Divided edge bundling for directional network data

The node-link diagram is an intuitive and venerable way to depict a graph. To reduce clutter and improve the readability of node-link views, Holten & van Wijk's force-directed edge bundling employs a physical simulation to spatially group graph edges. While both useful and aesthetic, this technique has shortcomings: it bundles spatially proximal edges regardless of direction, weight, or graph connectivity. As a result, high-level directional edge patterns are obscured. We present divided edge bundling to tackle these shortcomings. By modifying the forces in the physical simulation, directional lanes appear as an emergent property of edge direction. By considering graph topology, we only bundle edges related by graph structure. Finally, we aggregate edge weights in bundles to enable more accurate visualization of total bundle weights. We compare visualizations created using our technique to standard force-directed edge bundling, matrix diagrams, and clustered graphs; we find that divided edge bundling leads to visualizations that are easier to interpret and reveal both familiar and previously obscured patterns.     

Winding Roads: Routing edges into bundles

Visualizing graphs containing many nodes and edges efficiently is quite challenging. Drawings of such graphs generally suffer from visual clutter induced by the large amount of edges and their crossings. Consequently it is difficult to read the relationships between nodes and the high‐level edge patterns that may exist in standard node‐link diagram representations. Edge bundling techniques have been proposed to help solve this issue, which rely on high quality edge rerouting. In this paper, we introduce an intuitive edge bundling technique which efficiently reduces edge clutter in graphs drawings. Our method is based on the use of a grid built using the original graph to compute the edge rerouting. In comparison with previously proposed edge bundling methods, our technique improves both the level of clutter reduction and the computation performance. The second contribution of this paper is a GPU‐based rendering method which helps users perceive bundles densities while preserving edge color.     

Image‐Based Edge Bundles: Simplified Visualization of Large Graphs

We present a new approach aimed at understanding the structure of connections in edge‐bundling layouts. We combine the advantages of edge bundles with a bundle‐centric simplified visual representation of a graph's structure. For this, we first compute a hierarchical edge clustering of a given graph layout which groups similar edges together. Next, we render clusters at a user‐selected level of detail using a new image‐based technique that combines distance‐based splatting and shape skeletonization. The overall result displays a given graph as a small set of overlapping shaded edge bundles. Luminance, saturation, hue, and shading encode edge density, edge types, and edge similarity. Finally, we add brushing and a new type of semantic lens to help navigation where local structures overlap. We illustrate the proposed method on several real‐world graph datasets.     

Illustrative Parallel Coordinates

Illustrative parallel coordinates (IPC) is a suite of artistic rendering techniques for augmenting and improving parallel coordinate (PC) visualizations. IPC techniques can be used to convey a large amount of information about a multidimensional dataset in a small area of the screen through the following approaches: (a) edge‐bundling through splines; (b) visualization of “branched ” clusters to reveal the distribution of the data; (c) opacity‐based hints to show cluster density; (d) opacity and shading effects to illustrate local line density on the parallel axes; and (e) silhouettes, shadows and halos to help the eye distinguish between overlapping clusters. Thus, the primary goal of this work is to convey as much information as possible in a manner that is aesthetically pleasing and easy to understand for non‐experts.     

Visual Comparison of Hierarchically Organized Data

We provide a novel visualization method for the comparison of hierarchically organized data. Our technique visualizes a pair of hierarchies that are to be compared and simultaneously depicts how these hierarchies are related by explicitly visualizing the relations between matching subhierarchies. Elements that are unique to each hierarchy are shown, as well as the way in which hierarchy elements are relocated, split or joined. The relations between hierarchy elements are visualized using Hierarchical Edge Bundles (HEBs). HEBs reduce visual clutter, they visually emphasize the aforementioned splits, joins, and relocations of subhierarchies, and they provide an intuitive way in which users can interact with the relations. The focus throughout this paper is on the comparison of different versions of hierarchically organized software systems, but the technique is applicable to other kinds of hierarchical data as well. Various data sets of actual software systems are used to show how our technique can be employed to easily spot splits, joins, and relocations of elements, how sorting both hierarchies with respect to each other facilitates comparison tasks, and how user interaction is supported.     

