Splines and Multiresolution Analysis

Brigitte Forster ()
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Brigitte Forster: Universität Passau, Fakultät für Informatik und Mathematik

A chapter in Handbook of Mathematical Methods in Imaging, 2015, pp 1675-1716 from Springer

Abstract: Abstract Splines and multiresolution are two independent concepts, which – considered together – yield a vast variety of bases for image processing and image analysis. The idea of a multiresolution analysis is to construct a ladder of nested spaces that operate as some sort of mathematical looking glass. It allows to separate coarse parts in a signal or in an image from the details of various sizes. Spline functions are piecewise or domainwise polynomials in one dimension (1D) resp. nD. There is a variety of spline functions that generate multiresolution analyses. The viewpoint in this chapter is the modeling of such spline functions in frequency domain via Fourier decay to generate functions with specified smoothness in time domain resp. space domain. The mathematical foundations are presented and illustrated at the example of cardinal B-splines as generators of multiresolution analyses. Other spline models such as complex B-splines, polyharmonic splines, hexagonal splines, and others are considered. For all these spline families exist fast and stable multiresolution algorithms which can be elegantly implemented in frequency domain. The chapter closes with a look on open problems in the field.

Keywords: Multiresolution Analysis; Polyharmonic Splines; Dilation Matrix; Riesz Sequence; Riesz Basis (search for similar items in EconPapers)
Date: 2015
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DOI: 10.1007/978-1-4939-0790-8_28

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