536 research outputs found
Splines Are Universal Solutions of Linear Inverse Problems with Generalized TV Regularization
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, , 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
Periodic Splines and Gaussian Processes for the Resolution of Linear Inverse Problems
This paper deals with the resolution of inverse problems in a periodic
setting or, in other terms, the reconstruction of periodic continuous-domain
signals from their noisy measurements. We focus on two reconstruction
paradigms: variational and statistical. In the variational approach, the
reconstructed signal is solution to an optimization problem that establishes a
tradeoff between fidelity to the data and smoothness conditions via a quadratic
regularization associated to a linear operator. In the statistical approach,
the signal is modeled as a stationary random process defined from a Gaussian
white noise and a whitening operator; one then looks for the optimal estimator
in the mean-square sense. We give a generic form of the reconstructed signals
for both approaches, allowing for a rigorous comparison of the two.We fully
characterize the conditions under which the two formulations yield the same
solution, which is a periodic spline in the case of sampling measurements. We
also show that this equivalence between the two approaches remains valid on
simulations for a broad class of problems. This extends the practical range of
applicability of the variational method
Continuous-Domain Solutions of Linear Inverse Problems with Tikhonov vs. Generalized TV Regularization
We consider linear inverse problems that are formulated in the continuous
domain. The object of recovery is a function that is assumed to minimize a
convex objective functional. The solutions are constrained by imposing a
continuous-domain regularization. We derive the parametric form of the solution
(representer theorems) for Tikhonov (quadratic) and generalized total-variation
(gTV) regularizations. We show that, in both cases, the solutions are splines
that are intimately related to the regularization operator. In the Tikhonov
case, the solution is smooth and constrained to live in a fixed subspace that
depends on the measurement operator. By contrast, the gTV regularization
results in a sparse solution composed of only a few dictionary elements that
are upper-bounded by the number of measurements and independent of the
measurement operator. Our findings for the gTV regularization resonates with
the minimization of the norm, which is its discrete counterpart and also
produces sparse solutions. Finally, we find the experimental solutions for some
measurement models in one dimension. We discuss the special case when the gTV
regularization results in multiple solutions and devise an algorithm to find an
extreme point of the solution set which is guaranteed to be sparse
On the Uniqueness of Inverse Problems with Fourier-domain Measurements and Generalized TV Regularization
We study the super-resolution problem of recovering a periodic
continuous-domain function from its low-frequency information. This means that
we only have access to possibly corrupted versions of its Fourier samples up to
a maximum cut-off frequency. The reconstruction task is specified as an
optimization problem with generalized total-variation regularization involving
a pseudo-differential operator. Our special emphasis is on the uniqueness of
solutions. We show that, for elliptic regularization operators (e.g., the
derivatives of any order), uniqueness is always guaranteed. To achieve this
goal, we provide a new analysis of constrained optimization problems over Radon
measures. We demonstrate that either the solutions are always made of Radon
measures of constant sign, or the solution is unique. Doing so, we identify a
general sufficient condition for the uniqueness of the solution of a
constrained optimization problem with TV-regularization, expressed in terms of
the Fourier samples.Comment: 20 page
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