Variational Tensor-Product Splines

TL;DR

This paper introduces a novel regularization framework for multidimensional inverse problems using tensor-product splines, combining differential operator norms and bounded-variation norms to improve solution structure and localization.

Contribution

It proposes a new regularization method based on tensor-product splines, combining differential operator norms and bounded-variation norms, with theoretical insights on solution extremities and localization effects.

Findings

Extreme points are tensor product of one-dimensional splines.

Number of atoms in solutions is bounded by data points.

Regularization term adapts to localized data through bounded-variation norms.

Abstract

Multidimensional continuous-domain inverse problems are often solved by the minimization of a loss functional, formed as the sum of a data fidelity and a regularization. In this work, we present a new construction where the regularization is itself built as the sum of two terms: i) the M norm of the regularizing operator L1 b L2, with L1 and L2 being two one-dimensional differential operators; ii) a bounded-variation norm that regularizes on the infinite-dimensional nullspace of L1 b L2. In this construction, we show that the extreme points of the solution set are the tensor product of one-dimensional splines, with a number of atoms upper-bounded in term of the number of data points. Further, when the data of the inverse problem is localized, we reveal that the term ii) must take the form of a sum of bounded-variation norms, precomposed with partial derivative of different orders.