Abstract We propose a novel variational method to compute a highly accurate global signed distance function (SDF) to a given point cloud. To this end, the jump set of the gradient of the SDF, which coincides with the medial axis of the surface, is explicitly taken into account through a higher‐order variational formulation that enforces linear growth along the gradient direction away from this discontinuity set. The eikonal equation and the zero‐level set of the SDF are enforced as constraints. To make this variational problem computationally tractable, a phase field approximation of Ambrosio‐Tortorelli type is employed. The associated phase field function implicitly describes the medial axis. The method is implemented for surfaces represented by unoriented point clouds using neural network approximations of both the SDF and the phase field. Experiments demonstrate the method's accuracy both in the near field and globally. Quantitative and qualitative comparisons with other approaches show the advantages of the proposed method.

In this paper, we explore a scenario where a sender provides an information policy and a receiver, upon observing a realization of this policy, decides whether to take a particular action, such as making a purchase. The sender's objective is to maximize her utility derived from the receiver's action, and she achieves this by careful selection of the information policy. Building on the work of Kleiner et al., our focus lies specifically on information policies that are associated with power diagram partitions of the underlying domain. To address this problem, we employ entropy-regularized optimal transport, which enables us to develop an efficient algorithm for finding the optimal solution. We present experimental numerical results that highlight the qualitative properties of the optimal configurations, providing valuable insights into their structure. Furthermore, we extend our numerical investigation to derive optimal information policies for monopolists dealing with multiple products, where the sender discloses information about product qualities.

This paper develops a shape space framework for collision-aware geometric modeling, where basic geometric operations automatically avoid inter-penetration. Shape spaces are a powerful tool for surface modeling, shape analysis, nonrigid motion planning, and animation, but past formulations permit nonphysical intersections. Our framework augments an existing shape space using a repulsive energy such that collision avoidance becomes a first-class property, encoded in the Riemannian metric itself. In turn, tasks like intersection-free shape interpolation or motion extrapolation amount to simply computing geodesic paths via standard numerical algorithms. To make optimization practical, we develop an adaptive collision penalty that prevents mesh self-intersection, and converges to a meaningful limit energy under refinement. The final algorithms apply to any category of shape, and do not require a dataset of examples, training, rigging, nor any other prior information. For instance, to interpolate between two shapes we need only a single pair of meshes with the same connectivity. We evaluate our method on a variety of challenging examples from modeling and animation.

BACKGROUND: The efficient and robust statistical analysis of the shape of plant organs of different cultivars is an important investigation issue in plant breeding and enables a robust cultivar description within the breeding progress. Laserscanning is a highly accurate and high resolution technique to acquire the 3D shape of plant surfaces. The computation of a shape based principal component analysis (PCA) built on concepts from continuum mechanics has proven to be an effective tool for a qualitative and quantitative shape examination. RESULTS: The shape based PCA was used for a statistical analysis of 140 sugar beet roots of different cultivars. The calculation of the mean sugar beet root shape and the description of the main variations was possible. Furthermore, unknown and individual tap roots could be attributed to their cultivar by means of a robust classification tool based on the PCA results. CONCLUSION: The method demonstrates that it is possible to identify principal modes of root shape variations automatically and to quantify associated variances out of laserscanned 3D sugar beet tap root models. The introduced approach is not limited to the 3D shape description by laser scanning. A transfer to 3D MRI or radar data is also conceivable.

