A generalized class of fractal-wavelet transforms for image representation and compression

Abstract

The action of an affine fractal transform or (local) lterated Function System with gray-level Maps (IFSM) on a function f(x) induces a simple mapping on its expansion coefficients, cij, in the Haar wavelet basis. This is the basis of the discrete fractal-wavelet transform, where subtrees of the wavelet coefficient tree are scaled and copied to lower subtrees. Such transforms, which we shall also refer to as IFS on wavelet coefficients (IFSW), were introduced into image processing with other (compactly supported) wavelet basis sets in an attempt to remove the blocking artifacts that plague standard IFS block-encoding algorithms. In this paper a set of generalized 2-D fractal-wavelet transforms is introduced. Their primary difference from usual IFSW transforms lies in treating “horizontal,” “vertical” and “diagonal” quadtrees independently. This approach may seem expensive in terms of coding. However, the added flexibility provided by this method, resulting in a marked improvement in accuracy and low degradation with respect to quantization, makes it quite tractable for image compression. As in the one-dimensional case, the IFSW transforms are equivalent to recurrent IFSM with condensation functions. The net result of an affine IFSW is an extrapolation of high-frequency wavelet coefficients which grow or decay geometrically, according to the magnitudes of fractal scaling parameters α <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ij</inf> . This provides a connection between the α <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ij</inf> and the regularity/irregularity properties of regions of the image. IFSW extrapolation also makes possible “fractal zooming.” The results of computations, including some simple compression methods, are also presented.