Mathematical Sciences: One Higher Dimensional Wavelets fromFractal Interpolation Functions: Construction and Applications

数学科学：分形插值函数的一个高维小波：构造和应用

• 批准号: 9401352
• 负责人: Jeffrey Geronimo
• 金额: $ 6万
• 依托单位: Georgia Tech Research Corporation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1994
• 资助国家: 美国
• 起止时间: 1994-06-15 至 1997-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9401352&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Higher; Dimensional; Wavelets

9401352 Geronimo Work supported by this grant focuses on the application of wavelets to signal and image processing as well as to numerical analysis. The wavelet approach to problems in these areas has followed two basic paths: developing the wavelet from the solution of a scaling equation or via splines. A third procedure to be pursued in this work derives from recent construction of fractal interpolation functions. The construction has advantages of both of the previous methods, namely explicit regularity results, easily calculated inner products and linear phase as in the spline case and orthogonality, compact support and continuity as in the dilation equation case. The method also has a multidimensional generalization that gives wavelets which are not tensor products of univariate functions. This project continues investigations into the construction and application of wavelets using fractal interpolation functions in one and higher dimensions. In one dimension, the focus will be on constructing smoother (at least differentiable) compactly supported and orthogonal wavelets. In higher dimensions, the goal is to construct continuous, compactly supported orthogonal wavelets. Applications will be made to image processing by exploiting the unique properties (e.g. symmetry and regularity) of the wavelets. The study and applications of wavelets represents are relatively new branch of harmonic analysis in which single functions are used to generate Hilbert (or other) space bases by which functions may be represented. While this in itself is not too difficult, one seeks wavelets with additional properties such as smoothness, compact support, localization in time and frequency. Coupled with all of these demands, one also asks for computational simplicity. Remarkably, there are such functions. The goals of this research are to extend and refine basic knowledge in this field by combining existing techniques with those developed in parallel within t he field of fractal interpolation.

9401352 Geronimo 此项资助支持的工作重点是小波在信号和图像处理以及数值分析中的应用。 解决这些领域问题的小波方法遵循两条基本路径：从标度方程的解或通过样条来开发小波。 这项工作要追求的第三个程序源自最近分形插值函数的构造。 该构造具有前两种方法的优点，即显式正则性结果、易于计算的内积和线性相位（如样条情况）以及正交性、紧支撑和连续性（如膨胀方程情况）。 该方法还具有多维概括，给出的小波不是单变量函数的张量积。 该项目继续研究在一维和更高维度上使用分形插值函数构建和应用小波。 在一维中，重点将在于构造更平滑（至少可微分）的紧支撑和正交小波。 在更高维度中，目标是构造连续的、紧支撑的正交小波。通过利用小波的独特属性（例如对称性和规律性），将应用于图像处理。 小波表示的研究和应用是调和分析的相对较新的分支，其中使用单个函数来生成可以表示函数的希尔伯特（或其他）空间基。 虽然这本身并不太困难，但人们寻求具有附加属性的小波，例如平滑性、紧凑支持、时间和频率局部化。 除了所有这些需求之外，人们还要求计算简单性。 值得注意的是，有这样的功能。 这项研究的目标是通过将现有技术与分形插值领域内并行开发的技术相结合来扩展和完善该领域的基础知识。