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Journal Articles SIAM Journal on Imaging Sciences Year : 2023

Off-the-grid curve reconstruction through divergence regularisation: an extreme point result

Bastien Laville
Laure Blanc-Féraud
Gilles Aubert
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Abstract

We propose a new strategy for the reconstruction of curves in an image through an off-the-grid variational framework, inspired by the reconstruction of spikes in the literature. We introduce a new functional CROC on the space of 2-dimensional Radon measures with finite divergence denoted , and we establish several theoretical tools through the definition of a certificate. Our main contribution lies in the sharp characterisation of the extreme points of the unit ball of the-norm: there are exactly measures supported on 1-rectifiable oriented simple Lipschitz curves, thus enabling a precise characterisation of our functional minimisers and further opening a promising avenue for the algorithmic implementation.
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Dates and versions

hal-03658949 , version 1 (04-05-2022)
hal-03658949 , version 2 (28-07-2022)
hal-03658949 , version 3 (12-01-2023)

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Bastien Laville, Laure Blanc-Féraud, Gilles Aubert. Off-the-grid curve reconstruction through divergence regularisation: an extreme point result. SIAM Journal on Imaging Sciences, 2023, 16 (2), pp.867-885. ⟨10.1137/22M1494373⟩. ⟨hal-03658949v3⟩
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