The essence of fractal image denoising is to predict the fractal code of a noiseless image from its noisy observation. From the predicted fractal code, one can generate an estimate of the original image. We show how well fractal-wavelet denoising predicts parent wavelet subtress of the noiseless image. The performance of various fractal-wavelet denoising schemes (e.g., fixed partitioning, quadtree partitioning) is compared to that of some standard wavelet thresholding methods. We also examine the use of cycle spinning in fractal-based image denoising for the purpose enhancing the denoised estimates. Our experimental results show that these fractal-based image denoising methods are quite competitive with standard wavelet thresholding methods for image denoising. Finally, we compare the performance of the pixel- and wavelet-based fractal denoising schemes.

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.

We propose a simple yet effective fractal-wavelet scheme for edge-preserving smoothing of noisy images. Over the past decade, there has been significant interest in fractal coding for the purpose of image compression. Fractal-wavelet transforms were introduced in an effort to reduce the blockiness and computational complexity that are inherent in fractal image compression. Applications of fractal-based coding to other aspects of image processing, however, have received little attention. The authors proposed a simple yet effective fractal-based image denoising scheme that is applied in the spatial domain of the image. We extend the application of this fractal denoising scheme to the wavelet domain of the image. We find that when the wavelet transform of 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 wavelet transform of the original noise-free image from that of the wavelet transform of the noisy image. The use of the quadtree partitioning scheme for the purpose of fractal-wavelet coding results in 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. The main advantage of the wavelet-based fractal denoising scheme over the standard fractal denoising scheme is that it is computationally less expensive.