Abstract This paper introduces the construction of a low‐dimensional nonlinear space capturing the variability of a non‐rigid shape from a data set of example poses. The core of the approach is a Sparse Principal Geodesic Analysis (SPGA) on the Riemannian manifold of discrete shells, in which a pose of a non‐rigid shape is a point. The SPGA is invariant to rigid body motions of the poses and supports large deformation. Since the Riemannian metric measures the membrane and bending distortions of the shells, the sparsity term forces the modes to describe largely decoupled and localized deformations. This property facilitates the analysis of articulated shapes. The modes often represent characteristic articulations of the shape and usually come with a decomposing of the spanned subspace into low‐dimensional widely decoupled subspaces. For example, for human models, one expects distinct, localized modes for the bending of elbow or knee whereas some more modes are required to represent shoulder articulation. The decoupling property can be used to construct useful starting points for the computation of the nonlinear deformations via a superposition of shape submanifolds resulting from the decoupling. In a preprocessing stage, samples of the individual subspaces are computed, and, in an online phase, these are interpolated multilinearly. This accelerates the construction of nonlinear deformations and makes the method applicable for interactive applications. The method is compared to alternative approaches and the benefits are demonstrated on different kinds of input data.

Continuous image morphing is a classical task in image processing. The metamorphosis model proposed by Trouvé, Younes, and coworkers [M. I. Miller and L. Younes, Int. J. Comput. Vis., 41 (2001), pp. 61--84; A. Trouvé and L. Younes, Found. Comput. Math., 5 (2005), pp. 173--198] casts this problem in the frame of Riemannian geometry and geodesic paths between images. The associated metric in the space of images incorporates dissipation caused by a viscous flow transporting image intensities and its variations along motion paths. In many applications, images are maps from the image domain into a manifold (e.g., in diffusion tensor imaging (DTI), the manifold of symmetric positive definite matrices with a suitable Riemannian metric). In this paper, we propose a generalized metamorphosis model for manifold-valued images, where the range space is a finite-dimensional Hadamard manifold. A corresponding time discrete version was presented in [S. Neumayer, J. Persch, and G. Steidl, SIAM J. Imaging Sci., 11 (2018), pp. 1898--1930] based on the general variational time discretization proposed in [B. Berkels, A. Effland, and M. Rumpf, SIAM J. Imaging Sci., 8 (2015), pp. 1457--1488]. Here, we prove the Mosco--convergence of the time discrete metamorphosis functional to the proposed manifold-valued metamorphosis model, which implies the convergence of time discrete geodesic paths to a geodesic path in the (time continuous) metamorphosis model. In particular, the existence of geodesic paths is established. In particular, the existence of geodesic paths is established. In fact, images as maps into Hadamard manifold are not only relevant in applications, but it is also shown that the joint convexity of the distance function---which characterizes Hadamard manifolds---is a crucial ingredient to establish existence of the metamorphosis model.

Abstract We propose a novel approach for shape matching between triangular meshes that, in contrast to existing methods, can match crease features. Our approach is based on a hybrid optimization scheme, that solves simultaneously for an elastic deformation of the source and its projection on the target. The elastic energy we minimize is invariant to rigid body motions, and its non‐linear membrane energy component favors locally injective maps. Symmetrizing this model enables feature aligned correspondences even for non‐isometric meshes. We demonstrate the advantage of our approach over state of the art methods on isometric and non‐isometric datasets, where we improve the geodesic distance from the ground truth, the conformal and area distortions, and the mismatch of the mean curvature functions. Finally, we show that our computed maps are applicable for surface interpolation, consistent cross‐field computation, and consistent quadrangular remeshing of a set of shapes.

Abstract Important sources of shape variability, such as articulated motion of body models or soft tissue dynamics, are highly nonlinear and are usually superposed on top of rigid body motion which must be factored out. We propose a novel, nonlinear, rigid body motion invariant Principal Geodesic Analysis (PGA) that allows us to analyse this variability, compress large variations based on statistical shape analysis and fit a model to measurements. For given input shape data sets we show how to compute a low dimensional approximating submanifold on the space of discrete shells, making our approach a hybrid between a physical and statistical model. General discrete shells can be projected onto the submanifold and sparsely represented by a small set of coefficients. We demonstrate two specific applications: model‐constrained mesh editing and reconstruction of a dense animated mesh from sparse motion capture markers using the statistical knowledge as a prior.

