A Box-Spline Framework for Inverse Problems with Continuous-Domain Sparsity Constraints
M. Pourya, A. Boquet-Pujadas, M. Unser
IEEE Transactions on Computational Imaging, vol. 10, pp. 790–805, 2024.
The formulation of inverse problems in the continuum eliminates discretization errors and allows for the exact incorporation of priors. In this paper, we formulate a continuous-domain inverse problem over a search space of continuous and piecewise-linear functions parameterized by box splines. We present a numerical framework to solve those inverse problems with total variation (TV) or its Hessian-based extension (HTV) as regularizers. We show that the box-spline basis allows for exact and efficient convolution-based expressions for both TV and HTV. Our optimization strategy relies on a multiresolution scheme whereby we progressively refine the solution until its cost stabilizes. We test our framework on linear inverse problems and demonstrate its ability to effectively reach a stage beyond which the refinement of the search space no longer decreases the optimization cost.
@ARTICLE(http://bigwww.epfl.ch/publications/pourya2401.html,
AUTHOR="Pourya, M. and Boquet-Pujadas, A. and Unser, M.",
TITLE="A Box-Spline Framework for Inverse Problems with
Continuous-Domain Sparsity Constraints",
JOURNAL="{IEEE} Transactions on Computational Imaging",
YEAR="2024",
volume="10",
number="",
pages="790--805",
month="",
note="")