A spherical representation for recognition of free-form surfaces

Introduces a new surface representation for recognizing curved objects. The authors approach begins by representing an object by a discrete mesh of points built from range data or from a geometric model of the object. The mesh is computed from the data by deforming a standard shaped mesh, for example, an ellipsoid, until it fits the surface of the object. The authors define local regularity constraints that the mesh must satisfy. The authors then define a canonical mapping between the mesh describing the object and a standard spherical mesh. A surface curvature index that is pose-invariant is stored at every node of the mesh. The authors use this object representation for recognition by comparing the spherical model of a reference object with the model extracted from a new observed scene. The authors show how the similarity between reference model and observed data can be evaluated and they show how the pose of the reference object in the observed scene can be easily computed using this representation. The authors present results on real range images which show that this approach to modelling and recognizing 3D objects has three main advantages: (1) it is applicable to complex curved surfaces that cannot be handled by conventional techniques; (2) it reduces the recognition problem to the computation of similarity between spherical distributions; in particular, the recognition algorithm does not require any combinatorial search; and (3) even though it is based on a spherical mapping, the approach can handle occlusions and partial views     

On stochastic methods for surface reconstruction

In this article, we present and discuss three statistical methods for surface reconstruction. A typical input to a surface reconstruction technique consists of a large set of points that has been sampled from a smooth surface and contains uncertain data in the form of noise and outliers. We first present a method that filters out uncertain and redundant information yielding a more accurate and economical surface representation. Then we present two methods, each of which converts the input point data to a standard shape representation; the first produces an implicit representation while the second yields a triangle mesh.     

A new approach to solid modeling with trivariate T-splines based on mesh optimization

We present a new method to construct a trivariate T-spline representation of complex genus-zero solids for the application of isogeometric analysis. The proposed technique only demands a surface triangulation of the solid as input data. The key of this method lies in obtaining a volumetric parameterization between the solid and the parametric domain, the unitary cube. To do that, an adaptive tetrahedral mesh of the parametric domain is isomorphically transformed onto the solid by applying a mesh untangling and smoothing procedure. The control points of the trivariate T-spline are calculated by imposing the interpolation conditions on points sited both on the inner and on the surface of the solid. The distribution of the interpolating points is adapted to the singularities of the domain in order to preserve the features of the surface triangulation.     

The meccano method for isogeometric solid modeling and applications

We present a new method to construct a trivariate T-spline representation of complex solids for the application of isogeometric analysis. We take a genus-zero solid as a basis of our study, but at the end of the work we explain the way to generalize the results to any genus solids. The proposed technique only demands a surface triangulation of the solid as input data. The key of this method lies in obtaining a volumetric parameterization between the solid and the parametric domain, the unitary cube. To do that, an adaptive tetrahedral mesh of the parametric domain is isomorphically transformed onto the solid by applying a mesh untangling and smoothing procedure. The control points of the trivariate T-spline are calculated by imposing the interpolation conditions on points sited both on the inner and on the surface of the solid. The distribution of the interpolating points is adapted to the singularities of the domain to preserve the features of the surface triangulation. We present some results of the application of isogeometric analysis with T-splines to the resolution of Poisson equation in solids parameterized with this technique.     

Direct Reconstruction of a Displaced Subdivision Surface from Unorganized Points

In this paper we describe the generation of a displaced subdivision surface directly from a set of unorganized points. The displaced subdivision surface is an efficient mesh representation that defines a detailed mesh with a displacement map over a smooth domain surface and has many benefits including compression, rendering, and animation, which overcome limitations of an irregular mesh produced by an ordinary mesh reconstruction scheme. Unlike previous displaced subdivision surface reconstruction methods, our method does not rely on a highly detailed reconstructed mesh. Instead, we efficiently create a coarse base mesh, which is used to sample displacements directly from unorganized points, and this results in a simple process and fast calculation. We suggest a shrink-wrapping-like shape approximation and a point-based mesh simplification method that uses the distance between a set of points and a mesh as an error metric to generate a domain surface that optimally approximates the given points. We avoid time-consuming energy minimization by employing a local subdivision surface fitting scheme. Finally, we show several reconstruction results that demonstrate the usability of our algorithm.     

