Abstract Integral lines of the gradient flow are standard features in continuously differentiable scalar fields that enjoy some useful properties: They cover the domain densely, do not split, merge, or intersect, and are therefore invertible. For widely used discretizations of scalar fields, the corresponding polygonal approximations of integral lines do not enjoy these properties anymore. We analyze conditions for integral lines in 2D piecewise linear (PL) scalar fields to be invertible by identifying and classifying critical edges in the underlying triangulation. We show that under mild conditions, every 2D PL scalar field can be transformed into an arbitrarily close PL field with invertible integral lines. We present an algorithm that computes this transformation and apply it to a number of test data sets.

Abstract Flow maps are a fundamental concept for the visualization and analysis of time‐dependent flows. A common approach is to apply sampling and reconstruction – which comes with a number of problems: flow maps are high‐dimensional functions with locally large gradients which makes approximations very challenging. In general, one must apply large sampling resolutions and more advanced methods to achieve sufficiently good results. Existing methods for efficient, high‐quality sampling and approximation restrict to subsets of flow maps, e.g., with fixed start and end times, or at least fixed differences. In this paper, we propose the first method to handle the entire flow map by decreasing the dimensionality of the space‐time‐time domain. Essentially, it is a framework which splits the high‐dimensional flow map into few lower‐dimensional functions. This enables to achieve much larger resolutions than traditional sampling with the same memory consumption. We use our method in established applications and show that it can significantly decrease errors in the reconstruction by using these larger resolutions.

Principal Component Analysis (PCA) is perhaps the most popular linear projection technique for dimensionality reduction. We consider PCA under the assumption that the high-dimensional data points are equipped with Gaussian uncertainty. Several approaches to such uncertainty-aware PCA have been developed recently in the visualization community. Since PCA is a discontinuous map, a small uncertainty in the data points can result in a huge uncertainty in the projected points. We show that the uncertainty of the data points also creates uncertainty in the eigenvectors of the covariance matrix that defines the PCA projection. We present a closed-form expression to quantify eigenvector uncertainty. Based on this, we propose a 3D glyph that supports the decision whether existing solutions for uncertainty-aware PCA are sufficient, or whether a more expensive sampling-based approach is required. We apply our approach to several test data sets.

Abstract Finding projections of multidimensional data domains to the 2D screen space is a well‐known problem. Multidimensional data often comes with the property that the dimensions are measured in different physical units, which renders the ratio between dimensions, i.e., their scale, arbitrary. The result of common projections, like PCA, t‐SNE, or MDS, depends on this ratio, i.e., these projections are variant to scaling. This results in an undesired subjective view of the data, and thus, their projection. Simple solutions like normalization of each dimension are widely used, but do not always give high‐quality results. We propose to visually analyze the space of all scalings and to find optimal scalings w.r.t. the quality of the visualization. For this, we evaluate different quality criteria on scatter plots. Given a quality criterion, our approach finds scalings that yield good visualizations with little to no user input using numerical optimization. Simultaneously, our method results in a scaling invariant projection, proposing an objective view to the projected data. We show for several examples that such an optimal scaling can significantly improve the visualization quality.

Abstract Lagrangian coherent structures (LCS) is an important concept for the visualization of unsteady flows. They describe the boundaries of regions for which material transport stays mostly coherent over time which can help for a better understanding of dynamical systems. One of the most common techniques for their computation is the extraction of ridges from the finite‐time Lyapunov exponent (FTLE) field. FTLE ridges are challenging to extract, both in terms of accuracy and performance, because they expose strong gradients of the underlying field, tend to come close to each other and are dynamic with respect to different time parameters. We present a new method for extracting FTLE ridges for series of integration times which is able to show how coherent regions and their borders evolve over time. Our techniques mainly build on a particle system which is used for sampling the ridges uniformly. This system is highly optimized for the challenges of FTLE ridge extraction. Further, it is able to take advantage of the continuous evolvement of the ridges which makes their sampling for multiple integration times much faster. We test our method on multiple 3D datasets and compare it to the standard Marching Ridges technique. For the extraction examples our method is 13 to over 300 times faster, suggesting a significant advantage.

