Convex Curve Recovery in Phase Space Extreme Points and Exact Graph Discretization
Abstract
Recovering filaments, cracks, or vessels from indirect measurements calls for convex regularizers whose extreme points are curve measures. We first identify an obstruction in the image domain: under a natural concentration assumption, the closed convex hulls of length-normalized curve measures have only Dirac masses as extreme points or extreme rays. In contrast, fixedlength simple curves with uniformly bounded curvature are extreme without any lifting; the difficulty is therefore to express this geometric constraint in a tractable convex form. We address it by lifting measures to the position-orientation space Ω = X ×T. The proposed LACE functional combines length and total curvature through a conservation law involving an advance measure and an angular flux. Using Smirnov's decomposition, we place its extreme points among horizontal elementary solenoids and prove extremality for simple closed curves, doubled open arcs, and a class of self-intersecting curves. We then introduce a graph discretization for which the extreme rays of the lifted admissible cone are exactly the simple cycles, providing a near-perfect match with the continuous setting. The resulting convex inverse problems can be solved either by a primal-dual method or by a conditional-gradient algorithm whose oracle is a polynomial-time maximum-ratio-cycle problem. Experiments on deconvolution and partial Fourier sampling illustrate the recovery of curvilinear structures from highly incomplete measurements.
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BibTeX
@misc{weiss2026convex,
title = {{Convex Curve Recovery in Phase Space Extreme Points and Exact Graph Discretization}},
author = {Pierre Weiss},
howpublished = {HAL},
year = {2026},
month = sep,
url = {https://hal.science/hal-05743493v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Convex Curve Recovery in Phase Space Extreme Points and Exact Graph Discretization
AU - Pierre Weiss
PY - 2026
DA - 2026/09/09
PB - HAL
UR - https://hal.science/hal-05743493v1
AB - Recovering filaments, cracks, or vessels from indirect measurements calls for convex regularizers whose extreme points are curve measures. We first identify an obstruction in the image domain: under a natural concentration assumption, the closed convex hulls of length-normalized curve measures have only Dirac masses as extreme points or extreme rays. In contrast, fixedlength simple curves with uniformly bounded curvature are extreme without any lifting; the difficulty is therefore to express this geometric constraint in a tractable convex form. We address it by lifting measures to the position-orientation space Ω = X ×T. The proposed LACE functional combines length and total curvature through a conservation law involving an advance measure and an angular flux. Using Smirnov's decomposition, we place its extreme points among horizontal elementary solenoids and prove extremality for simple closed curves, doubled open arcs, and a class of self-intersecting curves. We then introduce a graph discretization for which the extreme rays of the lifted admissible cone are exactly the simple cycles, providing a near-perfect match with the continuous setting. The resulting convex inverse problems can be solved either by a primal-dual method or by a conditional-gradient algorithm whose oracle is a polynomial-time maximum-ratio-cycle problem. Experiments on deconvolution and partial Fourier sampling illustrate the recovery of curvilinear structures from highly incomplete measurements.
ER -