Some Research On Subdivision Schemes, Interpolation And Wavelet

This paper mainly studies a type of subdivision scheme that defined on a bounded interval and the matching scale function and wavelet. Deslauriers and Dubuc brought forward a 2-band subdivision scheme, which is named Deslauriers-Dubuc subdivision. This paper advances this scheme to 3-band. The idea behind the scheme is that: if the original control points fall on an odd-degree polynomial, then the newly generated control points must also lie on the same polynomial. Firstly, we make use of Langrange fundamental polynomial to structure the interpolatory subdivision scheme that is symmetric. In theorem4.3, we show that this scheme is convergent. Then we define a multiscale finite sequence of functions on a bounded interval, which are then proved to be refinable. Using this fact, in theorem4.6, the result adapts interpolatory subdivision scheme for finite sequences that is convergent. Corresponding interpolation wavelets on an interval are defined, and explicit formulations of the resulting decomposition and reconstruction algorithms are calculated.This paper is organized as follows:In chapter 1, the history background, the developing progress and the basic ideas of subdivision schemes as well as the improvement of this paper are introduced. The basic idea of subdivision scheme is to get smooth surface modeling from coarse and simple surface modeling through adding new vertices.In chapter 2, the paper shows the basic conception of subdivision scheme. For unvaried schemes are widely used, the method of analyzing the convergence and smoothness of unvaried schemes is shown.In chapter 3, using the connection between wavelets and subdivision schemes, the paper build up the basic wavelet by subdivision schemes.In chapter 4, the paper studies a type of 3-band subdivision scheme.