Abstract The smoothing of surface curves is an essential tool in mesh processing, important to applications that require segmenting and cutting surfaces such as surgical planning. Surface curves are typically designed by professionals to match certain surface features. For this reason, the smoothed curves should be close to the original and easily adjustable by the user in interactive tools. Previous methods achieve this desired behavior, e.g., by utilizing energy‐minimizing splines or generalizations of Bézier splines, which require a significant number of control points and may not provide interactive frame rates or numerical stability. This paper presents a new algorithm for robust smoothing of discrete surface curves on triangular surface meshes. By using a scalar penalty potential as the fourth coordinate, the given surface mesh is embedded into the 4D Euclidean space. Our method is based on finding geodesics in this lifted surface, which are then projected back onto the original 3D surface. The benefits of this approach include guaranteed convergence and good approximation of the initial curve. We propose a family of penalty potentials with one single parameter for adjusting the trade‐off between smoothness and similarity. The implementation of our method is straightforward as we rely on existing methods for computing geodesics and penalty fields. We evaluate our implementation and confirm its robustness and efficiency.

Abstract Interpolation between camera positions is a standard problem in computer graphics and can be considered the foundation of camera path planning. As the basis for a new interpolation method, we introduce a new Riemannian metric in camera space, which measures the 3D image flow under a small movement of the camera. Building on this, we define a linear interpolation between two cameras as shortest geodesic in camera space, for which we provide a closed‐form solution after a mild simplification of the metric. Furthermore, we propose a geodesic Catmull‐Rom interpolant for keyframe camera animation. We compare our approach with several standard camera interpolation methods and obtain consistently better camera paths especially for cameras with extremely varying scales.

Abstract Aortic dissection is a life‐threatening vascular disease characterized by abrupt formation of a new flow channel (false lumen) within the aortic wall. Survivors of the acute phase remain at high risk for late complications, such as aneurysm formation, rupture, and death. Morphologic features of aortic dissection determine not only treatment strategies in the acute phase (surgical vs. endovascular vs. medical), but also modulate the hemodynamics in the false lumen, ultimately responsible for late complications. Accurate description of the true and false lumen, any communications across the dissection membrane separating the two lumina, and blood supply from each lumen to aortic branch vessels is critical for risk prediction. Patient‐specific surface representations are also a prerequisite for hemodynamic simulations, but currently require time‐consuming manual segmentation of CT data. We present an aortic dissection cross‐sectional model that captures the varying aortic anatomy, allowing for reliable measurements and creation of high‐quality surface representations. In contrast to the traditional spline‐based cross‐sectional model, we employ elliptic Fourier descriptors, which allows users to control the accuracy of the cross‐sectional contour of a flow channel. We demonstrate (i) how our approach can solve the requirements for generating surface and wall representations of the flow channels, (ii) how any number of communications between flow channels can be specified in a consistent manner, and (iii) how well branches connected to the respective flow channels are handled. Finally, we discuss how our approach is a step forward to an automated generation of surface models for aortic dissections from raw 3D imaging segmentation masks.

Abstract The model of a Double Gyre flow by Shadden et al. is a standard benchmark data set for the computation of hyperbolic Lagrangian Coherent Structures (LCS) in flow data. While structurally extremely simple, it generates hyperbolic LCS of arbitrary complexity. Unfortunately, the Double Gyre does not come with a well‐defined ground truth: the location of hyperbolic LCS boundaries can only be approximated by numerical methods that usually involve the gradient of the flow map. We present a new benchmark data set that is a small but carefully designed modification of the Double Gyre, which comes with ground truth closed‐form hyperbolic trajectories. This allows for computing hyperbolic LCS boundaries by a simple particle integration without the consideration of the flow map gradient. We use these hyperbolic LCS as a ground truth solution for testing an existing numerical approach for extracting hyperbolic trajectories. In addition, we are able to construct hyperbolic LCS curves that are significantly longer than in existing numerical methods.

