Feature-aware manifold meshing and remeshing of point clouds and polyhedral surfaces with guaranteed smallest edge length

Computational paradigms for direct triangular surface remeshing

This thesis examines geometric process planning in four-axis rough machining. A review of existing literature provides a foundation for process planning in machining; efficiency (tool path length) is identified as a primary concern. Emergent structures (thin webs and strings) are proposed as a new metric of process robustness. Previous research efforts are contrasted to establish motivation for improvements in these areas in four-axis rough machining. The original research is presented as a journal article. This research develops a new methodology for quickly estimating the remaining stock during a plurality of 2 1⁄2 D cuts defined by their depth and orientation relative to a rotary fourth axis. Unlike existing tool path simulators, this method can be performed independently of (and thus prior to) tool path generation. The algorithms presented use polyhedral mesh surface input to create and analyze polygonal slices, which are again reconstructed into polyhedral surfaces. At the slice level, nearly all operations are Boolean in nature, allowing simple implementation. A novel heuristic for polyhedral reconstruction for this application is presented. Results are shown for sample components, showing a significant reduction in overall rough machining tool path length. The discussion of future work provides a brief discussion of how this new methodology can be applied to detecting thin webs and strings prior to tool path planning or machining. The methodology presented in this work provides a novel method of calculating remaining stock such that it can be performed during process planning, prior to committing to tool path generation.

Smooth surfaces are approximated by polyhedral surfaces both for display, and for other computational purposes. This is probably the most common surface representation in computer graphics, because polygons are what the current generation of graphics hardware can display efficiently. In computer vision this representation has not been very popular for a long time, probably because it was not seen as very appropriate for object recognition and positioning applications. But it is now being used in more applications, particularly in the medical domain, where polyhedral surface approximations of unknown surfaces are routinely generated with iso-surface construction algorithms. We look at polyhedral surfaces as a generalization of digital images, where the polygon takes the place of the pixel. The polygon is the fundamental surface element. Digital images are functions defined at the nodes of a regular rectangular grid. Discrete surface signals are functions defined at the vertices of a polyhedral surface of arbitrary topology. Signal processing operations, and Fourier analysis in particular, are the fundamental tools of low level computer vision. The lack of regularity and self-similarity of arbitrary polyhedral surface meshes, as opposed to the regular rectangular grids of digital images, complicates the analysis and processing of digital surface signals. Nevertheless, in this paper we generalize Fourier analysis to discrete surface signals. As a first application of this theory we consider the problem of surface smoothing, which corresponds to low-pass filtering within this framework. As in the classical cases of one-dimensional signals, and of digital images, the analysis of discrete surface signals is reduced to matrix analysis, and matrix computation techniques are used to achieve fast discrete surface signal processing operations. We intend to continue along this line of research in the near future, extending higher level computer vision operations to arbitrary polyhedral surfaces.

CyberTape: an interactive measurement tool on polyhedral surface

Polyhedral surfaces are used as representation model for CAM and process planning purposes because of its simplicity for data exchange and geometric computation. However, there are few tool path planning strategies for such surfaces but isoplanar method. This paper presents a contour offset approach to tool path generation for three-axis ball-end milling of polyhedral surfaces, based on a novel method for offsetting curves on polyhedral surfaces. One of its salient features is to reduce the task of removing complex interfering of offsets from 3D physical surfaces to 2D plane by flattening mesh surfaces and avoid costly 3D Boolean set operations and relatively expensive distance calculation. Moreover, in practical implement, the procedures of calculating offset points and removing interfering loops are merged and carried out simultaneously results in an efficient tool path generation method. Empirical examples illustrate the feasibility of the proposed method.

