We propose a containment query that is robust to the watertightness of regions bound by trimmed NURBS surfaces, as this property is difficult to guarantee for in-the-wild CAD models. Containment is determined through the generalized winding number (GWN), a mathematical construction that is indifferent to the arrangement of surfaces in the shape. Applying contemporary techniques for the 3D GWN to trimmed NURBS surfaces requires some form of geometric discretization, introducing computational inefficiency to the algorithm and even risking containment misclassifications near the surface. In contrast, our proposed method leverages properties of the 3D solid angle to solve the relevant surface integral using a boundary formulation with rapidly converging adaptive quadrature. Batches of queries are further accelerated by memoizing (i.e. caching and reusing) quadrature node positions and tangents as they are evaluated. We demonstrate that our GWN method is robust to complex trimming geometry in a CAD model, and is accurate up to arbitrary precision at arbitrary distances from the surface. The derived containment query is therefore robust to model non-watertightness while respecting all curved features of the input shape.

The generalized winding number (GWN) is a scalar field that supports robust containment queries on curved geometry, including non-watertight, overlapping, and nested boundary representations. While queries can be easily parallelized over samples, direct evaluation on parametric curves and surfaces remains costly for large and complex models. Fast, state-of-the-art GWN approaches leverage a spatial index to approximate the GWN, typically coupled with a Taylor expansion which approximates the GWN contribution for far clusters of geometric primitives. However, such methods operate only on discrete inputs such as triangle meshes and point clouds, and would introduce containment errors near boundaries if applied to curved input. We extend support for fast GWN evaluation over arbitrary collections of NURBS curves in 2D and trimmed NURBS patches in 3D via a Bounding Volume Hierarchy that stores efficiently precomputed moment data in the hierarchy nodes. When querying the hierarchy, approximations for far clusters are used alongside direct evaluation for nearby NURBS primitives, achieving sub-linear complexity while preserving the geometric features in the vicinity of the query point. Central to our performance improvements is an adaptive subdivision strategy for NURBS primitives during a preprocessing phase, creating better spatial partitions while retaining the same accuracy for containment decisions as a direct evaluation. We demonstrate the performance and accuracy of our approach across a large collection of 2D and 3D datasets.

Point containment queries for regions bound by watertight geometric surfaces, i.e., closed and without self-intersections, can be evaluated straightforwardly with a number of well-studied algorithms. When this assumption on domain geometry is not met, such methods are either unusable, or prone to misclassifications that can lead to cascading errors in downstream applications. More robust point classification schemes based on generalized winding numbers have been proposed, as they are indifferent to these imperfections. However, existing algorithms are limited to point clouds and collections of linear elements. We extend this methodology to encompass more general curved shapes with an algorithm that evaluates the winding number scalar field over unstructured collections of rational parametric curves. In particular, we evaluate the winding number for each curve independently, making the derived containment query robust to how the curves are arranged. We ensure geometric fidelity in our queries by treating each curve as equivalent to an adaptively constructed polyline that provably has the same generalized winding number at the point of interest. Our algorithm is numerically stable for points that are arbitrarily close to the model, and explicitly treats points that are coincident with curves. We demonstrate the improvements in computational performance granted by this method over conventional techniques as well as the robustness induced by its application.

Many foundational visualization techniques including isosurfacing, direct volume rendering and texture mapping rely on piecewise multilinear interpolation over the cells of a mesh. However, there has not been much focus within the visualization community on techniques that efficiently generate and encode globally continuous functions defined by the union of multilinear cells. Wavelets provide a rich context for analyzing and processing complicated datasets. In this paper, we exploit adaptive regular refinement as a means of representing and evaluating functions described by a subset of their nonzero wavelet coefficients. We analyze the dependencies involved in the wavelet transform and describe how to generate and represent the coarsest adaptive mesh with nodal function values such that the inverse wavelet transform is exactly reproduced via simple interpolation (subdivision) over the mesh elements. This allows for an adaptive, sparse representation of the function with on-demand evaluation at any point in the domain. We focus on the popular wavelets formed by tensor products of linear B-splines, resulting in an adaptive, nonconforming but crack-free quadtree (2D) or octree (3D) mesh that allows reproducing globally continuous functions via multilinear interpolation over its cells.

