Spline-Wavelet Frames in Computer Graphics and other Applications

Charles K Chui, Wenjie He, Joachim StoecklerU （Missouri, st . louis）摘要：通过矩阵扩展技术，对任意m阶（或m-1阶）的样条进行扩张和平移的标准仿射操作，生成具有比例因子2的紧坐标系，称为坐标系生成器。然而，无论使用多少个（样条）帧生成器，当遵循矩阵扩展方法时，至少有一个（样条）帧生成器只有一个消失时刻。在我们最近的工作中，我们引入了“消失力矩恢复”的概念，并引入了劳伦多项式因子S(z)来表明，对于任意阶的m，两个紧支撑紧框架生成器都可以实现消失力矩的最大数量m。此外，当紧框架被放松为兄弟框架时，劳伦多项式S(z)可以显式地表示；也就是说，两个帧生成器及其相应的对偶都是紧支持的相同阶次的基数样条。这些额外的消失矩对于有效地利用小波系数进行特征提取、噪声去除等是必不可少的。基数样条是具有等间距结序列的样条函数。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，需要具有任意结点的m阶样条，或者至少在感兴趣区间的一个或两个端点上具有m个堆叠结点。这个新的研究项目涉及洛朗多项式S(z)的矩阵等效Sk的公式，Sk的构造和相应的具有任意结点和m个消失矩的m阶紧支撑样条的紧（和更一般的兄弟）框架生成器，兄弟框架的正交性等重要特征的实现，相关框架算法与现有样条工具的开发和集成。在计算机图形学应用中增加稀疏化和编辑特性的样条-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.