Fractals/Computer graphic techniques/2D/optimisation

From Wikibooks, open books for an open world
Jump to navigation Jump to search
 "Your current attempt runs in about O(N^2). So even if you had enough memory, it won't finish in your lifetime." Alexander Yee[1]
   "premature optimization is the root of all evil" - Donald Knuth in paper "Structured Programming With Go To Statements"
           "good algorithms beat optimized code" Bruce Dawson ( Fractal extreme )

Software optimisation[edit | edit source]

  • Profiling code
  • benchmarking [2]

How to ask about optimisation ?[edit | edit source]

  • "If you've actually profiled the code, have specific snippets so that everyone can run the same code to see its performance, and you have this library published somewhere where it can be seen (GitHub, Bitbucket, etc.), then I don't necessarily see a problem with it going on Code Review. Including the results from your profiler of choice with identified bottlenecks would go a long way towards keeping it on-topic.
  • If you're just starting the code but have profiled a small amount of code that exhibits the aberrant performance, then asking it on Stack Overflow would be acceptable. Including the results from your profiler of choice with identified bottlenecks would go a long way towards keeping it on-topic.
  • If you're asking for people to just help you optimize the code, don't ask it anywhere. Getting a code dump and being asked to find the performance bottleneck is only something I do professionally, not on my free time." (Makoto)[3]

See also : code review[4]

Inner loop[edit | edit source]

  • inner loop in wikipedia [5]

Optimisation of numerical computations[edit | edit source]

Distance[edit | edit source]


Decomposition[edit | edit source]

  • quadtree decomposition
"I did a web-based quad-tree Mandelbrot once.[7]  I labelled the quadrants (eg a,b,c,d) the tile files were in directories (something like a.png b.png a/a.png a/b.png a/c/a/b.png), along with an sqlite database that said which tiles  were boring (all same colour), along with their period (0 for 100% exterior, 1 for 100% inside the main cardioid, 2 for 100% inside the main circle, etc).  The client was really simple, just a web page with tile paths; the server  would do a redirect to the relevant boring tile 0.png 1.png 2.png etc if any parent of the tile was boring, otherwise it'd try to serve the tile.png.  Meanwhile there was another process that just generated the tiles and populated the database.  I can't remember if I made 404 tile not found errors bump the priority of the tile for the renderer, but maybe.

Eventually i abandoned the project because recalculating from scratch is typically faster and more flexible for shallow zooms, and for deep zooms the storage is prohibitive." Claude Heiland-Allen [8]

Symmetry[edit | edit source]

Mirror symmetry[edit | edit source]

Parameter plane and the Mandelbrot set is symmetric with respect to the x-axis in the plane :[9]

colour(x,y) = colour(x,-y)

Here is a C code that shows how to divide symmetric image (with axis in the middle of its height) into 3 parts :

  • axis of symmetry
  • 2 symmetric parts : above and below axis
// fill array using mirror symmetry of image 
// uses global var :  ...
int FillArray(unsigned char data[] )
{
   
 unsigned char Color; // gray from 0 to 255

// draw axis of symmetry
iy = iyAxisOfSymmetry; 
for(ix=ixMin;ix<=ixMax;++ix) PlotPoint(ix, iy, GiveColor(ix, iy));

// draw symmetric parts : above and below axis 
for(iyAbove = iyAboveMin; iyAbove<=iyAboveMax; ++iyAbove) 
  for(ix=ixMin; ix<=ixMax; ++ix)

  { // above axis compute color and save it to the array
    iy = iyAxisOfSymmetry + iyAbove;
    Color = GiveColor(ix, iy);
    PlotPoint(ix, iy, Color ); 
    // below the axis only copy Color the same as above without computing it 
    PlotPoint(ix, iyAxisOfSymmetry - iyAbove , Color ); 
   }   
 return 0;
}

See also Pascal code ( Delphi) for general case [10]

Rotational symmetry[edit | edit source]

Dynamical plane and the Julia set have rotational symmetry. It can be used to speed up computing :

// fill array using symmetry of image 
// uses global var :  ...
int FillArraySymmetric(unsigned char data[] )
{
   
 unsigned char Color; // gray from 0 to 255

printf("axis of symmetry \n"); 
iy = iyAxisOfSymmetry;

for(ix=ixMin;ix<=ixMax;++ix) PlotPoint(ix, iy, GiveColor(ix, iy));

