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We present two representer theorems that provide the parametric form of the solution(s) of generic linear inverse problems with Tikhonov (p=2) vs. total-variation (p=1) regularization. Remarkably, the solutions in both cases are generalized splines that are tied to the underlying regularization operator L. For p=2, the knots are fixed with basis functions that are smoothed versions of the measurement operator. In the total variation scenario, the solutions are nonuniform L-splines with adaptive (and fewer) knots.
This presentation is part of Minisymposium “MS47 - Splines in Imaging (3 parts)”
organized by: Carolina Beccari (Dept. Mathematics, University of Bologna) , Virginie Uhlmann (EPFL, Lausanne) , Michael Unser (EPFL, Lausanne) .