Abstract Measured data often incorporates some amount of uncertainty, which is generally modeled as a distribution of possible samples. In this paper, we consider second‐order symmetric tensors with uncertainty. In the 3D case, this means the tensor data consists of 6 coefficients – uncertainty, however, is encoded by 21 coefficients assuming a multivariate Gaussian distribution as model. The high dimension makes the direct visualization of tensor data with uncertainty a difficult problem, which was until now unsolved. The contribution of this paper consists in the design of glyphs for uncertain second‐order symmetric tensors in 2D and 3D. The construction consists of a standard glyph for the mean tensor that is augmented by a scalar field that represents uncertainty. We show that this scalar field and therefore the displayed glyph encode the uncertainty comprehensively, i.e., there exists a bijective map between the glyph and the parameters of the distribution. Our approach can extend several classes of existing glyphs for symmetric tensors to additionally encode uncertainty and therefore provides a possible foundation for further uncertain tensor glyph design. For demonstration, we choose the well‐known superquadric glyphs, and we show that the uncertainty visualization satisfies all their design constraints.

We present a formal approach to the visual analysis of recirculation in flows by introducing recirculation surfaces for 3D unsteady flow fields. Recirculation surfaces are the loci where massless particle integration returns to its starting point after some variable, finite integration. We give a rigorous definition of recirculation surfaces as 2-manifolds embedded in 5D space and study their properties. Based on this we construct an algorithm for their extraction, which searches for intersections of a recirculation surface with lines defined in 3D. This reduces the problem to a repeated search for critical points in 3D vector fields. We provide a uniform sampling of the search space paired with a surface reconstruction and visualize results. This way, we present the first algorithm for a comprehensive feature extraction in the 5D flow map of a 3D flow. The problem of finding isolated closed orbits in steady vector fields occurs as a special case of recirculation surfaces. This includes isolated closed orbits with saddle behavior. We show recirculation surfaces for a number of artificial and real flow data sets.

Abstract The Parallel Vectors (PV) Operator extracts the locations of points where two vector fields are parallel. In general, these features are line structures. The PV operator has been used successfully for a variety of problems, which include finding vortex‐core lines or extremum lines. We present a new generic feature extraction method for multiple 3D vector fields: TheApproximate Parallel Vectors(APV) Operator extracts lines where all fields are approximately parallel. The definition of the APV operator is based on the application of PV for two vector fields that are derived from the given set of fields. The APV operator enables the direct visualization of features of vector field ensembles without processing fields individually and without causing visual clutter. We give a theoretical analysis of the APV operator and demonstrate its utility for a number of ensemble data.

Abstract The visual analysis of combustion processes is one of the challenges of modern flow visualization. In turbulent combustion research, the behaviour of the flame surface contains important information about the interactions between turbulence and chemistry. The extraction and tracking of this surface is crucial for understanding combustion processes. This is impossible to realize as a post‐process because of the size of the involved datasets, which are too large to be stored on disk. We present an on‐the‐fly method for tracking the flame surface directly during simulation and computing the local tangential surface deformation for arbitrary time intervals. In a massively parallel simulation, the data are distributed over many processes and only a single time step is in memory at any time. To satisfy the demands on parallelism and accuracy posed by this situation, we track the surface with independent micro‐patches and adapt their distribution as needed to maintain numerical stability. With our method, we enable combustion researchers to observe the detailed movement and deformation of the flame surface over extended periods of time and thus gain novel insights into the mechanisms of turbulence–chemistry interactions. We validate our method on analytic ground truth data and show its applicability on two real‐world simulations.

Abstract Vortices are important features in vector fields that show a swirling behavior around a common core. The concept of a vortex core line describes the center of this swirling behavior. In this work, we examine the extension of this concept to 3D second‐order tensor fields. Here, a behavior similar to vortices in vector fields can be observed for trajectories of the eigenvectors. Vortex core lines in vector fields were defined by Sujudi and Haimes to be the locations where stream lines are parallel to an eigenvector of the Jacobian. We show that a similar criterion applied to the eigenvector trajectories of a tensor field yields structurally stable lines that we calltensor core lines. We provide a formal definition of these structures and examine their mathematical properties. We also present a numerical algorithm for extracting tensor core lines in piecewise linear tensor fields. We find all intersections of tensor core lines with the faces of a dataset using a simple and robust root finding algorithm. Applying this algorithm to tensor fields obtained from structural mechanics simulations shows that it is able to effectively detect and visualize regions of rotational or hyperbolic behavior of eigenvector trajectories.

