Algorithm for Correct Computation of Reeb Spaces for PL Bivariate Fields

Core Concepts

The authors present an algorithm for computing Reeb spaces of PL bivariate fields, focusing on the homeomorphism between the Multi-Dimensional Reeb Graph and the Reeb space.

Abstract

The content discusses an algorithm for computing Reeb spaces for PL bivariate fields, emphasizing the relationship between the Multi-Dimensional Reeb Graph and the Reeb space. It covers topological changes, critical points, Jacobi sets, and Morse conditions in detail.

The Reeb space is a topological structure that extends the concept of the Reeb graph to multi-fields by generalizing contour topology.
Techniques for computing multi-field topology have been developed based on Jacobi sets, fibers, and Reeb spaces.
The authors introduce an algorithm for computing a net-like structure corresponding to the Reeb space of a generic PL bivariate field.
The content provides detailed insights into simplicial complexes, PL scalar fields, critical points, and topological changes in time-varying Reeb graphs.
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An Algorithm for Correct Computation of Reeb Spaces for PL Bivariate Fields

by Amit Chattop... at arxiv.org 03-12-2024

https://arxiv.org/pdf/2403.06564.pdf

Deeper Inquiries

How does the algorithm presented in this content contribute to advancing computational mathematics

The algorithm presented in the content significantly advances computational mathematics by providing a method for computing correct net-like approximations of Reeb spaces for PL bivariate fields. This algorithm is crucial as it allows for the computation of multi-dimensional Reeb graphs (MDRG) without requiring range quantization, which has been a challenging problem in previous methods. By proving the homeomorphism between MDRG and the Reeb space, this algorithm offers a novel approach to capturing topological features in data across various computational domains.

What are potential limitations or challenges associated with implementing the proposed algorithm in practical applications

There are several potential limitations or challenges associated with implementing the proposed algorithm in practical applications. One challenge could be related to the complexity of computations involved in determining critical points and detecting topological changes accurately. Additionally, ensuring that the algorithm can handle large datasets efficiently without compromising accuracy may pose another challenge. Moreover, validating the results obtained from this algorithm against real-world data sets to ensure its reliability and robustness would be essential before widespread adoption.

How might advancements in understanding multi-field topology impact other areas of mathematical research

Advancements in understanding multi-field topology have far-reaching implications for other areas of mathematical research. The insights gained from studying multi-field topology can contribute to advancements in fields such as machine learning, computer vision, robotics, and scientific computing. For example, improved techniques for analyzing complex data structures using multi-field topology can enhance pattern recognition algorithms and optimization strategies. Furthermore, these advancements may lead to new developments in geometric modeling, shape analysis, and topological data analysis across various disciplines within mathematics and beyond.