Splines and Multiresolution Analysis

Living reference work entry

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

Scaling Function Trigonometric Polynomial Multiresolution Analysis Riesz Basis Fourier Domain 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Fakultät für Informatik und MathematikUniversität PassauPassauGermany

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