// above and below axis 
for(iyAbove = iyAboveMin; iyAbove<=iyAboveMax; ++iyAbove) 
  {printf(" %d from %d\n", iyAbove, iyAboveMax); //info 
  for(ix=ixMin; ix<=ixMax; ++ix)

  { // above axis compute color and save it to the array
    iy = iyAxisOfSymmetry + iyAbove;
    Color = GiveColor(ix, iy);
    PlotPoint(ix, iy, Color ); 
    // below the axis only copy Color the same as above without computing it 
    PlotPoint(ixMax-ix, iyAxisOfSymmetry - iyAbove , Color ); 
   } 
}  
 return 0;
}

Complex numbers[edit | edit source]

Computations without explicit definition of complex numbers or without complex type are usually faster.

Parallel computing[edit | edit source]

In Mandelbrot and Julia sets each point/pixel is calculated independently, so it can be easly divided into an smaller tasks and carried out simultaneously.[11]

Types of parallel computing : Wikipedia description[12]

Vectorisation[edit | edit source]

C[edit | edit source]

c++[edit | edit source]

R[edit | edit source]

This image is made with use of vectorised code. Its creation time is very short.

Vectorisation [29][30]

Here is R code of above image with comments:[31]

## based on the R code by Jarek Tuszynski
## http://commons.wikimedia.org/wiki/File:Mandelbrot_Creation_Animation_%28800x600%29.gif

iMax=20; # number of frames and iterations

## world coordinate ( float )
cxmin = -2.2;
cxmax = 1.0;
cymax = 1.2;
cymin = -1.2;

## screen coordinate ( integer)

dx=800; dy=600 ;           # define grid size

## sequences of  values = rows and columns
cxseq = seq(cxmin, cxmax, length.out=dx);
cyseq = seq(cymin, cymax, length.out=dy);
## sequences of  values = rows * columns
cxrseq = rep(cxseq, each = dy);
cyrseq = rep(cyseq,  dx);
C = complex( real=cxrseq, imag = cyrseq); # complex vector
C = matrix(C,dy,dx)       # convert from vector to matrix
# Now C is a matrix of c points coordinates (cx,cy)

# allocate memory for all the frames
F = array(0, dim = c(dy,dx,iMax)) # dy*dx*iMax array filled with zeros

Z = 0                     # initialize Z to zero
for (i in 1:iMax)         # perform iMax iterations
{
  Z = Z^2+C               # iteration; uses n point to compute n+1 point
  
  F[,,i]  = exp(-abs(Z))              # an omitted index is used to represent an entire column or row
     # store magnitude of the complex number, scale it using an exponential function to emphasize fine detail, 
}

# color and save F array to the file 
library(caTools)          # load library with write.gif function
# mapped to color palette jetColors . Dark red is a very low number, dark blue is a very high number
jetColors = colorRampPalette(c("#00007F", "blue", "#007FFF", "cyan", "#7FFF7F", "yellow", "#FF7F00", "red", "#7F0000"))
write.gif(F, "Mandelbrot.gif", col=jetColors, delay=100, transparent=0)

Multicore[edit | edit source]

Image and src code using OpenMP and symmetry

"I always create as many worker threads as I have cores. More than that and your system spends too much time task switching" - Duncan C[32][33]

Using OpenMP :

// fill array using symmetry of image 
// uses global var :  ...
int FillArraySymmetric(unsigned char data[] )
{
   
 unsigned char Color; // gray from 0 to 255

printf("axis of symmetry \n"); 
iy = iyAxisOfSymmetry; 
#pragma omp parallel for schedule(dynamic) private(ix,Color) shared(ixMin,ixMax, iyAxisOfSymmetry)
for(ix=ixMin;ix<=ixMax;++ix) PlotPoint(ix, iy, GiveColor(ix, iy));

/*
The use of ‘shared(variable, variable2) specifies that these variables should be shared among all the threads.
The use of ‘private(variable, variable2)’ specifies that these variables should have a separate instance in each thread.
*/

#pragma omp parallel for schedule(dynamic) private(iyAbove,ix,iy,Color) shared(iyAboveMin, iyAboveMax,ixMin,ixMax, iyAxisOfSymmetry)

// above and below axis 
for(iyAbove = iyAboveMin; iyAbove<=iyAboveMax; ++iyAbove) 
  {printf(" %d from %d\n", iyAbove, iyAboveMax); //info 
  for(ix=ixMin; ix<=ixMax; ++ix)

