[Paper Review] Fractal Transformations in 2 and 3 Dimensions

Michael F. Barnsley, Brendan Harding|arXiv (Cornell University)|Aug 30, 2013

Mathematical Dynamics and Fractals3 references3 citations

TL;DR

This paper extends fractal transformations—previously applied in 2D—to 3D by adapting algorithms for masked iterated function systems (IFS) and demonstrating their use on voxel data and surface meshes. It introduces a method to generate continuous, homeomorphic mappings between 3D attractors using affine IFS on the unit cube, enabling realistic special effects and modeling in computer graphics.

ABSTRACT

We present some work relating to fractal transformations on masked iterated function systems and demonstrate how well known algorithms for generating fractal transformations can be modifed for these systems. We also demonstrate that these algorithms work equally well when applied to three dimensional data sets and suggest some possible applications to special effects and modelling.

Motivation & Objective

• To generalize fractal transformations to three-dimensional data sets using masked iterated function systems (IFS).
• To adapt existing 2D fractal transformation algorithms—such as the chaos game and pixel chaining—for 3D applications.
• To ensure continuity and homeomorphism in 3D fractal mappings through proper IFS construction and triangulation.
• To demonstrate practical applications in 3D modeling, special effects, and data transformation using voxel and mesh data.
• To provide a foundation for future development of real-time, parallelized, and interactive 3D fractal transformation systems.

Proposed method

• Uses a point-fibred IFS on a compact Hausdorff space to define attractors and coding maps, ensuring unique fixed-point attractors.
• Applies the chaos game and pixel chaining algorithms to generate 3D attractor sets, with modifications for masked IFS and higher-dimensional data.
• Employs affine transformations of the form $ f_i(x,y,z) = (s_1 x t_{i,1} + ((1-s_1)x + s_1)(1-t_{i,1}), ext{ and similar for } y,z) $ to define IFS on the unit cube.
• Requires mesh data to be triangulated into triangles to preserve planarity after transformation, avoiding non-planar distortions.
• Uses Drishti and OpenGL for 3D visualization of transformed voxel and mesh data, enabling pre- and post-transformation inspection.
• Constructs fractal homeomorphisms by mapping data embedded in one attractor (unit cube) to another via different parameter sets $\tilde{s} \in (0,1)^3$.

Experimental results

Research questions

• RQ1 Can fractal transformations, previously limited to 2D, be effectively extended to 3D data sets such as voxels and surface meshes?
• RQ2 What modifications to existing IFS algorithms (e.g., chaos game, pixel chaining) are required to ensure efficiency and accuracy in 3D?
• RQ3 Under what conditions do fractal transformations in 3D remain continuous or homeomorphic, particularly when applied to complex 3D structures?
• RQ4 How can 3D mesh data be preprocessed (e.g., triangulation) to preserve geometric integrity during fractal transformations?
• RQ5 What are the practical applications of 3D fractal transformations in modeling, special effects, and image synthesis?

Key findings

• Fractal transformations can be successfully extended to 3D using masked IFS and affine mappings on the unit cube, producing continuous and homeomorphic mappings.
• The proposed method enables the transformation of 3D voxel data and surface meshes using a family of 8 affine functions, with the unit cube as the unique attractor.
• Triangulation of 3D meshes is essential to preserve planarity after transformation, as fractal mappings may otherwise distort polygonal planes.
• Transformations of a solid sphere and a gourd mesh demonstrate realistic deformation effects, including internal structural changes and crumpling.
• Cross-sectional views of transformed spheres reveal that concentric layers are mapped non-uniformly, preserving internal structure while introducing complexity.
• The absence of ready-made software for 3D fractal transformations suggests a need for custom implementation, though high-level math software or custom code can be used effectively.