The space of images can be equipped with a Riemannian metric measuring both the cost of transport of image intensities and the variation of image intensities along motion lines. The resulting metamorphosis model was introduced and analyzed in [M. I. Miller and L. Younes, Int. J. Comput. Vis., 41 (2001), pp. 61--84; A. Trouvé and L. Younes, Found. Comput. Math., 5 (2005), pp. 173--198], and a variational time discretization for the geodesic interpolation was proposed in [B. Berkels, A. Effland, and M. Rumpf, SIAM J. Imaging Sci., 8 (2015), pp. 1457--1488]. In this paper, this time discrete model is expanded and an image extrapolation via a discretization of the geometric exponential map is consistently derived for the variational time discretization. For a given weakly differentiable initial image and an initial image variation, the exponential map allows one to compute a discrete geodesic extrapolation path in the space of images. It is shown that a time step of this shooting method can be formulated in the associated deformations only. For sufficiently small time steps, local existence and uniqueness are proved using a suitable fixed point formulation and the implicit function theorem. A spatial Galerkin discretization with cubic splines on coarse meshes for the deformations and piecewise bilinear finite elements on fine meshes for the image intensities are used to derive a fully practical algorithm. Different applications underline the efficiency and stability of the proposed approach.

The motion of a thin viscous film of fluid on a curved surface exhibits many intricate visual phenomena, which are challenging to simulate using existing techniques. A possible alternative is to use a reduced model, involving only the temporal evolution of the mass density of the film on the surface. However, in this model, the motion is governed by a fourth-order nonlinear PDE, which involves geometric quantities such as the curvature of the underlying surface, and is therefore difficult to discretize. Inspired by a recent variational formulation for this problem on smooth surfaces, we present a corresponding model for triangle meshes. We provide a discretization for the curvature and advection operators which leads to an efficient and stable numerical scheme, requires a single sparse linear solve per time step, and exactly preserves the total volume of the fluid. We validate our method by qualitatively comparing to known results from the literature, and demonstrate various intricate effects achievable by our method, such as droplet formation, evaporation, droplets interaction and viscous fingering. Finally, we extend our method to incorporate non-linear van der Waals forcing terms which stabilize the motion of the film and allow additional effects such as pearling.

Abstract Cubic splines in Euclidean space minimize the mean squared acceleration among all curves interpolating a given set of data points. We extend this observation to the Riemannian manifold of discrete shells in which the associated metric measures both bending and membrane distortion. Our generalization replaces the acceleration with the covariant derivative of the velocity. We introduce an effective time‐discretization for this novel paradigm for navigating shell space. Further transferring this concept to the space of triangular surface descriptors—edge lengths, dihedral angles, and triangle areas—results in a simplified interpolation method with high computational efficiency.

In this paper we describe how to perform Principal Components Analysis in "shell space". Thin shells are a physical model for surfaces with non-zero thickness whose deformation dissipates elastic energy. Thin shells, or their discrete counterparts, can be considered to reside in a shell space in which the notion of distance is given by the elastic energy required to deform one shape into another. It is in this setting that we show how to perform statistical analysis of a set of shapes (meshes in dense correspondence), providing a hybrid between physical and statistical shape modelling. The resulting models are better able to capture non-linear deformations, for example resulting from articulated motion, even when training data is very sparse compared to the dimensionality of the observation space.

