Studies On Subdivision Schemes And Wavelets Based On Generating Functions

Subdivision method is an important geometric design method in computer aided geo-metric design. In the past forty years, studies on the subdivision method have entered into a mature phase. Studying subdivision schemes from a unified framework is an important problem both in subdivision theories and applications. By using the generating functions of subdivision method, we studies the explicit formula of generating function, the relationship between constructing interpolatory subdivisions and subdivision wavelets from approximat-ing subdivisions, and generalized pseudo-Butterworth splines. The main work is shown as follows:1. We propose the explicit formula of a class of generating functions. Based on the idea of Lane-Riesenfeld algorithm, we propose the explicit formula of generating functions cor-responding to one strategy of constructing new interpolatory schemes from approximating schemes. The proposed explicit formula of generating function can also correspond to an approximating scheme by replacing the initial subsymbol with a specific subsymbol, and the formula still has significant geometric illustrations. We analyze the zero condition and poly-nomial reproduction properties of the explicit formula. We find that the zero condition order of the explicit formula is not only related to the zero condition order of the approximating scheme, but also related to the order of polynomial reproduction. Moreover, we introduce variant forms of the explicit formula, and construct some new subdivision schemes.2. We investigate the theoretical relationship between the constructions of interpola-tory subdivisions and subdivision wavelets from approximating subdivisions. The product of the generating function of an approximating scheme and a specific generating function can correspond to an interpolatory scheme. An affine combination plays a central role in the constructions of subdivision wavelets. We show that the coefficients of the specific generating function are exactly the affine combination parameters. The affine combina-tion parameters can character the existence of interpolatory subdivisions and subdivision wavelets. If the affine combination parameter vector satisfies the invertible conditions, then the existence of interpolatory subdivisions is equivalent to that of subdivision wavelets. The affine combination also makes it possible for us to apply the results of subdivision wavelets to interpolatory subdivisions, and vice versa. We use several examples to demonstrate the impacts of this relationship.3. We propose the generalized pseudo-Butterworth refinable functions. We use the trigonometric forms of the generating functions. Since the generalized pseudo-Butterworth refinable functions have more parameters than the pseudo-Butterworth refinable functions, they involve more refinable functions. They include pseudo-splines of type Ⅰ and Ⅱ, dual pseudo-splines, pseudo-Butterworth refinable functions, and almost all symmetric and causal fractional B-splines. Furthermore, based on the theory of Holder continuous generating func-tions, we prove the convergence of cascade algorithms associated with the new generating functions, and construct the Riesz wavelet bases in L2(R). We also analyze the regularity of the generalized pseudo-Butterworth refinable functions.