Glyphs are a powerful tool for visualizing second-order tensors in a variety of scientic data as they allow to encode physical behavior in geometric properties. Most existing techniques focus on symmetric tensors and exclude non-symmetric tensors where the eigenvectors can be non-orthogonal or complex. We present a new construction of 2d and 3d tensor glyphs based on piecewise rational curves and surfaces with the following properties: invariance to (a) isometries and (b) scaling, (c) direct encoding of all real eigenvalues and eigenvectors, (d) one-to-one relation between the tensors and glyphs, (e) glyph continuity under changing the tensor. We apply the glyphs to visualize the Jacobian matrix fields of a number of 2d and 3d vector fields.

Abstract In recent years, much work was devoted to the design of light editing methods such as relighting and light path editing. So far, little work addressed the target‐based manipulation and animation of caustics, for instance to a differently‐shaped caustic, text or an image. The aim of this work is the animation of caustics by blending towards a given target irradiance distribution. This enables an artist to coherently change appearance and style of caustics, e.g., for marketing applications and visual effects. Generating a smooth animation is nontrivial, as photon density and caustic structure may change significantly. Our method is based on the efficient solution of a discrete assignment problem that incorporates constraints appropriate to make intermediate blends plausibly resemble caustics. The algorithm generates temporally coherent results that are rendered with stochastic progressive photon mapping. We demonstrate our system in a number of scenes and show blends as well as a key frame animation.

Abstract The selection of meaningful lines for 3D line data visualization has been intensively researched in recent years. Most approaches focus on single line fields where one line passes through each domain point. This paper presents a selection approach for sets of line fields which is based on a global optimization of the opacity of candidate lines. For this, existing approaches for single line fields are modified such that significantly larger amounts of line representatives are handled. Furthermore, time coherence is addressed for animations, making this the first approach that solves the line selection problem for 3D time‐dependent flow. We apply our technique to visualize dense sets of pathlines, sets of magnetic field lines, and animated sets of pathlines, streaklines and masslines.

Abstract In flow visualization, integral surfaces rapidly tend to expand, fold and produce vast amounts of occlusion. While silhouette enhancements and local transparency mappings proved useful for semi‐transparent depictions, they still introduce visual clutter when surfaces grow more complex. An effective visualization of the flow requires a balance between the presentation of interesting surface parts and the avoidance of occlusions that hinder the view. In this paper, we extend the concept of opacity optimization to surfaces to obtain a global approach to the occlusion problem. Starting with a partition of the surfaces into patches, we compute per‐patch opacity as minimizer of a bounded‐variable least‐squares problem. For the final rendering, opacity is interpolated on the surfaces. The resulting visualization technique is interactive, frame‐coherent, view‐dependent and driven by domain knowledge.

Abstract Lagrangian coherent structures (LCSs) have become a widespread and powerful method to describe dynamic motion patterns in time‐dependent flow fields. The standard way to extract LCS is to compute height ridges in the finite‐time Lyapunov exponent field. In this work, we present an alternative method to approximate Lagrangian features for 2D unsteady flow fields that achieve subgrid accuracy without additional particle sampling. We obtain this by a geometric reconstruction of the flow map using additional material constraints for the available samples. In comparison to the standard method, this allows for a more accurate global approximation of LCS on sparse grids and for long integration intervals. The proposed algorithm works directly on a set of given particle trajectories and without additional flow map derivatives. We demonstrate its application for a set of computational fluid dynamic examples, as well as trajectories acquired by Lagrangian methods, and discuss its benefits and limitations.

Abstract Stream surfaces are a well‐studied and widely used tool for the visualization of 3D flow fields. Usually, stream surface seeding is carried out manually in time‐consuming trial and error procedures. Only recently automatic selection methods were proposed. Local methods support the selection of a set of stream surfaces, but, contrary to global selection methods, they evaluate only thequalityof the seeding lines but not the quality of the whole stream surfaces. Global methods, on the other hand, only support the selection of asingleoptimal stream surface until now. However, for certain flow fields a single stream surface is not sufficient to represent all flow features. In our work, we overcome this limitation by introducing a global selection technique for asetof stream surfaces. All selected surfaces optimize global stream surface quality measures and are guaranteed to be mutually distant, such that they can convey different flow features. Our approach is an efficient extension of the most recent global selection method for single stream surfaces. We illustrate its effectiveness on a number of analytical and simulated flow fields and analyze the quality of the results in a user study.

