rendering Infrastructure (DRI)
  11. wikipedia : Field line
  12. Curve sketching in wikipedia
  13. mathwords : simple_closed_curve
  14. matrixlab - line-detection
  15. Sobel Edge Detector by R. Fisher, S. Perkins, A. Walker and E. Wolfart.
  16. NVIDIA Forums , CUDA GPU Computing discussion by kr1_karin
  17. Sobel Edge by RoboRealm
  18. forum nvidia : Sobel Filter Don't understand a little thing in the SDK example
  19. Sobel Edge Detector by R. Fisher, S. Perkins, A. Walker and E. Wolfart.
  20. Image Magic doc
  21. Edge operator from Imagew Magic dosc
  22. imagemagick doc : morphology / edgein
  23. dilation at HIPR2 by Robert Fisher, Simon Perkins, Ashley Walker, Erik Wolfart
  24. imagemagick doc : morphology, dilate
  25. Fred's ImageMagick Scripts
  26. ON THE NUMERICAL CONSTRUCTION OF HYPERBOLIC STRUCTURES FOR COMPLEX DYNAMICAL SYSTEMS by Jennifer Suzanne Lynch Hruska
  27. "Images of Julia sets that you can trust" by Luiz Henrique de Figueiredo and Joao Batista Oliveira
  28. adaptive algorithms for generating guaranteed images of Julia sets by Luiz Henrique de Figueiredo
  29. Drawing Fractals With Interval Arithmetic - Part 1 by Dr Rupert Rawnsley
  30. Drawing Fractals With Interval Arithmetic - Part 2 by Dr Rupert Rawnsley
  31. Spatial anti aliasing at wikipedia
  32. fractalforums discussion : Antialiasing fractals - how best to do it?
  33. Supersampling at wikipedia
  34. Aptus ( python and c code ) by Ned Batchelder
  35. Pil function resize
  36. Java code by Josef Jelinek