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.

Numerical Methods in Shape Spaces and Optimal Branching Patterns

The contribution of this thesis is twofold. The main part deals with numerical methods in the context of shape space analysis, where the shape space at hand is considered as a Riemannian manifold. In detail, we apply and extend the time-discrete geodesic calculus (established by Rumpf and Wirth [WBRS11, RW15]) to the space of discrete shells, i.e. triangular meshes with fixed connectivity. The essential building block is a variational time-discretization of geodesic curves, which is based on a local approximation of the squared Riemannian distance on the manifold. On physical shape spaces this approximation can be derived e.g. from a dissimilarity measure. The dissimilarity measure between two shell surfaces can naturally be defined as an elastic deformation energy capturing both membrane and bending distortions. Combined with a non-conforming discretization of a physically sound thin shell model the time-discrete geodesic calculus applied to the space of discrete shells is shown to be suitable to solve important problems in computer graphics and animation. To extend the existing calculus, we introduce a generalized spline functional based on the covariant derivative along a curve in shape space whose minimizers can be considered as Riemannian splines. We establish a corresponding time-discrete functional that fits perfectly into the framework of Rumpf and Wirth, and prove this discretization to be consistent. Several numerical simulations reveal that the optimization of the spline functional—subject to appropriate constraints—can be used to solve the multiple interpolation problem in shape space, e.g. to realize keyframe animation. Based on the spline functional, we further develop a simple regression model which generalizes linear regression to nonlinear shape spaces. Numerical examples based on real data from anatomy and botany show the capability of the model. Finally, we apply the statistical analysis of elastic shape spaces presented by Rumpf and Wirth [RW09, RW11] to the space of discrete shells. To this end, we compute a Fréchet mean within a class of shapes bearing highly nonlinear variations and perform a principal component analysis with respect to the metric induced by the Hessian of an elastic shell energy. The last part of this thesis deals with the optimization of microstructures arising e.g. at austenite-martensite interfaces in shape memory alloys. For a corresponding scalar problem, Kohn and Müller [KM92, KM94] proved existence of a minimizer and provided a lower and an upper bound for the optimal energy. To establish the upper bound, they studied a particular branching pattern generated by mixing two different martensite phases. We perform a finite element simulation based on subdivision surfaces that suggests a topologically different class of branching patterns to represent an optimal microstructure. Based on these observations we derive a novel, low dimensional family of patterns and show—numerically and analytically—that our new branching pattern results in a significantly better upper energy bound

Geometric Surface Processing and Virtual Modeling

In this work we focus on two main topics "Geometric Surface Processing" and "Virtual Modeling". The inspiration and coordination for most of the research work contained in the thesis has been driven by the project New Interactive and Innovative Technologies for CAD (NIIT4CAD), funded by the European Eurostars Programme. NIIT4CAD has the ambitious aim of overcoming the limitations of the traditional approach to surface modeling of current 3D CAD systems by introducing new methodologies and technologies based on subdivision surfaces in a new virtual modeling framework. These innovations will allow designers and engineers to transform quickly and intuitively an idea of shape in a high-quality geometrical model suited for engineering and manufacturing purposes. One of the objective of the thesis is indeed the reconstruction and modeling of surfaces, representing arbitrary topology objects, starting from 3D irregular curve networks acquired through an ad-hoc smart-pen device. The thesis is organized in two main parts: "Geometric Surface Processing" and "Virtual Modeling". During the development of the geometric pipeline in our Virtual Modeling system, we faced many challenges that captured our interest and opened new areas of research and experimentation. In the first part, we present these theories and some applications to Geometric Surface Processing. This allowed us to better formalize and give a broader understanding on some of the techniques used in our latest advancements on virtual modeling and surface reconstruction. The research on both topics led to important results that have been published and presented in articles and conferences of international relevance

Total Generalized Variation for Manifold-valued Data

In this paper we introduce the notion of second-order total generalized variation (TGV) regularization for manifold-valued data in a discrete setting. We provide an axiomatic approach to formalize reasonable generalizations of TGV to the manifold setting and present two possible concrete instances that fulfill the proposed axioms. We provide well-posedness results and present algorithms for a numerical realization of these generalizations to the manifold setup. Further, we provide experimental results for synthetic and real data to further underpin the proposed generalization numerically and show its potential for applications with manifold-valued data

