Fractal image denoising

Abstract

Over the past decade, there has been significant interest in fractal coding for the purpose of image compression. However, applications of fractal-based coding to other aspects of image processing have received little attention. We propose a fractal-based method to enhance and restore a noisy image. If the noisy image is simply fractally coded, a significant amount of the noise is suppressed. However, one can go a step further and estimate the fractal code of the original noise-free image from that of the noisy image, based upon a knowledge (or estimate) of the variance of the noise, assumed to be zero-mean, stationary and Gaussian. The resulting fractal code yields a significantly enhanced and restored representation of the original noisy image. The enhancement is consistent with the human visual system where extra smoothing is performed in flat and low activity regions and a lower degree of smoothing is performed near high frequency components, e.g., edges, of the image. We find that, for significant noise variance (sigma > or = 20), the fractal-based scheme yields results that are generally better than those obtained by the Lee filter which uses a localized first order filtering process similar to fractal schemes. We also show that the Lee filter and the fractal method are closely related.

Continuous Evolution of Fractal Transforms and Nonlocal PDE Imaging

“…Assuming that the noise is additive, stationary and of zero mean, the spatial contraction/decimation that comprises the mapping of domain to range blocks contributes to the reduction of the variance of the noise. (We have shown [11] that the degree of denoising can be increased by improving the fractal code of the "denoised" attractor.) Alexander's scheme, however, capitalizes on the fact, mentioned earlier, that there are generally several domain blocks that match a given range block almost as well as the "optimal" block, i.e., the block yielding the lowest collage error.…”

Fractal Image Coding as Projections Onto Convex Sets

“…More details on fractal image coding can be found in many places [1,3,7,11], including one of our recent papers on fractal-based denoising [9].…”

“…If a noisy image is fractally coded, with little or no regard for compression, then the attractor produced by the fractal code often represents a significantly denoised version of the original image. (In fact, one can go some steps further and improve this procedure -see [9].) In such fractal image denoising algorithms, however, one performs the usual iteration procedure after the fractal code is obtained.…”

Analysis of Multi Resolution Image Denoising Scheme using Fractal Transform

“…For example, in medical image processing, artifacts can bring mistake in diagnosis. These artifacts can be reduced by nonlinear diffusion [12]. Denoising by combining wavelet transform and anisotropic diffusion to reduce these artifacts has been explored before and shown to give positive results [13].…”