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Splines in Higher Order TV Regularization

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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−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.

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References

  • Chambolle, A. 2004. An algorithm for total variation minimization and applications. Journal of Mathematical Imaging and Vision, (20):89–97.

    Article  MathSciNet  Google Scholar 

  • Chambolle, A. and Lions, P.-L. 1997 Image recovery via total variation minimization and related problems. Numerische Mathematik, 76:167–188.

    Article  MATH  MathSciNet  Google Scholar 

  • 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.

  • Chan, T. F., Golub, G. H., and Mulet, P. 1999. A nonlinear primal—dual method for total-variation based image restoration. SIAM Journal on Scientific Computing, 20(6):1964–1977.

    Article  MATH  MathSciNet  Google Scholar 

  • Chan, T. F., Marquina, A., and Mulet, P. 2000. High-order total variation-based image restoration. SIAM Journal on Scientific Computing, 22(2):503–516.

    Article  MATH  MathSciNet  Google Scholar 

  • Davies, P. L. and Kovac, A. 2001. Local extremes, runs, strings and multiresolution. Annals of Statistics, 29:1–65.

    Article  MATH  MathSciNet  Google Scholar 

  • Didas, S. 2004. Higher order variational methods for noise removal in signals and images. Diplomarbeit, Universität des Saarlandes.

  • 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 

  • Girosi, F. 1998. An equivalence between sparse approximation and support vector machines. Neural computation, 10(6):1455–1480.

    Article  Google Scholar 

  • Hinterberger, W., Hintermüller, M., Kunisch K., von Oehsen M., and Scherzer, O. 2003. Tube methods for BV regularization. Journal of Mathematical Imaging and Vision, 19:223–238.

    Article  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 

  • Hintermüller, W. and Kunisch, K. May 2004. Total bounded variation regularization as a bilaterally constrained optimization problem. SIAM Journal on Applied Mathematics, 64(4):1311–1333.

    Article  Google Scholar 

  • Hyman, J. M. and Shashkov, M. J. 1997. Natural discretizations for the divergence, gradient, and curl on logically rectangular grids. Comput. Math. Appl., 33(4):81–104.

    Article  MATH  MathSciNet  Google Scholar 

  • Kimmeldorf, G. S. and Wahba, G. 1971. Some results on Tchebycheffian spline functions. J. Anal. Appl., 33:82–95.

    Article  Google Scholar 

  • Lysaker, M., Lundervold, A., and Tai, X. 2003. Noise removal using fourth-order partial differential equations with applications to medical magnetic resonance images in space and time. IEEE Transactions on Image Processing, 12(12):1579–1590.

    Article  Google Scholar 

  • Mammen, E. and van de Geer, S. 1997. Locally adaptive regression splines. Annals of Statistics, 25(1):387–413.

    Article  MATH  MathSciNet  Google Scholar 

  • Mangasarian, O. L. and Schumaker, L. L. 1971. Mangasarian, O. L. and Schumaker, L. L. 1971. Discrete splines via mathematical programming. SIAM Journal on Control, 9(2):174–183.

    Article  MATH  MathSciNet  Google Scholar 

  • Mangasarian, O. L. and Schumaker, L. L. 1973. Best summation formulae and discrete splines via mathematical programming. SIAM Journal on Numerical Analysis, 10(3):448–459.

    Article  MATH  MathSciNet  Google Scholar 

  • Mehrotra, S. 1992. On the implementation of a primal-dual interior point method. SIAM Journal on Optimization, 2(4):575–601.

    Article  MATH  MathSciNet  Google Scholar 

  • Nielsen, M., Florack, L., and Deriche, R. 1997. Regularization, scale-space and edge detection filters. Journal of Mathematical Imaging and Vision, 7:291–307.

    Article  MathSciNet  Google Scholar 

  • Nikolova, M. 2004. A variational approach to remove outliers and impulse noise. Journal of Mathematical Imaging and Vision, 20:99–120.

    Article  MathSciNet  Google Scholar 

  • Rockafellar, R. T. 1970. Convex Analysis. Princeton University Press, Princeton.

    Google Scholar 

  • Rudin, L. I., Osher S., and Fatemi, E. 1992. Nonlinear total variation based noise removal algorithms. Physica D, 60:259–268.

    Article  MATH  Google Scholar 

  • Schnörr, C. 1998. A study of a convex variational diffusion approach for image segmentation and feature extraction. Journal of Mathematical Imaging and Vision, 8(3):271–292.

    Article  MATH  MathSciNet  Google Scholar 

  • Steidl, G. 2006. A note on the dual treatment of higher order regularization functionals. Computing, 76:135–148.

    Article  MATH  MathSciNet  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 

  • Steidl, G., Weickert, J., Brox, T., Mrázek, P., and Welk, M. 2004. On the equivalence of soft wavelet shrinkage, total variation diffusion, total variation regularization, and SIDEs. SIAM Journal on Numerical Analysis, 42(2):686–713.

    Article  MATH  MathSciNet  Google Scholar 

  • Unser, M. and Blu, T. 2005. Generalized smoothing splines and the optimal discretization of the Wiener filter. IEEE Transactions on Signal Processing, 53(6):2146–2159.

    Article  MathSciNet  Google Scholar 

  • Vapnik, V. N. 1998. Statistical Learning Theory. John Wiley and Sons, Inc.

  • 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.

  • Yip, A. M. and Park, F. 2003. Solution dynamics, causality, and critical behaviour of the regularization parameter in total variation denoising problems. UCLA Report.

  • You, Y.-L. and Kaveh, M. 2000. Fourth-order partial differential equations for noise removal. IEEE Transactions on Image Processing, 9(10):1723–1730.

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Faculty of Mathematics and Computer Science, University of Mannheim, 68131, Mannheim, Germany

    Gabriele Steidl & Julia Neumann

  2. Mathematical Image Analysis Group, Faculty of Mathematics and Computer Science, Saarland University, 66123, Saarbrücken, Germany

    Stephan Didas

Authors
  1. Gabriele Steidl
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  2. Stephan Didas
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  3. Julia Neumann
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Correspondence to Gabriele Steidl.

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Steidl, G., Didas, S. & Neumann, J. Splines in Higher Order TV Regularization. Int J Comput Vision 70, 241–255 (2006). https://doi.org/10.1007/s11263-006-8066-7

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  • DOI: https://doi.org/10.1007/s11263-006-8066-7

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