Numerical Methods for Hamilton-Jacobi-Bellman Equations with Applications

Hamilton-Jacobi-Bellman (HJB) equations are nonlinear controlled partial differential equations (PDEs). In this thesis, we propose various numerical methods for HJB equations arising from three specific applications. First, we study numerical methods for the HJB equation coupled with a Kolmogorov-Fokker-Planck (KFP) equation arising from mean field games. In order to solve the nonlinear discretized systems efficiently, we propose a multigrid method. The main novelty of our approach is that we subtract artificial viscosity from the direct discretization coarse grid operators, such that the coarse grid error estimations are more accurate. The convergence rate of the proposed multigrid method is mesh-independent and faster than the existing methods in the literature. Next, we investigate numerical methods for the HJB formulation that arises from the mass transport image registration model. We convert the PDE of the model (a Monge-Ampère equation) to an equivalent HJB equation, propose a monotone mixed discretization, and prove that it is guaranteed to converge to the viscosity solution. Then we propose multigrid methods for the mixed discretization, where we set wide stencil points as coarse grid points, use injection at wide stencil points as the restriction, and achieve a mesh-independent convergence rate. Moreover, we propose a novel periodic boundary condition for the image registration PDE, such that when two images are related by a combination of a translation and a non-rigid deformation, the numerical scheme recovers the underlying transformation correctly. Finally, we propose a deep neural network framework for the HJB equations emerging from the study of American options in high dimensions. We convert the HJB equation to an equivalent Backward Stochastic Differential Equation (BSDE), introduce the least squares residual of the BSDE as the loss function, and propose a new neural network architecture that utilizes the domain knowledge of American options. Our proposed framework yields American option prices and deltas on the entire spacetime, not only at a given point. The computational cost of the proposed approach is quadratic in dimension, which addresses the curse of dimensionality issue that state-of-the-art approaches suffer

Automatically Controlled Morphing of 2D Shapes with Textures

This paper deals with 2D image transformations from a perspective of a 3D heterogeneous shape modeling and computer animation. Shape and image morphing techniques have attracted a lot of attention in artistic design, computer animation, and interactive and streaming applications. We present a novel method for morphing between two topologically arbitrary 2D shapes with sophisticated textures (raster color attributes) using a metamorphosis technique called space-time blending (STB) coupled with space-time transfinite interpolation. The method allows for a smooth transition between source and target objects by generating in-between shapes and associated textures without setting any correspondences between boundary points or features. The method requires no preprocessing and can be applied in 2D animation when position and topology of source and target objects are significantly different. With the conversion of given 2D shapes to signed distance fields, we have detected a number of problems with directly applying STB to them. We propose a set of novel and mathematically substantiated techniques, providing automatic control of the morphing process with STB and an algorithm of applying those techniques in combination. We illustrate our method with applications in 2D animation and interactive applications

On Visualizing Branched Surface: an Angle/Area Preserving Approach

The techniques of surface deformation and mapping are useful tools for the visualization of medical surfaces, especially for highly undulated or branched surfaces. In this thesis, two algorithms are presented for flattened visualizations of multi-branched medical surfaces, such as vessels. The first algorithm is an angle preserving approach, which is based on conformal analysis. The mapping function is obtained by minimizing two Dirichlet functionals. On a triangulated representation of vessel surfaces, this algorithm can be implemented efficiently using a finite element method. The second algorithm adjusts the result from conformal mapping to produce a flattened representation of the original surface while preserving areas. It employs the theory of optimal mass transport via a gradient descent approach. A new class of image morphing algorithms is also considered based on the theory of optimal mass transport. The mass moving energy functional is revised by adding an intensity penalizing term, in order to reduce the undesired "fading" effects. It is a parameter free approach. This technique has been applied on several natural and medical images to generate in-between image sequences.Ph.D.Allen Tannenbaum Committee Chair Anthony J. Yezzi, Committee Member; James Gruden, Committee Member; May D. Wang, Committee Member; Oskar Skrinjar, Committee Membe

