Abstract. A physically motivated approach is presented to compute a shape average of a given number of shapes. An elastic deformation is assigned to each shape. The shape average is then described as the common image under all elastic deformations of the given shapes, which minimizes the total elastic energy stored in these deformations. The underlying nonlinear elastic energy measures the local change of length, area, and volume. It is invariant under rigid body motions, and isometries are local minimizers. The model is relaxed involving a further energy which measures how well the elastic deformation image of a particular shape matches the average shape, and a suitable shape prior can be considered for the shape average. Shapes are represented via their edge sets, which also allows for an application to averaging image morphologies described via ensembles of edge sets. To make the approach computationally tractable, sharp edges are approximated via phase fields, and a corresponding variational phase field model is derived. Finite elements are applied for the spatial discretization, and a multi-scale alternating minimization approach allows the efficient computation of shape averages in 2D and 3D. Various applications, e. g. averaging the shape of feet or human organs, underline the qualitative properties of the presented approach.

Abstract. The management of intra-fractional respiratory motion is becoming increasingly important in radiation therapy. Based on in advance acquired accurate 3D CT data and intra-fractionally recorded noisy timeof-flight (ToF) range data an improved treatment can be achieved. In this paper, a variational approach for the joint registration of the thorax surface extracted from a CT and a ToF image and the denoising of the ToF image is proposed. This enables a robust intra-fractional full torso surface acquisition and deformation tracking to cope with variations in patient pose and respiratory motion. Thereby, the aim is to improve radiotherapy for patients with thoracic, abdominal and pelvic tumors. The approach combines a Huber norm type regularization of the ToF data and a geometrically consistent treatment of the shape mismatch. The algorithm is tested and validated on synthetic and real ToF/CT data and then evaluated on real ToF data and 4D CT phantom experiments. 1

In its most widespread imaging and vision applications, Ambrosio and Tortorelli (AT) phase field is a technical device for applying gradient descent to Mumford and Shah simultaneous segmentation and restoration functional or its extensions. As such, it forms a diffuse alternative to sharp interfaces or level sets and parametric techniques. The functionality of the AT field, however, is not limited to segmentation and restoration applications. We demonstrate the possibility of coding parts – features that are higher level than edges and boundaries – after incorporating higher level influences via distances and averages. The iteratively extracted parts using the level curves with double point singularities are organized as a proper binary tree. Inconsistencies due to non-generic configurations for level curves as well as due to visual changes such as occlusion are successfully handled once the tree is endowed with a probabilistic structure. As a proof of concept, we present 1) the most probable configurations from our randomized trees; and 2) correspondence matching results between illustrative shape pairs. The work is a significant step towards establishing exponentially decaying diffuse distance fields as bridges between low level visual processing and shape computations.

We demonstrate the possibility of coding parts, features that are higher level than boundaries, using a modified AT field after augmenting the interaction term of the AT energy with a non-local term and weakening the separation into boundary/not-boundary phases. The iteratively extracted parts using the level curves with double point singularities are organized as a proper binary tree. Inconsistencies due to non-generic configurations for level curves as well as due to visual changes such as occlusion are successfully handled once the tree is endowed with a probabilistic structure. The work is a step in establishing the AT function as a bridge between low and high level visual processing.

Fig. 1. Time-discrete geodesics between a cat and a lion and the letters A and B. Geodesic distance is measured on the basis of viscous dissipation inside the objects (color-coded in the middle row from blue, low dissipation, to red, high dissipation), which is induced by a pairwise 1-1 deformation map between consecutive shapes along the discrete geodesic path. Shapes are represented via level set functions, whose level lines are texture-coded in the bottom row for the 2D example.

Abstract. In this chapter we are concerned with variational methods in image analysis. Special attention is paid on free discontinuity approaches of Mumford Shah type and their application in segmentation, matching and motion analysis. We study combined approaches, where one simultaneously relaxes a functional with respect to multiple unknowns. Examples are the simultaneously extraction of edges in two different images for joint image segmentation and image registration or the joint estimation of motion, moving object, and object intensity map. In these approaches the identification of one of the unknowns improves the capability to extract the other as well. Hence, combined methods turn out to be very powerful approaches. Indeed, fundamental tasks in image processing are highly interdependent: Registration of image morphology significantly benefits from previous denoising and structure segmentation. On the other hand, combined information of different image modalities makes shape segmentation significantly more robust. Furthermore, robustness in motion extraction of shapes can be significantly enhanced via a coupling with the detection of edge surfaces in space time and a corresponding feature sensitive space time smoothing. Furthermore, one of the key tools throughout most of the methods to be presented is nonlinear elasticity based on hyperelastic and polyconvex energy functional. Based on first principles from continuum mechanics this allows a flexible description of shape correspondences and in many cased enables to establish existence results and one-to-one mapping properties. Numerical experiments underline the robustness of the presented methods and show applications on medical images and biological experimental data. This chapter is based on a couple of recent articles [8,49,29,30,63] published

Many problems in imaging are actually inverse problems. One reason for this is that conditions and parameters of the physical processes underlying the actual image acquisition are usually not known. Examples for this are the inhomogeneities of the magnet field in magnetic resonance images leading to

The analysis of shapes as elements in a frequently infinite-dimensional space of shapes has attracted increasing attention over the last decade. There are pioneering contributions in the theoretical foundation of shape space as a Riemannian manifold as well as path-breaking applications to quantitative shape comparison, shape recognition, and shape statistics. The aim of this chapter is to adopt a primarily physical perspective on the space of shapes and to relate this to the prevailing geometric perspective. Indeed, we here consider shapes given as boundary contours of volumetric objects, which consist either of a viscous fluid or an elastic solid. In the first case, shapes are transformed into each other via viscous transport of fluid material, and the flow naturally generates a connecting path in the space of shapes. The viscous dissipation rate—the rate at which energy is converted into heat due to friction—can be defined as a metric on an associated Riemannian manifold. Hence, via the computation of shortest transport paths one defines a distance measure between shapes.

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