We adapt multilevel, force-directed graph layout techniques to visualizing dynamic graphs in which vertices and edges are added and removed in an online fashion (i.e., unpredictably). We maintain multiple levels of coarseness using a dynamic, randomized coarsening algorithm. To ensure the vertices follow smooth trajectories, we employ dynamics simulation techniques, treating the vertices as point particles. We simulate fine and coarse levels of the graph simultaneously, coupling the dynamics of adjacent levels. Projection from coarser to finer levels is adaptive, with the projection determined by an affine transformation that evolves alongside the graph layouts. The result is a dynamic graph visualizer that quickly and smoothly adapts to changes in a graph.
Offline graph animation tools develop an animation from a sequence of key frames (layouts of static graphs). Such systems find layouts for key frame graphs, and then interpolate between the key frames in an appropriate way (e.g. @cite_0 ).
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First we introduce the concept of graph animations as a sequence of evolving graphs and a generic algorithm which computes a Foresighted Layout for dynamically drawing these graphs while preserving the mental map. The algorithm is generic in the sense that it takes a static graph drawing algorithm as a parameter. In other words, trees can be animated with a static tree layouter, graphs with a static Sugiyama-style layouter or a spring embedder, etc. Second we discuss applications of Foresighted Layout in algorithm animation and visualization of navigation behaviour. We present a novel hybrid representation for character animation from 4D Performance Capture (4DPC) data which combines skelet al control with surface motion graphs. 4DPC data are temporally aligned 3D mesh sequence reconstructions of the dynamic surface shape and associated appearance from multiple-view video. The hybrid representation supports the production of novel surface sequences which satisfy constraints from user-specified key-frames or a target skelet al motion. Motion graph path optimisation concatenates fragments of 4DPC data to satisfy the constraints while maintaining plausible surface motion at transitions between sequences. Space-time editing of the mesh sequence using a learned part-based Laplacian surface deformation model is performed to match the target skelet al motion and transition between sequences. The approach is quantitatively evaluated for three 4DPC datasets with a variety of clothing styles. Results for key-frame animation demonstrate production of novel sequences that satisfy constraints on timing and position of less than 1p of the sequence duration and path length. Evaluation of motion-capture-driven animation over a corpus of 130 sequences shows that the synthesised motion accurately matches the target skelet al motion. The combination of skelet al control with the surface motion graph extends the range and style of motion which can be produced while maintaining the natural dynamics of shape and appearance from the captured performance. We propose a heuristic for dynamic hierarchical graph drawing. Applications include incremental graph browsing and editing, display of dynamic data structures and networks, and browsing large graphs. The heuristic is an on-line interpretation of the static layout algorithm of Sugiyama, Togawa and Toda. It incorporates topological and geometric information with the objective of making layout animations that are incrementally stable and readable through long editing sequences. We measured the performance of a prototype implementation. Abstract Splines and subdivision curves are flexible tools in the design and manipulation of curves in Euclidean space. In this paper we study generalizations of interpolating splines and subdivision schemes to the Riemannian manifold of shell surfaces in which the associated metric measures both bending and membrane distortion. The shells under consideration are assumed to be represented by Loop subdivision surfaces. This enables the animation of shells via the smooth interpolation of a given set of key frame control meshes. Using a variational time discretization of geodesics efficient numerical implementations can be derived. These are based on discrete geodesic interpolation, a discrete geometric logarithm, a discrete exponential map, and discrete parallel transport. With these building blocks at hand discrete Riemannian cardinal splines and three different types of discrete, interpolatory subdivision schemes are defined. Numerical results for two different subdivision shell models underline the potential of this approach in key frame animation.