Abstract
We introduce an algorithm to solve linear inverse problems regularized with the total (gradient) variation in a gridless manner. Contrary to most existing methods, that produce an approximate solution which is piecewise constant on a fixed mesh, our approach exploits the structure of the solutions and consists in iteratively constructing a linear combination of indicator functions of simple polygons.
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Notes
- 1.
A repository containing a complete implementation of Algorithm 1 can be found online at https://github.com/rpetit/tvsfw.
- 2.
Details on the first variations of perimeter and area can be found in [12, Sect. 17.3].
- 3.
This only occurs when \(\lambda \) is small enough. For higher values of \(\lambda \), the output is similar to (d) or (e).
- 4.
Assumption 1 is for example satisfied by \(\varphi :x\mapsto \text {exp}\left( -||x||^2/(2\sigma ^2)\right) \) for any \(\sigma >0\).
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Acknowledgments
This work was supported by a grant from Région Ile-De-France and by the ANR CIPRESSI project, grant ANR-19-CE48-0017-01 of the French Agence Nationale de la Recherche. RP warmly thanks the owners of Villa Margely for their hospitality during the writing of this paper.
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De Castro, Y., Duval, V., Petit, R. (2021). Towards Off-the-grid Algorithms for Total Variation Regularized Inverse Problems. In: Elmoataz, A., Fadili, J., Quéau, Y., Rabin, J., Simon, L. (eds) Scale Space and Variational Methods in Computer Vision. SSVM 2021. Lecture Notes in Computer Science(), vol 12679. Springer, Cham. https://doi.org/10.1007/978-3-030-75549-2_44
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