Generalized Parametric Slant Transforms With Applications In Image Compression

Different orthogonal transforms (Fourier, DCT, Walsh, Haar, etc.) and signal transforms relative to them are used widely in the control and communication theory, digital signal and image processing, and analysis, synthsis and testing of digital circuits. Orthogonal transforms provide the foundation for the large-scale scientific computation, the growing significance of their applications has been marked in different areas for many years. They can be divided into sinusoidal transforms and nonsinusoidal transforms. Walsh transforms and Haar transforms are the most important members among nonsinusoidal transforms.Digital images exhibit a phenomenon characteristic of having approximately constant or uniformly changing gray levels over a considerable distance or area. The slant transforms and parametric Slant transforms (based on the Hadamard-Walsh transform) are specifically defined for the efficient representation of such images. Now a new class of generalized parametric Slant Transforms (GPST) has been constructed, which extends the classical Walsh transforms of order 2n to order k n, where n, k are arbitrary positive integers. In this paper, we introduce the basic knowledge of GPST, and design fast algorithms from different views, and apply them in image compression. In the numerical experiments, we show that the compression effect can be optimized by choosing the proper parameters in GPST and using the transform of proper order, and get the range of the best parameters'value. We also show that GPST is superior to DCT for compression of the test images in the aspects of running time and compression ratio, but inferior to DCT in the value of PSNR (Peak of Signal-Noise Ratio). To minimize the block effect of GPST, we design a new form of morphological wavelet called GPST morphological wavelet, which turns out to have better compression effect compared to GPST and the classical Haar morphological wavelet.