In this article, we study the regularization of quadratic energies that are integrated over discrete domains. This is a fairly general setting, often found in, but not limited to, geometry processing. The standard Tikhonov regularization is widely used such that, for instance, a low-pass filter enforces smoothness of the solution. This approach, however, is independent of the energy and the concrete problem, which leads to artifacts in various applications. Instead, we propose a regularization that enforces a low variation of the energy and is problem specific by construction. Essentially, this approach corresponds to minimization with respect to a different norm. Our construction is generic and can be plugged into any quadratic energy minimization, is simple to implement, and has no significant runtime overhead. We demonstrate this for a number of typical problems and discuss the expected benefits.

This paper applies Poisson-based methods to assist in interactive exploration of steady flow fields. Using data-driven deformations we obtain flow-orthogonal and flow-tangential surfaces by a flux-based optimization. Surfaces are positioned interactively and deformed in real-time according to local flow. The deformed surfaces are particularly useful for defining seed structures. We show how the same gradient-based computational framework can be applied to obtain parametrizations of flow-aligned surfaces. This way it is easy to define nontrivial seed structures for integration-based flow visualization methods. Additionally, the flow-aligned parametrizations are employed for view-independent surface-based LIC visualizations. We apply our method to a number of data sets to show the effectiveness of our deformations and parametrization-based seed extraction methods for interactive flow exploration.

Abstract Stream surfaces are well‐known and widely‐used structures for 3D flow visualization. A single surface can be sufficient to represent important global flow characteristics. Unfortunately, due to the huge space of possible stream surfaces, finding the globally most representative stream surface turns out to be a hard task that is usually performed by time‐consuming manual trial and error exploration using slight modifications of seed geometries. To assist users we propose a new stream surface selection method that acts as an automatic preprocessing step before data analysis. We measure stream surface relevance by a novel surface‐based quality measure that prefers surfaces where the flow is aligned with principal curvature directions. The problem of seed structure selection can then be reduced to the computation of simple minimal paths in a weighted graph spanning the domain. We apply a simulated annealing‐based optimization method to find smooth seed curves of globally near‐optimal stream surfaces. We illustrate the effectiveness of our method on a series of synthetic and real‐world data sets.

For the visualization of dense line fields, the careful selection of lines to be rendered is a vital aspect. In this paper, we present a global line selection approach that is based on an optimization process. Starting with an initial set of lines that covers the domain, all lines are rendered with a varying opacity, which is subject to the minimization of a bounded-variable least-squares problem. The optimization strives to keep a balance between information presentation and occlusion avoidance. This way, we obtain view-dependent opacities of the line segments, allowing a real-time free navigation while minimizing the danger of missing important structures in the visualization. We compare our technique with existing local and greedy approaches and apply it to data sets in flow visualization, medical imaging, physics, and computer graphics.

The Finite Time Lyapunov Exponent (FTLE) has become a widespread tool for analyzing unsteady flow behavior. For its computation, several numerical methods have been introduced, which provide trade-offs between performance and accuracy. In order to decide which methods and parameter settings are suitable for a particular application, an evaluation of the different FTLE methods is necessary. We propose a general benchmark for FTLE computation, which consists of a number of 2D time-dependent flow fields and error measures. Evaluating the accuracy of a numerically computed FTLE field requires a ground truth, which is not available for realistic flow data sets, since such fields can generally not be described in a closed form. To overcome this, we introduce approaches to create non-trivial vector fields with a closed-form formulation of the FTLE field. Using this, we introduce a set of benchmark flow data sets that resemble relevant geometric aspects of Lagrangian structures, but have an analytic solution for FTLE. Based on this ground truth, we perform a comparative evaluation of three standard FTLE concepts. We suggest error measures based on the variance of both, the fields and the extracted ridge structures.

We define APAP surfaces, surfaces that are as perpendicular as possible to steady 3D vector fields, and present a method to construct discrete representations of them. Since, in general, a perfectly perpendicular surface to a vector field does not exist, we propose and minimize an error metric to enforce perpendicularity as much as possible. Our algorithm constructs an APAP surface by deforming a seed surface anchored in a domain point. In the discrete setting this minimization results in iteratively solving linear least-squares problems and integrating a locally scaled version of the vector field. The definition of the error metric and its numerical minimization guarantee that the minimum zero is attained for the perfectly perpendicular surface if it exists. Otherwise, the minimization converges to the same local minimum independent of the seed configuration, and the resulting surface is - in a least-squares sense - as perpendicular as possible to the flow. We apply these APAP surfaces as an interactive flow visualization tool which we demonstrate on a number of synthetic and real flow data sets.