We consider the problem of efficient computing and simplifying Morse complexes on a Triangulated Irregular Network (TIN) based on discrete Morse theory. We develop a compact encoding for the discrete Morse gradient field, defined by the terrain elevation, by attaching it to the triangles of the TIN. This encoding is suitable to be combined with any TIN data structure storing just its vertices and triangles. We show how to compute such gradient field from the elevation values given at the TIN vertices, and how to simplify it effectively in order to reduce the number of critical elements. We demonstrate the effectiveness and scalability of our approach over large terrains by developing algorithms for extracting the cells of the Morse complexes as well as the graph joining the critical elements from the discrete gradient field. We compare implementations of our approach on a widely-used and compact adjacency-based topological data structure for a TIN and on a compact spatio-topological data structure that we have recently developed, the PR-star quadtree.

Abstract We consider the problem of computing discrete Morse and Morse‐Smale complexes on an unstructured tetrahedral mesh discretizing the domain of a 3D scalar field. We use a duality argument to define the cells of the descending Morse complex in terms of the supplied (primal) tetrahedral mesh and those of the ascending complex in terms of its dual mesh. The Morse‐Smale complex is then described combinatorially as collections of cells from the intersection of the primal and dual meshes. We introduce a simple compact encoding for discrete vector fields attached to the mesh tetrahedra that is suitable for combination with any topological data structure encoding just the vertices and tetrahedra of the mesh. We demonstrate the effectiveness and scalability of our approach over large unstructured tetrahedral meshes by developing algorithms for computing the discrete gradient field and for extracting the cells of the Morse and Morse‐Smale complexes. We compare implementations of our approach on an adjacency‐based topological data structure and on the PR‐star octree, a compact spatio‐topological data structure.

We propose the PR-star octree as a combined spatial data structure for performing efficient topological queries on tetrahedral meshes. The PR-star octree augments the Point Region octree (PR Octree) with a list of tetrahedra incident to its indexed vertices, i.e. those in the star of its vertices. Thus, each leaf node encodes the minimal amount of information necessary to locally reconstruct the topological connectivity of its indexed elements. This provides the flexibility to efficiently construct the optimal data structure to solve the task at hand using a fraction of the memory required for a corresponding data structure on the global tetrahedral mesh. Due to the spatial locality of successive queries in typical GIS applications, the construction costs of these runtime data structures are amortized over multiple accesses while processing each node. We demonstrate the advantages of the PR-star octree representation in several typical GIS applications, including detection of the domain boundaries, computation of local curvature estimates and mesh simplification.

Abstract Hierarchical spatial decompositions are a basic modelling tool in a variety of application domains. Several papers on this subject deal with hierarchical simplicial decompositions generated throughregular simplex bisection. Such decompositions, originally developed for finite elements, are extensively used as the basis for multi‐resolution models of scalar fields, such as terrains, and static or time‐varying volume data. They have also been used as an alternative to quadtrees and octrees as spatial access structures. The primary distinction among all such approaches is whether they treat the simplex or clusters of simplices, called diamonds, as the modelling primitive. This leads to two classes of data structures and to different query approaches. We present the hierarchical models in a dimension‐independent manner, and organize the description of the various applications, primarily interactive terrain rendering and isosurface extraction, according to the dimension of the domain.

We consider a model of a 3D image obtained by discretizing it into a multiresolution tetrahedral mesh known as a hierarchy of diamonds. This model enables us to extract crack-free approximations of the 3D image at any uniform or variable resolution, thus reducing the size of the data set without reducing the accuracy. A 3D intensity image is a scalar field (the intensity field) defined at the vertices of a 3D regular grid and thus the graph of the image is a hypersurface in R4. We measure the discrete distortion, a generalization of the notion of curvature, of the transformation which maps the tetrahedralized 3D grid onto its graph in R4. We evaluate the use of a hierarchy of diamonds to analyze properties of a 3D image, such as its discrete distortion, directly on lower resolution approximations. Our results indicate that distortion-guided extractions focus the resolution of approximated images on the salient features of the intensity image.

