A KNOTTED MINIMAL TREE

We present a method to extract polyhedral structures from a three-dimensional set of points, even if these structures are embedded in a perturbed background. The method is based on a family of affine diagrams which is an extension of the Voronoi diagram. These diagrams, namely anisotropic diagrams, are defined by using a parameterized distance whose unit ball is an ellipsoidal one. The parameters, upon which this distance depends, control the elongation and the orientation of the associated ellipsoidal ball. The triangulations, dual to the anisotropic diagrams, have the property to connect points that are not neighbors in the Voronoi diagram. Based on these triangulations, we define a family of three-dimensional anisotropic α-shapes. Unlike Euclidean α-shapes, anisotropic ones allow us to detect linear and planar structures in a given direction. The detection of more general polyhedral structures is obtained by merging several anisotropic α-shapes computed for different orientations.

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THREE-DIMENSIONAL SHAPES OF A FINITE SET OF POINTS

International Journal of Pattern Recognition and Artificial Intelligence ◽

10.1142/s0218001403002368 ◽

2003 ◽

Vol 17 (02) ◽

pp. 301-318 ◽

Cited By ~ 3

Author(s):

MAHMOUD MELKEMI

Keyword(s):

Voronoi Diagram ◽

Delaunay Triangulation ◽

Efficient Algorithm ◽

Point Cloud ◽

Three Dimensional ◽

Levels Of Detail ◽

Finite Set ◽

Set Of Points ◽

Different Levels ◽

Different Parts

The three-dimensional [Formula: see text]-shape is based on a mathematical formalism which determines exact relationships between points and shapes. It reconstructs surface and volume and detects 3D dot patterns for a given point cloud. [Formula: see text]-shape of a set of points is a sub-complex of Delaunay triangulation of this set. It generates a family of shapes according to the selected [Formula: see text] (a set of points). A method to compute the positions of the points of [Formula: see text] is proposed. These points are selected from the vertices of Voronoi diagram by analyzing the form of the polytopes; their elongation. This method allows the [Formula: see text]-shape to reflect different levels of detail in different parts of space. An efficient algorithm computing the three-dimensional [Formula: see text]-shape is presented, the [Formula: see text]-shape of a set of points is derived from the Delaunay triangulation of the same set. The speed of the algorithm is determined by the speed of the algorithm computing the Delaunay triangulation.

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A Randomized Approach to Volume Constrained Polyhedronization Problem

Journal of Computing and Information Science in Engineering ◽

10.1115/1.4029559 ◽

2015 ◽

Vol 15 (1) ◽

Cited By ~ 1

Author(s):

Jiju Peethambaran ◽

Amal Dev Parakkat ◽

Ramanathan Muthuganapathy

Keyword(s):

Three Dimensional ◽

Greedy Heuristic ◽

Point Sets ◽

Maximal Volume ◽

Simple Polyhedron ◽

Practical Algorithms ◽

Finite Set ◽

Potential Applications ◽

Brute Force Method ◽

Set Of Points

Given a finite set of points in R3, polyhedronization deals with constructing a simple polyhedron such that the vertices of the polyhedron are precisely the given points. In this paper, we present randomized approximation algorithms for minimal volume polyhedronization (MINVP) and maximal volume polyhedronization (MAXVP) of three dimensional point sets in general position. Both, MINVP and MAXVP, problems have been shown to be NP-hard and to the best of our knowledge, no practical algorithms exist to solve these problems. It has been shown that for any point set S in R3, there always exists a tetrahedralizable polyhedronization of S. We exploit this fact to develop a greedy heuristic for MINVP and MAXVP constructions. Further, we present an empirical analysis on the quality of the approximation results of some well defined point sets. The algorithms have been validated by comparing the results with the optimal results generated by an exhaustive searching (brute force) method for MINVP and MAXVP for some well chosen point sets of smaller sizes. Finally, potential applications of minimum and maximum volume polyhedra in 4D printing and surface lofting, respectively, have been discussed.

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Global structure of solutions for equations of Brezis–Nirenberg type on the unit ball

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500001037 ◽

2001 ◽

Vol 131 (3) ◽

pp. 647-665 ◽

Cited By ~ 5

Author(s):

Yoshitsugu Kabeya ◽

Eiji Yanagida ◽

Shoji Yotsutani

Keyword(s):

Boundary Condition ◽

Unit Ball ◽

Existence And Uniqueness ◽

Three Dimensional ◽

Critical Value ◽

Global Structure ◽

Radial Solutions ◽

Structure Of Solutions ◽

Dimensional Unit ◽

Robin Condition

The Brezis–Nirenberg equation and the scalar field equation on the three-dimensional unit ball are studied. Under the Robin condition, we show the existence and uniqueness of radial solutions in a unified way. In particular, it is shown that the global structure of solutions changes qualitatively when a parameter in the boundary condition exceeds a certain critical value.

