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Combined ℓ2 data and gradient fitting in conjunction with ℓ1 regularization

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Abstract

We are interested in minimizing functionals with ℓ2 data and gradient fitting term and ℓ1 regularization term with higher order derivatives in a discrete setting. We examine the structure of the solution in 1D by reformulating the original problem into a contact problem which can be solved by dual optimization techniques. The solution turns out to be a ’smooth’ discrete polynomial spline whose knots coincide with the contact points while its counterpart in the contact problem is a discrete version of a spline with higher defect and contact points as knots. In 2D we modify Chambolle’s algorithm to solve the minimization problem with the ℓ1 norm of interacting second order partial derivatives as regularization term. We show that the algorithm can be implemented efficiently by applying the fast cosine transform. We demonstrate by numerical denoising examples that the ℓ2 gradient fitting term can be used to avoid both edge blurring and staircasing effects.

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Authors and Affiliations

  1. Faculty of Mathematics and Computer Science, Saarland University, 66041, Saarbrücken, Germany

    Stephan Didas

  2. Institute of Mathematics, University of Mannheim, 68131, Mannheim, Germany

    Simon Setzer & Gabriele Steidl

Authors
  1. Stephan Didas
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  2. Simon Setzer
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  3. Gabriele Steidl
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Correspondence to Gabriele Steidl.

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Communicated by Juan Manuel Peña.

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Didas, S., Setzer, S. & Steidl, G. Combined ℓ2 data and gradient fitting in conjunction with ℓ1 regularization. Adv Comput Math 30, 79–99 (2009). https://doi.org/10.1007/s10444-007-9061-4

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  • DOI: https://doi.org/10.1007/s10444-007-9061-4

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