Abstract The generation of discrete stream surfaces is an important and challenging task in scientific visualization, which can be considered a particular instance of geometric modeling. The quality of numerically integrated stream surfaces depends on a number of parameters that can be controlledlocally, such as time step or distance of adjacent vertices on the front line. In addition there is a parameter that cannot be controlled locally: stream surface meshes tend to show high quality, well‐shaped elements only if the current front line is “globally” approximately perpendicular to the flow direction. We analyze the impact of this geometric property and present a novel solution – a stream surface integrator that forces the front line to be perpendicular to the flow and that generates quad‐dominant meshes with well‐shaped and well‐aligned elements. It is based on the integration of a scaled version of the flow field, and requires repeated minimization of an error functional along the current front line. We show that this leads to computing the 1‐dimensional kernel of a bidiagonal matrix: a linear problem that can be solved efficiently. We compare our method with existing stream surface integrators and apply it to a number of synthetic and real world data sets.

We propose a new technique for visual exploration of streamlines in 3D vector fields. We construct a map from the space of all streamlines to points in IR(n) based on the preservation of the Hausdorff metric in streamline space. The image of a vector field under this map is a set of 2-manifolds in IR(n) with characteristic geometry and topology. Then standard clustering methods applied to the point sets in IR(n) yield a segmentation of the original vector field. Our approach provides a global analysis of 3D vector fields which incorporates the topological segmentation but yields additional information. In addition to a pure segmentation, the established map provides a natural "parametrization” visualized by the manifolds. We test our approach on a number of synthetic and real-world data sets.

We present a dense 3D correspondence finding method that enables spatio-temporally coherent reconstruction of surface animations from multi-view video data. Given as input a sequence of shape-from-silhouette volumes of a moving subject that were reconstructed for each time frame individually, our method establishes dense surface correspondences between subsequent shapes independently of surface discretization. This is achieved in two steps: first, we obtain sparse correspondences from robust optical features between adjacent frames. Second, we generate dense correspondences which serve as map between respective surfaces. By applying this procedure subsequently to all pairs of time steps we can trivially align one shape with all others. Thus, the original input can be reconstructed as a sequence of meshes with constant connectivity and small tangential distortion. We exemplify the performance and accuracy of our method using several synthetic and captured real-world sequences.

We present an efficient approach for solving the spherical parameterization problem. The essence of the approach is to look for a solution in the curvilinear coordinate system without requiring the additional spherical constraints usually needed in Cartesian formulations. This setup allows us to take full advantage of some existing techniques originally developed for planar parameterization. Our results substantiate the efficiency of the method and confirm its robustness. Meshes of non-trivial geometry with tens of thousands of triangles are processed in a few seconds, always yielding bijective maps. This computational achievement bridges a so far wide gap in performance between spherical and planar parameterization

In the last decade, surface mesh parameterization has emerged as a standard technique in computer graphics. The ever increasing need for processing large and highly detailed data sets fosters the development of efficient parameterization techniques that can capture the geometry of the input meshes and produce low distortion planar maps. We present a set of novel techniques allowing for low distortion parameterization. In particular, we address one of the major shortcomings of linear methods by allowing the parametric representation to evolve freely on the plane without any fixed boundary vertices. Our method consists of several simple steps, each solving a linear problem. Our results exhibit a fair balance between high-quality and computational efficiency.

We introduce new linear operators for surface parameterization. Given an initial mapping from the parametric plane onto a surface mesh, we establish a secondary map from the plane onto itself that mimics the initial one. The resulting low-distortion parameterization is smooth as it stems from solving a quasi-harmonic equation. Our parameterization method is robust and independent of (the quality of) the initial map.

