Continuous Evolution of Fractal Transforms and Nonlocal PDE Imaging

Abstract

Traditional fractal image coding seeks to approximate an image function u as a union of spatially-contracted and greyscale-modified copies of itself, i.e., u ≈Tu, where T is a contractive fractal transform operator on an appropriate space of functions. Consequently u is well approximated by $\bar u$, the unique fixed point of T, which can then be constructed by the discrete iteration procedure u _n + 1 = T _n .

In a previous work, we showed that the evolution equation y _t = Oy – y produces a continuous evolution y(x,t) to $\bar y$, the fixed point of a contractive operator O. This method was applied to the discrete fractal transform operator, in which case the evolution equation takes the form of a nonlocal differential equation under which regions of the image are modified according to information from other regions.

In this paper we extend the scope of this evolution equation by introducing additional operators, e.g., diffusion or curvature operators, that “compete” with the fractal transform operator. As a result, the asymptotic limiting function y _∞ is a modification of the fixed point $\bar u$ of the original fractal transform. The modification can be viewed as a replacement of traditional postprocessing methods that are employed to “touch up” the attractor function $\bar{u}$.