Wavelet and fractal transforms for image compression

Fractal coding is one of the promising techniques for image compression. Its main advantage is to present a good reconstructed image quality even at very low bit rates, i.e., smaller than 0.1 bit per pixel, where the JPEG standard gives very poor results. This is, however, a lossy technique that introduces distortions in the reconstructed images. In particular, the image high frequency components are poorly coded. This may be visually annoying, with effects such as globally blurred images, artifacts at edges, loss of precision in texture rendering and detail coding. Wavelet transform is an efficient way to perform multiresolution signal decomposition thanks to some very interesting properties of the wavelet functions such as good time and frequency resolution and a simple generation of families of functions to generate bases of the space of square summable sequences. Indeed, the basis functions used in the wavelet transform are all affine transformed versions of an original function. The compactly supported wavelets are defined from a scaling function that is the solution of a fractal-like equation. These remarks suggest that relationships between fractal coding and wavelet transform are worth investigating. The theoretical background about the general fractal coding scheme and the way wavelet transform may be incorporated to it are outlined. As a direct application, a compression algorithm combining wavelet and fractal transforms is presented, numerical results given, and further investigations are suggested