The Wavelet IDR Center Research Subteam

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Images are most often defined on a rectangular uniform domain. Many other types of data sets cannot be described in such uniform way. We refer to such problem as scattered data problems. Scattered data problems are often harder to analyze; algorithms for such problems that possess a sufficient degree of robustness are often correspondingly hard to design and to implement; often these difficulties are compounded by the large size of the problems of interest. This makes it sometimes hard to transfer and exploit knowledge accumulated during the treatment of one such problem towards the resolution of another. The primary objective of this subteam is to seek a unified platform for scattered data problems. Such a platform is necessary for the development of general tools, and in particular, general multi-scale techniques that do not rely heavily on the specific characteristics of the problem, and therefore may be portable from one problem to another. Furthermore, developing a theory that distinguishes the core from the details is necessary when one attempts to assess the potential of an already established well-tested technique for a new problem, hence to allow the exchange of knowledge across the different scientific disciplines.

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Two main approaches are currently examined: mesh-oriented techniques and meshless techniques. In the mesh-oriented techniques, the domain is partitioned subject to the given vertices, in a way that reflects the geometry and topology of the vertices and of the source of the scientific data. Multiscale methods then follow the paradigm of extracting coarser versions of that mesh, and projecting the data onto the vertices of the coarse grid. In the meshless situation, the multiscale is determined by the vertices only. The goal is to establish a theory that identifies suitable `scaling functions', and determines consistent methods for extracting the scale spaces and the wavelets.

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Approximation by radial basis functions is one the leading approaches in the areas of meshless techniques. We currently work on the conversion of tools developed in the study of shift-invariant spaces [BDR94]. The approach is based on the general conversion method developed by N. Dyn and A. Ron [DR95]. Based on this approach, an algorithm and code for the approximation of scattered data on bounded planar domains was developed by Jungho Yoon. Numerical Experiments suggest that the algorithm is superior to other known techniques, as the example below indicates.

LEFT: Contour lines of an approximand. MIDDLE: Contour lines of the thin-plate smoothing spline (TPSS) fitted to noisy scattered function values. RIGHT: Contour lines of our approximant to the same input values as those used by the TPSS.

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Approximation Theory on spheres and other general manifolds Our approach in Euclidean domains is based on the well-established theory of shift-invariant spaces, which in turn is based on the existence of uniform lattices (such as the integer lattice) in the Euclidean domain. Such approach is invalid on the sphere. A general theory for approximation on spheres is a prerequisite to any further progress. In the Euclidean case, the existing theory unables us to identify the very few basis functions that are `suitable' (such as thin-plate splines and multiquadrics). In the absence of such theory on the sphere, even the first step, of selecting the suitable space of approximants, is in jeopardy.
Multiscale Analysis of radial basis function approximation It is not clear at present how to define the `wavelet spaces', and how to decompose the approximand (i.e., the scattered data) into `frequency levels'. Let alone, there is no fast transform that implements the wavelet decomposition.