In this paper the space of images is considered as a Riemannian manifold using the metamorphosis approach (see [M. I. Miller and L. Younes, Int. J. Comput. Vis., 41 (2001), pp. 61--84; A. Trouvé and L. Younes, SIAM J. Math. Anal., 37 (2005), pp. 17--59; and A. Trouvé and L. Younes, Found. Comput. Math., 5 (2005), pp. 173--198]), where the underlying Riemannian metric simultaneously measures the cost of image transport and intensity variation. A robust and effective variational time discretization of geodesics paths is proposed. This requires minimizing a discrete path energy consisting of a sum of consecutive image matching functionals over a set of image intensity maps and pairwise matching deformations. For square-integrable input images the existence of discrete, connecting geodesic paths defined as minimizers of this variational problem is shown. Furthermore, $\Gamma$-convergence of the underlying discrete path energy to the continuous path energy is proved. This includes a diffeomorphism property for the induced transport and the existence of a square-integrable weak material derivative in space and time. A spatial discretization via finite elements combined with an alternating descent scheme in the set of image intensity maps and the set of matching deformations is presented to approximate discrete geodesic paths numerically. Computational results underline the efficiency of the proposed approach and demonstrate important qualitative properties.

Abstract We prove both in the smooth and discrete setting that the Hessian of an elastic deformation energy results in a proper Riemannian metric on the space of shells (modulo rigid body motions). Based on this foundation we develop a time‐ and space‐discrete geodesic calculus. In particular we show how to shoot geodesics with prescribed initial data, and we give a construction for parallel transport in shell space. This enables, for example, natural extrapolation of paths in shell space and transfer of large nonlinear deformations from one shell to another with applications in animation, geometric, and physical modeling. Finally, we examine some aspects of curvature on shell space.

Based on a local approximation of the Riemannian distance on a manifold by a computationally cheap dissimilarity measure, a time discrete geodesic calculus is developed and applications to shape space are explored. The dissimilarity measure is derived from a deformation energy whose Hessian reproduces the underlying Riemannian metric, and it is used to define length and energy of discrete paths in shape space. The notion of discrete geodesics defined as energy minimizing paths gives rise to a discrete logarithmic map, a variational definition of a discrete exponential map, and a time discrete parallel transport. This new concept is developed in the context of shape spaces with shapes that are described via deformations of a given reference shape, and it is applied to a particular shape space in which shapes are considered as boundary contours of physical objects consisting of viscous material. The flexibility and computational efficiency of the approach is demonstrated for topology preserving shape morphing, the representation of paths in shape space via local shape variations as path generators, shape extrapolation via discrete geodesic flow, and the transfer of geometric features.

Abstract Building on concepts from continuum mechanics, we offer a computational model for geodesics in the space of thin shells, with a metric that reflects viscous dissipation required to physically deform a thin shell. Different from previous work, we incorporate bending contributions into our deformation energy on top of membrane distortion terms in order to obtain a physically sound notion of distance between shells, which does not require additional smoothing. Our bending energy formulation depends on the so‐called relative Weingarten map, for which we provide a discrete analogue based on principles of discrete differential geometry. Our computational results emphasize the strong impact of physical parameters on the evolution of a shell shape along a geodesic path.

A physically motivated approach is presented for computing 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 multiscale alternating minimization approach allows the efficient computation of shape averages in two and three dimensions. Various applications, e.g., averaging the shape of feet or human organs, underline the qualitative properties of the presented approach.

The problem of motion estimation and restoration of objects in a blurred video sequence is addressed in this paper. Fast movement of the objects, together with the aperture time of the camera, result in a motion-blurred image. The direct velocity estimation from this blurred video is inaccurate. On the other hand, an accurate estimation of the velocity of the moving objects is critical for restoration of motion-blurred video. Therefore, restoration needs accurate motion estimation and vice versa, and a joint process is called for. To address this problem we derive a novel model of the blurring process and propose a Mumford-Shah type of variational framework, acting on consecutive frames, for joint object deblurring and velocity estimation. The proposed procedure distinguishes between the moving object and the background and is accurate also close to the boundary of the moving object. Experimental results both on simulated and real data show the importance of this joint estimation and its superior performance when compared to the independent estimation of motion and restoration.

