arXiv subjects

Sheng-Gwo Chen

Publications and source records attributed to Sheng-Gwo Chen.

8 recordsLinked to original sources

arXiv2026

A Neural Network Framework for Geodesic-Like Curve Computation on Parametric Surfaces

The concept of geodesic-like curves was introduced by Chen in 2010 as a method for estimating shortest paths (geodesics) on parametric surfaces, with its convergence established theoretically. However, an efficient numerical computational framework has not yet been developed. In this paper, we propose an elegant and efficient approach for computing geodesic-like curves by leveraging deep learning and Physics-Informed Neural Networks (PINNs). Under the proposed framework, not only can single parametric surfaces be handled efficiently, but a broad class of complex parametric surfaces including multi-surface systems with $C^0$ or higher continuity and surfaces of revolution can also be robustly addressed.

cs.CG↗

arXiv2013

High-order algorithms for solving eigenproblems over discrete surfaces

The eigenvalue problem of the Laplace-Beltrami operators on curved surfaces plays an essential role in the convergence analysis of the numerical simulations of some important geometric partial differential equations which involve this operator. In this note we shall combine the local tangential lifting (LTL) method with the configuration equation to develop a new effective and convergent algorithm to solve the eigenvalue problems of the Laplace-Beltrami operators acting on functions over discrete surfaces. The convergence rates of our algorithms of discrete Laplace-Beltrami operators over surfaces is $O(r^n)$, $n \geq 1$, where $r$ represents the size of the mesh of discretization of the surface. The problem of high-order accuracies will also be discussed and used to compute geometric invariants of the underlying surfaces. Some convergence tests and eigenvalue computations on the sphere, tori and a dumbbell are presented.

math.NA↗

arXiv2011

A secure solution on hierarchical access control

Hierarchical access control is an important and traditional problem in information security. In 2001, Wu et.al. proposed an elegant solution for hierarchical access control by the secure-filter. Jeng and Wang presented an improvement of Wu et. al.'s method by the ECC cryptosystem. However, secure-filter method is insecure in dynaminc access control. Lie, Hsu and Tripathy, Paul pointed out some secure leaks on the secure-filter and presented some improvements to eliminate these secure flaws. In this paper, we revise the secure-filter in Jeng-Wang method and propose another secure solutions in hierarchical access control problem. CA is a super security class (user) in our proposed method and the secure-filter of $u_i$ in our solutions is a polynomial of degree $n_i+1$ in $\mathbb{Z}_p^*$, $f_i(x)=(x-h_i)(x-a_1)...(x-a_{n_i})+L_{l_i}(K_i)$. Although the degree of our secure-filter is larger than others solutions, our solution is secure and efficient in dynamics access control.

cs.CR↗

arXiv2011

Discrete Conservation Law on Curved Surfaces

In this paper we shall introduce a simple, effective numerical method for finding differential operators for scalar and vector-valued functions on surfaces. The key idea of our algorithm is to develop an intrinsic and unified way to compute directly the partial derivatives of functions defined on triangular meshes which are the discretization of regular surfaces under consideration. Most importantly, the divergence theorem and conservation laws on triangular meshes are fulfilled.

cs.CG↗

arXiv2010

Computation of the shortest path between two curves on a parametric surface by geodesic-like method

In this paper, we present the geodesic-like algorithm for the computation of the shortest path between two objects on NURBS surfaces and periodic surfaces. This method can improve the distance problem not only on surfaces but in $\mathbb{R}^3$. Moreover, the geodesic-like algorithm also provides an efficient approach to simulate the minimal geodesic between two holes on a NURBS surfaces.

cs.CG↗

arXiv2010

Convergent discrete Laplace-Beltrami operators over surfaces

The convergence problem of the Laplace-Beltrami operators plays an essential role in the convergence analysis of the numerical simulations of some important geometric partial differential equations which involve the operator. In this note we present a new effective and convergent algorithm to compute discrete Laplace-Beltrami operators acting on functions over surfaces. We prove a convergence theorem for our discretization. To our knowledge, this is the first convergent algorithm of discrete Laplace-Beltrami operators over surfaces for functions on general surfaces. Our algorithm is conceptually simple and easy to compute. Indeed, the convergence rate of our new algorithm of discrete Laplace-Beltrami operators over surfaces is $O(r)$ where r represents the size of the mesh of discretization of the surface.

cs.CG↗

arXiv2009

A new intrinsic numerical method for PDE on surfaces

In this note we shall introduce a simple, effective numerical method for solving partial differential equations for scalar and vector-valued data defined on surfaces. Even though we shall follow the traditional way to approximate the regular surfaces under consideration by triangular meshes, the key idea of our algorithm is to develop an intrinsic and unified way to compute directly the partial derivatives of functions defined on triangular meshes. We shall present examples in computer graphics and image processing applications.

cs.CG↗

arXiv2005

A simple effective method for curvatures estimation on triangular meshes

To definite and compute differential invariants, like curvatures, for triangular meshes (or polyhedral surfaces) is a key problem in CAGD and the computer vision. The Gaussian curvature and the mean curvature are determined by the differential of the Gauss map of the underlying surface. The Gauss map assigns to each point in the surface the unit normal vector of the tangent plane to the surface at this point. We follow the ideas developed in Chen and Wu \cite{Chen2}(2004) and Wu, Chen and Chi\cite{Wu}(2005) to describe a new and simple approach to estimate the differential of the Gauss map and curvatures from the viewpoint of the gradient and the centroid weights. This will give us a much better estimation of curvatures than Taubin's algorithm \cite{Taubin} (1995).

cs.CG↗