Wolfgang Ring

A Second Order Shape Optimization Approach for Image Segmentation

The problem of segmentation of a given image using the active contour technique is considered. An abstract calculus to find appropriate speed functions for active contour models in image segmentation or related problems based on variational principles is presented. The speed method from shape sensitivity analysis is used to derive speed functions which correspond to gradient or Newton-type directions for the underlying optimization problem. The Newton-type speed function is found by solving an elliptic problem on the current active contour in every time step. Numerical experiments comparing the classical gradient method with Newtons method are presented.

A Mumford-Shah level-set approach for the inversion and segmentation of X-ray tomography data

A level-set based approach for the determination of a piecewise constant density function from data of its Radon transform is presented. Simultaneously, a segmentation of the reconstructed density is obtained. The segmenting contour and the corresponding density are found as minimizers of a Mumford-Shah like functional over the set of admissible contours and - for a fixed contour - over the space of piecewise constant densities which may be discontinuous across the contour. Shape sensitivity analysis is used to find a descent direction for the cost functional which leads to an update formula for the contour in the level-set framework. The descent direction can be chosen with respect to different metrics. The use of an L^2-type and an H^1-type metric is proposed and the corresponding steepest descent flow equations are derived. A heuristic approach for the insertion of additional components of the density is presented. The method is tested for several data sets including synthetic as well as real-world data. It is shown that the method works especially well for large data noise (~10% noise). The choice of the H^1-metric for the determination of the descent direction is found to have positive effect on the number of level-set steps necessary for finding the optimal contours and densities.

An Inexact Newton-CG-Type Active Contour Approach for the Minimization of the Mumford-Shah Functional

The problem of segmentation of a given gray scale image by minimization of the Mumford-Shah functional is considered. The minimization problem is formulated as a shape optimization problem where the contour which separates homogeneous regions is the (geometric) optimization variable. Expressions for first and second order shape sensitivities are derived using the speed method from classical shape sensitivity calculus. Second order information (the shape Hessian of the cost functional) is used to set up a Newton-type algorithm, where a preconditioning operator is applied to the gradient direction to obtain a better descent direction. The issue of positive definiteness of the shape Hessian is addressed in a heuristic way. It is suggested to use a positive definite approximation of the shape Hessian as a preconditioner for the gradient direction. The descent vector field is used as speed vector field in the level set formulation for the propagating contour. The implementation of the algorithm is discussed in some detail. Numerical experiments comparing gradient and Newton-type flows for different images are presented.

A Mumford–Shah Level‐Set Approach for Geometric Image Registration

A new method for nonrigid registration of multimodal images is presented. Due to the large interdependence of segmentation and registration, the approach is based on simultaneous segmentation and edge alignment. The two processes are directly coupled and thus benefit from using complementary information of the entire underlying data set. The approach is formulated as a bivariate, variational, free discontinuity problem in the Mumford–Shah framework. A geometric variable describing the contour set and a functional variable which represents the underlying deformation are simultaneously identified. The contour set is represented by a level‐set function. We derive a regularized gradient flow and describe an efficient numerical implementation using finite element discretization and multigrid techniques. Finally, we illustrate the method in several applications, such as multimodal intrapatient registration and reconstruction by registration to a reference object.

Medical Image Registration and Interpolation by Optical Flow with Maximal Rigidity

In this paper a variational method for registering or mapping like points in medical images is proposed and analyzed. The proposed variational principle penalizes a departure from rigidity and thereby provides a natural generalization of strictly rigid registration techniques used widely in medical contexts. Difficulties with finite displacements are elucidated, and alternative infinitesimal displacements are developed for an optical flow formulation which also permits image interpolation. The variational penalty against non-rigid flows provides sufficient regularization for a well-posed minimization and yet does not rule out irregular registrations corresponding to an object excision. Image similarity is measured by penalizing the variation of intensity along optical flow trajectories. The approach proposed here is also independent of the order in which images are taken. For computations, a lumped finite element Eulerian discretization is used to solve for the optical flow. Also, a Lagrangian integration of the intensity along optical flow trajectories has the advantage of prohibiting diffusion among trajectories which would otherwise blur interpolated images. The subtle aspects of the methods developed are illustrated in terms of simple examples, and the approach is finally applied to the registration of magnetic resonance images.

Regularization of nonlinear illposed problems with closed operators

In this paper Tikhonov regularization for nonlinear illposed problems is investigated. The regularization term is characterized by a closed linear operator, permitting seminorm regularization in applications. Results for existence, stability, convergence and con- vergence rates of the solution of the regularized problem in terms of the noise level are given. An illustrating example involving parameter estimation for a one dimensional stationary heat equation is given.

Optimal Control for the Thermistor Problem

This paper is concerned with the state-constrained optimal control of the two-dimensional thermistor problem, a quasi-linear coupled system of a parabolic and elliptic PDE with mixed boundary conditions. This system models the heating of a conducting material by means of direct current. Existence, uniqueness, and continuity for the state system are derived by employing maximal elliptic and parabolic regularity. By similar arguments the linearized state system is discussed, while the adjoint system involving measures is investigated using a duality argument. These results allow us to derive first-order necessary conditions for the optimal control problem.

A level set approach for the solution of a state-constrained optimal control problem

Summary.State constrained optimal control problems for linear elliptic partial differential equations are considered. The corresponding first order optimality conditions in primal-dual form are analyzed and linked to a free boundary problem resulting in a novel algorithmic approach with the boundary (interface) between the active and inactive sets as optimization variable. The new algorithm is based on the level set methodology. The speed function involved in the level set equation for propagating the interface is computed by utilizing techniques from shape optimization. Encouraging numerical results attained by the new algorithm are reported on.

Regularization of ill-posed Mumford―Shah models with perimeter penalization

The problems of existence, stability and convergence of minimizers of a Mumford–Shah functional for the simultaneous reconstruction and segmentation of a distributed parameter in an ill-posed operator equation are considered. An appropriate set of regular partitions of the domain of the distributed parameter is introduced and a distance for partitions is constructed. Pairs of partitions and coefficient vectors form the space of feasible variables for the Mumford–Shah functional. Each such pair defines a piecewise constant function. The relation between convergence of geometric and vectorial variables on the one hand and convergence of the associated piecewise constant functions in L1 on the other is investigated under the assumption of injectivity of the operator equation on the set of piecewise constant parameters. Compactness and semi-continuity results for regular partitions are presented and used to prove existence, stability with respect to perturbations in the data and convergence of the solution if the error level goes to zero. Numerical experiments in a computerized tomography setting demonstrate the validity of the derived convergence result.

Shape design with great geometrical deformations using continuously moving finite element nodes

In this paper design sensitivity analysis is applied to solve the TEAM workshop problem 25. In order to justify the use of a gradient method, it is necessary to assume a continuously differentiable dependence of the stiffness matrix on the design parameters. Since design sensitivity analysis is mainly applicable to optimization problems, where the geometrical parameters undergo small changes only - which is not the case for the problem investigated in this paper-a procedure is proposed, which allows this method to be applied also when the changes in geometry are significant.