M.Sc Thesis

M.Sc StudentGruzd Marina
SubjectRegularity of Spline-Based Wavelets and their Application
to Image Compression
DepartmentDepartment of Applied Mathematics
Supervisors PROFESSOR EMERITUS Yuri Brudnyi
DR. Valery Zheludev


The thesis is devoted to an investigation of properties of the waveforms generated by a new family of spline-based biorthogonal wavelet transforms, recently developed by A.Averbuch and V.Zheludev, and to applications of the transforms to image compression. In our experiments we used the newly constructed transforms and compared their performance with such one of

the Daubechies's  9/7 transform, accepted in the JPEG 2000 image compression standard. To complete a compression process we used the SPIHT coding algorithm with different compression ratios. The experiments demonstrated that these transforms are competitive with the 9/7 transform in the performance as well as in the computational complexity.

     The main difference from the well known wavelet transforms is that new ones use IIR filters. The produced wavelets are not compactly supported, but the scaling functions and wavelets generated by these transforms decay exponentially. The rational structure of the transfer functions enables us to implement transforms via fast recursive filtering, and thus their computational complexity is not greater than the implementation complexity of the conventional wavelet transforms that use FIR filters.

     In our work we investigated the scaling functions generated by new transforms. All spline-based filters of these transforms are associated with the interpolatory subdivision schemes. Therefore methods of the analysis of convergence and of regularity of limit functions, which are developed in the theory of subdivision schemes can be applied to the analysis of scaling functions.

Our analysis of convergence and regularity of the presented subdivision schemes is based on a technique developed by N.Dyn, J.Gregony and D.Levin with some modifications. Unlike the existing subdivision schemes, we study the schemes with infinite but exponentially decaying masks. We established the convergence of the subdivision schemes, which correspond to the spline-based wavelet transforms, and analyzed the regularity of their basic limit functions, which are the scaling functions of the wavelet transforms.