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Optimality of Splines for the Resolution of Linear Inverse Problems with Tikhonov or Total-Variation Regularization

H. Gupta, J. Fageot, M. Unser

SIAM Conference on Imaging Science (SIS'18), Bologna, Italian Republic, June 5-8, 2018, session MS47-1.



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.


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