This paper describes some fundamental issues for robust implementations of progressively refined tetrahedralizations generated through sequences of edge collapses. We address the definition of appropriate cost functions and explain on various tests which are necessary to preserve the consistency of the mesh when collapsing edges. Although being considered a special case of progressive simplicial complexes [10], the results of our method are of high practical importance and can be used in many different applications, such as finite element meshing, scattered data interpolation, or rendering of unstructured volume data. CR Categories: I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling -- surfaces and object representations. Keywords: mesh simplification, multiresolution, level-ofdetail, unstructured meshes, mesh generation. 1 INTRODUCTION Progressive meshes [7] and its generalizations to higher dimensions [10] proofed to be an extremely powerful notion for the effic...

Multiresolution methods are becoming increasingly important tools for the interactive visualization of very large data sets. Multiresolution isosurface visualization allows the user to explore volume data using simplified and coarse representations of the isosurface for overview images, and finer resolution in areas of high interest or when zooming into the data. Ideally, a coarse isosurface should have the same topological structure as the original. The topological genus of the isosurface is one important property which is often neglected in multiresolution algorithms. This results in uncontrolled topological changes which can occur whenever the level-of-detail is changed. The scope of this paper is to propose an efficient technique which allows preservation of topology as well as controlled topology simplification in multiresolution isosurface extraction. CR Categories: G.1.2 [Numerical Analysis]: Approximation--- Approximation of Surfaces and Contours I.3.5 [Computer Graphics ]: Co...

A system to represent and visualize scalar volume data at multiple resolution is presented. The system is built on a multiresolution model based on tetrahedral meshes with scattered vertices that can be obtained from any initial dataset. The model is built off-line through data simplification techniques, and stored in a compact data structure that supports fast on-line access. The system supports interactive visualization of a representation at an arbitrary level of resolution through isosurface and projective methods. The user can interactively adapt the quality of visualization to requirements of a specific application task, and to the performance of a specific hardware platform. Representations at different resolutions can be used together to enhance further interaction and performance through progressive and multiresolution rendering. Index Terms -- Volume data visualization, multiresolution representation, tetrahedral meshes. I. Introduction Volume datasets used in current applic...

Nowadays, multiresolution visualization methods become an indispensable ingredient of real time interactive post processing. We will here present an efficient approach for tetrahedral grids recursively generated by bisection, which is based on a more general method for arbitrary nested grids. It especially applies to regular grids, the hexahedra of which are procedurally subdivided into tetrahedra. Besides different types of error indicators, we especially focus on improving the algorithm's performance and reducing the memory requirements. Furthermore, parallelization combined with an appropriate load balancing on multiprocessor workstations is discussed. 1 Introduction A variety of multiresolution visualization methods has been designed to serve as tools for interactive visualization of large data sets. The local resolution of the generated visual objects, such as isosurfaces, is thereby steered by error indicators which measure the error due to a locally coarser approximation of the...

levels of detail are widely used for large-scale two- and three-dimensional data sets. We present a four-dimensional multiresolution approach for time-varying volume data. This approach supports a hierarchy with spatial and temporal scalability.

Topological volume skeletons represent level-set graphs of 3D scalar fields, and have recently become crucial to visualizing the global isosurface transitions in the volume. However, it is still a time-consuming task to extract them especially when input volumes are large-scale data and/or prone to small-amplitude noise. This paper presents an efficient method for accelerating the computation of such skeletons using adaptive tetrahedralization. The present tetrahedralization is a top-down approach to linear interpolation of the scalar fields in that it selects tetrahedra to be subdivided adaptively using several criteria. As the criteria, the method employs a topological criterion as well as a geometric one in order to pursue all the topological isosurface transitions that may contribute to the global skeleton of the volume. The tetrahedralization also allows us to avoid unnecessary tracking of minor degenerate features that hide the global skeleton. Experimental results are included to demonstrate that the present method smoothes out the original scalar fields effectively without missing any significant topological features. 1

). The approximation error to the 2 original function in the uniform norm is color coded (from red ≥ 0.075 to blue=0) for the standard trilinear model (left) and our new quadratic super splines (right). The center image shows a visually perfect reconstruction using our model for (4 × 41) 3 samples. The maximum error is 0.0065 (center) compared to 0.088 (right) which illustrates that the quasi-interpolating spline yields nearly optimal approximation order. 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 ∆. 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 tetrahedron of ∆, 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-Bézier 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 raycasting. 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.

Multiresolution meshes are a common basis for building representations of a geometric shape at dierent levels of detail. The use of the term multiresolution depends on the remark that the accuracy (or, level of detail) of a mesh in approximating a shape is related to the mesh resolution, i.e., to the density (size and number) of its cells. A multiresolution mesh provides several alternative meshbased approximations of a spatial object (e.g., a surface describing the boundary of a solid object, or the graph of a scalar eld).

We present a tetrahedral mesh generation algorithm designed for the Lagrangian simulation of deformable bodies. The algorithm’s input is a level set (i.e., a signed distance function on a Cartesian grid or octree). First a bounding box of the object is covered with a uniform lattice of subdivision-invariant tetrahedra. The level set is then used to guide a red green adaptive subdivision procedure that is based on both the local curvature and the proximity to the object boundary. The final topology is carefully chosen so that the connectivity is suitable for large deformation and the mesh approximates the desired shape. Finally, this candidate mesh is compressed to match the object boundary. To maintain element quality during this compression phase we relax the positions of the nodes using finite elements, masses and springs, or an optimization procedure. The resulting mesh is well suited for simulation since it is highly structured, has topology chosen specifically for large deformations, and is readily refined if required during subsequent simulation. We then use this algorithm to generate meshes for the simulation of skeletal muscle from level set representations of the anatomy. The geometric complexity of biological materials makes it very difficult to generate these models procedurally and as a result we obtain most if not all data from an actual human subject. Our current method involves using voxelized data from the Visible Male [1] to create level set representations of muscle and bone geometries. Given this representation, we use simple level set operations to rebuild and repair errors in the segmented data as well as to smooth aliasing inherent in the voxelized data.

Nowadays computing and post processing of simulation data is often based on efficient hierarchical methods. While multigrid methods are already established standards for fast simulation codes, multiresolution visualization methods have only recently become an important ingredient of real--time interactive post processing. Both methodologies use local error indicators which serve as criteria where to refine the data representation on the physical domain. In this article we give an overview on different types of error measurement on nested grids and compare them for selected applications in 2D as well as in 3D. Furthermore, it is pointed out that a certain saturation of the considered error indicator plays an important role in multilevel visualization and computing on implicitly defined adaptive grids. 1 Introduction Today, efficient adaptive multigrid methods are available for a large variety of simulation problems [11]. Following this frontier a variety of multiresolution visualizati...