Splines in Higher Order TV Regularization

Issue Title: Special Issue: The 5th International Conference on Scale-Space and PDE Methods in Computer Vision 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 stri...

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Vydané v:	International journal of computer vision Ročník 70; číslo 3; s. 241 - 255
Hlavní autori:	Steidl, Gabriele, Didas, Stephan, Neumann, Julia
Médium:	Journal Article Konferenčný príspevok..
Jazyk:	English
Vydavateľské údaje:	Heidelberg Springer 01.12.2006 Springer Nature B.V
Predmet:	Applied sciences Artificial intelligence Computer science; control theory; systems Exact sciences and technology Studies higher order TV regularization Scale space Image processing Sobolev space Total variation Minimization Regression analysis splines Legendre polynomial Spline approximation support vector regression Legendre-Fenchel dualization taut-string algorithm Television Character string Vector support machine Sparse representation
ISSN:	0920-5691, 1573-1405
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Abstract	Issue Title: Special Issue: The 5th International Conference on Scale-Space and PDE Methods in Computer Vision 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-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 λ. 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 2λ 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 ^sub 2,0^ ^sup m^. From this point of view, the solution of our minimization problem has a sparse representation in terms of discrete fundamental splines.[PUBLICATION ABSTRACT]
AbstractList	Issue Title: Special Issue: The 5th International Conference on Scale-Space and PDE Methods in Computer Vision 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-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 λ. 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 2λ 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 ^sub 2,0^ ^sup m^. From this point of view, the solution of our minimization problem has a sparse representation in terms of discrete fundamental splines.[PUBLICATION 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 rn - 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 A. 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 2lambda 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 Wn20. From this point of view, the solution of our minimization problem has a sparse representation in terms of discrete fundamental splines.
Author	Didas, Stephan Steidl, Gabriele Neumann, Julia
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Keywords	higher order TV regularization Scale space Image processing Sobolev space Total variation Minimization Regression analysis splines Legendre polynomial Spline approximation support vector regression Legendre-Fenchel dualization taut-string algorithm Television Character string Vector support machine Sparse representation
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Title	Splines in Higher Order TV Regularization
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