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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Vydáno v:	International journal of computer vision Ročník 70; číslo 3; s. 241 - 255
Hlavní autoři:	Steidl, Gabriele, Didas, Stephan, Neumann, Julia
Médium:	Journal Article Konferenční příspěvek
Jazyk:	angličtina
Vydáno:	Heidelberg Springer 01.12.2006 Springer Nature B.V
Témata:	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	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. 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]
Author	Didas, Stephan Steidl, Gabriele Neumann, Julia
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Cites_doi	10.1214/aos/996986501 10.1137/S0036142903422429 10.1137/0802028 10.1007/s002110050258 10.1162/089976698300017269 10.1214/aos/1034276635 10.1023/A:1008282127190 10.1137/S1064827596299767 10.1137/1.9781611970128 10.1137/1.9780898717570 10.1007/s00607-005-0129-z 10.1137/0710039 10.1137/S0036139903422784 10.1109/TSP.2005.847821 10.1007/11408031_52 10.1515/9781400873173 10.1137/0309015 10.1023/B:JMIV.0000011920.58935.9c 10.1023/A:1026276804745 10.1016/0167-2789(92)90242-F 10.1016/S0898-1221(97)00009-6 10.1109/TIP.2003.819229 10.1023/A:1008278718907 10.1023/B:JMIV.0000011320.81911.38 10.1109/83.869184 10.1016/0022-247X(71)90184-3 10.1137/S1064827598344169
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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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