Curvature estimation for segmentation of triangulated surfaces (original) (raw)
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A Geometric Approach to Curvature Estimation on Triangulated 3D Shapes
We present a geometric approach to define discrete normal, principal, Gaussian and mean curvatures, that we call Ccurvature. Our approach is based on the notion of concentrated curvature of a polygonal line and a simulation of rotation of the normal plane of the surface at a point. The advantages of our approach is its simplicity and its natural meaning. A comparison with widely-used discrete methods is presented.
Lecture Notes in Computer Science, 2006
For effective application of laser or X-ray CT scanned mesh models in design, analysis, and inspection etc, it is preferable that they are segmented into desirable regions as a pre-processing. Engineering parts are commonly covered with analytic surfaces, such as planes, cylinders, spheres, cones, and tori. Therefore, the portions of the part's boundary where each can be represented by a type of analytic surface have to be extracted as regions from the mesh model. In this paper, we propose a new mesh segmentation method for this purpose. We use the mesh curvature estimation with sharp edge recognition, and the non-iterative region growing to extract the regions. The proposed mesh curvature estimation is robust for measurement noise. Moreover, our proposed region growing enables to find more accurate boundaries of underlying surfaces, and to classify extracted analytic surfaces into higher-level classes of surfaces: fillet surface, linear extrusion surface and surface of revolution than those in the existing methods.
Curvature estimation for unstructured triangulations of surfaces
In this work, a survey of several curvature estimation methods for surface meshes was conducted, and a comparison of two types of curvature estimation techniques was conducted based on convergence studies. As a result of this work, a new and improved method was proposed as an extension of one of the surveyed methods. The new method robustly estimates normals, principal curvatures, mean curvatures and Gaussian curvatures at vertices of general unstructured triangulations. The method has been tested on complex meshes and has provided good results.
Re-triangulation of existing surface meshes with high curvatures
This work describes an automatic algorithm for unstructured mesh regeneration on arbitrarily shaped threedimensional surfaces. The arbitrary surface may be: a triangulated mesh, a set of points, or an analytical surface (such as a collection of NURBS patches). To be generic, the algorithm requires the implementation of three abstract methods. The first, given a point location, returns the desired characteristic size of a triangular element at this position. The second method, given the current edge in the boundary contraction algorithm, locates the ideal apex point that forms a triangle with this edge. And the third method, given a point in space and a projection direction, returns the closest point on the geometrical supporting surface. This work also describes the implementation of these three methods to re-mesh an existing triangulated mesh that might present regions of high curvature. In order to test the efficiency of the proposed algorithm of surface mesh generation and implementation of the three abstract methods, results of performance and quality of generated triangular element examples are presented.
A CURVATURE-SENSITIVE PARAMETERIZATION-INDEPENDENT TRIANGULATION ALGORITHM
Triangulations of a connected subset F of parametric surfaces S (u, v) (with continuity C 2 or higher) are required because a C 0 approximation of such F (called a FACE) is widely required for finite element analysis, rendering, manufacturing, design, reverse engineering, etc. The triangulation T is such an approximation, when its piecewise linear subsets are triangles (which, on the other hand, is not a compulsory condition for being C 0 ). A serious obstacle for algorithms which triangulate in the parametric space u − v is that such a space may be extremely warped, and the distances in parametric space be dramatically different of the distances in R 3 . Recent publications have reported parameter -independent triangulations, which triangulate in R 3 space. However, such triangulations are not sensitive to the curvature of the S(u, v). The present article presents an algorithm to obtain parameter-independent, curvature-sensitive triangulations. The invariant of the algorithm is that a vertex v of the triangulation if identified, and a quasiequilateral triangulation around v is performed on the plane Π tangent to S(u, v) at v. The size of the triangles incident to v is a function of K(v), the curvature of S (u, v) at v. The algorithm was extensively and successfully tested, rendering short running times, with very demanding boundary representations.
