Markus Grasmair

Sparse regularization with lq penalty term

We consider the stable approximation of sparse solutions to nonlinear operator equations by means of Tikhonov regularization with a subquadratic penalty term. Imposing certain assumptions, which for a linear operator are equivalent to the standard range condition, we derive the usual convergence rate of the regularized solutions in dependence of the noise level δ. Particular emphasis lies on the case, where the true solution is known to have a sparse representation in a given basis. In this case, if the differential of the operator satisfies a certain injectivity condition, we can show that the actual convergence rate improves up to O(δ).

The Equivalence of the Taut String Algorithm and BV-Regularization

It is known that discrete BV-regularization and the taut string algorithm are equivalent. In this paper we extend this result to the continuous case. First we derive necessary equations for the solution of both BV-regularization and the taut string algorithm by computing suitable Gateaux derivatives. The equivalence then follows from a uniqueness result.

Locally Adaptive Total Variation Regularization

We introduce a locally adaptive parameter selection method for total variation regularization applied to image denoising. The algorithm iteratively updates the regularization parameter depending on the local smoothness of the outcome of the previous smoothing step. In addition, we propose an anisotropic total variation regularization step for edge enhancement. Test examples demonstrate the capability of our method to deal with varying, unknown noise levels.

A non-convex PDE scale space

For image filtering applications, it has been observed recently that both diffusion filtering and associated regularization models provide similar filtering properties. The comparison has been performed for regularization functionals with convex penalization functional. In this paper we discuss the relation between non-convex regularization functionals and associated time dependent diffusion filtering techniques (in particular the Mean Curvature Flow equation). Here, the general idea is to approximate an evolution process by a sequence of minimizers of iteratively convexified energy (regularization) functionals.

Generalizations of the Taut String Method

The taut string method is classically used in statistical applications to obtain a sparse estimation for a density given by point measurements. Mostly, a discrete formulation is employed that interpretes the data and the output as piecewise constant splines. This paper deals with the continuous formulation of this algorithm. We show that it is able to deal with continuous data as well as with discrete data interpreted as Dirac measures. In fact, any one-dimensional finite signed Radon measure is suited as input for the method. Moreover, we study the usage of tubes of nonconstant diameter. Examples indicate that such tubes can be useful in various applications. An existence and uniqueness theorem is given for the continuous formulation of the taut string algorithm with arbitrary tubes of nonnegative diameter.

Relaxation of Nonlocal Singular Integrals

ABSTRACT In this paper we study well-posedness of a class of nonconvex variational principles arising in regularization theory for denoising of data with sampling errors and level set regularization methods for inverse problems. These models result in minimization of nonconvex, singular functionals involving (possibly) non-local operators.