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

VoroCrust: Voronoi Meshing Without Clipping

Polyhedral meshes are increasingly becoming an attractive option with particular advantages over traditional meshes for certain applications. What has been missing is a robust polyhedral meshing algorithm that can handle broad classes of domains exhibiting arbitrarily curved boundaries and sharp features. In addition, the power of primal-dual mesh pairs, exemplified by Voronoi-Delaunay meshes, has been recognized as an important ingredient in numerous formulations. The VoroCrust algorithm is the first provably-correct algorithm for conforming polyhedral Voronoi meshing for non-convex and non-manifold domains with guarantees on the quality of both surface and volume elements. A robust refinement process estimates a suitable sizing field that enables the careful placement of Voronoi seeds across the surface circumventing the need for clipping and avoiding its many drawbacks. The algorithm has the flexibility of filling the interior by either structured or random samples, while preserving all sharp features in the output mesh. We demonstrate the capabilities of the algorithm on a variety of models and compare against state-of-the-art polyhedral meshing methods based on clipped Voronoi cells establishing the clear advantage of VoroCrust output.Comment: 18 pages (including appendix), 18 figures. Version without compressed images available on https://www.dropbox.com/s/qc6sot1gaujundy/VoroCrust.pdf. Supplemental materials available on https://www.dropbox.com/s/6p72h1e2ivw6kj3/VoroCrust_supplemental_materials.pd

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 in following in a literal way each sampled point -- 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 C1 curve C(u) : R → R3, whose velocity vector never vanishes -- One of the methods articulates in an entirely new way Principal Component Analysis (statistical) and Voronoi-Delaunay (deterministic) approaches -- It uses these two methods to calculate the best possible tape-shaped polygon covering the planarised point set, and then approximates the manifold by the medial axis of such a polygon -- The other method applies Principal Component Analysis 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 -- It turns out that the method solely based on Principal Component Analysis is 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 Integration of meshes originated in range images of an art piece -- Such an application reaches the point of complete reconstruction of a unified mes

Robust Surface Reconstruction from Point Clouds

The problem of generating a surface triangulation from a set of points with normal information arises in several mesh processing tasks like surface reconstruction or surface resampling. In this paper we present a surface triangulation approach which is based on local 2d delaunay triangulations in tangent space. Our contribution is the extension of this method to surfaces with sharp corners and creases. We demonstrate the robustness of the method on difficult meshing problems that include nearby sheets, self intersecting non manifold surfaces and noisy point samples

Surface Reconstruction from Unorganized Point Cloud Data via Progressive Local Mesh Matching

This thesis presents an integrated triangle mesh processing framework for surface reconstruction based on Delaunay triangulation. It features an innovative multi-level inheritance priority queuing mechanism for seeking and updating the optimum local manifold mesh at each data point. The proposed algorithms aim at generating a watertight triangle mesh interpolating all the input points data when all the fully matched local manifold meshes (umbrellas) are found. Compared to existing reconstruction algorithms, the proposed algorithms can automatically reconstruct watertight interpolation triangle mesh without additional hole-filling or manifold post-processing. The resulting surface can effectively recover the sharp features in the scanned physical object and capture their correct topology and geometric shapes reliably. The main Umbrella Facet Matching (UFM) algorithm and its two extended algorithms are documented in detail in the thesis. The UFM algorithm accomplishes and implements the core surface reconstruction framework based on a multi-level inheritance priority queuing mechanism according to the progressive matching results of local meshes. The first extended algorithm presents a new normal vector combinatorial estimation method for point cloud data depending on local mesh matching results, which is benefit to sharp features reconstruction. The second extended algorithm addresses the sharp-feature preservation issue in surface reconstruction by the proposed normal vector cone (NVC) filtering. The effectiveness of these algorithms has been demonstrated using both simulated and real-world point cloud data sets. For each algorithm, multiple case studies are performed and analyzed to validate its performance

Efficient Reconstruction From Scattered Points

Most algorithms that reconstruct surface from sample points rely on computationally demanding operations to derive the reconstruction, beside this, most of the classical algorithm use a kind of three-dimensional structure to derive a two-dimensional one. In this paper we introduce an innovative approach for generating two-dimensional piecewise linear approximations from sample points in R3 that simplify significantly the numerical calculation and the memory usage in the reconstruction process. The approach proposed here is an advancing front approach that uses rigid movements in the three-dimensional space and a bidimensional Delaunay triangulation as the main tools for the algorithm. The principal idea is to use a combination of rotations and translations in order to simplify the calculations and avoid the three-dimensional structure used by the most of the algorithms. Avoiding those structures, this approach can reduce the computational cost and numerical instabilities typically associated with the classical algorithm reconstructions

Local Neighborhoods for Shape Classification and Normal Estimation

We introduce the concept of local neighborhoods, a generalization of the one-ring on a mesh to unlabeled 3D data points arising from sampling a 2D surface embedded in 3D. The local neighborhood supports both local shape classification and robust normal estimation. In particular, local neighborhoods out-perform traditional approaches in unevenly sampled, curved regions. We show that the local neighborhood can be used in place of a full mesh structure for applications such as smoothing, moving least-squares reconstruction, and parameterization. Longer version of paper submitted to CAG

Gap Processing for Adaptive Maximal Poisson-Disk Sampling

In this paper, we study the generation of maximal Poisson-disk sets with varying radii. First, we present a geometric analysis of gaps in such disk sets. This analysis is the basis for maximal and adaptive sampling in Euclidean space and on manifolds. Second, we propose efficient algorithms and data structures to detect gaps and update gaps when disks are inserted, deleted, moved, or have their radius changed. We build on the concepts of the regular triangulation and the power diagram. Third, we will show how our analysis can make a contribution to the state-of-the-art in surface remeshing.Comment: 16 pages. ACM Transactions on Graphics, 201