Polyhedral surface has become a popular representation for computer-aided design, manufacturing, and inspection systems due to its simplicity for geometric computation and product shape communication. However, little attempts have been made in relation to inspection planning strategy of freeform surface-based polyhedral models. The intention of the work presented in this article attempts to demonstrate the efficiency and flexibility of a measurement planning method for the inspection of components with polyhedral surface representations. Based on the one-step inverse method, the triangular mesh surface can be flattened onto a plane, which makes it possible to select sampling points and to generate inspection paths in a 2-D parametric domain. Subsequently, a curvature-oriented point distribution strategy is proposed for measurement path generation. Finally, all these 2-D sample points are inverse-mapped onto the original mesh surface. The advantage of this approach is that it reduces complex intersection operations in the 3-D space and strikes a balance between accuracy and efficiency. The effectivity and feasibility of the proposed method are verified by simulation experiments.

We propose a method to build a three-dimensional adapted surface mesh with respect to a mesh size map driven by surface curvature. The data needed to optimize the mesh have been reduced to an initial mesh. The building of a local geometrical model but continuous over the whole domain is based on a local Hermite diffuse interpolation calculated from the nodes of the initial mesh and from the normal vectors to the surface. The optimization procedures involve extracting from the surface mesh sets of triangles sharing the same node or the same edge and then remeshing the outer contour to a higher criterion (size or shape). These procedures may be used in order to refine or coarsen the mesh but also in a final step to enhance the shape quality of the elements. Examples demonstrate the ability of the method to create adapted meshes of complex surfaces while meeting high-quality standards and a good respect of the geometrical surface. Copyright © 2000 John Wiley & Sons, Ltd.

We propose a method to build a three-dimensional adapted surface mesh with respect to a mesh size map driven by surface curvature. The data needed to optimize the mesh have been reduced to an initial mesh. The building of a local geometrical model but continuous over the whole domain is based on a local Hermite diffuse interpolation calculated from the nodes of the initial mesh and from the normal vectors to the surface. The optimization procedures involve extracting from the surface mesh sets of triangles sharing the same node or the same edge and then remeshing the outer contour to a higher criterion (size or shape). These procedures may be used in order to refine or coarsen the mesh but also in a final step to enhance the shape quality of the elements. Examples demonstrate the ability of the method to create adapted meshes of complex surfaces while meeting high-quality standards and a good respect of the geometrical surface. Copyright © 2000 John Wiley & Sons, Ltd.

Accelerating surface remeshing through GPU-based computation of the restricted tangent face

As we have seen in Chap. 1, a Euclidean triangulated surface (T l , M) characterizes a polyhedral metric with conical singularities associated with the vertices of the triangulation. In this chapter we show that around any such a vertex we can introduce complex coordinates in terms of which we can write down the conformal conical metric, locally parametrizing the singular structure of (T l , M). This makes available a powerful dictionary between 2-dimensional triangulations and complex geometry. It must be noted that, both in the mathematical and in the physical applications of the theory, the connection between Riemann surfaces and triangulations typically emphasizes only the role of ribbon graphs and of the associated metric. The conical geometry of the polyhedral surface (T l , M) is left aside and seems to play no significant a role. Whereas this attitude can be motivated by the observation (due to M. Troyanov, On the moduli space of singular Euclidean surfaces. In: Papadopoulos A (ed) Handbook of teichmuller theory, vol I. IRMA lectures in mathematics and theoretical physics, vol 11. European Mathematical Society (EMS), Zurich, pp 507–540, arXiv:math/0702666v2 [math.DG], 2007) that the conformal structure does not see the conical singularities (see below, for details), it gives a narrow perspective of the much wider role that the theory has to offer. Thus, it is more appropriate to connect a polyhedral surface (T l , M) to a corresponding Riemann surface by taking fully into account the conical geometry of (T l , M). This connection is many–faceted and exploits a vast repertoire of notion ranging from complex function theory to algebraic geometry. We start by defining the barycentrically dual polytope (P T , M) associated with a polyhedral surface (T l , M) and discuss the geometry of the corresponding ribbon graph. Then, by adapting to our case the elegant approach in Mulase and Penkava (Asian J Math 2(4):875–920, 1998. math-ph/9811024 v2), we explicitly construct the Riemann surface associated with (P T , M). This directly bring us to the analysis of Troyanov’s singular Euclidean structures and to the construction of the bijective map between the moduli space \(\mathfrak {M}_{g,N_0}\) of Riemann surfaces (M, N0) with N0 marked points, decorated with conical angles, and the space of polyhedral structures. In particular the first Chern class of the line bundles naturally defined over \(\mathfrak {M}_{g,N_0}\) by the cotangent space at the i-th marked point is related with the corresponding Euler class of the circle bundles over the space of polyhedral surfaces defined by the conical cotangent spaces at the i-th vertex of (T l , M). Whereas this is not an unexpected connection, the analogy with Witten–Kontsevich theory being obvious, we stress that the conical geometry adds to this property the possibility of a deep and explicit characterization of the Weil–Petersson form in terms of the edge–lengths of (T l , M). This result (Carfora et al., JHEP 0612:017, 2006. arXiv:hep-th/0607146) is obtained by a subtle interplay between the geometry of (T l , M) and 3-dimensional hyperbolic geometry, and it will be discussed in detail in Chap. 3 since it explicitly hints to the connection between polyhedral surfaces and quantum geometry in higher dimensions.

