Fractals/Computer graphic techniques/2D

All tasks (image processing[1]) can be done using:

One can use free graphic libraries:

graphic workstationEdit


Creating graphicEdit

Here are 3 targets / tasks:

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

Graphic fileEdit

Graphic files

Memory arrayEdit

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 pixelsEdit

glxinfo | grep OpenGL
glxinfo | grep "direct rendering"

DRIEdit

Direct Rendering Infrastructure (DRI2)[10]

ColorEdit

 
Example of interesting color gradient
 
Palette graphics, palette replacement mechanism

CurveEdit

Field linesEdit

Field line[11]

TracingEdit

 
Tracing a curve on a triangular grid

Tracing curve[12]

Methods

  • general, (analytic or systematic) = curve sketching[13]
  • local method


Three Curve Tracing Models[14]

  • Pixel-by-Pixel tracing
  • The bipartite receptive field operator
  • The zoom lens operator


Images


Problems:

Examples

Curve rasterisationEdit

RayEdit

Ray can be parametrised with radius (r)

Closed curveEdit

Simple closed curve ("a connected curve that does not cross itself and ends at the same point where it begins"[16] = having no endpoints) can be parametrized with angle (t).

Edge detectionEdit

Sobel filterEdit

Short introductionEdit

Sobel filter G consist of 2 filters (masks):

  • Gh for horizontal changes.
  • Gv for vertical changes.
Sobel kernelsEdit
 
8-point neighborhood on a 2D grid
 
2D Convolution Animation

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:

 

The Sobel kernel for computing vertical changes is:

 

Note that:

  • sum of weights of kernels are zero

 

 

  • One kernel is simply the other rotated by 90 degrees.[18]
  • 3 weights in each kernal are zero.
Pixel kernelEdit

Pixel kernel A containing central pixel   with its 3x3 neighbourhood:

 

Other notations for pixel kernel:

 

where:[19]

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:[20]

 

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

So central (tested) pixel is:

 

Sobel filtersEdit

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

 
 

Because 3 weights in each kernal are zero so there are only 6 products.[21]

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

Result is computed (magnitude of gradient):

 

It is a color of tested pixel.

One can also approximate result by sum of 2 magnitudes:

 

which is much faster to compute.[22]

AlgorithmEdit

  • 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

ProgrammingEdit

 
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:

ProblemsEdit

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

Edge position:

In ImageMagick 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."[23]

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."[24]
  • lines are not meeting in good points.

See also new operators from 6 version of ImageMagick: EdgeIn and EdgeOut from Morphology[25]

Edge thickeningEdit

dilation[26][27][28]

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

Filling contourEdit

 
Filling contour - simple procedure in c

Quality of imageEdit

  • " using 100 samples per pixel, so rendered the full image 100 times with small random jitter (movement at sub pixel size) and accumulate the result, gives really nice antialiasing." 3DickUlus
  • Tips for making better images by Paul Bourke
    • do the fractal creation not in 8 bit, but use 16 bit or floating point for each pixel ( pfm file, kfb and exr ( OpenEXR) from Kalles Fraktaler or array with floating point values )
    • apply antialiasing by supersampling each final pixel ( render a 30,000 pixel version of the image )
    • do all colour external to the fractal creation using gradient maps...allows you to make appearance decisions independent to the creation

Interval arithmeticEdit

AntialiasingEdit

 
Aliased chessboard - image and c src code

SupersamplingEdit

 
example of supersampled image
 
Cpp code of supersampling

Other names:

  • antigrain geometry
  • Supersampling (downsampling)[37][38]
  • downsizing
  • downscaling[39]
  • 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[40] (see aptuscmd.py) is using a high-quality downsampling filter thru PIL function resize[41]
  • Java code by Josef Jelinek:[42] 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 ImageMagick:
convert big.ppm -resize 2000x2000 m.png

See also:

PlaneEdit

Description is here.

OptimizationEdit

Optimisation is described here

ReferencesEdit

  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. GraphicsMagick
  8. OpenCv
  9. OpenImageIO
  10. w:Direct Rendering Infrastructure (DRI)
  11. wikipedia: Field line
  12. Curve sketching in wikipedia
  13. slides from MALLA REDDY ENGINEERING COLLEGE
  14. Predicting the shape of distance functions in curve tracing: Evidence for a zoom lens operator by PETER A. McCORMICK and PIERRE JOLICOEUR
  15. stackoverflow question: line-tracking-with-matlab
  16. mathwords: simple_closed_curve
  17. matrixlab - line-detection
  18. Sobel Edge Detector by R. Fisher, S. Perkins, A. Walker and E. Wolfart.
  19. NVIDIA Forums, CUDA GPU Computing discussion by kr1_karin
  20. Sobel Edge by RoboRealm
  21. nvidia forum: Sobel Filter Don't understand a little thing in the SDK example
  22. Sobel Edge Detector by R. Fisher, S. Perkins, A. Walker and E. Wolfart.
  23. ImageMagick doc
  24. Edge operator from ImageMagick docs
  25. ImageMagick doc: morphology / EdgeIn
  26. dilation at HIPR2 by Robert Fisher, Simon Perkins, Ashley Walker, Erik Wolfart
  27. ImageMagick doc: morphology, dilate
  28. Fred's ImageMagick Scripts
  29. Images of Julia sets that you can trust Luiz Henrique de Figueiredo
  30. ON THE NUMERICAL CONSTRUCTION OF HYPERBOLIC STRUCTURES FOR COMPLEX DYNAMICAL SYSTEMS by Jennifer Suzanne Lynch Hruska
  31. "Images of Julia sets that you can trust" by Luiz Henrique de Figueiredo and Joao Batista Oliveira
  32. adaptive algorithms for generating guaranteed images of Julia sets by Luiz Henrique de Figueiredo
  33. Drawing Fractals With Interval Arithmetic - Part 1 by Dr Rupert Rawnsley
  34. Drawing Fractals With Interval Arithmetic - Part 2 by Dr Rupert Rawnsley
  35. Spatial anti aliasing at wikipedia
  36. fractalforums discussion: Antialiasing fractals - how best to do it?
  37. Supersampling at wikipedia
  38. ImageMagick v6 Examples -- Resampling Filters
  39. What is the best image downscaling algorithm (quality-wise)?
  40. Aptus (python and c code) by Ned Batchelder
  41. Pil function resize
  42. Java code by Josef Jelinek
  43. ImageMagick: resize_gamma
  44. A Cheritat wiki: see image showing gamma-correct downscale of dense part of Mandelbrot set