A simplified interpolation and conversion method of contour surface model to mesh surface model

Contour line type representation is useful for understanding the surface structure qualitatively to a human. However, for the computer handling, a contour line model may not be suitable and it is required to be converted to the other type model such as a mesh surface model. It may be a problem how to decide mesh data on a mesh line where there are fewer points intersecting with contour lines. In this article, the authors propose a new method to convert a contour line model to a mesh surface model with minimum errors. Mesh lines (equivalent to the planes intersected with contour surface) located on the contour surface and intersected with contour lines are calculated. A mesh line which has maximum number of effective sampling points (that means maximum number of intersections with contour lines, and hereafter referred to as latter) is selected and a sectional shape along this mesh line is decided. The sectional shape is represented by spline curve with parameters, so that the mesh point data on the mesh line can be determined easily, and these decided mesh points are regarded as equivalent as intersection with contour lines. The above processes are repeated until all mesh lines which have intersection points or obtained mesh data have been chosen and calculated. Thus we can convert a contour model to a mesh surface model with minimum loss of contour line informations.     

Robust multi‐level partition of unity implicits from triangular meshes

This paper presents a new robust multi‐level partition of unity (MPU) method, which constructs an implicit surface from a triangular mesh via the new error metric between the mesh and the implicit surface. The new error metric employs a weighted function of inner points and vertices of a triangle to fit an implicit surface, which can control the approximation error between the surface and vertices of the triangle. Furthermore, it is applied to the MPU method by utilizing the dual graph of a triangular mesh, and the general quadric implicit surface is used for surface representation. Compared with the MPU method, the new method generates fewer subdivision cells with the same approximation error and performs more steadily especially when given triangular mesh with fewer vertices. Copyright © 2013 John Wiley & Sons, Ltd.     

A new vector field distance transform and its application to mesh processing from 3D scanned data

In this paper we define a new 3D vector field distance transform to implicitly represent a mesh surface. We show that this new representation is more accurate than the classic scalar field distance transform by comparing both representations with an error metric evaluation. The widely used marching cube triangulation algorithm is adapted to the new vector field distance transform to correctly reconstruct the resulting explicit surface. In the reconstruction process of 3D scanned data, the useful mesh denoising operation is extended to the new vector field representation, which enables adaptive and selective filtering features. Results show that mesh processing with this new vector field representation is more accurate than with the scalar field distance transform and that it outperforms previous mesh filtering algorithms. Future work is discussed to extend this new vector field representation to other mesh useful operations and applications.     

Resampling Feature and Blend Regions in Polygonal Meshes for Surface Anti-Aliasing

Efficient surface reconstruction and reverse engineering techniques are usually based on a polygonal mesh representation of the geometry: the resulting models emerge from piecewise linear interpolation of a set of sample points. The quality of the reconstruction not only depends on the number and density of the sample points but also on their alignment to sharp and rounded features of the original geometry. Bad alignment can lead to severe alias artifacts. In this paper we present a sampling pattern for feature and blend regions which minimizes these alias errors. We show how to improve the quality of a given polygonal mesh model by resampling its feature and blend regions within an interactive framework. We further demonstrate sophisticated modeling operations that can be implemented based on this resampling technique.     

基于SOM的散乱数据点集的B样条曲面重建

利用自组织映射神经网络(SOM)技术对散乱数据点集进行B样条曲面重建时,往往存在网络学习时间过长和学习效果不理想等问题。提出了一种新的神经元初始化方法和分块学习算法,该算法首先运用主元素分析方法(PCA)对散乱数据进行分块,将拓扑结构为四边形的输出层神经元初始化在每块散乱数据的最小二乘平面上进行网络学习和训练,将分块学习得到的各网格曲面拼接成一个整体;然后对该整体网格曲面的边界和内部单独学习,得到一张逼近待重建曲面的双线性B样条曲面;最后对该B样条曲面误差进行了修正。实例证明,该算法可以明显地减少SOM网络学习时间,并改善网络学习效果。     

