Study On Nonlinear Wavelet Transform, Multiresolution And Their Applications In Image And Signal Processing

Wavelet analysis and applications are a novel research field in the world, in particular, which are a rapidly developing subject in signal processing. In the pass of study of the past decade years, important mathematical basis and fundamental theory frames have been established. With increasing perfection of basic theory and widely deepening of applications, nonlinear wavelet methods are developing hotspots and difficulties at present. As an important type of wavelet methods, nonlinear wavelet methods have been successfully applied in some fields, but there are many basic theory and potential powers of applications, which need to be perfected and exploited. It is under this background that the dissertation deeply studies the theory and applications of nonlinear wavelet methods. The main results include:When dealing with nonlinear filtering algorithms for images denoising problems, there are two crucial aspects, namely, the choice of the thresholding parameter λ and the use of a proper filter function. Both greatly influence the quality of the resulting denoised image. The third chapter proposes two new filters, which are a piecewise degree n and an exponential function of λ, respectively, and we prove how they can be successfully used instead of the classical Donoho and Johnstone's Soft thresholding filter. We exploit the increased regularity and flexibility of the new filter to improve the quality of the images. Moreover, we prove that our filtered approximation is a near-minimizer of the functional which has to be minimized to solve the denoising problem. We also show that the piecewise degree n filter yield good results if we choose λ as the Donoho and Johnstone universal threshold and the limit value of the degree n filter is an ideal lowpass filter, while the exponential one is more suitable if we use the recently proposed H-curve criterion.The fourth chapter discusses in detail a novel method for edge feature detection of images based on wavelet decomposition and reconstruction by means of interval biorthogonal wavelet. By applying the wavelet decomposition technique, an image becomes a wavelet representation, i.e. the image is decomposed into a set of wavelet approximation coefficients and wavelet detail coefficients. Discarding wavelet approximation, the edge extraction is implemented by means of the wavelet reconstruction technique.The fifth chapter develops a lifting scheme based on the recursive filtering of thesignals. The lifting scheme we propose operates with IIR filters contrary to the conventional lifting scheme. The lifting construction exploits predictor and update operator design which is based on the interpolatory discrete splines in a spatial domain. The proposed scheme is based on interpolation and, as such, it involves only samples of signals and it does not require any use of quadrature formulas and is more suitable for signal processing. In addition, we established explicit formulas for the construction of wavelets with arbitrary number of vanishing moments.The sixth chapter we first discuss design of a bank of four tight frame FIR filters taking on the form Hi(z) = H0(-z) and h2(n)= hx{m-\-n) using spectral factorization, and then design symmetric tight frame filters by means of resulting filters. These filters are smoother than those resulting from orthogonal filters of comparable length. In addition, the filters discussed in this chapter exhibit a degree of near orthogonality. The scaling function of tight frame allows for minimal length and a improved smoothness, in comparison with orthonormal wavelets bases, while the wavelets obtained are nearly shift invariant.