Linear time recognition algorithms for topological invariants in 3D

@article{Chen2008LinearTR,
  title={Linear time recognition algorithms for topological invariants in 3D},
  author={Li M. Chen and Yongwu Rong},
  journal={2008 19th International Conference on Pattern Recognition},
  year={2008},
  pages={1-4}
}
  • Li M. ChenYongwu Rong
  • Published 11 April 2008
  • Mathematics
  • 2008 19th International Conference on Pattern Recognition
In this paper, we design linear time algorithms to recognize and determine topological invariants such as genus and homology groups in 3D. These invariants can be used to identify patterns in 3D image recognition and medical image analysis. Our method is based on cubical images with direct adjacency, also called (6,26)-connectivity images in discrete geometry. According to the fact that there are only six types of local surface points in 3D and a discrete version of the well-known Gauss-Bonnett… 

Figures from this paper

Algorithms for Computing Topological Invariants in 2D and 3D Digital Spaces

This paper designed fast algorithms for topological invariants such as connected components, hole counting in 2D and boundary surface genus for 3D and also O(n) time algorithm to get genus of the closed surface.

Algorithms for Computing Topological Invariants in Digital Spaces

This paper includes fast algorithms and implementations for topological invariants such as connected components, hole counting in 2D, and boundary surface genus for 3D and a linear time algorithm to solve the hole counting problem.

Digital Curvatures Applied to 3D Object Analysis and Recognition: A Case Study

It is found that Gaussian curvatures mainly describe the global features and average characteristics such as the five regions of a human face, however, mean curvatures can be used to find local features and extreme points such as nose in 3D facial data.

Some Results on Simplical Homology Groups of 2D Digital Images.

In this paper we study some results related to the simplicial homology groups of 2D digital images. We show that if a bounded digital image Z X ⊂ is nonempty and κ -connected, then its homology

On Some Local Topological Properties of Naive Discrete Sphere

Novel results on the local topological properties of the naive model of discrete sphere are presented, which follow from the bijection of each quadraginta octant of naive sphere with its projection map on the corresponding functional plane and from the characterization of certain jumps in the f-map.

Contributions to Topological Data Analysis for Scientific Visualization

This thesis presents new research directions, supported by recent preliminary results for the topological analysis of uncertain and bivariate scalar fields, and discusses practical challenges that recently arose with the ongoing development of high performance computing resources.

Topology Verification for Isosurface Extraction

A framework for verification of isosurfacing implementations to check topological properties is extended and stratified Morse theory and digital topology are used to design algorithms which verify topological invariants.

Reproducing Expert-Like Motion in Deformable Environments Using Active Learning and IOC

This work hypothesizes that the relative sensitivities of deformable objects are encoded in the expert’s demonstrated motion, and presents a framework which is able to imitate an expert's behavior by learning a sensitivity-based cost function under which the expert's motion is optimal.

Enabling Motion Planning and Execution for Tasks Involving Deformation and Uncertainty

This work focuses on two important areas of poorly controlled robotic manipulation: motion planning for deformable objects and in deformable environments; and manipulation with uncertainty, which incorporates contact with the environment and compliance of the robot to generate motion policies which are then adapted during execution to reflect actual robot behavior.

References

SHOWING 1-10 OF 17 REFERENCES

Computing Homology Generators for Volumes Using Minimal Generalized Maps

An algorithm for computing efficiently homology generators of 3D subdivided orientable objects which can contain tunnels and cavities by reducing the number of cells using simplification operations and preserving homology.

Border and SurfaceTracing - Theoretical Foundations

  • V. E. BrimkovR. Klette
  • Mathematics, Computer Science
    IEEE Transactions on Pattern Analysis and Machine Intelligence
  • 2008
This work appears to be the first one on digital manifolds based on a graph-theoretical definition of dimension, and provides (in particular) a general theoretical basis for curve or surface tracing in picture analysis.

3D well-composed pictures

A very natural definition of a continuous analog for well-composed digital pictures leads to regular properties of surfaces, which allows us to give a simple proof of a digital version of the 3D Jordan–Brouwer separation theorem.

An incremental algorithm for Betti numbers of simplicial complexes on the 3-sphere

Polyhedral Surfaces of Constant Mean Curvature

way, say, by assigning a length to each edge which fulÞlls the triangle identity on each triangle. In a locally Euclidean metric the distance between two points is measured along curves whose length

Computing homology groups of simplicial complexes in R3

A new approach to analyze simplicial complexes in Euclidean 3-space by using methods from topology to analyze triangulated 3-manifolds and determining homology groups and concrete representations of their generators for a given complex.

Cubical approximation and computation of homology

The algorithm used in the homology computations is based on a local reduction procedure, and a subquadratic estimate of its computational complexity is given.

Topological Algorithms for Digital Image Processing

Algebraic Topology

The focus of this paper is a proof of the Nielsen-Schreier Theorem, stating that every subgroup of a free group is free, using tools from algebraic topology.