Variational Methods in Shape Space

This dissertation deals with the application of variational methods in spaces of geometric shapes. In particular, the treated topics include shape averaging, principal component analysis in shape space, computation of geodesic paths in shape space, as well as shape optimisation. Chapter 1 provides a brief overview over the employed models of shape space. Geometric shapes are identified with two- or three-dimensional, deformable objects. Deformations will be described via physical models; in particular, the objects will be interpreted as consisting of either a hyperelastic solid or a viscous liquid material. Furthermore, the description of shapes via phase fields or level sets is briefly introduced. Chapter 2 reviews different and related approaches to shape space modelling. References to related topics in image segmentation and registration are also provided. Finally, the relevant shape optimisation literature is introduced. Chapter 3 recapitulates the employed concepts from continuum mechanics and phase field modelling and states basic theoretical results needed for the later analysis. Chapter 4 addresses the computation of shape averages, based on a hyperelastic notion of shape dissimilarity: The dissimilarity between two shapes is measured as the minimum deformation energy required to deform the first into the second shape. A corresponding phase-field model is introduced, analysed, and finally implemented numerically via finite elements. A principal component analysis of shapes, which is consistent with the previously introduced average, is considered in Chapter 5. Elastic boundary stresses on the average shape are used as representatives of the input shapes in a linear vector space. On these linear representatives, a standard principal component analysis can be performed, where the employed covariance metric should be properly chosen to depend on the input shapes. Chapter 6 interprets shapes as belonging to objects made of a viscous liquid and correspondingly defines geodesic paths between shapes. The energy of a path is given as the total physical dissipation during the deformation of an object along the path. A rigid body motion invariant time discretisation is achieved by approximating the dissipation along a path segment by the deformation energy of a small solid deformation. The numerical implementation is based on level sets. Chapter 7 is concerned with the optimisation of the geometry and topology of solid structures that are subject to a mechanical load. Given the load configuration, the structure rigidity, its volume, and its surface area shall be optimally balanced. A phase field model is devised and analysed for this purpose. In this context, the use of nonlinear elasticity allows to detect buckling phenomena which would be ignored in linearised elasticity

Signal Processing on Textured Meshes

In this thesis we extend signal processing techniques originally formulated in the context of image processing to techniques that can be applied to signals on arbitrary triangles meshes. We develop methods for the two most common representations of signals on triangle meshes: signals sampled at the vertices of a finely tessellated mesh, and signals mapped to a coarsely tessellated mesh through texture maps. Our first contribution is the combination of Lagrangian Integration and the Finite Elements Method in the formulation of two signal processing tasks: Shock Filters for texture and geometry sharpening, and Optical Flow for texture registration. Our second contribution is the formulation of Gradient-Domain processing within the texture atlas. We define a function space that handles chart discontinuities, and linear operators that capture the metric distortion introduced by the parameterization. Our third contribution is the construction of a spatiotemporal atlas parameterization for evolving meshes. Our method introduces localized remeshing operations and a compact parameterization that improves geometry and texture video compression. We show temporally coherent signal processing using partial correspondences

Complexity Reduction in Image-Based Breast Cancer Care

The diversity of malignancies of the breast requires personalized diagnostic and therapeutic decision making in a complex situation. This thesis contributes in three clinical areas: (1) For clinical diagnostic image evaluation, computer-aided detection and diagnosis of mass and non-mass lesions in breast MRI is developed. 4D texture features characterize mass lesions. For non-mass lesions, a combined detection/characterisation method utilizes the bilateral symmetry of the breast s contrast agent uptake. (2) To improve clinical workflows, a breast MRI reading paradigm is proposed, exemplified by a breast MRI reading workstation prototype. Instead of mouse and keyboard, it is operated using multi-touch gestures. The concept is extended to mammography screening, introducing efficient navigation aids. (3) Contributions to finite element modeling of breast tissue deformations tackle two clinical problems: surgery planning and the prediction of the breast deformation in a MRI biopsy device

Mathematical Imaging and Surface Processing

Within the last decade image and geometry processing have become increasingly rigorous with solid foundations in mathematics. Both areas are research fields at the intersection of different mathematical disciplines, ranging from geometry and calculus of variations to PDE analysis and numerical analysis. The workshop brought together scientists from all these areas and a fruitful interplay took place. There was a lively exchange of ideas between geometry and image processing applications areas, characterized in a number of ways in this workshop. For example, optimal transport, first applied in computer vision is now used to define a distance measure between 3d shapes, spectral analysis as a tool in image processing can be applied in surface classification and matching, and so on. We have also seen the use of Riemannian geometry as a powerful tool to improve the analysis of multivalued images. This volume collects the abstracts for all the presentations covering this wide spectrum of tools and application domains