Generalized B-spline subdivision-surface wavelets for geometry compression

We present a new construction of lifted biorthogonal wavelets on surfaces of arbitrary two-manifold topology for compression and multiresolution representation. Our method combines three approaches: subdivision surfaces of arbitrary topology, B-spline wavelets, and the lifting scheme for biorthogonal wavelet construction. The simple building blocks of our wavelet transform are local lifting operations performed on polygonal meshes with subdivision hierarchy. Starting with a coarse, irregular polyhedral base mesh, our transform creates a subdivision hierarchy of meshes converging to a smooth limit surface. At every subdivision level, geometric detail is expanded from wavelet coefficients and added to the surface. We present wavelet constructions for bilinear, bicubic, and biquintic B-spline subdivision. While the bilinear and bicubic constructions perform well in numerical experiments, the biquintic construction turns out to be unstable. For lossless compression, our transform is computed in integer arithmetic, mapping integer coordinates of control points to integer wavelet coefficients. Our approach provides a highly efficient and progressive representation for complex geometries of arbitrary topology.     

Direct boolean intersection between acquired and designed geometry

In this paper, a new shape modeling approach that can enable direct Boolean intersection between acquired and designed geometry without model conversion is presented. At its core is a new method that enables direct intersection and Boolean operations between designed geometry (objects bounded by NURBS and polygonal surfaces) and scanned geometry (objects represented by point cloud data).We use the moving least-squares (MLS) surface as the underlying surface representation for acquired point-sampled geometry. Based on the MLS surface definition, we derive closed formula for computing curvature of planar curves on the MLS surface. A set of intersection algorithms including line and MLS surface intersection, curvature-adaptive plane and MLS surface intersection, and polygonal mesh and MLS surface intersection are successively developed. Further, an algorithm for NURBS and MLS surface intersection is then developed. It first adaptively subdivides NURBS surfaces into polygonal mesh, and then intersects the mesh with the MLS surface. The intersection points are mapped to the NURBS surface through the Gauss-Newton method.Based on the above algorithms, a prototype system has been implemented. Through various examples from the system, we demonstrate that direct Boolean intersection between designed geometry and acquired geometry offers a useful and effective means for the shape modeling applications where point-cloud data is involved.     

Adaptive mesh refinement in stress-constrained topology optimization

We present a topology structural optimization framework with adaptive mesh refinement and stress-constraints. Finite element approximation and geometry representation benefit from such refinement by enabling more accurate stress field predictions and greater resolution of the optimal structural boundaries. We combine a volume fraction filter to impose a minimum design feature size, the RAMP penalization to generate “black-and-white designs” and a RAMP-like stress definition to resolve the “stress singularity problem.” Regions with stress concentrations dominate the optimized design. As such, rigorous simulations are required to accurately approximate the stress field. To achieve this goal, we invoke a threshold operation and mesh refinement during the optimization. We do so in an optimal fashion, by applying adaptive mesh refinement techniques that use error indicators to refine and coarsen the mesh as needed. In this way, we obtain more accurate simulations and greater resolution of the design domain. We present results in two dimensions to demonstrate the efficiency of our method.     

On discrete boundaries and solution accuracy in anisotropic adaptive meshing

In the adaptive mesh generation, the space mesh should be adequate to the surface mesh. When the analytical surface representation is not known, additional surface information may be extracted from triangular surface meshes. We describe a new surface reconstruction method which uses approximate Hessian of a piecewise linear function representing the discrete surface. Efficiency of the proposed method is illustrated with two CFD applications.     

Shape reconstruction on a varying mesh

A central class of image understanding problems is concerned with reconstructing a shape from an incomplete data set, such as fitting a surface to (partially) given contours. A new theory for solving such problems is presented. Unlike the current heuristic methods, the method used starts from fundamental principles that should be followed by any reconstruction method, regardless of its mathematical or physical implementation. A mathematical procedure which conforms to these principles is presented. One major advantage of the method is the ability to handle shapes containing both smooth and sharp parts without using thresholds. A sharp variation, such as a corner, requires a high-resolution mesh for adequate representation, while slowly varying sections can be represented with sparser mesh points. Unlike current methods, this procedure fits the surface on a varying mesh. The mesh is constructed automatically to be more dense at parts of the image that have more rapid variation. Analytical examples are given in simple cases, followed by numerical experiments     

