Spline-Wavelet Frames in Computer Graphics and other Applications

建议0098331密苏里州圣路易斯市的Charles K Chui，Wejie He和Joachim StoeckraerU摘要：通过应用矩阵扩展技术，可以很容易地为任意m阶样条(或m-1次)构造具有比例因子2的紧框架，该框架由两个紧支撑的基样条的伸缩和平移的标准仿射运算生成，称为框架生成器。在我们最近的工作中，引入了“消失矩恢复”的概念，引入了洛朗多项式因子S(Z)，证明了对于任意阶m，两个紧支集的紧框架生成器都可以获得最大消失矩m。此外，当紧框架松弛为兄弟框架时，洛朗多项式S(Z)可以显式地表示出来；也就是说，这两个帧生成器及其对应的对偶都是紧支撑的同阶m次基数样条线。这些附加的消失矩对于有效地利用小波系数进行特征提取、去噪等是必不可少的。基数样条线是具有等间距结点序列的样条函数。然而，在大多数实际应用中，感兴趣的区间是有界的，数据样本可能不是均匀分布的。因此，需要具有任意节点的m阶样条，或者至少在感兴趣区间的一个或两个端点处具有m个堆叠节点。这一新的研究项目涉及到洛朗多项式S(Z)的矩阵等价Sk的提出，Sk的构造以及具有任意结点和m个消失矩的m阶紧支样条曲线的相应的紧(更一般的兄弟)框架生成器，实现了兄弟框架的互正交性，相关框架算法的开发和与现有样条工具的集成，用于计算机图形学应用的添加稀疏和编辑功能的样条小波框架工具的研究，以及便携式软件库的开发。

Proposal 0098331 Charles K Chui, Wenjie He, and Joachim StoecklerU of Missouri, Saint LouisAbstract: Tight frames with scaling factor 2, generated by the standard affine operations of dilation and translation of two compactly supported cardinal splines, called frame generators, can be easily constructed for any spline order m (or degree m-1), by applying matrix extension techniques. However, regardless of the number of (spline) frame generators being used, at least one of them has only one vanishing moment, when the matrix extension approach is followed.In our recent work, we introduced the notion of "vanishing-moment recovery" Laurent polynomial factors S(z) is introduced to show that the maximum number m of vanishing moments can be achieved by both compactly supported tight frame generators, for any order m. Furthermore, the Laurent polynomials S(z) can be formulated explicitly when tight frames are relaxed to be sibling frames; that is, both frame generators, together with their corresponding duals, are compactly supported cardinal splines of the same order m. These additional vanishing moments are essential for effective use of the wavelet coefficients for feature extraction, noise removal, etc.Cardinal splines are spline functions with an equally spaced knot sequence extending from. However, in most practical applications, the intervals of interest are bounded and data samples may not be uniformly distributed. Hence, mth order splines with arbitrary knots, or at least with m stacked knots at one or both end-points of the interval of interest, are needed. This new research project is concerned with formulation of the matrix equivalent Sk of the Laurent polynomials S(z), construction of Sk and the corresponding tight (and more generally sibling) frame generators of mth order compactly supported splines with arbitrary knots and with m vanishing moments, achievement of such important features as inter-orthogonality for sibling frames, development and integration of the associated frame algorithms with the existing spline tools, investigation of spline-wavelet frame tools for adding sparsification and editing fearures for applications in computer graphics, and development of a portable software library.