| In this thesis, we study subdivision schemes, biorthogonal wavelets and wavelet-based image compression. Subdivision schemes are important in computer aided geometric design to generate curves and high dimensional surfaces. First, we shall characterize the Lp convergence of any subdivision scheme with a finitely supported refinement mask in multivariate case. Then we shall study the error behaviour of any subdivision scheme if there is a round-off of its refinement mask. Next, we study a special kind of subdivision schemes—interpolatory subdivision schemes. We shall analyze the optimal properties, such as sum rules of an interpolatory refinement mask and the smoothness of its associated refinable function, of any interpolatory subdivision scheme. A general construction of optimal interpolatory subdivision schemes is presented. Next, we shall study biorthogonal multivariate wavelets since there is a well known close relation between interpolatory subdivision schemes and biorthogonal wavelets. We shall study the optimal approximation and smoothness properties of any biorthogonal wavelet. More importantly, a general and easy way (CBC algorithm) is presented to construct multivariate biorthogonal wavelets. As an example, a family of optimal bivariate biorthogonal wavelets is given. Finally, we try to apply our results in image compression and a 2-D wavelet transform C++ program is established which can use a library of bivariate biorthogonal wavelet filters. |
