Curve reconstruction in inverse problems: From divergence-measure vector fields to density Lebesgue measures
Abstract
We present a novel variational framework for curve reconstruction in inverse problems, formulated through vector field optimization. The problem is defined in a space of vector-valued measures absolutely continuous with respect to the Lebesgue measure, with Sobolev-type densities. Within this setting, we establish an equivalence, in terms of minimizers, between the proposed functional and a previously defined one within the space of divergence-measure fields. This formulation enables the use of standard convex optimization algorithms for numerical implementation. Furthermore, we propose an acquisition model specifically adapted to blurred curves in microscopy images, leading to promising numerical results.
Fields of Study & Subjects
Numerical methods in inverse problems
Statistical and numerical algorithms
Publication details
- Journal / source
- Inverse Problems and Imaging
- Volume
- 26
- Issue
- 0
- Pages
- 172–188
- Publication date
- 30 March 2026
- Language
- EN