Adaptive construction of wavelets for image compression

i I affirm, that I have created this diploma thesis on my own, with the exclusive usage of the refer-enced literature. Contents Introduction 1 1 Mathematical background 3 1. A Test images 77 iii iv CONTENTS Introduction The discrete wavelet transformation (see section 1.1 for details) has become a very popular tool to preprocess images to improve the performance of many tasks in the field of analysis and compression of signals. In this work we focus on image data and the enhancement of the compression results. Before the JPEG-2000 standard the block-wise FOURIER transformation was used for image pre-processing in the JPEG File Interchange Format (JFIF) standard of the Joint Photographic Expert Group (JPEG) for a long time. The purpose of both wavelet and FOURIER transformation is to emphasize the important details of an image and suppress those which could be disregarded. The preprocessing itself is reversible in both cases, but using a FOURIER transformation algorithm it is harder to avoid rounding errors than in case of wavelets. That is because one dimensional wavelets can generally be computed with the lifting scheme which prevents from rounding errors. [4] Once the transformation on an image is completed the compression is performed. It can be either lossless, which means that it converts to a more compact data representation only, or it can be lossy, in which case details are canceled up to a given threshold. One can verify intuitively, that the wavelet decomposition is a more natural description of images than block-wise FOURIER transform: Imagine you get a picture A and the same picture B digitized at the double resolution. When lossily compressing A and B to the same file size, the reconstructed images will differ quite much if packed with JFIF because the blocks used for the FOURIER transform have different relative sizes compared to the sizes of A and B. Thus the blocks are relatively smaller in B and the scope for detecting structures is smaller. The advantage of wavelet transformation is that since images are decomposed into informations about structures on each scale, one can easily obtain an image from B similar to A by ignoring information about small scales. That is one possibility to achieve compression in a lossy way. In this work a further attempt is started to improve the image transformation for better compression results. The properties whose optimization is considered here are 1. High correlation …