Discrete Riemannian Calculus and A Posteriori Error Control on Shape Spaces

In this thesis, a novel discrete approximation of the curvature tensor on Riemannian manifolds is derived, efficient methods to interpolate and extrapolate images in the context of the time discrete metamorphosis model are analyzed, and an a posteriori error estimator for the binary Mumford–Shah model is examined. Departing from the variational time discretization on (possibly infinite-dimensional) Riemannian manifolds originally proposed by Rumpf and Wirth, in which a consistent time discrete approximation of geodesic curves, the logarithm, the exponential map and parallel transport is analyzed, we construct the discrete curvature tensor and prove its convergence under certain smoothness assumptions. To this end, several time discrete parallel transports are applied to suitably rescaled tangent vectors, where each parallel transport is computed using Schild’s ladder. The associated convergence proof essentially relies on multiple Taylor expansions incorporating symmetry and scaling relations. In several numerical examples we validate this approach for surfaces. The by now classical flow of diffeomorphism approach allows the transport of image intensities along paths in time, which are characterized by diffeomorphisms, and the brightness of each image particle is assumed to be constant along each trajectory. As an extension, the metamorphosis model proposed by Trouvé, Younes and coworkers allows for intensity variations of the image particles along the paths, which is reflected by an additional penalization term appearing in the energy functional that quantifies the squared weak material derivative. Taking into account the aforementioned time discretization, we propose a time discrete metamorphosis model in which the associated time discrete path energy consists of the sum of squared L2-mismatch functionals of successive square-integrable image intensity functions and a regularization functional for pairwise deformations. Our main contributions are the existence proof of time discrete geodesic curves in the context of this model, which are defined as minimizers of the time discrete path energy, and the proof of the Mosco-convergence of a suitable interpolation of the time discrete to the time continuous path energy with respect to the L2-topology. Using an alternating update scheme as well as a multilinear finite element respectively cubic spline discretization for the images and deformations allows to efficiently compute time discrete geodesic curves. In several numerical examples we demonstrate that time discrete geodesics can be robustly computed for gray-scale and color images. Taking into account the time discretization of the metamorphosis model we define the discrete exponential map in the space of images, which allows image extrapolation of arbitrary length for given weakly differentiable initial images and variations. To this end, starting from a suitable reformulation of the Euler–Lagrange equations characterizing the one-step extrapolation a fixed point iteration is employed to establish the existence of critical points of the Euler–Lagrange equations provided that the initial variation is small in L2. In combination with an implicit function type argument requiring H1-closeness of the initial variation one can prove the local existence as well as the local uniqueness of the discrete exponential map. The numerical algorithm for the one-step extrapolation is based on a slightly modified fixed point iteration using a spatial Galerkin scheme to obtain the optimal deformation associated with the unknown image, from which the unknown image itself can be recovered. To prove the applicability of the proposed method we compute the extrapolated image path for real image data. A common tool to segment images and shapes into multiple regions was developed by Mumford and Shah. The starting point to derive a posteriori error estimates for the binary Mumford–Shah model, which is obtained by restricting the original model to two regions, is a uniformly convex and non-constrained relaxation of the binary model following the work by Chambolle and Berkels. In particular, minimizers of the binary model can be exactly recovered from minimizers of the relaxed model via thresholding. Then, applying duality techniques proposed by Repin and Bartels allows deriving a consistent functional a posteriori error estimate for the relaxed model. Afterwards, an a posteriori error estimate for the original binary model can be computed incorporating a suitable cut-out argument in combination with the functional error estimate. To calculate minimizers of the relaxed model on an adaptive mesh described by a quadtree structure, we employ a primal-dual as well as a purely dual algorithm. The quality of the error estimator is analyzed for different gray-scale input images

