Splines Are Universal Solutions of Linear Inverse Problems with Generalized TV Regularization

Abstract

Abstract. Splines come in a variety of flavors that can be characterized in terms of some differential operator L. The simplest piecewise-constant model corresponds to the derivative operator. Likewise, one can extend the traditional notion of total variation by considering more general operators than the derivative. This results in the definitions of a generalized total variation seminorm and its corresponding native space, which is further identified as the direct sum of two Banach spaces. We then prove that the minimization of the generalized total variation (gTV), subject to some arbitrary (convex) consistency constraints on the linear measurements of the signal, admits nonuniform L-spline solutions with fewer knots than the number of measurements. This shows that nonuniform splines are universal solutions of continuous-domain linear inverse problems with LASSO, L 1 , or total-variationlike regularization constraints. Remarkably, the type of spline is fully determined by the choice of L and does not depend on the actual nature of the measurements.

Sparsity of solutions for variational inverse problems with finite-dimensional data

“…We can furthermore characterize the extremal points of the ball associated to φ N according to the following theorem. Note that a similar result was also obtained by [31] and [23] in different settings and more restrictive hypotheses. For this purpose, for x ∈ R d , denote by G x the fundamental solution G translated by x, i.e., such that LG x = δ x .…”

“…It also justifies once more the usage of locally convex spaces in the abstract theory. In this setting, as an application of our main theorem, we are able to recover the same result as in [31] and [23].…”

“…In this section we consider the case where φ(u) = Lu M , namely the Radon norm of a linear, translation-invariant scalar differential operator L. This was already treated in [31] and in [23] in different settings. Our goal is to show that our theory applies straightforwardly to this case.…”

Directionality guided non linear diffusion compressed sensing MR image reconstruction

“…One can consider an additional system of n neighborhoods i p , i = 1, … , n, each separated by an angle θ and all neighbors located at the same distance d from p. Following this convention, Figure 2B shows the neighborhood 1 p and Figure 2C illustrate the collection of all such n + 1 neighborhoods of p. These additional neighbors (U(x, y) in Figure 2D, for example) can be estimated from the known neighboring pixel values (red and black pixels) using standard interpolators such as the bilinear interpolator or bicubic spline interpolator. [29][30][31][32][33] For detailed steps, the reader may refer to the Extended Neighborhoods section in the Supporting Information file.…”

Image Edges Resolved Well When Using an Overcomplete Piecewise-Polynomial Model

“…Proximal splitting algorithms are suitable for such a class of tasks. We utilize the Condat proximal algorithm [26], which is able to solve (among more general formulations) problems of the type The connection between (15) and (12) is provided via assigning h 1 = · 21 , h 2 = λ · 21 , L 1 = L v , L 2 = L h reshape(·). To fit the formulation (15), the feasible set is recast in the unconstrained form using the indicator function h 3 = ι {z: y−z 2≤δ} , where ι C denotes the indicator function of a convex set C [27].…”