Carlos A. Cadavid

Technical Section: 2D shape similarity as a complement for Voronoi-Delone methods in shape reconstruction

In surface reconstruction from planar cross sections it is necessary to build surfaces between 2D contours in consecutive cross sections. This problem has been traditionally attacked by (i) direct reconstruction based on local geometric proximity between the contours, and (ii) classification of topological events between the cross sections. These approaches have been separately applied with limited success. In case (i), the resulting surfaces may have overstretched or unnatural branches. These arise from local contour proximity which does not reflect global similarity between the contours. In case (ii), the topological events are identified but are not translated into the actual construction of a surface. This article presents an integration of the approaches (i) and (ii). Similarity between the composite 2D regions bounded by the contours in consecutive cross sections is used to: (a) decide whether a surface should actually relate two composite 2D regions, (b) identify the type and location of topological transitions between cross sections and (c) drive the surface construction for the regions found to be related in step (a). The implemented method avoids overstretched or unnatural branches, rendering a surface which is both geometrically intuitive and topologically faithful to the cross sections of the original object. The presented method is a good alternative in cases in which correct reproduction of the topology of the surface (e.g. simulation of flow in conduits) is more important than its geometry (e.g. assessment of tumor mass in radiation planning).

Principal component and Voronoi skeleton alternatives for curve reconstruction from noisy point sets

Surface reconstruction from noisy point samples must take into consideration the stochastic nature of the sample. In other words, geometric algorithms reconstructing the surface or curve should not insist on matching each sampled point precisely. Instead, they must interpret the sample as a “point cloud” and try to build the surface as passing through the best possible (in the statistical sense) geometric locus that represents the sample. This work presents two new methods to find a piecewise linear approximation from a Nyquist-compliant stochastic sampling of a quasi-planar C 1 curve C(u):R → R 3, whose velocity vector never vanishes. One of the methods combines principal component analysis (PCA) (statistical) and Voronoi-Delaunay (deterministic) approaches in an entirely new way. It uses these two methods to calculate the best possible tape-shaped polygon covering the flattened point set, and then approximates the manifold using the medial axis of such a polygon. The other method applies PCA to find a direct piecewise linear approximation of C(u). A complexity comparison of these two methods is presented, along with a qualitative comparison with previously developed ones. The results show that the method solely based on PCA is both simpler and more robust for non-self-intersecting curves. For self-intersecting curves, the Voronoi-Delaunay based medial axis approach is more robust, at the price of higher computational complexity. An application is presented in the integration of meshes created from range images of a sculpture to form a complete unified mesh.

Hessian Eigenfunctions for Triangular Mesh Parameterization

Hessian Locally Linear Embedding (HLLE) is an algorithm that computes the nullspace of a Hessian functional H for Dimensionality Reduction (DR) of a sampled manifold M. This article presents a variation of classic HLLE for parameterization of 3D triangular meshes. Contrary to classic HLLE which estimates local Hessian nullspaces, the proposed approach follows intuitive ideas from Differential Geometry where the local Hessian is estimated by quadratic interpolation and a partition of unity is used to join all neighborhoods. In addition, local average triangle normals are used to estimate the tangent plane TxM at x 2 M instead of PCA, resulting in local parameterizations which reflect better the geometry of the surface and perform better when the mesh presents sharp features. A high frequency dataset (Brain) is used to test our algorithm resulting in a higher rate of success (96:63%) compared to classic HLLE (76:4%).

Gabriel-constrained Parametric Surface Triangulation

The Boundary Representation of a 3D manifold contains FACES (connected subsets of a parametric surface S : R2−R3) -- In many science and engineering applications it is cumbersome and algebraically difficult to deal with the polynomial set and constraints (LOOPs) representing the FACE -- Because of this reason, a Piecewise Linear (PL) approximation of the FACE is needed, which is usually represented in terms of triangles (i.e. 2-simplices) -- Solving the problem of FACE triangulation requires producing quality triangles which are: (i) independent of the arguments of S, (ii) sensitive to the local curvatures, and (iii) compliant with the boundaries of the FACE and (iv) topologically compatible with the triangles of the neighboring FACEs -- In the existing literature there are no guarantees for the point (iii) -- This article contributes to the topic of triangulations conforming to the boundaries of the FACE by applying the concept of parameter independent Gabriel complex, which improves the correctness of the triangulation regarding aspects (iii) and (iv) -- In addition, the article applies the geometric concept of tangent ball to a surface at a point to address points (i) and (ii) -- Additional research is needed in algorithms that (i) take advantage of the concepts presented in the heuristic algorithm proposed and (ii) can be proved correct

Mesh segmentation and texture mapping for dimensional inspection in Web3D

Traditionally, the data generated by industrial metrology software is stored as static reports that metrology experts produce for engineering and production departments. Nevertheless, industry demands new approaches that provide ubiquitous and real time access to overall geometry, manufacturing and other data. Web3D technologies can help to improve the traditional metrology methods and offer new ways to convey this information in web-based continuous friendly manner. However, enriched point clouds may be massive, thus presenting transmission and display limitations. To partially overcome these limitations, this article presents an algorithm that computes efficient metrology textures, which are then transferred and displayed through Web3D standards. Texture coordinates are computed only once for the reference CAD mesh on the server using in-house thermal-based segmentation and Hessian-based parameterization algorithms. The metrology data is then encoded in a texture file, which becomes available instantly for interactive visual inspection through the Web3D platform.

Hybrid geometry / topology based mesh segmentation for reverse engineering

Abstract Mesh segmentation and parameterization are crucial for Reverse Engineering (RE). Bijective parameterizations of the sub-meshes are a sine-qua-non test for segmentation. Current segmentation methods use either (1) topologic or (2) geometric criteria to partition the mesh. Reported topology-based segmentations produce large sub-meshes which reject parameterizations. Geometry-based segmentations are very sensitive to local variations in dihedral angle or curvatures, thus producing an exaggerated large number of small sub-meshes. Although small sub-meshes accept nearly isometric parameterizations, this significant granulation defeats the intent of synthesizing a usable Boundary Representation (compulsory for RE). In response to these limitations, this article presents an implementation of a hybrid geometry / topology segmentation algorithm for mechanical workpieces. This method locates heat transfer constraints (topological criterion) in low frequency neighborhoods of the mesh (geometric criterion) and solves for the resulting temperature distribution on the mesh. The mesh partition dictated by the temperature scalar map results in large, albeit parameterizable, sub-meshes. Our algorithm is tested with both benchmark repository and physical piece scans data. The experiments are successful, except for the well - known cases of topological cylinders, which require a user - introduced boundary along the cylinder generatrices.

Limits of quotients of polynomial functions in three variables

A method for computing limits of quotients of real analytic functions in two variables was developed in [4]. In this article we generalize the results obtained in that paper to the case of quotients q = f(x, y, z)/g(x, y, z) of polynomial functions in three variables with rational coefficients. The main idea consists in examining the behavior of the function q along certain real variety X(q) (the discriminant variety associated to q). The original problem is then solved by reducing to the case of functions of two variables. The inductive step is provided by the key fact that any algebraic curve is birationally equivalent to a plane curve. Our main result is summarized in Theorem 2. In Section 4 we describe an effective method for computing such limits. We provide a high level description of an algorithm that generalizes the one developed in [4], now available in Maple as the <code>limit/multi</code> command.