Extraction Of Feature Lines On Surface Meshes Based On Discrete Morse Theory

We present an approach for extracting extremal feature lines of scalar indicators on surface meshes, based on discrete Morse Theory. By computing initial Morse‐Smale complexes of the scalar indicators of the mesh, we obtain a candidate set of extremal feature lines of the surface. A hierarchy of Morse‐Smale complexes is computed by prioritizing feature lines according to a novel criterion and applying a cancellation procedure that allows us to select the most significant lines. Given the scalar indicators on the vertices of the mesh, the presented feature line extraction scheme is interpolation free and needs no derivative estimates. The technique is insensitive to noise and depends only on one parameter: the feature significance. We use the technique to extract surface features yielding impressive, non photorealistic images.     

Hierarchical Convex Approximation of 3D Shapes for Fast Region Selection

Given a 3D solid model S represented by a tetrahedral mesh, we describe a novel algorithm to compute a hierarchy of convex polyhedra that tightly enclose S. The hierarchy can be browsed at interactive speed on a modern PC and it is useful for implementing an intuitive feature selection paradigm for 3D editing environments. Convex parts often coincide with perceptually relevant shape components and, for their identification, existing methods rely on the boundary surface only. In contrast, we show that the notion of part concavity can be expressed and implemented more intuitively and efficiently by exploiting a tetrahedrization of the shape volume. The method proposed is completely automatic, and generates a tree of convex polyhedra in which the root is the convex hull of the whole shape, and the leaves are the tetrahedra of the input mesh. The algorithm proceeds bottom‐up by hierarchically clustering tetrahedra into nearly convex aggregations, and the whole process is significantly fast. We prove that, in the average case, for a mesh of n tetrahedra O(n log² n) operations are sufficient to compute the whole tree.     

Visualisation of Sensor Data from Animal Movement

A new area of biological research is identifying and grouping patterns of behaviour in wild animals by analysing data obtained through the attachment of tri-axial accelerometers. As these recording devices become smaller and less expensive their use has increased. Currently acceleration data are visualised as 2D time series plots, and analyses are based on summary statistics and the application of Fourier transforms. We develop alternate visualisations of this data so as to analyse, explore and present new patterns of animal behaviour. Our visualisations include interactive spherical scatterplots, spherical histograms, clustering methods, and feature-based state diagrams of the data. We study the application of these visualisation methods to accelerometry data from animal movement. The reaction of biologists to these visualisations is also reported.     

Visual Inspection of Multivariate Graphs

Most graph visualization techniques focus on the structure of graphs and do not offer support for dealing with node attributes and edge labels. To enable users to detect relations and patterns in terms of data associated with nodes and edges, we present a technique where this data plays a more central role. Nodes and edges are clustered based on associated data. Via direct manipulation users can interactively inspect and query the graph. Questions that can be answered include, “which edge types are activated by specific node attributes?” and, “how and from where can I reach specific types of nodes?” To validate our approach we contrast it with current practice. We also provide several examples where our method was used to study transition graphs that model real‐world systems.     

Abstractive Representation and Exploration of Hierarchically Clustered Diffusion Tensor Fiber Tracts

Diffusion tensor imaging (DTI) has been used to generate fibrous structures in both brain white matter and muscles. Fiber clustering groups the DTI fibers into spatially and anatomically related tracts. As an increasing number of fiber clustering methods have been recently developed, it is important to display, compare, and explore the clustering results efficiently and effectively. In this paper, we present an anatomical visualization technique that reduces the geometric complexity of the fiber tracts and emphasizes the high‐level structures. Beginning with a volumetric diffusion tensor image, we first construct a hierarchical clustering representation of the fiber bundles. These bundles are then reformulated into a 3D multi‐valued volume data. We then build a set of geometric hulls and principal fibers to approximate the shape and orientation of each fiber bundle. By simultaneously visualizing the geometric hulls, individual fibers, and other data sets such as fractional anisotropy, the overall shape of the fiber tracts are highlighted, while preserving the fibrous details. A rater with expert knowledge of white matter structure has evaluated the resulting interactive illustration and confirmed the improvement over straightforward DTI fiber tract visualization.     