  { // above axis compute color and save it to the array
    iy = iyAxisOfSymmetry + iyAbove;
    Color = GiveColor(ix, iy);
    PlotPoint(ix, iy, Color ); 
    // below the axis only copy Color the same as above without computing it 
    PlotPoint(ixMax-ix, iyAxisOfSymmetry - iyAbove , Color ); 
   } 
}  
 return 0;
}

Cell[edit | edit source]

"The Sony PlayStation 3 is one of the cheapest parallel computers available on the consumer market."[34][35]

GPGPU[edit | edit source]

Canvas : list of supporting browsers

Effects[edit | edit source]

Effects on 2 core CPU ( with 1 thread per core )

Method time [minutes] relative speed
CPU & no optimisation 80 1
CPU & symmetry 39 2
CPU & symmetry & OpenMP 24 3
GPU ? ( to do )

GPU should be:

  • 62 time faster than CPU [38]
  • 760 times faster [39]

Perturbation algorithm with series approximation ( 100 times faster on zoom levels around e100! ) [40]


Using WinXP Intel 2.9 GHz CPU (1 CPU used) with a GTX 480 GPU I get the following using 1000x1000 plot with 1000 max iterations :[41]

type time description
gpu 0.07s gpu is a pure CUDA solution on the GPU
gpuarray 3.4s gpuarray uses a numpy-like CUDA wrapper in Python on the GPU
numpy 43.4s numpy is a pure Numpy (C-based) solution on the CPU
python (serial) 1605.6s python is a pure Python solution on the CPU with numpy arrays

References[edit | edit source]

  1. stackoverflow question: what-is-the-fastest-way-to-calculate-e-to-2-trillion-digits
  2. The Computer Language Benchmarks Game
  3. Stack overflow meta : Is it okay to ask code optimization help?
  4. stack exchanfge : codereview
  5. Inner loop in wikipedia
  6. Adaptive super-sampling using distance estimate by Claude Heiland-Allen
  7. Distance estimation by Claude Heiland-Allen
  8. fractalforums.org: creating-super-resolution-fractal-images-really-fast-with-quad-trees
  9. wikipedia : Reflection symmetry
  10. Mirror symmetry around x axis at fraktal.republika.pl
  11. embarrassingly parallel problem
  12. wikipedia : Parallel_computers - Classes_of_parallel_computers
  13. Mandelbrot using OpenCl and Parallella
  14. wikipedia : Intel MIC
  15. parallel mandelbrot set (C Code with Message Passing Interface (MPI) library) by Omar U. Florez
  16. Guide into OpenMP: Easy multithreading programming for C++ by Joel Yliluoma
  17. Parallel Mandelbrot with OpenMP by dcfrogle
  18. claudiusmaximus  : exponential mapping and openmp
  19. Part 1: OpenCL™ – Portable Parallelism By manythreads
  20. A Mandelbrot Set on the GPU in Matlab by Loren Shure
  21. progressive-julia-fractal using webgl by Felix Woitzel
  22. glsl sandbox at heroku.com
  23. gcc  : Vector Extensions
  24. First code example using gcc vector support by bert hubert
  25. Mandelbrot calculation using SIMD by EJRH Edmund Horner
  26. The Computer Language Benchmarks Game
  27. Intel Advanced Vector Extensions
  28. ICC and Mandelbrot by Tim Horton
  29. Vectorization (parallel computing)
  30. Matlab - Code Vectorization Guide
  31. mandelbrot set in r by J.D. Long
  32. /programming/what-is-the-general-approach-to-threading-for-2d-plotting/ Fractal forums discussion : What is the general approach to threading for 2d plotting
  33. fractalforums discussion : most-powerful-computer-possible-for-a-reasonable-price/?PHPSESSID=7db41da33f499eb25ef85f46866438f2
  34. Algorithms for Specific Hardware -Mathematical Image Analysis Group Saarland University, Germany
  35. Jenks Parallel Programming Blog
  36. Programming high-performance applications on the Cell BE processor, Part 1: An introduction to Linux on the PLAYSTATION 3
  37. par4all
  38. MandelCPU vs MandelGPU Written by David Bucciarelli
  39. mathworks : llustrating-three-approaches-to-gpu-computing-the-mandelbrot-set
  40. Kalles Fraktaler 2 by Bernard Geiger
  41. Mandelbrot calculate using GPU, Serial numpy and faster numpy Use to show the speed difference between CPU and GPU calculations by Ian Ozsvald ianozsvald.com July 2010