The geometry of fractal sets
Fractals everywhere
Scale-Space and Edge Detection Using Anisotropic Diffusion
IEEE Transactions on Pattern Analysis and Machine Intelligence
Fractal image compression
Nonlinear total variation based noise removal algorithms
Proceedings of the eleventh annual international conference of the Center for Nonlinear Studies on Experimental mathematics : computational issues in nonlinear science: computational issues in nonlinear science
Geometric partial differential equations and image analysis
Geometric partial differential equations and image analysis
Fractal Imaging
A Non-Local Algorithm for Image Denoising
CVPR '05 Proceedings of the 2005 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR'05) - Volume 2 - Volume 02
IEEE Transactions on Image Processing
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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 un+1 = Tn. In a previous work, we showed that the evolution equation yt = 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 $\bar{u}$.