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We study that the $n-$graphs defining by smooth map $f:\Om\subset \ir{n}\to \ir{m}, m\ge 2,$ in $\ir{m+n}$ of the prescribed mean curvature and the Gauss image. We derive the interior curvature estimates $$\sup_{D_R(x)}|B|^2\le\f{C}{R^2}$$…

The combination of words ``discrete curvature'' is only an apparent contradiction. In this survey we describe curvature notions associated with polygons, polyhedral surfaces, and with abstract polyhedral manifolds. Several theorems about…

DeepAngle is a machine learning-based method to determine the contact angles of different phases in the tomography images of porous materials. Measurement of angles in 3--D needs to be done within the surface perpendicular to the angle…

In this paper we establish new simple local geometric criteria for discrete entropic curvature introduced in [47] that are powerful enough to capture many geometric properties of complex models arising in mathematical physics. These results…

We study theoretically the self-propulsion dynamics of a small droplet on general curved surfaces by a variational approach. A new reduced model is derived based on careful computations for the capillary energy and the viscous dissipation…

Point-cloud is an efficient way to represent 3D world. Analysis of point-cloud deals with understanding the underlying 3D geometric structure. But due to the lack of smooth topology, and hence the lack of neighborhood structure, standard…

Orienting surface normals correctly and consistently is a fundamental problem in geometry processing. Applications such as visualization, feature detection, and geometry reconstruction often rely on the availability of correctly oriented…

Conventional manual surveys of rock mass fractures usually require large amounts of time and labor; yet, they provide a relatively small set of data that cannot be considered representative of the study region. Terrestrial laser scanners…

This paper addresses the problem of generating dense point clouds from given sparse point clouds to model the underlying geometric structures of objects/scenes. To tackle this challenging issue, we propose a novel end-to-end learning-based…

We present a deep learning method that propagates point-wise feature representations across shapes within a collection for the purpose of 3D shape segmentation. We propose a cross-shape attention mechanism to enable interactions between a…

3D point cloud interpretation is a challenging task due to the randomness and sparsity of the component points. Many of the recently proposed methods like PointNet and PointCNN have been focusing on learning shape descriptions from point…

Point cloud analysis has drawn broader attentions due to its increasing demands in various fields. Despite the impressive performance has been achieved on several databases, researchers neglect the fact that the orientation of those point…

Point clouds are essential for storage and transmission of 3D content. As they can entail significant volumes of data, point cloud compression is crucial for practical usage. Recently, point cloud geometry compression approaches based on…

In this paper, we study the computation of curvatures at the singular points of algebraic curves and surfaces. The idea is to convert the problem to compute the curvatures of the corresponding regular parametric curves and surfaces, which…

Point cloud anomaly detection under the anomaly-free setting poses significant challenges as it requires accurately capturing the features of 3D normal data to identify deviations indicative of anomalies. Current efforts focus on devising…

The present work illustrates a difficulty with the level-set method to accurately capture the curvature of interfaces in regions that are of equal distance to two or more interfaces. Such regions are characterized by kinks in the level-set…

This paper addresses the problem of generating uniform dense point clouds to describe the underlying geometric structures from given sparse point clouds. Due to the irregular and unordered nature, point cloud densification as a generative…

A successful point cloud registration often lies on robust establishment of sparse matches through discriminative 3D local features. Despite the fast evolution of learning-based 3D feature descriptors, little attention has been drawn to the…

Efficient point cloud compression is fundamental to enable the deployment of virtual and mixed reality applications, since the number of points to code can range in the order of millions. In this paper, we present a novel data-driven…

Duality principle for approximation of geometrical objects (also known as Eudoxus exhaustion method) was extended and perfected by Archimedes in his famous tractate "Measurement of circle". The main idea of the approximation method by…