Multimodal image registration significantly benefits from previous denoising and structure segmentation and vice versa. In particular combined information of different image modalities makes segmentation significantly more robust. Indeed, fundamental tasks in image processing are highly interdependent. A variational approach is presented, which combines the detection of corresponding edges, an edge preserving denoising and the morphological registration via a non-rigid deformation for a pair of images with structural correspondence. The morphology of an image function is split into a singular part consisting of the edge set and a regular part represented by the field of normals on the ensemble of level sets. A Mumford-Shah type free discontinuity problem is applied to treat the singular morphology and the matching of corresponding edges under the deformation. The matching of the regular morphology is quantified by a second contribution which compares deformed normals and normals at deformed positions. Finally, a nonlinear elastic energy controls the deformation itself and ensures smoothness and injectivity. A multi scale approach that is based on a phase field approximation leads to an effective and efficient algorithm. Numerical experiments underline the robustness of the presented approach and show applications on medical images.

This paper presents a new algorithm based on the Mumford-Shah model for simultaneously detecting the edge features of two images and jointly estimating a consistent set of transformations to match them. Compared to the current asymmetric methods in the literature, this fully symmetric method allows one to determine one-to-one correspondences between the edge features of two images. The entire variational model is realized in a multiscale framework of the finite element approximation. The optimization process is guided by an estimation minimization-type algorithm and an adaptive generalized gradient flow to guarantee a fast and smooth relaxation. The algorithm is tested on T1 and T2 magnetic resonance image data to study the parameter setting. We also present promising results of four applications of the proposed algorithm: interobject monomodal registration, retinal image registration, matching digital photographs of neurosurgery with its volume data, and motion estimation for frame interpolation.

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

Based on continuous surface models one can derive variational problems and geometric evolutions problems for various surface processing applications. In a second step these models can then be discretized by finite element techniques. This concept is demonstrated for surface fairing, surface restoration, surface generation via subdivision, and finally surface parametrization.

We present a novel multiscale approach for flow visualization. We define a local alignment tensor that encodes a measure for alignment to the direction of a given flow field. This tensor induces an anisotropic differential operator on the flow domain, which is discretized with a standard finite element technique. The entries of the corresponding stiffness matrix represent the anisotropically weighted couplings of adjacent nodes of the domain mesh. We use an algebraic multigrid algorithm to generate a hierarchy of fine to coarse descriptions for the above coupling data. This hierarchy comprises a set of coarse grid nodes, a multiscale of basis functions and their corresponding supports. We use these supports to obtain a multilevel decomposition of the flow structure. Standard streamline icons are used to visualize this decomposition at any user-selected level of detail. The method provides a single framework for vector field decomposition independent on the domain dimension or mesh type. Applications are shown in 2D, for flow fields on curved surfaces, and for 3D volumetric flow fields.

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.

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.

We present a novel variational method to nonrigid registration of multimodal data. A suitable deformation will be determined via the minimization of a morphological, i.e., contrast invariant, matching functional along with an appropriate regularization energy. Here we want to give special focus on the practical issues involving scale-space methods, regularization of the corresponding gradient flow and hyperelastic regularization.

Implicit active contours are a very flexible technique in the segmentation of digital images. A novel type of hardware implementation is presented here to approach real time applications We propose to exploit the high performance of modern graphics cards for numerical computations. Vectors are regarded as images and linear algebraic operations on vectors are realized by the graphics operations of image blending. Thus, the performance benefits from the high memory bandwidth and the economy of command transfers, while the restricted precision does not infect the qualitative behavior of the level set propagation Here, we pick up a first order solver for the basic implicit level set model and present an implementation performing at 2 ms for an explicit timestep on a 128/sup 2/ image.

The visualization of time-dependent flow is an important and challenging topic in scientific visualization. Its aim is to represent transport phenomena governed by time-dependent vector fields in an intuitively understandable way, using images and animations. Here we pick up the recently presented anisotropic diffusion method, expand and generalize it to allow a multiscale visualization of long-term, complex transport problems. Instead of streamline type patterns generated by the original method now streakline patterns are generated and advected. This process obeys a nonlinear transport diffusion equation with typically dominant transport. Starting from some noisy initial image, the diffusion actually generates and enhances patterns which are then transported in the direction of the flow field. Simultaneously the image is again sharpened in the direction orthogonal to the flow field. A careful adjustment of the models parameters is derived to balance diffusion and transport effects in a reasonable way. Properties of the method can be discussed for the continuous model, which is solved by an efficient upwind finite element discretization. As characteristic for the class of multiscale image processing methods, we can in advance select a suitable scale for representing the flow field.

