【文章內(nèi)容簡介】 tion, signals, time series, movies, color images, etc. Thus, applications of the wavelet idea include big parts of signal and image processing, data pression, fingerprint encoding, and many other fields of science and engineering. This thesis focuses on the processing of color images with the use of custom designed wavelet algorithms, and mathematical threshold filters. Although there have been a number of recent papers on the operator theory of wavelets, there is a need for a tutorial which explains some applied tends from scratch to operator theorists. Wavelets as a subject is highly interdisciplinary and it draws in crucial ways on ideas from the outside world. We aim to outline various connections between Hilbert space geometry and image processing. Thus, we hope to help students and researchers from one area understand what is going on in the other. One difficulty with municating across areas is a vast difference in lingo,jargon, and mathematical terminology. With handson experiments, our paper is meant to help create a better understanding of links between the two sides, math and images. It is a delicate balance deciding what to include. In choosing, we had in mind students in operator theory,stressing explanations that are not easy to find in the journal literature. Our paper results extend what was previously known, and we hope yields new insight into scaling and of representation of color images。 especially, we have aimed for better algorithms. The paper concludes with a set of puter generated images which serve to illustrate our ideas and our algorithms, and also with the resulting pressed images.. Overview. Wavelet Image Processing enables puters to store an image in many scales of resolutions, thus deposing an image into various levels and types of details and approximation with different valued resolutions. Hence, making it possible to zoom in to obtain more detail of the trees, leaves and even a monkey on top of the tree. Wavelets allow one to press the image using less storage space with more details of the advantage of deposing images to approximate and detail parts as in is that it enables to isolate and manipulate the data with specific properties. With this, it is possible to determine whether to preserve more specific details. For instance, keeping more vertical detail instead of keeping all the horizontal, diagonal and vertical details of an image that has more vertical aspects. This would allowthe image to lose a certain amount of horizontal and diagonal details, but would not affect the image in human perception. As mathematically illustrated in , an image can be deposed into approximate, horizontal, vertical and diagonal details. N levels of deposition is done. After that, quantization is done on the deposed image where different quantization maybe done on different ponents thus maximizing the amount of needed details and ignoring ‘notsowanted’ details. This is done by thresholding where some coefficient values for pixels in images are ‘thrown out’ or set to zero or some ‘smoothing’ effect is done on the image matrix. This process is used in JPEG2000.. Motivation. In many papers and books, the topics in wavelets and image processing are discussed in mostly in one extreme, namely in terms of engineering aspects of it or wavelets are discussed in terms operators without being specifically mentioned how it is being used in its application in engineering. In this paper, the author adds onto [Sko01], [Use01] and [Vet01] more insights about mathematical properties such as properties from Operator Theory, Functional Analysis, etc. of wavelets playing a major role in results in wavelet image pression. Our paper aims in establishing if not already established or improve the connection between the mathematical aspects of wavelets and its application in image processing. Also,our paper discuss on how the images are implemented with puter program,and how wavelet deposition is done on the digital images in terms of puter program, and in terms of mathematics, in the hope that the munication between mathematics and engineering will improve, thus will bring greater benefits to mathematicians and engineers.2 Wavelet Color Image Compression The whole process of wavelet image pression is performed as follows: An input image is taken by the puter, forward wavelet transform is performed on the digital image, thresholding is done on the digital image, entropy coding is done on the image where necessary, thus the pression of image is done on the puter. Then with the pressed image, reconstruction of wavelet transformed image is done, then inverse wavelet transform is performed on the image, thus image is reconstructed. In some cases, zerotree algorithm [Sha93] is used and it is known to have better pression with zerotree algorithm but it was not implemented here. Forward Wavelet Transform. Various wavelet transforms are used in this step. Namely, Daubechies wavelets, Coiflets, biorthogonal wavelets, and Symlets. These various transforms are implemented to observe how various mathematical properties such as symmetry, number of vanishing moments and orthogonality differ the result of pressed image. Advantages short support is that it preserves locality. The Daubechies wavelets used are orthogonal, so do Coiflets. Symlets have the property of being close to symmetric. The biorthogonal wavelets are not orthogonal but not having to be orthogonal gives more options to a variety of filters such as symmetric filters thus allowing them to possess the symmetric property. MATLAB has a subroutine called wavedec2 which performs the deposition of the image for you up to the given desired level (N) with the given desired wavelet(wname). Since there are three ponents to deal with, the wavelet transform was applied ponentwise. “wavedec” is a twodimensional wavelet analysis function. [C,S] = wavedec2(X,N,‘wname’) returns the wavelet depo