Exact Discretization of Continuous-Domain Linear Inverse Problems with Generalized TV Regularization Using B-Splines​
Thomas Debarre, EPFL STI LIB

Meeting • 24 August 2017

Abstract
We study continuous-domain linear inverse problems with generalized Total-Variation (gTV), expressed in terms of a regularization operator L. It has recently been proved that such inverse problems have sparse spline solutions, with fewer coefficients than the number of measurements. Moreover, the type of spline solely depends on L (L-splines), and is independent of the measurements. For computational feasibility, the continuous-domain inverse problem can be recast as a discrete, finite-dimensional problem by enforcing the spline knots to be located on a grid. However, expressing the L-spline coefficients in the dictionary basis of the Green's function of L is ill-suited for practical problems due to its infinite support. Instead, we propose to formulate the problem in the B-spline dictionary basis, which leads to better-conditioned system matrices. We therefore define a discrete linear inverse problem in the B-spline basis and propose an algorithmic scheme to compute its sparse solutions. We demonstrate that the latter is computationally feasible for 1D signals when L is an ordinary differential operator.