A new method for the simplification of flow fields is presented. It is based on continuous clustering. A well-known physical clustering model, the Cahn-Hilliard (1958) model, which describes phase separation, is modified to reflect the properties of the data to be visualized. Clusters are defined implicitly as connected components of the positivity set of a density function. An evolution equation for this function is obtained as a suitable gradient flow of an underlying anisotropic energy functional, where time serves as the scale parameter. The evolution is characterized by a successive coarsening of patterns, during which the underlying simulation data specifies preferable pattern boundaries. We introduce specific physical quantities in the simulation to control the shape, orientation and distribution of the clusters as a function of the underlying flow field. In addition, the model is expanded, involving elastic effects. In the early stages of the evolution, a shear-layer-type representation of the flow field can thereby be generated, whereas, for later stages, the distribution of clusters can be influenced. Furthermore, we incorporate upwind ideas to give the clusters an oriented drop-shaped appearance. We discuss the applicability of this new type of approach mainly for flow fields, where the cluster energy penalizes cross-streamline boundaries. However, the method also carries provisions for other fields as well. The clusters can be displayed directly as a flow texture. Alternatively, the clusters can be visualized by iconic representations, which are positioned by using a skeletonization algorithm.

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 new method for the simplification of flow fields is presented. It is based on continuous clustering. A well-known physical clustering model, the Cahn Hilliard model (J. Cahn and J. Hilliard, 1958), which describes phase separation, is modified to reflect the properties of the data to be visualized. Clusters are defined implicitly as connected components of the positivity set of a density function. An evolution equation for this function is obtained as a suitable gradient flow of an underlying anisotropic energy functional. Here, time serves as the scale parameter. The evolution is characterized by a successive coarsening of patterns: the actual clustering, and meanwhile the underlying simulation data specifies preferable pattern boundaries. The authors discuss the applicability of this new type of approach mainly for flow fields, where the cluster energy penalizes cross streamline boundaries, but the method also carries provisions in other fields as well. The clusters are visualized via iconic representations. A skeletonization algorithm is used to find suitable positions for the icons.

Vector field visualization is an important topic in scientific visualization. Its aim is to graphically represent field data on two and three-dimensional domains and on surfaces in an intuitively understandable way. Here, a new approach based on anisotropic nonlinear diffusion is introduced. It enables an easy perception of vector field data and serves as an appropriate scale space method for the visualization of complicated flow pattern. The approach is closely related to nonlinear diffusion methods in image analysis where images are smoothed while still retaining and enhancing edges. Here, an initial noisy image intensity is smoothed along integral lines, whereas the image is sharpened in the orthogonal direction. The method is based on a continuous model and requires the solution of a parabolic PDE problem. It is discretized only in the final implementational step. Therefore, many important qualitative aspects can already be discussed on a continuous level. Applications are shown for flow fields in 2D and 3D, as well as for principal directions of curvature on general triangulated surfaces. Furthermore, the provisions for flow segmentation are outlined.

Vector field visualization is an important topic in scientific visualization. Its aim is to graphically represent field data in an intuitively understandable and precise way. Here a new approach based on anisotropic nonlinear diffusion is introduced. It enables an easy perception of flow data and serves as an appropriate scale space method for the visualization of complicated flow patterns. The approach is closely related to nonlinear diffusion methods in image analysis where images are smoothed while still retaining and enhancing edges. An initial noisy image is smoothed along streamlines, whereas the image is sharpened in the orthogonal direction. The method is based on a continuous model and requires the solution of a parabolic PDE problem. It is discretized only in the final implementational step. Therefore, many important qualitative aspects can already be discussed on a continuous level. Applications are shown in 2D and 3D and the provisions for flow segmentation are outlined.

