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Continuous-Domain Signal Reconstruction Using Lp-Norm Regularization

P. Bohra, M. Unser

IEEE Transactions on Signal Processing, vol. 68, pp. 4543-4554, August 3, 2020.



We focus on the generalized-interpolation problem. There, one reconstructs continuous-domain signals that honor discrete data constraints. This problem is infinite-dimensional and ill-posed. We make it well-posed by imposing that the solution balances data fidelity and some Lp-norm regularization. More specifically, we consider p ≥ 1 and the multi-order derivative regularization operator L = DN0. We reformulate the regularized problem exactly as a finite-dimensional one by restricting the search space to a suitable space of polynomial splines with knots on a uniform grid. Our splines are represented in a B-spline basis, which results in a well-conditioned discretization. For a sufficiently fine grid, our search space contains functions that are arbitrarily close to the solution of the underlying problem where our constraint that the solution must live in a spline space would have been lifted. This remarkable property is due to the approximation power of splines. We use the alternating-direction method of multipliers along with a multiresolution strategy to compute our solution. We present numerical results for spatial and Fourier interpolation. Through our experiments, we investigate features induced by the Lp-norm regularization, namely, sparsity, regularity, and oscillatory behavior.


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