Self-Quantized Wavelet Subtrees: A Wavelet-Based Theory for Fractal Image Compression

We describe a novel adaptive wavelet-based compression scheme for images which takes advantage of image redundancy across scales. As with standard wavelet transform coders, our compressed image representation consists of a set of quantized wavelet coeecients and quantized wavelet subtrees. Instead of having a xed sub-tree codebook, however, we construct a codebook from the image being compressed. Subtrees are quantized to contracted isometries of coarser scale subtrees. This codebook drawn from the image is eeective for quantizing locally smooth regions and locally straight edges. We prove that this self-quantization enables us to recover the ne scale wavelet coeecients of an image given its coarse scale coeecients. 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 greatly simpliies the analysis of fractal compression schemes and places fractal compression in the context of existing wavelet subtree coding schemes. We obtain a simple convergence proof which strengthens existing fractal compression results, we describe a new reconstruction algorithm which requires O(N) operations for an N pixel image, and we derive an improved means of estimating the error incurred in decoding fractal compressed images. We show the eeect of self-quantization for a test image.