As we have seen in Chap. 1, a Euclidean triangulated surface (T l , M) characterizes a polyhedral metric with conical singularities associated with the vertices of the triangulation. In this chapter we show that around any such a vertex we can introduce complex coordinates in terms of which we can write down the conformal conical metric, locally parametrizing the singular structure of (T l , M). This makes available a powerful dictionary between 2-dimensional triangulations and complex geometry. It must be noted that, both in the mathematical and in the physical applications of the theory, the connection between Riemann surfaces and triangulations typically emphasizes only the role of ribbon graphs and of the associated metric. The conical geometry of the polyhedral surface (T l , M) is left aside and seems to play no significant a role. Whereas this attitude can be motivated by the observation (due to M. Troyanov, On the moduli space of singular Euclidean surfaces. In: Papadopoulos A (ed) Handbook of teichmuller theory, vol I. IRMA lectures in mathematics and theoretical physics, vol 11. European Mathematical Society (EMS), Zurich, pp 507–540, arXiv:math/0702666v2 [math.DG], 2007) that the conformal structure does not see the conical singularities (see below, for details), it gives a narrow perspective of the much wider role that the theory has to offer. Thus, it is more appropriate to connect a polyhedral surface (T l , M) to a corresponding Riemann surface by taking fully into account the conical geometry of (T l , M). This connection is many–faceted and exploits a vast repertoire of notion ranging from complex function theory to algebraic geometry. We start by defining the barycentrically dual polytope (P T , M) associated with a polyhedral surface (T l , M) and discuss the geometry of the corresponding ribbon graph. Then, by adapting to our case the elegant approach in Mulase and Penkava (Asian J Math 2(4):875–920, 1998. math-ph/9811024 v2), we explicitly construct the Riemann surface associated with (P T , M). This directly bring us to the analysis of Troyanov’s singular Euclidean structures and to the construction of the bijective map between the moduli space \(\mathfrak {M}_{g,N_0}\) of Riemann surfaces (M, N0) with N0 marked points, decorated with conical angles, and the space of polyhedral structures. In particular the first Chern class of the line bundles naturally defined over \(\mathfrak {M}_{g,N_0}\) by the cotangent space at the i-th marked point is related with the corresponding Euler class of the circle bundles over the space of polyhedral surfaces defined by the conical cotangent spaces at the i-th vertex of (T l , M). Whereas this is not an unexpected connection, the analogy with Witten–Kontsevich theory being obvious, we stress that the conical geometry adds to this property the possibility of a deep and explicit characterization of the Weil–Petersson form in terms of the edge–lengths of (T l , M). This result (Carfora et al., JHEP 0612:017, 2006. arXiv:hep-th/0607146) is obtained by a subtle interplay between the geometry of (T l , M) and 3-dimensional hyperbolic geometry, and it will be discussed in detail in Chap. 3 since it explicitly hints to the connection between polyhedral surfaces and quantum geometry in higher dimensions.

