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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Shrnutí: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]
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ISSN:0920-5691
1573-1405
DOI:10.1007/s11263-006-8066-7