Optimality of splines for the resolution of linear inverse problems with Tikhonov or total-variation regularizationMS47

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) .

Michael Unser (EPFL, Lausanne)
Gupta Harshit (EPFL, Lausanne)
Fageot Julien (EPFL, Lausanne)
computed tomography, image reconstruction, inverse problems, splines