Polyhedral surfaces are used as representation model for CAM and process planning purposes because of its simplicity for data exchange and geometric computation. However, there is few tool path planning strategies for such surfaces but the iso-plane method. In this paper, contour parallel path are generated for three-axis ball-end milling. This tool path is based on a novel algorithm for offsetting curves on polyhedral surfaces presented in this paper. It reduces the task of removing complex interfering of offset curve from 3D surface to 2D plane by flattening mesh surface, and avoids costly 3D Boolean set operations and expensive distance calculation. This results in an efficient tool-path generation. Empirical examples illustrate the feasibility the proposed method.

The task of explicit surface reconstruction is to generate a surface mesh by interpolating a given point cloud. Explicit surface reconstruction is necessary when the point cloud is required to appear exactly on the surface. However, for a non-perfect input, such as lack of normals, low density, irregular distribution, thin and tiny parts, and high genus, a robust explicit reconstruction method that can generate a high-quality manifold triangulation is missing. We propose a robust explicit surface reconstruction method that starts from an initial simple surface mesh, alternately performs a Filmsticking step and a Sculpting step of the initial mesh, and converges when the surface mesh interpolates all input points (except outliers) and remains stable. The Filmsticking is to minimize the geometric distance between the surface mesh and the point cloud through iteratively performing a restricted Voronoi diagram technique on the surface mesh, whereas the Sculpting is to bootstrap the Filmsticking iteration from local minima by applying appropriate geometric and topological changes of the surface mesh. Our algorithm is fully automatic and produces high-quality surface meshes for non-perfect inputs that are typically considered to be challenging for prior state of the art. We conducted extensive experiments on simulated scans and real scans to validate the effectiveness of our approach.

A common representation of surfaces with complicated topology and geometry is through composite parametric surfaces. This is the case for most CAD modelers. The majority of these models focus on having a good approximation of the surface itself, but they are usually built without taking into account a subsequent mesh generation. Indeed they are often characterized by too many patches which are not logically connected and make a standard mesh generator fail. In this work, we present a novel mesh generation strategy that can handle such “bad” input data and produces an anisotropic curvature-adapted surface mesh. There are two main ingredients to achieve this goal. First of all, we define a new and fast way to project point on an input model which overcomes the presence of non-connected patches. Then we consider the higher embedding strategy to build the final anisotropic surface mesh.

Many applications rely on scanned data, which can come from a variety of sources: optical scanners, coordinate measuring machines, or medical imaging. We assume that the data input to these applications is an unorganized point cloud or mesh of vertices. The objective may be to find particular features (medical diagnostics or reverse engineering) or comparison to some reference geometry (e.g. dimensional metrology). This paper focuses on the feature fitting of a segmented point cloud, specifically for branched, organic structures or structural frames, and targets non-monolithic geometries. In this paper, a methodology is presented for the automated recovery of cross-sectional shapes using centrally located curves. We assume a triangulated surface mesh is generated from the scanned point cloud. This surface mesh is the starting point for our methodology. We then find the curve skeleton of the part which abstractly describes the global geometry and topology. Next after segmenting the curve skeleton into non-branching segments, orthogonal planes to the curve skeleton segments, at preset or adaptive intervals, make slices through the surface mesh edges. The intersection points are extracted creating a 2D point cloud of the cross section. A number of application specific post-processing operations can be performed after obtaining the 2D point cloud cross sections and the curve skeleton paths including: calculations such as area or area moments of inertia, feature fitting or recognition, and digital shape reconstruction. Case studies are presented to demonstrate capabilities and limitations, and to provide insight into appropriate uses and adaptations for the presented methodology. Highlights Automated cross-sectional extraction for branching structures is presented. Methodology utilized skeletonization of object as reference for sampling planes. Surface mesh is sliced to extract a 2D point cloud. Filter algorithm for exclusion of peripheral slicing is presented. Several case studies demonstrate capabilities and limitations of the method.