A fully geometric approach for developable cloth deformation simulation

We present a new method for simulation of inextensible cloth subjected to a conservative force (e.g., the gravity) and collision-free constraint. Traditional algorithms for cloth simulation are all physically-based in which cloth is treated as an elastic material with some stiffness coefficient(s). These algorithms break down ultimately if one tries to set this stiffness coefficient to infinite which corresponds to inextensible cloth. The crux of the method is an algorithm for interpolating a given set of arbitrary points or space curves by a smooth developable mesh surface. We formulate this interpolation problem as a mesh deformation process that transforms an initial developable mesh surface, e.g., a planar figure, to a final mesh surface that interpolates the given points (called anchor points). During the deformation process, all the triangle elements in the intermediate meshes are kept isometric to their initial shapes, while the potential energy due to the conservative force is reduced gradually. The collision problem is resolved by introducing dynamic anchor points owing to the collision during the deformation. Notwithstanding its simplicity, the proposed method has shown some promising efficacy for simulation of inextensible cloth.     

Compression and Importance Sampling of Near‐Field Light Sources

This paper presents a method for compressing measured datasets of the near‐field emission of physical light sources (represented by raysets). We create a mesh on the bounding surface of the light source that stores illumination information. The mesh is augmented with information about directional distribution and energy density. We have developed a new approach to smoothly generate random samples on the illumination distribution represented by the mesh, and to efficiently handle importance sampling of points and directions. We will show that our representation can compress a 10 million particle rayset into a mesh of a few hundred triangles. We also show that the error of this representation is low, even for very close objects.     

Surface Reconstruction based on Lower Dimensional Localized Delaunay Triangulation

We present a fast, memory efficient algorithm that generates a manifold triangular mesh S passing through a set of unorganized points P R ³. Nothing is assumed about the geometry, topology or presence of boundaries in the data set except that P is sampled from a real manifold surface. The speed of our algorithm is derived from a projection-based approach we use to determine the incident faces on a point. We define our sampling criteria to sample the surface and guarantee a topologically correct mesh after surface reconstruction for such a sampled surface. We also present a new algorithm to find the normal at a vertex, when the surface is sampled according our given criteria. We also present results of our surface reconstruction using our algorithm on unorganized point clouds of various models.     

An analytical representation of conformal mapping for genus-zero implicit surfaces and its application to surface shape similarity assessment

This paper develops an analytical representation of conformal mapping for genus-zero implicit surfaces based on algebraic polynomial functions, and its application to surface shape similarity assessment. Generally, the conformal mapping often works as a tool of planar or spherical parameterization for triangle mesh surfaces. It is further exploited for implicit surface matching in this study. The method begins with discretizing one implicit surface by triangle mesh, where a discrete harmonic energy model related to both the mesh and the other implicit surface is established based on a polynomial-function mapping. Then both the zero-center constraint and the landmark constraints are added to the model to ensure the uniqueness of mapping result with the Möbius transformation. By searching optimal polynomial coefficients with the Lagrange–Newton method, the analytical representation of conformal mapping is obtained, which reveals all global and continuous one-to-one correspondent point pairs between two implicit surfaces. Finally, a shape similarity assessment index for (two) implicit surfaces is proposed through calculating the differences of all the shape index values among those corresponding points. The proposed analytical representation method of conformal mapping and the shape assessment index are both verified by the simulation cases for the closed genus-zero implicit surfaces. Experimental results show that the method is effective for genus-zero implicit surfaces, which will offer a new way for object retrieval and manufactured surface inspection.     

Quasi-Developable Mesh Surface Interpolation via Mesh Deformation

We present a new algorithm for finding a most "developable" smooth mesh surface to interpolate a given set of arbitrary points or space curves. Inspired by the recent progress in mesh editing that employs the concepts of preserving the Laplacian coordinates and handle-based shape editing, we formulate the interpolation problem as a mesh deformation process that transforms an initial developable mesh surface, such as a planar figure, to a final mesh surface that interpolates the given points and/or curves. During the deformation, the developability of the intermediate mesh is maintained by means of preserving the zero-valued Gaussian curvature on the mesh. To treat the high nonlinearity of the geometric constrains owing to the preservation of Gaussian curvature, we linearize those nonlinear constraints using Taylor expansion and eventually construct a sparse and over-determined linear system which is subsequently solved by a robust least-squares solution. By iteratively performing this procedure, the initial mesh is gradually and smoothly "dragged" to the given points and/or curves. The initial experimental data has shown some promising aspects of the proposed algorithm as a general quasi-developable surface interpolation tool.