The Computation of Spline Wavelets

• Main Authors: Jang, Jyh-Cheng, 鄭智成
• Other Authors: Hwang Chyi
• Format: Others
• Language: zh-TW
• Published: 1997
• Online Access: http://ndltd.ncl.edu.tw/handle/76815366920991798180

碩士 === 國立中正大學 === 化學工程研究所 === 85 === In recent years, there has been a great development in wavelets for bounded domain, andmany interval wavelets have been constructed. In particular, Chui and coworkersare among the first to construct interval $B$-wavelets associated with the $B$-splinefunctions. The interval spline-wavelets are piecewise polynomials and have the propertiesof compact support, semi- orthogonal and symmetry. The $B$-spline approach also allowsus to consider vastly irrgular knot sequence.The construction of $B$-wavelets depends closely on the evaluation of the inner products of $B$-spline functions. Aiming directly at this requirement, an efficientalgorithm for the evaluation of the inner products of $B$-splines is presented in this thesis.Based on representing the monomial $t^i$ over a bounded interval($i=0, \cdots,m-1$) in a series of $B$-spline functions of order $m$ and applying the order recursive formulas of $B$-splines, we can derive a recursive algorithm for computing the moments and inner products for $B$-splines. Moreover, based on the two-scale relation of $B$-splines, the inner products of $B$-spline functions with different resolution levels can be represented in the same resolution level.Therefore, all the inner products of $B$-splines which are required for constructing the associated $B$-wavelets can be obtained.As an application of spline- wavelets, the numerical inverse of Laplace transform is alsopresented. Applying the notions of direct-sum decompositions of $L^{2}(R)$ andmultiresolution analysis, the inverted function over a bounded domain can be representedin a series of spline- wavelets or a $B$-spline series. The expansion coefficientsare determined such that the integral of square error of approximation is minimized. By exploiting the Laplace domain properties of the spline wavelets, a system of linear equations is formed for the unknown wavelet-series coefficients. The method has the advantage that the accurate inverted function over a bounded domain can be computed in a recursive manner.