TV-based Reconstruction of Periodic Functions

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.

arXiv comment31 pages, 10 figures

References in corpus (8)

• The Convex Geometry of Linear Inverse Problems
• Harnessing Sparsity over the Continuum: Atomic Norm Minimization for Super Resolution
• The Sliding Frank-Wolfe Algorithm and its Application to Super-Resolution Microscopy
• Splines are Universal Solutions of Linear Inverse Problems with Generalized-TV regularization
• Continuous-Domain Solutions of Linear Inverse Problems with Tikhonov vs. Generalized TV Regularization
• On the linear convergence rates of exchange and continuous methods for total variation minimization
• Deep Neural Networks with Trainable Activations and Controlled Lipschitz Constant
• Periodic Splines and Gaussian Processes for the Resolution of Linear Inverse Problems

Cited by in corpus (5)

• CPGD: Cadzow Plug-and-Play Gradient Descent for Generalised FRI
• Sampling and Reconstruction of Sparse Signals in Shift-Invariant Spaces: Generalized Shannon's Theorem Meets Compressive Sensing
• Functional Penalised Basis Pursuit on Spheres
• A Fast and Scalable Polyatomic Frank-Wolfe Algorithm for the LASSO
• On the Uniqueness of Solutions for the Basis Pursuit in the Continuum