Ulrich Clarenz

Anisotropic geometric diffusion in surface processing

A new multiscale method in surface processing is presented which combines the image processing methodology based on nonlinear diffusion equations and the theory of geometric evolution problems. Its aim is to smooth discretized surfaces while simultaneously enhancing geometric features such as edges and corners. This is obtained by an anisotropic curvature evolution, where time is the multiscale parameter. Here, the diffusion tensor depends on the shape operator of the evolving surface. A spatial finite element discretization on arbitrary unstructured triangular meshes and a semi-implicit finite difference discretization in time are the building blocks of the easy to code algorithm presented. The systems of linear equations in each timestep are solved by appropriate, preconditioned iterative solvers. Different applications underline the efficiency and flexibility of the presented type of surface processing tool.

A finite element method for surface restoration with smooth boundary conditions

In surface restoration usually a damaged region of a surface has to be replaced by a surface patch which restores the region in a suitable way. In particular one aims for C1-continuity at the patch boundary. The Willmore energy is considered to measure fairness and to allow appropriate boundary conditions to ensure continuity of the normal field. The corresponding L2-gradient flow as the actual restoration process leads to a system of fourth order partial differential equations, which can also be written as a system of two coupled second order equations. As it is well known, fourth order problems require an implicit time discretization. Here a semi-implicit approach is presented which allows large time steps. For the discretization of the boundary condition, two different numerical methods are introduced. Finally, we show applications to different surface restoration problems.

Robust feature detection and local classification for surfaces based on moment analysis

The stable local classification of discrete surfaces with respect to features such as edges and corners or concave and convex regions, respectively, is as quite difficult as well as indispensable for many surface processing applications. Usually, the feature detection is done via a local curvature analysis. If concerned with large triangular and irregular grids, e.g., generated via a marching cube algorithm, the detectors are tedious to treat and a robust classification is hard to achieve. Here, a local classification method on surfaces is presented which avoids the evaluation of discretized curvature quantities. Moreover, it provides an indicator for smoothness of a given discrete surface and comes together with a built-in multiscale. The proposed classification tool is based on local zero and first moments on the discrete surface. The corresponding integral quantities are stable to compute and they give less noisy results compared to discrete curvature quantities. The stencil width for the integration of the moments turns out to be the scale parameter. Prospective surface processing applications are the segmentation on surfaces, surface comparison, and matching and surface modeling. Here, a method for feature preserving fairing of surfaces is discussed to underline the applicability of the presented approach.

Axioms and variational problems in surface parameterization

For a surface patch on a smooth, two-dimensional surface in R^3, low-distortion parameterizations are described in terms of minimizers of suitable energy functionals. Appropriate distortion measures are derived from principles of rational mechanics, closely related to the theory of non-linear elasticity. The parameterization can be optimized with respect to the varying importance of conformality, length preservation and area preservation. A finite element discretization is introduced and a constrained Newton method is used to minimize a corresponding discrete energy. Results of the new approach are compared with other recent parameterization methods.

Processing textured surfaces via anisotropic geometric diffusion

A multiscale method in surface processing is presented which carries over image processing methodology based on nonlinear diffusion equations to the fairing of noisy, textured, parametric surfaces. The aim is to smooth noisy, triangulated surfaces and accompanying noisy textures-as they are delivered by new scanning technology-while enhancing geometric and texture features. For an initial textured surface a fairing method is described which simultaneously processes the texture and the surface. Considering an appropriate coupling of the two smoothing processes one can take advantage of the frequently present strong correlation between edge features in the texture and on the surface edges. The method is based on an anisotropic curvature evolution of the surface itself and an anisotropic diffusion on the processed surface applied to the texture. Here, the involved diffusion tensors depends on a regularized shape operator of the evolving surface and on regularized texture gradients. A spatial finite element discretization on arbitrary unstructured triangular grids and a semi-implicit finite difference discretization in time are the building blocks of the corresponding numerical algorithm. A normal projection is applied to the discrete propagation velocity to avoid tangential drifting in the surface evolution. Different applications underline the efficiency and flexibility of the presented surface processing tool.

Finite elements on point based surfaces

framework efficiently and effectively brings well-known PDE-based processing techniques to the field of point-based surfaces. Our method is based on the construction of local tangent planes and a local Delaunay triangulation of adjacent points projected onto this plane. The definition of tangent spaces relies on moment-based computation with proven scaling and stability properties. Once local couplings are obtained, we are able to easily assemble PDE-specific mass and stiffness matrices and solve corresponding linear systems by standard iterative solvers. We demonstrate our framework by different classes of PDE-based surface processing applications, such as texture synthesis and processing, geometric fairing, and segmentation.

Computational methods for nonlinear image registration

1 Institut fur Mathematik, Gerhard-Mercator Universitat Duisburg, Lotharstrase 63/65, 47048 Duisburg, Germany {clarenz|droske|rumpf}@math.uni-duisburg.de. 2 Lehrstuhl fur Mathematische Optimierung, Mathematisches Institut, Heinrich-Heine Universitat Dusseldorf, Universitatsstrase 1, D-40225 Dusseldorf, Germany. henn@am.uni-duesseldorf.de 3 Lehrstuhl fur Angewandte Mathematik, Mathematisches Institut, Heinrich-Heine Universitat Dusseldorf, Universitatsstrase 1, D-40225 Dusseldorf, Germany. witsch@am.uni-duesseldorf.de Summary. Image registration is the process of the alignment of two or more data sets recorded with the same or different imaging machineries. Especially nonlinear image registration techniques allow the alignment of data sets that are mismatched in a nonuniform manner. Mathematically, this yields a nonlinear ill–conditioned inverse problem. In this presentation, we introduce several computational methods based on variational PDE approaches to obtain an approximate solution of the nonlinear registration problem. In each approach we have to solve a sequence of subproblems. Each subproblem has to be well-posed and should be efficiently solvable.

Fairing of point based surfaces

We present a framework for processing point-based surfaces via partial differential equations (PDEs). Our framework allows an efficient and effective way to bring well-established PDE-based surface processing techniques to the field of point-based representations. We demonstrate the method by a PDE-based surface fairing application

On Generalized Mean Curvature Flow in Surface Processing

Geometric evolution problems for curves and surfaces and especially curvature flow problems are an exciting and already classical mathematical research field. They lead to interesting systems of nonlinear partial differential equations and allow the appropriate mathematical modeling of physical processes such as material interface propagation, fluid free boundary motion, crystal growth.

On Level Set Formulations for Anisotropic Mean Curvature Flow and Surface Diffusion

Anisotropic mean curvature motion and in particular anisotropic surface diffusion play a crucial role in the evolution of material interfaces. This evolution interacts with conservations laws in the adjacent phases on both sides of the interface and are frequently expected to undergo topological chances. Thus, a level set formulation is an appropriate way to describe the propagation. Here we recall a general approach for the integration of geometric gradient flows over level set ensembles and apply it to derive a variational formulation for the level set representation of anisotropic mean curvature motion and anisotropic surface flow. The variational formulation leads to a semi-implicit discretization and enables the use of linear finite elements.