An Approach to Integer Wavelet Transformations for Lossless Image Compression

We developed a general way to create integer wavelet transformations that can be used in lossless (reversible) compression of images with arbitrary size. The method, based on some updating techniques such as lifting and correction, allows us to generate a series of reversible integer transformations which have the similar features with the corresponding biorthogonal wavelet transforms and some non-orthogonal wavelet transforms ; but need only be calculated with integer addition and bit-shift operations. In addition, the integer wavelet transforms created in this paper possess a property of precision preservation (PPP). This property is very useful, in lossless compression, for conserving memory in both compression and decompression, and speeding up the computational process. x1. Introduction The wavelet transform has been proven to be one of the most powerful tools in the eld of image compression. In general, wavelet based image compression can be divided into three steps: wavelet transform, quantization and entropy coding. Theoretically, the wavelet transformation part is considered lossless since the transformation is reversible in the sense of mathematics. However, most transformations are lossy in practice, because all computers have only nite precision, even if we use oating point calculations. This fact is a limitation for lossless image compression by wavelet transform based algorithms. We are going to develop a general way to create reversible integer transforms, especially wavelets. All of wavelet transforms created here possess the property of precision preservation (PPP), which we will explain later in this paper. There are several reasons for creating this kind of reversible wavelets: The rst reason is, of course, for the purpose of lossless image compression. Lossless image compression is very important for images found in such applications as medical and space science. In such situations, the designers of the compression algorithm always try to compress the information as much as possible; and they also must be very careful to avoid discarding any information that may be required or even useful at some later point. Some of lossless image compression algorithms are based on the so called predictive transforms 5, 8-9], which are widely used in practice. However, this method