Embedded Implicit Stand-Ins for Animated Meshes: A Case of Hybrid Modelling

In this paper, we address shape modelling problems, encountered in computer animation and computer games development that are difficult to solve just using polygonal meshes. Our approach is based on a hybrid-modelling concept that combines polygonal meshes with implicit surfaces. A hybrid model consists of an animated polygonal mesh and an approximation of this mesh by a convolution surface stand-in that is embedded within it or is attached to it. The motions of both objects are synchronised using a rigging skeleton. We model the interaction between an animated mesh object and a viscoelastic substance, which is normally represented in an implicit form. Our approach is aimed at achieving verisimilitude rather than physically based simulation. The adhesive behaviour of the viscous object is modelled using geometric blending operations on the corresponding implicit surfaces. Another application of this approach is the creation of metamorphosing implicit surface parts that are attached to an animated mesh. A prototype implementation of the proposed approach and several examples of modelling and animation with near real-time preview times are presented.     

Fitting Sharp Features with Loop Subdivision Surfaces

Various methods have been proposed for fitting subdivision surfaces to different forms of shape data (e.g., dense meshes or point clouds), but none of these methods effectively deals with shapes with sharp features, that is, creases, darts and corners. We present an effective method for fitting a Loop subdivision surface to a dense triangle mesh with sharp features. Our contribution is a new exact evaluation scheme for the Loop subdivision with all types of sharp features, which enables us to compute a fitting Loop subdivision surface for shapes with sharp features in an optimization framework. With an initial control mesh obtained from simplifying the input dense mesh using QEM, our fitting algorithm employs an iterative method to solve a nonlinear least squares problem based on the squared distances from the input mesh vertices to the fitting subdivision surface. This optimization framework depends critically on the ability to express these distances as quadratic functions of control mesh vertices using our exact evaluation scheme near sharp features. Experimental results are presented to demonstrate the effectiveness of the method.     

An Operator Approach to Tangent Vector Field Processing

In this paper, we introduce a novel coordinate‐free method for manipulating and analyzing vector fields on discrete surfaces. Unlike the commonly used representations of a vector field as an assignment of vectors to the faces of the mesh, or as real values on edges, we argue that vector fields can also be naturally viewed as operators whose domain and range are functions defined on the mesh. Although this point of view is common in differential geometry it has so far not been adopted in geometry processing applications. We recall the theoretical properties of vector fields represented as operators, and show that composition of vector fields with other functional operators is natural in this setup. This leads to the characterization of vector field properties through commutativity with other operators such as the Laplace‐Beltrami and symmetry operators, as well as to a straight‐forward definition of differential properties such as the Lie derivative. Finally, we demonstrate a range of applications, such as Killing vector field design, symmetric vector field estimation and joint design on multiple surfaces.     

Conformal Flattening by Curvature Prescription and Metric Scaling

We present an efficient method to conformally parameterize 3D mesh data sets to the plane. The idea behind our method is to concentrate all the 3D curvature at a small number of select mesh vertices, called cone singularities, and then cut the mesh through those singular vertices to obtain disk topology. The singular vertices are chosen automatically. As opposed to most previous methods, our flattening process involves only the solution of linear systems of Poisson equations, thus is very efficient. Our method is shown to be faster than existing methods, yet generates parameterizations having comparable quasi‐conformal distortion.     

