Dyadic Biorthogonal Wavelets Construction Via Lifting Scheme And Its Application In Image Processing

Wavelets Analysis is an interesting mathematical theory which was proposed in the middle of 1980's. As an important theory in applied mathematical and a versatile tool in signal processing, wavelets analysis attracts many researchers'attention from both two fields. It is known that such properties as short support length, high order vanishing moments, linear phase, high regularity are desirable and critical properties in signal processing. Consequently, how to construct wavelets owning these properties simultaneously is a hot topic.Traditional wavelets construction techniques are based on Fourier analysis. It is proved that any useful wavelet can be constructed by these techniques. However, these techniques rely on complicated mathematical derivations. It is difficult to understand and to generalize for researchers who do not have a strong background in Fourier analysis. The Lifting Scheme proposed by Sweldens does not depend on Fourier analysis, Compare with traditional techniques, Lifting scheme has fixed construction formulas, and it is a very natural way to understand the construction of wavelets, which also have advantages such as generality, flexibility and efficient implementation.In this thesis, we investigate the relationship between lifting scheme and properties of initial filterbanks and generated filterbanks systematically. Consequently, a general algorithm framework of constructing wavelets filterbank with short support length, high order vanishing moments, linear phase, high regularity simultaneously, is proposed. On the other hand, the choices of initial filterbanks, and the choices of lifting strategies, and how these two notions affect the properties of generated filterbanks, is discussed in detail. Furthermore, the notion of Shortest Lifting Scheme is developed for the first time.Numerous examples can be given. Some famous dyadic wavelets, such as Biorthogonal Spline Wavelet series, Deslauriers-Dubuc biorthogonal wavelets series, are constructed in this algorithm framework. Furthermore, two classes biorthogonal wavelets with high performance—Shortest Lifting Scheme Wavelet (SLSW) series and the High Regularity Symmetric Biorthogonal Wavelets (HRSBW) series—are firstly introduced, both of which have dyadic fractions coefficients.It is proved that some wavelet based image processing applications benefit from the destined construction. Experiments of applying the newly constructed wavelets to image compression and denoising have shown that several odd-ordered SLSW outperform some popular wavelets in both MSE and PSNR.