Splines in Higher Order TV Regularization
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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 2,0 m . From this point of view, the solution of our minimization problem has a sparse representation in terms of discrete fundamental splines.
Keywordshigher order TV regularization splines support vector regression Legendre-Fenchel dualization taut-string algorithm
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- Chan, R. H., Ho, C. W., and Nikolova, M. Salt-and-pepper noise removal by median noise detectors and detail preserving regularization. IEEE Transactions on Image Processing, page to appear.Google Scholar
- Didas, S. 2004. Higher order variational methods for noise removal in signals and images. Diplomarbeit, Universität des Saarlandes.Google Scholar
- Duchon, J. 1997. Splines minimizing rotation-invariant seminorms in sobolev spaces. In Constructive Theory of Functions of Several Variables, pages 85–100, Berlin, Springer—Verlag.Google Scholar
- Hinterberger, W. and Scherzer, O. 2003. Variational methods on the space of functions of bounded Hessian for convexification and denoising. Technical report, University of Innsbruck, Austria.Google Scholar
- Rockafellar, R. T. 1970. Convex Analysis. Princeton University Press, Princeton.Google Scholar
- Steidl, G., Didas, S., and Neumann, J. 2005. Relations between higher order TV regularization and support vector regression. In R. Kimmel, N. Sochen, and J. Weickert, editors, Scale-Space and PDE Methods in Computer Vision, volume 3459 of Lecture Notes in Computer Science, pages 515–527. Springer, Berlin.Google Scholar
- Vapnik, V. N. 1998. Statistical Learning Theory. John Wiley and Sons, Inc.Google Scholar
- Vogel, C. R. 2002. Computational Methods for Inverse Problems. SIAM, Philadelphia.Google Scholar
- Wahba, G. 1990. Spline Models for Observational Data. SIAM, Philadelphia.Google Scholar
- Welk, M., Weickert, J., and Steidl, G. 2005. A four-pixel scheme for singular differential equations. In R. Kimmel, N. Sochen, and J. Weickert, editors, Scale-Space and PDE Methods in Computer Vision, Lecture Notes in Computer Science. Springer, Berlin, to appear.Google Scholar
- Yip, A. M. and Park, F. 2003. Solution dynamics, causality, and critical behaviour of the regularization parameter in total variation denoising problems. UCLA Report.Google Scholar