Fully Automatic Visualisation of Overlapping Sets

Visualisation of taxonomies and sets has recently become an active area of research. Many application fields now require more than a strict classification of elements into a hierarchy tree. Euler diagrams, one of the most natural ways of depicting intersecting sets, may provide a solution to these problems.
In this paper, we present an approach for the automatic generation of Euler-like diagrams. This algorithm differs from previous approaches in that it has no undrawable instances of input, allowing it to be used in systems where the output is always required. We also improve the readability of Euler diagrams through the use of Bézier curves and transparent coloured textures. Our approach has been implemented using the Tulip platform. Both the source and executable program used to generate the results are freely available.     

Automatic Conversion of Mesh Animations into Skeleton-based Animations

Recently, it has become increasingly popular to represent animations not by means of a classical skeleton‐based model, but in the form of deforming mesh sequences. The reason for this new trend is that novel mesh deformation methods as well as new surface based scene capture techniques offer a great level of flexibility during animation creation. Unfortunately, the resulting scene representation is less compact than skeletal ones and there is not yet a rich toolbox available which enables easy post‐processing and modification of mesh animations. To bridge this gap between the mesh‐based and the skeletal paradigm, we propose a new method that automatically extracts a plausible kinematic skeleton, skeletal motion parameters, as well as surface skinning weights from arbitrary mesh animations. By this means, deforming mesh sequences can be fully‐automatically transformed into fullyrigged virtual subjects. The original input can then be quickly rendered based on the new compact bone and skin representation, and it can be easily modified using the full repertoire of already existing animation tools.     

Estimating the Laplace-Beltrami Operator by Restricting 3D Functions

We present a novel approach for computing and solving the Poisson equation over the surface of a mesh. As in previous approaches, we define the Laplace-Beltrami operator by considering the derivatives of functions defined on the mesh. However, in this work, we explore a choice of functions that is decoupled from the tessellation. Specifically, we use basis functions (second-order tensor-product B-splines) defined over 3D space, and then restrict them to the surface. We show that in addition to being invariant to mesh topology, this definition of the Laplace-Beltrami operator allows a natural multiresolution structure on the function space that is independent of the mesh structure, enabling the use of a simple multigrid implementation for solving the Poisson equation.     

Guided Image Filtering for Interactive High‐quality Global Illumination

Interactive computation of global illumination is a major challenge in current computer graphics research. Global illumination heavily affects the visual quality of generated images. It is therefore a key attribute for the perception of photo‐realistic images. Path tracing is able to simulate the physical behaviour of light using Monte Carlo techniques. However, the computational burden of this technique prohibits interactive rendering times on standard commodity hardware in high‐quality. Trying to solve the Monte Carlo integration with fewer samples results in characteristic noisy images. Global illumination filtering methods take advantage of the fact that the integral for neighbouring pixels may be very similar. Averaging samples of similar characteristics in screen‐space may approximate the correct integral, but may result in visible outliers. In this paper, we present a novel path tracing pipeline based on an edge‐aware filtering method for the indirect illumination which produces visually more pleasing results without noticeable outliers. The key idea is not to filter the noisy path traced images but to use it as a guidance to filter a second image composed from characteristic scene attributes that do not contain noise by default. We show that our approach better approximates the Monte Carlo integral compared to previous methods. Since the computation is carried out completely in screen‐space it is therefore applicable to fully dynamic scenes, arbitrary lighting and allows for high‐quality path tracing at interactive frame rates on commodity hardware.     

Separatrix Persistence: Extraction of Salient Edges on Surfaces Using Topological Methods

Salient edges are perceptually prominent features of a surface. Most previous extraction schemes utilize the notion of ridges and valleys for their detection, thereby requiring curvature derivatives which are rather sensitive to noise. We introduce a novel method for salient edge extraction which does not depend on curvature derivatives. It is based on a topological analysis of the principal curvatures and salient edges of the surface are identified as parts of separatrices of the topological skeleton. Previous topological approaches obtain results including non-salient edges due to inherent properties of the underlying algorithms. We extend the profound theory by introducing the novel concept of separatrix persistence , which is a smooth measure along a separatrix and allows to keep its most salient parts only. We compare our results with other methods for salient edge extraction.