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Global structure of solutions for equations of Brezis-Nirenberg type on the unit ball

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210501000282 ◽

2001 ◽

Vol 131 (3) ◽

pp. 647-665

Author(s):

Yoshitsugu Kabeya ◽

Eiji Yanagida ◽

Shoji Yotsutani

Keyword(s):

Boundary Condition ◽

Unit Ball ◽

Existence And Uniqueness ◽

Three Dimensional ◽

Critical Value ◽

Global Structure ◽

Radial Solutions ◽

Structure Of Solutions ◽

Dimensional Unit ◽

Robin Condition

The Brezis-Nirenberg equation and the scalar field equation on the three-dimensional unit ball are studied. Under the Robin condition, we show the existence and uniqueness of radial solutions in a unified way. In particular, it is shown that the global structure of solutions changes qualitatively when a parameter in the boundary condition exceeds a certain critical value.

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Estimation of 6D Object Pose Using a 2D Bounding Box

Sensors ◽

10.3390/s21092939 ◽

2021 ◽

Vol 21 (9) ◽

pp. 2939

Author(s):

Yong Hong ◽

Jin Liu ◽

Zahid Jahangir ◽

Sheng He ◽

Qing Zhang

Keyword(s):

Neural Network ◽

Loss Function ◽

Three Dimensional ◽

Unit Vector ◽

Prediction Algorithm ◽

Computational Time ◽

Bounding Box ◽

Dimensional Unit ◽

Bounding Boxes ◽

Rgb Image

This paper provides an efficient way of addressing the problem of detecting or estimating the 6-Dimensional (6D) pose of objects from an RGB image. A quaternion is used to define an object′s three-dimensional pose, but the pose represented by q and the pose represented by -q are equivalent, and the L2 loss between them is very large. Therefore, we define a new quaternion pose loss function to solve this problem. Based on this, we designed a new convolutional neural network named Q-Net to estimate an object’s pose. Considering that the quaternion′s output is a unit vector, a normalization layer is added in Q-Net to hold the output of pose on a four-dimensional unit sphere. We propose a new algorithm, called the Bounding Box Equation, to obtain 3D translation quickly and effectively from 2D bounding boxes. The algorithm uses an entirely new way of assessing the 3D rotation (R) and 3D translation rotation (t) in only one RGB image. This method can upgrade any traditional 2D-box prediction algorithm to a 3D prediction model. We evaluated our model using the LineMod dataset, and experiments have shown that our methodology is more acceptable and efficient in terms of L2 loss and computational time.

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On extremums of sums of powered distances to a finite set of points

Geometriae Dedicata ◽

10.1007/s10711-012-9804-3 ◽

2012 ◽

Vol 167 (1) ◽

pp. 69-89 ◽

Cited By ~ 3

Author(s):

Nikolai Nikolov ◽

Rafael Rafailov

Keyword(s):

Finite Set ◽

Set Of Points

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STEREO VISION METHOD APPLICATION TO ROAD INSPECTION

The Baltic Journal of Road and Bridge Engineering ◽

10.3846/bjrbe.2017.05 ◽

2017 ◽

Vol 12 (1) ◽

pp. 38-47 ◽

Cited By ~ 19

Author(s):

Marcin Staniek

Keyword(s):

Stereo Vision ◽

Dimensional Space ◽

Three Dimensional ◽

Corner Detection ◽

Stereo Images ◽

Processing Technologies ◽

The Road ◽

Road Pavement ◽

Road Inspection ◽

Set Of Points

The paper presents the stereo vision method for the mapping of road pavement. The mapped road is a set of points in three-dimensional space. The proposed method of measurement and its implementation make it possible to generate a precise mapping of a road surface with a resolution of 1 mm in transverse, longitudinal and vertical dimensions. Such accurate mapping of the road is the effect of application of stereo images based on image processing technologies. The use of matching measure CoVar, at the stage of images matching, help eliminate corner detection and filter stereo images, maintaining the effectiveness of the algorithm mapping. The proper analysis of image-based data and application of mathematical transformations enable to determine many types of distresses such as potholes, patches, bleedings, cracks, ruts and roughness. The paper also aims at comparing the results of proposed solution and reference test-bench. The statistical analysis of the differences permits the judgment of error types.

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