Efficient multiresolution representations for isosurfaces and interval volumes are becoming increasingly important as the gap between volume data sizes and processing speed continues to widen. Our multiresolution scalar field model is a hierarchy of tetrahedral clusters generated by longest edge bisection that we call a hierarchy of diamonds. We propose two multiresolution models for representing isosurfaces, or interval volumes, extracted from a hierarchy of diamonds which exploit its regular structure. These models are defined by subsets of diamonds in the hierarchy that we call isodiamonds, which are enhanced with geometric and topological information for encoding the relation between the isosurface, or interval volume, and the diamond itself. The first multiresolution model, called a relevant isodiamond hierarchy, encodes the isodiamonds intersected by the isosurface, or interval volume, as well as their nonintersected ancestors, while the second model, called a minimal isodiamond hierarchy, encodes only the intersected isodiamonds. Since both models operate directly on the extracted isosurface or interval volume, they require significantly less memory and support faster selective refinement queries than the original multiresolution scalar field, but do not support dynamic isovalue modifications. Moreover, since a minimal isodiamond hierarchy only encodes intersected isodiamonds, its extracted meshes require significantly less memory than those extracted from a relevant isodiamond hierarchy. We demonstrate the compactness of isodiamond hierarchies by comparing them to an indexed representation of the mesh at full resolution.

Abstract Nested simplicial meshes generated by the simplicial bisection decomposition proposed by Maubach [ Mau95 ] have been widely used in 2D and 3D as multi‐resolution models of terrains and three‐dimensional scalar fields, They are an alternative to octree representation since they allow generating crack‐free representations of the underlying field. On the other hand, this method generates conforming meshes only when all simplices sharing the bisection edge are subdivided concurrently. Thus, efficient representations have been proposed in 2D and 3D based on a clustering of the simplices sharing a common longest edge in what is called a diamond. These representations exploit the regularity of the vertex distribution and the diamond structure to yield an implicit encoding of the hierarchical and geometric relationships among the triangles and tetrahedra, respectively. Here, we analyze properties ofd‐dimensional diamonds to better understand the hierarchical and geometric relationships among the simplices generated by Maubach's bisection scheme and derive closed‐form equations for the number of vertices, simplices, parents and children of each type of diamond. We exploit these properties to yield an implicit pointerless representation ford‐dimensional diamonds and reduce the number of required neighbor‐finding accesses fromO(d!) toO(d).

Volumetric datasets are often modeled using a multiresolution approach based on a nested decomposition of the domain into a polyhedral mesh. Nested tetrahedral meshes generated through the longest edge bisection rule are commonly used to decompose regular volumetric datasets since they produce highly adaptive crack-free representations. Efficient representations for such models have been achieved by clustering the set of tetrahedra sharing a common longest edge into a structure called a diamond. The alignment and orientation of the longest edge can be used to implicitly determine the geometry of a diamond and its relations to the other diamonds within the hierarchy. We introduce the supercube as a high-level primitive within such meshes that encompasses all unique types of diamonds. A supercube is a coherent set of edges corresponding to three consecutive levels of subdivision. Diamonds are uniquely characterized by the longest edge of the tetrahedra forming them and are clustered in supercubes through the association of the longest edge of a diamond with a unique edge in a supercube. Supercubes are thus a compact and highly efficient means of associating information with a subset of the vertices, edges and tetrahedra of the meshes generated through longest edge bisection. We demonstrate the effectiveness of the supercube representation when encoding multiresolution diamond hierarchies built on a subset of the points of a regular grid. We also show how supercubes can be used to efficiently extract meshes from diamond hierarchies and to reduce the storage requirements of such variable-resolution meshes.

Bintrees based on longest edge bisection and hierarchies of diamonds are popular multiresolution techniques on regularly sampled terrain datasets. In this work, we consider Sparse Terrain Pyramids as a compact multiresolution representation for terrain datasets whose samples are a subset of those lying on a regular grid. While previous diamond-based approaches can efficiently represent meshes built on a complete grid of resolution (2k +1)2, this is not suitable when the field values are uniform in large areas or simply non-existent. We explore properties of diamonds to simplify an encoding of the implicit dependency relationship between diamonds. Additionally, we introduce a diamond clustering technique to further reduce the geometric and topological overhead of such representations. We demonstrate the coherence of our clustering technique as well as the compactness of our representation.

In this paper, we report our newly developed 3D face modeling system with arbitrary expressions in a high level of detail using the topographic analysis and mesh instantiation process. Given a sequence of images of facial expressions at frontal views, we automatically generate 3D expressions at arbitrary views. Our face modeling system consists of two major components: facial surface representation using topographic analysis and generic model individualization based on labeled surface features and surface curvatures. The realism of the generated individual model is demonstrated through 3D views of facial expressions in videos. This work targets the accurate modeling of face and face expression for human computer interaction and 3D face recognition.