Splines in Higher Order TV Regularization

Authors:
Gabriele Steidl;Stephan Didas;Julia Neumann
Affiliations:
Faculty of Mathematics and Computer Science, University of Mannheim, Mannheim, Germany 68131;Mathematical Image Analysis Group, Faculty of Mathematics and Computer Science, Saarland University, Saarbrücken, Germany 66123;Faculty of Mathematics and Computer Science, University of Mannheim, Mannheim, Germany 68131
Venue:
International Journal of Computer Vision
Year:
2006

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Abstract

Splines play an important role as solutions of various interpolation and approximation problems that minimize special functionals in some smoothness spaces. In this paper, we show in a strictly discrete setting that splines of degree m驴1 solve also a minimization problem with quadratic data term and m-th order total variation (TV) regularization term. In contrast to problems with quadratic regularization terms involving m-th order derivatives, the spline knots are not known in advance but depend on the input data and the regularization parameter 驴. More precisely, the spline knots are determined by the contact points of the m---th discrete antiderivative of the solution with the tube of width 2驴 around the m-th discrete antiderivative of the input data. We point out that the dual formulation of our minimization problem can be considered as support vector regression problem in the discrete counterpart of the Sobolev space W 2,0 m . From this point of view, the solution of our minimization problem has a sparse representation in terms of discrete fundamental splines.