TV-based reconstruction of periodic functions. (23rd October 2020)

• Record Type: Journal Article
• Authors: Fageot, Julien; Simeoni, Matthieu
• Is Part Of: Inverse problems. Volume 36:Number 11(2020)
• Publication Date: 2020-10-23
• Subjects: periodic operators -- total variation norm -- splines -- optimization on measure spaces -- representer theorem -- native spaces; Inverse problems (Differential equations) -- Periodicals; 515.357
• DOI: 10.1088/1361-6420/abbd7e
• Languages: English
• ISSNs: 0266-5611

Abstract

Abstract: We introduce a general framework for the reconstruction of periodic multivariate functions from finitely many and possibly noisy linear measurements. The reconstruction task is formulated as a penalized convex optimization problem, taking the form of a sum between a convex data fidelity functional and a sparsity-promoting total variation based penalty involving a suitable spline-admissible regularizing operator L. In this context, we establish a periodic representer theorem, showing that the extreme-point solutions are periodic L-splines with less knots than the number of measurements. The main results are specified for the broadest classes of measurement functionals, spline-admissible operators, and convex data fidelity functionals. We exemplify our results for various regularization operators and measurement types (e.g., spatial sampling, Fourier sampling, or square-integrable functions). We also consider the reconstruction of both univariate and multivariate periodic functions.