Fractals/Iterations in the complex plane/tsa

Algorithm for computing images of polynomial Julia sets with mathemathical guarantees against sampling artifacts and rounding errors in floating points arithmethic.

name

• True Shape Algorithm ( TSA) 

theory

algorithm computes a decomposition of the complex plane into three regions:

• a white region, which is contained in exterior of Julia set
• a black region, which is contained in the interior of Julia set
• a gray region, which cannot be classified as white or gray. It is a region which contains julia set J

It gives adaptive approximation of the Julia set. Spatial resolution limited by available memory.

K is contained in the escape circle of radius R = max(|c|, 2) centered at the origin,

Image that is 100% guaranteed correct:

• white pixels are guaranteed to contain no interior or boundary points
• black pixels are guaranteed to contain only interior points
• red pixels have not been able to be decided at the current resolution (they might contain both interior or boindary and exterior points).

Unfortunately in practice the images have a lot of red pixels. It means that this algorithm is for proving that images are mathemathically correct. Point-sampling method is for creating images. If one sees that image is not smooth ( boundaries of components are smooth curves) then takes a bigger resolution ( subpixel accuracy) to get proper = smooth image. If one take good point sampled image, then TSA algorithm will not find a wrong pixel in it.

key words

• complex plane
  • region = union of the cell with the same label
  • cell = rectangle in the complex plane
• tree
  • quadtree
• graph
  • directed graph
  • cell graph that represents the cell mapping
• IA = Interval Arithmetic
• dyadic fraction : working with exactly double-representable numbers by using a near dyadic fraction

algorithms

• goal of the algorithm is mathematical correctness of the image
• the price is slow and not beatiful images. TSA is very slow and not suitable for creating images with aesthetic appearance.
• algorithm is designed for for Julia sets

algorithm avoids :

• point sampling ( 1-pixel aproximation): what happens between samples ?
• function iteration in the escape-time algorithm ( do not use it)
• Floating-point rounding errors ( squaring needs double digits )
• partial orbits ( program cannot run forever)

Main features of the algorithm:

• "cell mapping to reliably classify entire rectangles " in the complex plane, not just a finite sample of points
• "it handles orbits by using color propagation in graphs induced by cell mapping" = "tracks the fate of complex orbit by inspecting the directed graph induced by cell mapping"
• "The numbers processed by the algorithm are dyadic fractions that are restricted in range and precision and the algorithm uses error-free fixed-point arithmetic whose precision depends only on the spatial resolution of the image. "

decomposition

a quadtree decomposition of the complex plane

refine

adaptive refinement = Subdivide each gray leaf cell into four new gray subcells

cell mapping

• Simple Cell Mapping (SCM)
• Generalized Cell Mapping (GCM)

label propagation

• label propagation in graph

code

• C++ code by Marc Meidlinger, July-October 2019
• Haskell code by Claude
• CMlib is a library of cell mapping algorithms and utility functions and classes written in c++ by Gergely Gyebroszki

People

• Luiz Henrique de Figueiredo

Papers

• "Images of Julia sets that you can trust" by Luiz-Henrique de Figueiredo, Diego Nehab, Jofge Stolfi, Joao Batista Oliveira- 2013
• "Rigorous bounds for polynomial Julia sets" by Luiz-Henrique de Figueiredo, Diego Nehab, Jofge Stolfi, Joao Batista Oliveira

Video

• First Palis-Balzan Symposium on Dynamical Systems L. H. FIGUEIREDO - Images of Julia sets that you can trust

references

• fractalforums : trustworthy-2d3d-julia-set-source-code
• Trustworthy fractals by Claude Heiland-Allen
• fractalforums.org : julia-sets-true-shape-and-escape-time