Understanding and advancing PDE-based image compression

This thesis is dedicated to image compression with partial differential equations (PDEs). PDE-based codecs store only a small amount of image points and propagate their information into the unknown image areas during the decompression step. For certain classes of images, PDE-based compression can already outperform the current quasi-standard, JPEG2000. However, the reasons for this success are not yet fully understood, and PDE-based compression is still in a proof-of-concept stage. With a probabilistic justification for anisotropic diffusion, we contribute to a deeper insight into design principles for PDE-based codecs. Moreover, by analysing the interaction between efficient storage methods and image reconstruction with diffusion, we can rank PDEs according to their practical value in compression. Based on these observations, we advance PDE-based compression towards practical viability: First, we present a new hybrid codec that combines PDE- and patch-based interpolation to deal with highly textured images. Furthermore, a new video player demonstrates the real-time capacities of PDE-based image interpolation and a new region of interest coding algorithm represents important image areas with high accuracy. Finally, we propose a new framework for diffusion-based image colourisation that we use to build an efficient codec for colour images. Experiments on real world image databases show that our new method is qualitatively competitive to current state-of-the-art codecs.Diese Dissertation ist der Bildkompression mit partiellen Differentialgleichungen (PDEs, partial differential equations) gewidmet. PDE-Codecs speichern nur einen geringen Anteil aller Bildpunkte und transportieren deren Information in fehlende Bildregionen. In einigen Fällen kann PDE-basierte Kompression den aktuellen Quasi-Standard, JPEG2000, bereits schlagen. Allerdings sind die Gründe für diesen Erfolg noch nicht vollständig erforscht, und PDE-basierte Kompression befindet sich derzeit noch im Anfangsstadium. Wir tragen durch eine probabilistische Rechtfertigung anisotroper Diffusion zu einem tieferen Verständnis PDE-basierten Codec-Designs bei. Eine Analyse der Interaktion zwischen effizienten Speicherverfahren und Bildrekonstruktion erlaubt es uns, PDEs nach ihrem Nutzen für die Kompression zu beurteilen. Anhand dieser Einsichten entwickeln wir PDE-basierte Kompression hinsichtlich ihrer praktischen Nutzbarkeit weiter: Wir stellen einen Hybrid-Codec für hochtexturierte Bilder vor, der umgebungsbasierte Interpolation mit PDEs kombiniert. Ein neuer Video-Dekodierer demonstriert die Echtzeitfähigkeit PDE-basierter Interpolation und eine Region-of-Interest-Methode erlaubt es, wichtige Bildbereiche mit hoher Genauigkeit zu speichern. Schlussendlich stellen wir ein neues diffusionsbasiertes Kolorierungsverfahren vor, welches uns effiziente Kompression von Farbbildern ermöglicht. Experimente auf Realwelt-Bilddatenbanken zeigen die Konkurrenzfähigkeit dieses Verfahrens auf

Understanding and advancing PDE-based image compression

This thesis is dedicated to image compression with partial differential equations (PDEs). PDE-based codecs store only a small amount of image points and propagate their information into the unknown image areas during the decompression step. For certain classes of images, PDE-based compression can already outperform the current quasi-standard, JPEG2000. However, the reasons for this success are not yet fully understood, and PDE-based compression is still in a proof-of-concept stage. With a probabilistic justification for anisotropic diffusion, we contribute to a deeper insight into design principles for PDE-based codecs. Moreover, by analysing the interaction between efficient storage methods and image reconstruction with diffusion, we can rank PDEs according to their practical value in compression. Based on these observations, we advance PDE-based compression towards practical viability: First, we present a new hybrid codec that combines PDE- and patch-based interpolation to deal with highly textured images. Furthermore, a new video player demonstrates the real-time capacities of PDE-based image interpolation and a new region of interest coding algorithm represents important image areas with high accuracy. Finally, we propose a new framework for diffusion-based image colourisation that we use to build an efficient codec for colour images. Experiments on real world image databases show that our new method is qualitatively competitive to current state-of-the-art codecs.Diese Dissertation ist der Bildkompression mit partiellen Differentialgleichungen (PDEs, partial differential equations) gewidmet. PDE-Codecs speichern nur einen geringen Anteil aller Bildpunkte und transportieren deren Information in fehlende Bildregionen. In einigen Fällen kann PDE-basierte Kompression den aktuellen Quasi-Standard, JPEG2000, bereits schlagen. Allerdings sind die Gründe für diesen Erfolg noch nicht vollständig erforscht, und PDE-basierte Kompression befindet sich derzeit noch im Anfangsstadium. Wir tragen durch eine probabilistische Rechtfertigung anisotroper Diffusion zu einem tieferen Verständnis PDE-basierten Codec-Designs bei. Eine Analyse der Interaktion zwischen effizienten Speicherverfahren und Bildrekonstruktion erlaubt es uns, PDEs nach ihrem Nutzen für die Kompression zu beurteilen. Anhand dieser Einsichten entwickeln wir PDE-basierte Kompression hinsichtlich ihrer praktischen Nutzbarkeit weiter: Wir stellen einen Hybrid-Codec für hochtexturierte Bilder vor, der umgebungsbasierte Interpolation mit PDEs kombiniert. Ein neuer Video-Dekodierer demonstriert die Echtzeitfähigkeit PDE-basierter Interpolation und eine Region-of-Interest-Methode erlaubt es, wichtige Bildbereiche mit hoher Genauigkeit zu speichern. Schlussendlich stellen wir ein neues diffusionsbasiertes Kolorierungsverfahren vor, welches uns effiziente Kompression von Farbbildern ermöglicht. Experimente auf Realwelt-Bilddatenbanken zeigen die Konkurrenzfähigkeit dieses Verfahrens auf

