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TV-based Reconstruction of Periodic Functions

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Document pages: 31 pages

Abstract: We introduce a general framework for the reconstruction of periodicmultivariate functions from finitely many and possibly noisy linearmeasurements. The reconstruction task is formulated as a penalized convexoptimization problem, taking the form of a sum between a convex data fidelityfunctional and a sparsity-promoting total variation based penalty involving asuitable spline-admissible regularizing operator L. In this context, weestablish a periodic representer theorem, showing that the extreme-pointsolutions are periodic L-splines with less knots than the number ofmeasurements. The main results are specified for the broadest classes ofmeasurement functionals, spline-admissible operators, and convex data fidelityfunctionals. We exemplify our results for various regularization operators andmeasurement types (e.g., spatial sampling, Fourier sampling, orsquare-integrable functions). We also consider the reconstruction of bothunivariate and multivariate periodic functions.

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