Abstract not found

Abstract not found

We present an algorithmical concept for modeling and simulation with partial differential equations (PDEs) in image based computing where the computational geometry is defined through previously segmented image data. Such problems occur in applications from biology and medicine where the underlying image data has been acquired through, e.g. computed tomography (CT), magnetic resonance imaging (MRI) or electron microscopy (EM). Based on a level-set description of the computational domain, our approach is capable of automatically providing suitable composite finite element functions that resolve the complicated shapes in the medical/biological data set. It is efficient in the sense that the traversal of the grid (and thus assembling matrices for finite element computations) inherits the efficiency of uniform grids away from complicated structures. The method’s efficiency heavily depends on precomputed lookup tables in the vicinity of the domain boundary or interface. A suitable multigrid method is used for an efficient solution of the systems of equations resulting from the composite finite element discretization. The paper focuses on both algorithmical and implementational details. Scalar and vector valued model problems as well as real applications underline the usability of our approach.

Abstract—The Mumford-Shah functional has had a major impact on a variety of image analysis problems including image segmentation and filtering and, despite being introduced over two decades ago, it is still in widespread use. Present day optimization of the Mumford-Shah functional is predominated by active contour methods. Until recently, these formulations necessitated optimization of the contour by evolving via gradient descent, which is known for its overdependence on initialization and the tendency to produce undesirable local minima. In order to reduce these problems, we reformulate the corresponding Mumford-Shah functional on an arbitrary graph and apply the techniques of combinatorial optimization to produce a fast, lowenergy solution. In contrast to traditional optimization methods, use of these combinatorial techniques necessitates consideration of the reconstructed image outside of its usual boundary, requiring additionally the inclusion of regularization for generating these values. The energy of the solution provided by this graph formulation is compared with the energy of the solution computed via traditional gradient descent-based narrow-band level set methods. This comparison demonstrates that our graph formulation and optimization produces lower energy solutions than the traditional gradient descent based contour evolution methods in significantly less time. Finally, we demonstrate the usefulness of the graph formulation to apply the Mumford-Shah functional to new applications such as point clustering and filtering of non-uniformly sampled images. Index Terms—Level sets, active contours, piecewise smooth Mumford-Shah, combinatorial optimization, graph reformulation I.

gemeinschaft getragenen Sonderforschungsbereichs 611 an der Universität

The goal of this paper is to study optimal transportation problems and gradient flows of probability measures on the Wiener space, based on and extending fundamental results of Feyel-Üstünel. Carrying out the program of Ambrosio-Gigli-Savaré, we present a complete characterization of the derivative processes for certain class of absolutely continuous curves. We prove existence of the gradient flow curves for the relative entropy w.r.t. the Wiener measure and identify these gradient flow curves with solutions of the Ornstein-Uhlenbeck evolution equation.

Summary. This paper is concerned with the treatment of essential boundary conditions in meshfree methods. In particular we focus on the particle–partition of unity method (PPUM). However, the proposed technique is applicable to any partition of unnity based approach. We present an efficient scheme for the automatic construction of a direct splitting of a PPUM function space into the degrees of freedom suitable for the approximation of the Dirichlet data and the degrees of freedom that remain for the approximation of the PDE by simple linear algebra. Notably, our approach requires no restrictions on the distribution of the discretization points nor on the employed (local) approximation spaces. We attain the splitting of the global function space from the respective direct splittings of the employed local approximation spaces. Hence, the global splitting can be computed with (sub-)linear complexity. Due to this direct splitting of the meshfree PPUM function space we can implement a conforming local treatment of essential boundary data so that the realization of Dirichlet boundary values in the meshfree PPUM is straightforward. The presented approach yields an optimally convergent scheme which is demonstrated by the presented numerical results. Key words: meshfree method, partition of unity method, essential boundary conditions, Nitsche’s method 1

gemeinschaft getragenen Sonderforschungsbereichs 611 an der Universität Bonn entstanden und als Manuskript vervielfältigt worden. Bonn, März 2008 Synchronizability of a stochastic version of FitzHugh-Nagumo type neural oscillator networks

Abstract—Generating 3-D images of the heart during interven-tional procedures is a significant challenge. In addition to real time fluoroscopy, angiographic C-arm systems can also now be used to generate 3-D/4-D CT images on the same system. One protocol for cardiac CT uses ECG triggered multisweep scans. A 3-D volume of the heart at a particular cardiac phase is then reconstructed by applying Feldkamp (FDK) reconstruction to the projection images with retrospective ECG gating. In this work we introduce a unified framework for heart motion estimation and dynamic cone-beam reconstruction using motion corrections. The benefits of motion correction are 1) increased temporal and spatial resolution by removing cardiac motion which may still exist in the ECG gated data sets and 2) increased signal-to-noise ratio (SNR) by using more projection data than is used in standard ECG gated methods. Three signal-enhanced reconstruction methods

gemeinschaft getragenen Sonderforschungsbereichs 611 an der Universität Bonn entstanden und als Manuskript vervielfältigt worden. Bonn, April 2008 A phase field model for phospholipid surfactant