Processing Elastic Surfaces and Related Gradient Flows

Surface processing tools and techniques have a long history in the fields of computer graphics, computer aided geometric design and engineering. In this thesis we consider variational methods and geometric evolution problems for various surface processing applications including surface fairing, surface restoration and surface matching. Geometric evolution problems are often based on the gradient flow of geometric energies. The Willmore functional, defined as the integral of the squared mean curvature over the surface, is a geometric energy that measures the deviation of a surface from a sphere. Therefore, it is a suitable functional for surface restoration, where a destroyed surface patch is replaced by a smooth patch defined as the minimizer of the Willmore functional with boundary conditions for the position and the normal at the patch boundary. However, using the Willmore functional does not lead to satisfying results if an edge or a corner of the surface is destroyed. The anisotropic Willmore energy is a natural generalization of the Willmore energy which has crystal-shaped surfaces like cubes or octahedra as minimizers. The corresponding L2-gradient flow, the anisotropic Willmore flow, leads to a fourth-order partial differential equation that can be written as a system of two coupled second second order equations. Using linear Finite Elements, we develop a semi-implicit scheme for the anisotropic Willmore flow with boundary conditions. This approach suffer from significant restrictions on the time step size. Effectively, one usually has to enforce time steps smaller than the squared spatial grid size. Based on a natural approach for the time discretization of gradient flows we present a new scheme for the time and space discretization of the isotropic and anisotropic Willmore flow. The approach is variational and takes into account an approximation of the L2-distance between the surface at the current time step and the unknown surface at the new time step as well as a fully implicity approximation of the anisotropic Willmore functional at the new time step. To evaluate the anisotropic Willmore energy on the unknown surface of the next time step, we first ask for the solution of an inner, secondary variational problem describing a time step of anisotropic mean curvature motion. The time discrete velocity deduced from the solution of the latter problem is regarded as an approximation of the anisotropic mean curvature vector and enters the approximation of the actual anisotropic Willmore functional. The resulting two step time discretization of the Willmore flow is applied to polygonal curves and triangular surfaces and is independent of the co-dimension. Various numerical examples underline the stability of the new scheme, which enables time steps of the order of the spatial grid size. The Willmore functional of a surface is referred to as the elastic surface energy. Another interesting application of modeling elastic surfaces as minimizers of elastic energies is surface matching, where a correspondence between two surfaces is subject of investigation. There, we seek a mapping between two surfaces respecting certain properties of the surfaces. The approach is variational and based on well-established matching methods from image processing in the parameter domains of the surfaces instead of finding a correspondence between the two surfaces directly in 3D. Besides the appropriate modeling we analyze the derived model theoretically. The resulting deformations are globally smooth, one-to-one mappings. A physically proper morphing of characters in computer graphic is capable with the resulting computational approach