Spline-Wavelet Frames in Computer Graphics and other Applications

计算机图形学和其他应用中的样条小波框架

• 批准号: 0098331
• 负责人: Charles Chui
• 金额: $ 29.2万
• 依托单位: University of Missouri-Saint Louis
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-08-15 至 2005-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0098331&HistoricalAwards=false
• 关键词: Spline; Wavelet; Frames; Computer; Graphics

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.

提案 0098331 Charles K Chui、Wenjie He 和 Joachim StoecklerU（密苏里州圣路易斯）摘要：缩放因子为 2 的紧框架是通过两个紧支撑基数样条的膨胀和平移的标准仿射操作（称为框架生成器）生成的，可以通过应用矩阵扩展技术轻松地为任何样条阶数 m（或度 m-1）构造。 然而，无论使用多少个（样条）框架生成器，当遵循矩阵扩展方法时，至少其中一个只有一个消失矩。在我们最近的工作中，我们引入了“消失矩恢复”的概念，引入了洛朗多项式因子 S(z)，以表明对于任何阶 m，两个紧支撑紧框架生成器都可以实现最大数量 m 的消失矩。 此外，当紧框架放松为同级框架时，可以明确地制定洛朗多项式S(z)；也就是说，两个框架生成器及其相应的对偶生成器都是同阶 m 的紧支撑基数样条。 这些额外的消失矩对于有效使用小波系数进行特征提取、噪声消除等至关重要。基数样条是具有等距结序列延伸的样条函数。 然而，在大多数实际应用中，感兴趣的区间是有界的，并且数据样本可能不是均匀分布的。 因此，需要具有任意结的 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 个堆叠结。 这个新的研究项目涉及洛朗多项式 S(z) 的矩阵等效 Sk 的制定、Sk 的构造以及具有任意结和 m 个消失矩的 m 阶紧支撑样条的相应紧（更一般的兄弟）框架生成器、实现兄弟框架的相互正交性等重要特征、相关框架算法与现有样条工具的开发和集成， 研究为计算机图形学应用添加稀疏化和编辑功能的样条小波框架工具，并开发便携式软件库。