Recently multiresolution visualization methods have become an indispensable ingredient of real time interactive postprocessing. The enormous databases, typically coming along with some hierarchical structure, are locally resolved on different levels of detail to achieve a significant savings of CPU and rendering time. The method of adaptive projection and the corresponding operators on data functions, respectively are introduced. They are defined and discussed as mathematically rigorous foundations for multiresolution data analysis. Keeping in mind data from efficient numerical multigrid methods, this approach applies to hierarchical nested grids consisting of elements which are any tensor product of simplices, generated recursively by an arbitrary, finite set of refinement rules from some coarse grid. The corresponding visualization algorithms, e.g., color shading on slices or isosurface rendering, are confined to an appropriate depth first traversal of the grid hierarchy. A continuous projection of the data onto an adaptive, extracted subgrid is thereby calculated recursively. The presented concept covers different methods of local error measurement, time dependent data which have to be interpolated from a sequence of key frames, and a tool for local data focusing. Furthermore, it allows for a continuous level of detail.

The interest in particle methods and gridless discretizations has been increasing enormously in recent times: a development which is especially boosted by the rapid progress of hardware capabilities and the design of new highly sophisticated algorithms, thanks to which nowadays systems of several millions degrees of freedom can be simulated. Consequently there is a high demand for interactive visualization tools to explore the huge amount of data. Several techniques for visualizing particle systems are presented. These methods are based on efficient algorithms and models and make frequent use of the texture mapping capabilities of modern graphics hardware to improve the efficiency of scene display and, for example, to display properly illuminated particle traces. The main stress of the discussion is put on the presentation of a multiresolutional concept that enables the visualization of systems which even consist of several millions of particles. Its basic idea is to display a density function instead of resolving regions of high particle density particlewise. Therefore we construct two adaptive hierarchical trees, one of which is to store the particle information, whereas the other one contains complete density information. For the sake of economical memory management, both trees are efficiently stored in hash tables.

Recently, multiresolution visualization methods have become an indispensable ingredient of real-time interactive postprocessing. The enormous databases, typically coming along with some hierarchical structure, are locally resolved on different levels of detail to achieve a significant savings of CPU and rendering time. In this paper, the method of adaptive projection and the corresponding operators on data functions, respectively, are introduced. They are defined and discussed as mathematically rigorous foundations for multiresolution data analysis. Keeping in mind data from efficient numerical multigrid methods, this approach applies to hierarchical nested grids consisting of elements which are any tensor product of simplices, generated recursively by an arbitrary, finite set of refinement rules from some coarse grid. The corresponding visualization algorithms, e.g. color shading on slices or isosurface rendering, are confined to an appropriate depth-first traversal of the grid hierarchy. A continuous projection of the data onto an adaptive, extracted subgrid is thereby calculated recursively. The presented concept covers different methods of local error measurement, time-dependent data which have to be interpolated from a sequence of key frames, and a tool for local data focusing. Furthermore, it allows for a continuous level of detail.

Abstract Most of the rendering tools in scientific visualization are restricted to special data structures which differ substantially from the data formats used in numerical applications. Trying to close this gap, we present an interface between data from numerical methods on general types of grids ‐ like cuboidal, prismatic, simplicial, parametric, mixed, or hierarchical meshes — and general visualization routines. It is based on a procedural approach managing a collection of arbitrary elements and a set of functions describing each element type. No mapping of (an in general enormous amount of) numerical data onto new data structures is necessary; a user may use his own data structures and only has to provide this small set of procedures and functions. The visualization tools will then use these routines to access (temporarily and locally) data of interest, like information about a single element. Compared with display routines on a specialized data structure, this general interface does not produce much cpu overhead.