A Curve LASSO reconstruction in 2D imaging inverse problems

We introduce a novel framework for curve reconstruction in off-the-grid inverse problems. First, we establish a new Smirnov-type decomposition showing that every 2-dimensional Radon measure with finite divergence can be decomposed into elementary measures supported on simple regular curves. Building on this framework, we propose the CLASSO (Curve-LASSO) variational functional for curve reconstruction, which can be seen as the analog of the well-known BLASSO (Beurling-LASSO) for spike reconstruction. Minimizers of the CLASSO functional can be expressed as finite linear combinations of measures supported on such curves. We therefore formalize the Sliding Frank-Wolfe algorithm to recover these curve solutions and provide numerical simulations illustrating curve reconstruction from blurred and noisy images.