Principal Curvature-Driven Segmentation of Mesh Models: A Preliminary Assessment
2021
Three methods for triangle mesh segmentation, based on precomputed principal curvature values and using a region growing algorithm to label the vertices defining distinct surface regions, were developed, aiming at supporting the /ater manipulation of mesh models. Examples are presented, using different models, to illustrate their behavior. Results are promising but, in some cases, there is a clear need for a further post-processing step to refine the boundaries between adjoining regions and eliminate segmentation artifacts.
Farthest sampling segmentation of triangulated surfaces
Cornell University - arXiv, 2020
In this paper we introduce Farthest Sampling Segmentation (FSS), a new method for segmentation of triangulated surfaces, which consists of two fundamental steps: the computation of a submatrix W k of the affinity matrix W and the application of the k-means clustering algorithm to the rows of W k. The submatrix W k is obtained computing the affinity between all triangles and only a few special triangles: those which are farthest in the defined metric. This is equivalent to select a sample of columns of W without constructing it completely. The proposed method is computationally cheaper than other segmentation algorithms, since it only calculates few columns of W and it does not require the eigendecomposition of W or of any submatrix of W. We prove that the orthogonal projection of W on the space generated by the columns of W k coincides with the orthogonal projection of W on the space generated by the k eigenvectors computed by Nyström's method using the columns of W k as a sample of W. Further, it is shown that for increasing size k, the proximity relationship among the rows of W k tends to faithfully reflect the proximity among the corresponding rows of W. The FSS method does not depend on parameters that must be tuned by hand and it is very flexible, since it can handle any metric to define the distance between triangles. Numerical experiments with several metrics and a large variety of 3D triangular meshes show that the segmentations obtained computing less than the 10% of columns W are as good as those obtained from clustering the rows of the full matrix W .
Segmenting point-sampled surfaces
2010
Extracting features from point-based representations of geometric surface models is becoming increasingly important for purposes such as model classification, matching, and exploration. In an earlier paper, we proposed a multiphase segmentation process to identify elongated features in point-sampled surface models without the explicit construction of a mesh or other surface representation. The preliminary results demonstrated the strength and potential of the segmentation process, but the resulting segmentations were still of low quality, and the segmentation process could be slow. In this paper, we describe several algorithmic improvements to overcome the shortcomings of the segmentation process. To demonstrate the improved quality of the segmentation and the superior time efficiency of the new segmentation process, we present segmentation results obtained for various point-sampled surface models. We also discuss an application of our segmentation process to extract ridge-separated features in point-sampled surfaces of CAD models.
Optimizing 3D triangulations using discrete curvature analysis
2000
A tool for constructing a "good" 3D triangulation of a given set of vertices in 3D is developed and studied. The constructed triangulation is "optimal" in the sense that it locally minimizes a cost function which measures a certain discrete curvature over the resulting triangle mesh. The algorithm for obtaining the optimal triangulation is that of swapping edges sequentially, such that the cost function is reduced maximally by each swap. In this paper three easy-to-compute cost functions are derived using a simple model for defining discrete curvatures of triangle meshes. The results obtained by the different cost functions are compared. Operating on data sampled from simple 3D models, we compare the approximation error of the resulting optimal triangle meshes to the sampled model in various norms. The conclusion is that all three cost functions lead to similar results, and none of them can be said to be superior to the others. The triangle meshes generated by our algorithm, when serving as initial triangle meshes for the butterfly subdivision scheme, are found to improve significantly the limit butterfly-surfaces compared to arbitrary initial triangulations of the given sets of vertices. Based upon this observation, we believe that any algorithm operating on triangle meshes such as subdivision, finite element solution of PDE, or mesh simplification, can obtain better results if applied to a "good" triangle mesh with small discrete curvatures. Thus our algorithm can serve for modelling surfaces from sampled data as well as for initialization of other triangle mesh based algorithms.