Continuous-Domain Solutions of Linear Inverse Problems with Tikhonov vs. Generalized TV Regularization

02/05/2018
by   Harshit Gupta, et al.
0

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 l_1 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.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
09/19/2019

Non-smooth variational regularization for processing manifold-valued data

Many methods for processing scalar and vector valued images, volumes and...
research
02/14/2023

Sparse Bayesian Inference with Regularized Gaussian Distributions

Regularization is a common tool in variational inverse problems to impos...
research
12/11/2018

Convex Regularization and Representer Theorems

We establish a result which states that regularizing an inverse problem ...
research
03/02/2019

A unifying representer theorem for inverse problems and machine learning

The standard approach for dealing with the ill-posedness of the training...
research
12/14/2019

A subspace-accelerated split Bregman method for sparse data recovery with joint l1-type regularizers

We propose a subspace-accelerated Bregman method for the linearly constr...
research
06/26/2018

On Representer Theorems and Convex Regularization

We establish a general principle which states that regularizing an inver...
research
02/10/2018

A generalized matrix Krylov subspace method for TV regularization

This paper presents an efficient algorithm to solve total variation (TV)...

Please sign up or login with your details

Forgot password? Click here to reset