Splines and Multiresolution Analysis

Brigitte Forster
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Brigitte Forster: Technische Universität München

Chapter 28 in Handbook of Mathematical Methods in Imaging, 2011, pp 1231-1270 from Springer

Abstract: Abstract Splines and multiresolution are two independent concepts, which – consideredtogether – 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 spacesthat operate as some sort of mathematical looking glass. It allows to separatecoarse 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: Scaling Function; Trigonometric Polynomial; Hexagonal Lattice; Space Domain; Multiresolution Analysis (search for similar items in EconPapers)
Date: 2011
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DOI: 10.1007/978-0-387-92920-0_28

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