Self-quantized wavelet subtrees: a wavelet-based theory for fractal image compression

Abstract

We describe an adaptive wavelet-based compression scheme for images. We decompose an image into a set of quantized wavelet coefficients and quantized wavelet subtrees. The vector codebook used for quantizing the subtrees is drawn from the image. Subtrees are quantized to contracted isometries of coarser scale subtrees. This codebook drawn from the contracted image is effective for quantizing locally smooth regions and locally straight edges. We prove that this self-quantization enables us to recover the fine scale wavelet coefficients of an image given its coarse scale coefficients. We show that this self-quantization algorithm is equivalent to a fractal image compression scheme when the wavelet basis is the Haar basis. The wavelet framework places fractal compression schemes in the context of existing wavelet subtree coding schemes. We obtain a simple convergence proof which strengthens existing fractal compression results considerably, derive an improved means of estimating the error incurred in decoding fractal compressed images, and describe a new reconstruction algorithm which requires O(N) operations for an N pixel image.