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

9401352这笔赠款支持的Geronimo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。对于这些领域中的问题，小波方法遵循两条基本路径：从标度方程的解发展小波或通过样条线发展小波。这项工作中要追求的第三个过程来自于最近构造的分形内插函数。这种构造方法具有前两种方法的优点，即正则性结果明确、内积和线性相位易于计算(如在样条法中)和正交性、紧支性和连续性(如在伸缩方程情况下)。该方法还具有多维推广，给出了不是单变量函数张量积的小波。本项目继续研究使用一维及更高维分形插值函数的小波的构造和应用。在一个维度上，重点将是构造更光滑(至少是可微的)紧支集和正交小波。在更高维中，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性质(例如对称性和正则性)，将在图像处理中得到应用。小波表示的研究和应用是调和分析的一个相对较新的分支，在调和分析中，单个函数被用来产生可以表示函数的希尔伯特(或其他)空间基。虽然这本身并不太难，但人们寻找具有其他性质的小波，如光滑性、紧支性、在时间和频率上的局部化。除了所有这些要求外，人们还要求计算简单。值得注意的是，有这样的功能。本研究的目的是通过将现有的技术与在分形插值法领域内并行发展的技术相结合来扩展和提炼该领域的基本知识。

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.