SGP09: Eurographics Symposium on Geometry Processing
https://diglib.eg.org:443/handle/10.2312/438
Thu, 20 Jun 2019 06:26:44 GMT2019-06-20T06:26:44ZSeparatrix Persistence: Extraction of Salient Edges on Surfaces Using Topological Methods
https://diglib.eg.org:443/handle/10.2312/CGF.v28i5pp1519-1528
Separatrix Persistence: Extraction of Salient Edges on Surfaces Using Topological Methods
Weinkauf, T.; Guenther, D.
Salient edges are perceptually prominent features of a surface. Most previous extraction schemes utilize the notion of ridges and valleys for their detection, thereby requiring curvature derivatives which are rather sensitive to noise. We introduce a novel method for salient edge extraction which does not depend on curvature derivatives. It is based on a topological analysis of the principal curvatures and salient edges of the surface are identified as parts of separatrices of the topological skeleton. Previous topological approaches obtain results including non-salient edges due to inherent properties of the underlying algorithms. We extend the profound theory by introducing the novel concept of separatrix persistence, which is a smooth measure along a separatrix and allows to keep its most salient parts only. We compare our results with other methods for salient edge extraction.
Thu, 01 Jan 2009 00:00:00 GMThttps://diglib.eg.org:443/handle/10.2312/CGF.v28i5pp1519-15282009-01-01T00:00:00ZEstimating the Laplace-Beltrami Operator by Restricting 3D Functions
https://diglib.eg.org:443/handle/10.2312/CGF.v28i5pp1475-1484
Estimating the Laplace-Beltrami Operator by Restricting 3D Functions
Chuang, Ming; Luo, Linjie; Brown, Benedict J.; Rusinkiewicz, Szymon; Kazhdan, Michael
We present a novel approach for computing and solving the Poisson equation over the surface of a mesh. As in previous approaches, we define the Laplace-Beltrami operator by considering the derivatives of functions defined on the mesh. However, in this work, we explore a choice of functions that is decoupled from the tessellation. Specifically, we use basis functions (second-order tensor-product B-splines) defined over 3D space, and then restrict them to the surface. We show that in addition to being invariant to mesh topology, this definition of the Laplace-Beltrami operator allows a natural multiresolution structure on the function space that is independent of the mesh structure, enabling the use of a simple multigrid implementation for solving the Poisson equation.
Thu, 01 Jan 2009 00:00:00 GMThttps://diglib.eg.org:443/handle/10.2312/CGF.v28i5pp1475-14842009-01-01T00:00:00ZApproximating Gradients for Meshes and Point Clouds via Diffusion Metric
https://diglib.eg.org:443/handle/10.2312/CGF.v28i5pp1497-1508
Approximating Gradients for Meshes and Point Clouds via Diffusion Metric
Luo, Chuanjiang; Safa, Issam; Wang, Yusu
The gradient of a function defined on a manifold is perhaps one of the most important differential objects in data analysis. Most often in practice, the input function is available only at discrete points sampled from the underlying manifold, and the manifold is approximated by either a mesh or simply a point cloud. While many methods exist for computing gradients of a function defined over a mesh, computing and simplifying gradients and related quantities such as critical points, of a function from a point cloud is non-trivial.In this paper, we initiate the investigation of computing gradients under a different metric on the manifold from the original natural metric induced from the ambient space. Specifically, we map the input manifold to the eigenspace spanned by its Laplacian eigenfunctions, and consider the so-called diffusion distance metric associated with it. We show the relation of gradient under this metric with that under the original metric. It turns out that once the Laplace operator is constructed, it is easier to approximate gradients in the eigenspace for discrete inputs (especially point clouds) and it is robust to noises in the input function and in the underlying manifold. More importantly, we can easily smooth the gradient field at different scales within this eigenspace framework. We demonstrate the use of our new eigen-gradients with two applications: approximating / simplifying the critical points of a function, and the Jacobi sets of two input functions (which describe the correlation between these two functions), from point clouds data.
Thu, 01 Jan 2009 00:00:00 GMThttps://diglib.eg.org:443/handle/10.2312/CGF.v28i5pp1497-15082009-01-01T00:00:00ZDiscrete Critical Values: a General Framework for Silhouettes Computation
https://diglib.eg.org:443/handle/10.2312/CGF.v28i5pp1509-1518
Discrete Critical Values: a General Framework for Silhouettes Computation
Chazal, F.; Lieutier, A.; Montana, N.
Many shapes resulting from important geometric operations in industrial applications such as Minkowski sums or volume swept by a moving object can be seen as the projection of higher dimensional objects. When such a higher dimensional object is a smooth manifold, the boundary of the projected shape can be computed from the critical points of the projection. In this paper, using the notion of polyhedral chains introduced by Whitney, we introduce a new general framework to define an analogous of the set of critical points of piecewise linear maps defined over discrete objects that can be easily computed. We illustrate our results by showing how they can be used to compute Minkowski sums of polyhedra and volumes swept by moving polyhedra.
Thu, 01 Jan 2009 00:00:00 GMThttps://diglib.eg.org:443/handle/10.2312/CGF.v28i5pp1509-15182009-01-01T00:00:00Z