We develop the first approach Tor interactive volume visualization based on a sophisticated rendering method of shear-warp type, wavelet data encoding techniques, and a trivariate spline model, which has been introduced recently. As a first step of our algorithm, we apply standard wavelet expansions to represent and decimate the given gridded three-dimensional data. Based on this data encoding, we give a sophisticated version of the shear-warp based volume rendering method. Our new algorithm visits each voxel only once taking advantage of the particular data organization of octrees. In addition, the hierarchies of the data guide the local (re)construction of the quadratic super-spline models, which we apply as a pure visualization tool. The low total degree of the polynomial pieces allows to numerically approximate the volume rendering integral efficiently. Since the coefficients of the splines are almost immediately available from the given data, Bernstein-Bezier techniques can be fully employed in our algorithms. In this way, we demonstrate that these models can be successfully applied to full volume rendering of hierarchically organized data. Our computational results show that (even when hierarchical approximations are used) the new approach leads to almost artifact-free visualizations of high quality for complicated and noise-contaminated volume data sets, while the computational effort is considerable low, i.e. our current implementation yields 1-2 frames per second for parallel perspective rendering a 2563 volume data set (using simple opacity transfer functions) in a 5122 view-port.

We present an interactive method for applying deformations to a surface mesh while preserving its global shape and local properties. Two surface editing scenarios are discussed, which conceptually differ in the specification of deformations: Either interpolation constraints are imposed explicitly, e.g., by dragging a subset of vertices, or, deformation of a reference surface is mimicked. The contribution of this paper is a novel approach for interpolation of local deformations over the manifold and for efficiently establishing correspondence to a reference surface from only few pairs of markers. As a general tool for both scenarios, a harmonic field is constructed to guide the interpolation of constraints and to find correspondence required for deformation transfer. We show that our approach fits nicely in a unified mathematical framework, where the same type of linear operator is applied in all phases, and how this approach can be used to create an intuitive and interactive editing tool. Figure 1: A simple edit: The visualized harmonic field is used as guidance for bending the cactus (left). Here, the field is defined by one source (red) at the tip of the left arm and one sink (blue) below the middle of the trunk. The result is shown in the center image. Notice the different propagation of the rotation compared to the edit on the right, where three sources on all arms were chosen (without picture). 1.

We introduce a new technique for estimating the curvature tensor of a triangular mesh. The input of the algorithm is only a single triangle equipped with its (exact or estimated) vertex normals. This way we get a smooth junction of the curvature tensor inside each triangle of the mesh. We show that the error of the new method is comparable with the error of a cubic fitting approach if the incorporated normals are estimated. If the exact normals of the underlying surface are available at the vertices, the error drops significantly. We demonstrate the applicability of the new estimation at a rather complex data set.

One of the main challenges in editing a mesh is to retain the visual appearance of the surface after applying various modifications. In this paper we advocate the use of linear differential coordinates as means to preserve the high-frequency detail of the surface. The differential coordinates represent the details and are defined by a linear transformation of the mesh vertices. This allows the reconstruction of the edited surface by solving a linear system that satisfies the reconstruction of the local details in least squares sense. Since the differential coordinates are defined in a global coordinate system they are not rotation-invariant. To compensate for that, we rotate them to agree with the rotation of an approximated local frame. We show that the linear least squares system can be solved fast enough to guarantee interactive response time thanks to a precomputed factorization of the coefficient matrix. We demonstrate that our approach enables to edit complex detailed meshes while keeping the shape of the details in their natural orientation.

We propose a new approach to reconstruct nondiscrete models from gridded volume samples. As a model, we use quadratic trivariate super splines on a uniform tetrahedral partition. We discuss the smoothness and approximation properties of our model and compare to alternative piecewise polynomial constructions. We observe, as a nonstandard phenomenon, that the derivatives of our splines yield optimal approximation order for smooth data, while the theoretical error of the values is nearly optimal due to the averaging rules. Our approach enables efficient reconstruction and visualization of the data. As the piecewise polynomials are of the lowest possible total degree two, we can efficiently determine exact ray intersections with an isosurface for ray-casting. Moreover, the optimal approximation properties of the derivatives allow us to simply sample the necessary gradients directly from the polynomial pieces of the splines. Our results confirm the efficiency of the quasi-interpolating method and demonstrate high visual quality for rendered isosurfaces.

Topological simplification techniques and topology preserving compression approaches for 2D vector fields have been developed quite independently of each other. In this paper we propose a combination of both approaches: a vector field should be compressed in such a way that its important topological features (both critical points and separatrices) are preserved while its unimportant features are allowed to collapse and disappear. To do so, a number of new solutions and modifications of pre-existing algorithms are presented. We apply the approach to a flow data set, which is both large and topologically complex, and achieve significant compression ratios there.

