Wavelets (Mathematics)

This dissertation is concerned with the development of a bandwidth extrapolation technique that performs maximum entropy estimations over wavelet subspaces. Bandwidth extrapolation techniques have been used in radar applications to improve range and cross range resolution of radar cross section (RCS) images. Comparisons are made of the performance of conventional maximum entropy estimation to maximum entropy estimation over wavelet subspaces. A least squares prediction error measure is used to compare original measured RCS data to extrapolated data. Then a relative error is defined as the ratio of prediction error using conventional maximum entropy to prediction error using maximum entropy over wavelet subspaces. Application of the bandwidth extrapolation technique is to measured RCS data of two objects. The first object consists of two 3/8" diameter conducting spheres placed 4" apart. Measurements used are for vertical polarization and 0 degree aspect angle covering a frequency range of 8.0 to 12.3827 GHz. The second object is a 1.6 meter aluminum cone. Measurements used are for vertical polarization and 0 degree aspect angle (nose on) covering a frequency range of 4.64 to 18.00 GHz. Results are shown for extrapolate measured data plus the original data with Gaussian white noise added to noise ratios of 25 dB, 20 dB, 15 dB, and 10 dB.

A measure of the potential of a receiver for detection is detectability. Detectability is a function of the signal and noise, and given any one of them the detectability is fixed. In addition, complete transforms of the signal and noise cannot change detectability. Throughout this work we show that "Subspace methods" as defined here can improve detectability in specific subspaces, resulting in improved Receiver Operating Curves (ROC) and thus better detection in arbitrary noise environments. Our method is tested and verified on various signals and noises, both simulated and real. The optimum detection of signals in noise requires the computation of noise eigenvalues and vectors (EVD). This process neither is a trivial one nor is it computationally cheap, especially for non-stationary noise and can result in numerical instabilities when the covariance matrix is large. This work addresses this problem and provides solutions that take advantage of the subspace structure through plane rotations to improve on existing algorithms for EVD by improving their convergence rate and reducing their computational expense for given thresholds.

This work is an attempt of incorporating the latest advances in vision research and signal processing into the field of image coding. The scope of the dissertation is twofold. Firstly, it sets up a framework of the wavelet color image coder and makes optimizations of its performance. Secondly, it investigates the human vision models and implements human visual properties into the wavelet color image coder. A wavelet image coding framework consisting of image decomposition, coefficients quantization, data representation, and entropy coding is first set up, and then a couple of unsolved issues of wavelet image coding are studied and the consequent optimization schemes are presented and applied to the basic framework. These issues include the best wavelet bases selection, quantizer optimization, adaptive probability estimation in arithmetic coding, and the explicit transmission of significant map of wavelet data. Based on the established wavelet image coding framework, a human visual system (HVS) based adaptive color image coding scheme is proposed. Compared with the non-HVS-based coding methods, our method results in a superior performance without any cost of additional side information. As the rudiments of the proposed HVS-based coding scheme, the visual properties of the early stage of human vision are investigated first, especially the contrast sensitivity, the luminance adaptation, and the complicated simultaneous masking and crossed masking effects. To implement these visual properties into the wavelet image coding, the suitable estimation of local background luminance and contrast in the wavelet domain is also re-investigated. Based upon these prerequisite works, the effects of contrast sensitivity weighting and luminance adaptation are incorporated into our coding scheme. Furthermore, the mechanisms of all kinds of masking effects in color image, e.g., the self-masking, the neighbor masking, the crossbands masking, and the luminance-chrominance crossed-masking, are also studied and properly utilized into the coding scheme through an adaptive quantization scheme. Owing to elaborate arrangement and integration of the different parts of the perception based quantization scheme, the coefficient-dependent adaptive quantization step size can be losslessly restored during the decoding without any overhead of side information.