Optimal iso-surfaces

Since the publication of the original Marching Cubes algorithm, numerous variations have been proposed for guaranteeing water-tight constructions of triangulated approximations of iso-surfaces. Most approaches divide the 3D space into cubes that each occupies the space between eight neighboring samples of a regular lattice. The portion of the iso-surface inside a cube may be computed independently of what happens in the other cubes, provided that the constructions for each pair of neighboring cubes agree along their common face. The portion of the iso-surface associated with a cube may consist of one or more connected components, which we call sheets. We distinguish three types of decisions in the construction of the iso-surface connectivity: (1) how to split the X-faces, which have alternating in/out samples, (2) how many sheets to use in a cube, and (3) how to triangulate each sheet. Previously reported techniques make these decisions based on local criteria, often using pre-computed look-up tables or simple construction rules. Instead, we propose global strategies for optimizing several topological and combinatorial measures of the isosurfaces: triangle count, genus, and number of shells. We describe efficient implementations of these optimizations and the auxiliary data structures developed to support them.Postprint (updated version

A topological comparison of surface extraction algorithms

In many application areas, it is useful to convert the discrete information stored in the nodes of a regular grid into a continuous boundary model. Isosurface extraction algorithms differ on how the discrete information in the grid is generated, on what information does the grid store and on the properties of the output surface. Recent algorithms offer different solutions for the disambiguation problem and for controlling the final topology. Based on a number of properties of the grid’s grey cells and of the reconstruction algorithms, a characterization of several surface extraction strategies is proposed. The classification presented shows the inherent limitations of the different algorithms concerning global topology control and reconstruction of local features like thin portions of the volume and almost non-manifold regions. These limitations can be observed and are illustrated with some practical examples. We review in light of this classification some of the relevant papers in the literature, and see that they cluster in some areas of the proposed hierarchy, making a case for where it might be more interesting to focus in future research.Preprin

A topological comparison of surface extraction algorithms

In many application areas, it is useful to convert the discrete information stored in the nodes of a regular grid into a continuous boundary model. Isosurface extraction algorithms di er on how the discrete information in the grid is generated, on what information does the grid store and on the properties of the output surface.Preprin

Solid modelling for manufacturing: from Voelcker's boundary evaluation to discrete paradigms

Herb Voelcker and his research team laid the foundations of Solid Modelling, on which Computer-Aided Design is based. He founded the ambitious Production Automation Project, that included Constructive Solid Geometry (CSG) as the basic 3D geometric representation. CSG trees were compact and robust, saving a memory space that was scarce in those times. But the main computational problem was Boundary Evaluation: the process of converting CSG trees to Boundary Representations (BReps) with explicit faces, edges and vertices for manufacturing and visualization purposes. This paper presents some glimpses of the history and evolution of some ideas that started with Herb Voelcker. We briefly describe the path from “localization and boundary evaluation” to “localization and printing”, with many intermediate steps driven by hardware, software and new mathematical tools: voxel and volume representations, triangle meshes, and many others, observing also that in some applications, voxel models no longer require Boundary Evaluation. In this last case, we consider the current research challenges and discuss several avenues for further research.Project TIN2017-88515-C2-1-R funded by MCIN/AEI/10.13039/501100011033/FEDER‘‘A way to make Europe’’Peer ReviewedPostprint (published version

Large bichromatic point sets admit empty monochromatic 4-gons

We consider a variation of a problem stated by Erd˝os and Szekeres in 1935 about the existence of a number fES(k) such that any set S of at least fES(k) points in general position in the plane has a subset of k points that are the vertices of a convex k-gon. In our setting the points of S are colored, and we say that a (not necessarily convex) spanned polygon is monochromatic if all its vertices have the same color. Moreover, a polygon is called empty if it does not contain any points of S in its interior. We show that any bichromatic set of n ≥ 5044 points in R2 in general position determines at least one empty, monochromatic quadrilateral (and thus linearly many).Postprint (published version