Reconstruction of freeform surfaces for metrology

The application of freeform surfaces has increased since their complex shapes closely express a product's functional specifications and their machining is obtained with higher accuracy. In particular, optical surfaces exhibit enhanced performance especially when they take aspheric forms or more complex forms with multi-undulations. This study is mainly focused on the reconstruction of complex shapes such as freeform optical surfaces, and on the characterization of their form. The computer graphics community has proposed various algorithms for constructing a mesh based on the cloud of sample points. The mesh is a piecewise linear approximation of the surface and an interpolation of the point set. The mesh can further be processed for fitting parametric surfaces (Polyworks® or Geomagic®). The metrology community investigates direct fitting approaches. If the surface mathematical model is given, fitting is a straight forward task. Nonetheless, if the surface model is unknown, fitting is only possible through the association of polynomial Spline parametric surfaces. In this paper, a comparative study carried out on methods proposed by the computer graphics community will be presented to elucidate the advantages of these approaches. We stress the importance of the pre-processing phase as well as the significance of initial conditions. We further emphasize the importance of the meshing phase by stating that a proper mesh has two major advantages. First, it organizes the initially unstructured point set and it provides an insight of orientation, neighbourhood and curvature, and infers information on both its geometry and topology. Second, it conveys a better segmentation of the space, leading to a correct patching and association of parametric surfaces.EMR

Topological Data Analysis with Bregman Divergences

Given a finite set in a metric space, the topological analysis generalizes hierarchical clustering using a 1-parameter family of homology groups to quantify connectivity in all dimensions. The connectivity is compactly described by the persistence diagram. One limitation of the current framework is the reliance on metric distances, whereas in many practical applications objects are compared by non-metric dissimilarity measures. Examples are the Kullback-Leibler divergence, which is commonly used for comparing text and images, and the Itakura-Saito divergence, popular for speech and sound. These are two members of the broad family of dissimilarities called Bregman divergences. We show that the framework of topological data analysis can be extended to general Bregman divergences, widening the scope of possible applications. In particular, we prove that appropriately generalized Cech and Delaunay (alpha) complexes capture the correct homotopy type, namely that of the corresponding union of Bregman balls. Consequently, their filtrations give the correct persistence diagram, namely the one generated by the uniformly growing Bregman balls. Moreover, we show that unlike the metric setting, the filtration of Vietoris-Rips complexes may fail to approximate the persistence diagram. We propose algorithms to compute the thus generalized Cech, Vietoris-Rips and Delaunay complexes and experimentally test their efficiency. Lastly, we explain their surprisingly good performance by making a connection with discrete Morse theory

The Topology ToolKit

This system paper presents the Topology ToolKit (TTK), a software platform designed for topological data analysis in scientific visualization. TTK provides a unified, generic, efficient, and robust implementation of key algorithms for the topological analysis of scalar data, including: critical points, integral lines, persistence diagrams, persistence curves, merge trees, contour trees, Morse-Smale complexes, fiber surfaces, continuous scatterplots, Jacobi sets, Reeb spaces, and more. TTK is easily accessible to end users due to a tight integration with ParaView. It is also easily accessible to developers through a variety of bindings (Python, VTK/C++) for fast prototyping or through direct, dependence-free, C++, to ease integration into pre-existing complex systems. While developing TTK, we faced several algorithmic and software engineering challenges, which we document in this paper. In particular, we present an algorithm for the construction of a discrete gradient that complies to the critical points extracted in the piecewise-linear setting. This algorithm guarantees a combinatorial consistency across the topological abstractions supported by TTK, and importantly, a unified implementation of topological data simplification for multi-scale exploration and analysis. We also present a cached triangulation data structure, that supports time efficient and generic traversals, which self-adjusts its memory usage on demand for input simplicial meshes and which implicitly emulates a triangulation for regular grids with no memory overhead. Finally, we describe an original software architecture, which guarantees memory efficient and direct accesses to TTK features, while still allowing for researchers powerful and easy bindings and extensions. TTK is open source (BSD license) and its code, online documentation and video tutorials are available on TTK's website

Curve counting, instantons and McKay correspondences

We survey some features of equivariant instanton partition functions of topological gauge theories on four and six dimensional toric Kahler varieties, and their geometric and algebraic counterparts in the enumerative problem of counting holomorphic curves. We discuss the relations of instanton counting to representations of affine Lie algebras in the four-dimensional case, and to Donaldson-Thomas theory for ideal sheaves on Calabi-Yau threefolds. For resolutions of toric singularities, an algebraic structure induced by a quiver determines the instanton moduli space through the McKay correspondence and its generalizations. The correspondence elucidates the realization of gauge theory partition functions as quasi-modular forms, and reformulates the computation of noncommutative Donaldson-Thomas invariants in terms of the enumeration of generalized instantons. New results include a general presentation of the partition functions on ALE spaces as affine characters, a rigorous treatment of equivariant partition functions on Hirzebruch surfaces, and a putative connection between the special McKay correspondence and instanton counting on Hirzebruch-Jung spaces.Comment: 79 pages, 3 figures; v2: typos corrected, reference added, new summary section included; Final version to appear in Journal of Geometry and Physic

Surface reconstruction by computing restricted Voronoi cells in parallel

International audienceWe present a method for reconstructing a 3D surface triangulation from an input point set. The main component of the method is an algorithm that computes the restricted Voronoi diagram. In our specific case, it corresponds to the intersection between the 3D Voronoi diagram of the input points and a set of disks centered at the points and orthogonal to the estimated normal directions. The method does not require coherent normal orientations (just directions). Our algorithm is based on a property of the restricted Voronoi cells that leads to an embarrassingly parallel implementation. We experimented our algorithm with scanned point sets with up to 100 million vertices that were processed within few minutes on a standard computer. The complete implementation is provided