We develop a new approach to reconstruct non-discrete models from gridded volume samples. As a model, we use quadratic trivariate super splines on a uniform tetrahedral partition /spl Delta/. The approximating splines are determined in a natural and completely symmetric way by averaging local data samples, such that appropriate smoothness conditions are automatically satisfied. On each tetra-hedron of /spl Delta/ , the quasi-interpolating spline is a polynomial of total degree two which provides several advantages including efficient computation, evaluation and visualization of the model. We apply Bernstein-Bezier techniques well-known in CAGD to compute and evaluate the trivariate spline and its gradient. With this approach the volume data can be visualized efficiently e.g., with isosurface ray-casting. Along an arbitrary ray the splines are univariate, piecewise quadratics and thus the exact intersection for a prescribed isovalue can be easily determined in an analytic and exact way. Our results confirm the efficiency of the quasi-interpolating method and demonstrate high visual quality for rendered isosurfaces.

Abstract In this paper we introduce a new compression technique for 2D vector fields which preserves the complete topology, i.e., the critical points and the connectivity of the separatrices. As the theoretical foundation of the algorithm, we show in a theorem that for local modifications of a vector field, it is possible to decide entirely by a local analysis whether or not the global topology is preserved. This result is applied in a compression algorithm which is based on a repeated local modification of the vector field ‐ namely a repeated edge collapse of the underlying piecewise linear domain. We apply the compression technique to a number of data sets with a complex topology and obtain significantly improved compression ratios in comparison to pre‐existing topology‐preserving techniques.

We propose a divide and conquer algorithm for the single resolution encoding of triangle mesh connectivity. Starting from a boundary edge we grow a zig-zag strip which divides the mesh into two submeshes which are encoded separately in a recursive process. We introduce a novel data structure for triangle mesh connectivity encoding, a binary tree with positive integer weights assigned to its nodes. The length of the initial strip is stored in the root of the binary tree, while the encoding of the left and right submesh are stored in the left and right subtree, respectively. We find a simple criterion determining which objects of this data structure correspond to triangle meshes. As the algorithm implicitly traverses the triangles of the mesh, it can be classified into the family of Edgebreaker like encoding schemes. Hence, the compression ratios, both in the form of theoretical upper bounds and practical results are similar to the Edgebreaker's, while the simplicity and flexibility of the algorithm makes it particularly suitable for applications where the connectivity encoding is only a small part of the problem at hand.

Due to their simplicity triangle meshes are often used to represent geometric surfaces. Their main drawback is the large number of triangles that are required to represent a smooth surface. This problem has been addressed by a large number of mesh simplification algorithms which reduce the number of triangles and approximate the initial mesh. Hierarchical triangle mesh representations provide access to a triangle mesh at a desired resolution, without omitting any information. In this paper we present an infrastructure for mesh decimation, geometric mesh smoothing, and interactive multiresolution editing of arbitrary unstructured triangle meshes. In particular, we demonstrate how mesh reduction and geometric mesh smoothing can be combined to provide a powerful and numerically efficient multiresolution smoothing and editing paradigm.

Remeshing artifacts are a fundamental problem when converting a given geometry into a triangle mesh. We propose a new remeshing technique that is sensitive to features. First, the resolution of the mesh is iteratively adapted by a global restructuring process which additionally optimizes the connectivity. Then a particle system approach evenly distributes the vertices across the original geometry. To exactly find the features we extend this relaxation procedure by an effective mechanism to attract the vertices to feature edges. The attracting force is imposed by means of a hierarchical curvature field and does not require any thresholding parameters to classify the features.

We present an interactive system for computer aided generation of line art drawings to illustrate 3D models that are given as triangulated surfaces. In a preprocessing step, an enhanced 2D view of the scene is computed by sampling for every pixel the shading, the normal vectors and the principal directions obtained from discrete curvature analysis. Then streamlines are traced in the 2D direction fields and are used to define line strokes. In order to reduce noise artifacts, the user may interactively select sparse reference lines and the system will automatically fill in additional strokes. By exploiting the special structure of the streamlines, an intuitive and simple tone mapping algorithm can be derived to generate the final rendering.