Abstract
In inverse problems, the use of an \(\ell _{12}\) analysis regularizer induces a bias in the estimated solution. We propose a general refitting framework for removing this artifact while keeping information of interest contained in the biased solution. This is done through the use of refitting block penalties that only act on the co-support of the estimation. Based on an analysis of related works in the literature, we propose a new penalty that is well suited for refitting purposes. We also present an efficient algorithmic method to obtain the refitted solution along with the original (biased) solution for any convex refitting block penalty. Experiments illustrate the good behavior of the proposed block penalty for refitting.
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Notes
- 1.
As in [2], extended support \(\Vert \hat{z}_i \Vert =\lambda \) can be tackled by testing \(\Vert (\hat{z}^k+\sigma \varGamma \hat{v}^k )_i \Vert \geqslant \lambda \).
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Acknowledgement
This project has been carried out with support from the French State, managed by the French National Research Agency (ANR-16-CE33-0010-01). This work was supported by the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 777826.
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Deledalle, CA., Papadakis, N., Salmon, J., Vaiter, S. (2019). Refitting Solutions Promoted by \(\ell _{12}\) Sparse Analysis Regularizations with Block Penalties. In: Lellmann, J., Burger, M., Modersitzki, J. (eds) Scale Space and Variational Methods in Computer Vision. SSVM 2019. Lecture Notes in Computer Science(), vol 11603. Springer, Cham. https://doi.org/10.1007/978-3-030-22368-7_11
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