WIAS Preprint No. 2689, (2020)

Dualization and automatic distributed parameter selection of total generalized variation via bilevel optimization


  • Hintermüller, Michael
    ORCID: 0000-0001-9471-2479
  • Papafitsoros, Kostas
    ORCID: 0000-0001-9691-4576
  • Rautenberg, Carlos N.
    ORCID: 0000-0001-9497-9296
  • Sun, Hongpeng

2010 Mathematics Subject Classification

  • 94A08 68U10 49K20 49Q20 65K15 26A45


  • Image restoration, image denoising, total generalized variation, spatially distributed regularization, weight, bilevel optimization




Total Generalized Variation (TGV) regularization in image reconstruction relies on an infimal convolution type combination of generalized first- and second-order derivatives. This helps to avoid the staircasing effect of Total Variation (TV) regularization, while still preserving sharp contrasts in images. The associated regularization effect crucially hinges on two parameters whose proper adjustment represents a challenging task. In this work, a bilevel optimization framework with a suitable statistics-based upper level objective is proposed in order to automatically select these parameters. The framework allows for spatially varying parameters, thus enabling better recovery in high-detail image areas. A rigorous dualization framework is established, and for the numerical solution, two Newton type methods for the solution of the lower level problem, i.e. the image reconstruction problem, and two bilevel TGV algorithms are introduced, respectively. Denoising tests confirm that automatically selected distributed regularization parameters lead in general to improved reconstructions when compared to results for scalar parameters.

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