Efficient Geometric Shape Recognition from Noisy Point Cloud Data using Core and Alpha-Core Bifiltrations

Conceitos Básicos

This paper introduces the core and alpha-core bifiltrations as efficient alternatives to the multicover bifiltration for recognizing geometric shapes from noisy point cloud data. The proposed bifiltrations are interleaved with the multicover bifiltration and enjoy similar stability properties.

Resumo

The paper introduces two new bifiltrations, the core bifiltration and the alpha-core bifiltration, which are inspired by the HDBSCAN clustering algorithm and the multicover bifiltration.

Key highlights:

1. The core and alpha-core bifiltrations are interleaved with the multicover bifiltration, allowing the transfer of stability results from the multicover bifiltration to the new bifiltrations.
2. The core bifiltration admits a cover consisting of metric balls, enabling the use of the nerve lemma to show that it is homotopy equivalent to the core Čech bifiltration.
3. Similar arguments are made for the alpha-core bifiltration, showing its relationship to the core bifiltration and the multicover bifiltration.
4. Stability results are established for the core and alpha-core bifiltrations, similar to the multicover stability results.
5. Experiments are performed on synthetic point cloud datasets, computing the persistent homology of the alpha-core bifiltration along horizontal and sloped lines in the parameter space. The bottleneck distance between the persistence diagrams of noisy and clean samples is used to evaluate the robustness to noise.

The paper provides a comprehensive analysis of the core and alpha-core bifiltrations, establishing their theoretical properties and demonstrating their practical utility for geometric shape recognition from noisy point cloud data.

Estatísticas

The paper does not contain any explicit numerical data or statistics. The experiments section discusses the performance of the alpha-core bifiltration on various synthetic point cloud datasets, but the results are presented in the form of persistence diagrams and bottleneck distances rather than numerical tables.

Citações

"The motivation of this paper is to recognize a geometric shape from a noisy sample in the form of a point cloud."
"We introduce the core bifiltration together with a variant built on the alpha-complex, the alpha-core bifiltration, both of which are interleaved with the multicover bifiltration."
"Using an approach similar to Blumberg and Lesnick [2] we get a Prohorov stability result for the core and alpha-core bifiltrations."

Principais Insights Extraídos De

Core Bifiltration

by Nello Blaser... às arxiv.org 05-03-2024

https://arxiv.org/pdf/2405.01214.pdf

Perguntas Mais Profundas

How can the core and alpha-core bifiltrations be extended to handle higher-dimensional parameter spaces beyond the bifiltered case

To extend the core and alpha-core bifiltrations to higher-dimensional parameter spaces beyond the bifiltered case, we can consider generalizing the concept of bifiltrations to handle multiple parameters in a more flexible manner. One approach could be to incorporate additional parameters into the dissimilarities used to define the core and alpha-core bifiltrations. By allowing for more parameters in the dissimilarities, we can construct a multi-parameter core and alpha-core bifiltration that captures the geometric shape of a point cloud in a higher-dimensional parameter space. This extension would involve defining dissimilarities that consider multiple aspects of the data simultaneously, leading to a more comprehensive analysis of the underlying structure.

What are the limitations of the core and alpha-core bifiltrations, and in what scenarios would the multicover bifiltration be a more suitable choice

The core and alpha-core bifiltrations have certain limitations that may impact their applicability in certain scenarios. One limitation is that these bifiltrations are based on specific geometric constructions, such as Voronoi cells and core distances, which may not always capture the complex structure of high-dimensional data. In cases where the data exhibits intricate topological features or noise that is not well-handled by these constructions, the multicover bifiltration may be a more suitable choice. The multicover bifiltration offers a more flexible approach to capturing the topology of data by considering multiple covers of the space, allowing for a more robust analysis of complex datasets with varying densities and shapes.

Can the core and alpha-core bifiltrations be applied to real-world datasets beyond synthetic point clouds, and how would their performance compare to other topological data analysis techniques

The core and alpha-core bifiltrations can be applied to real-world datasets beyond synthetic point clouds, providing insights into the underlying topology of the data. These techniques can be particularly useful in analyzing datasets from various fields such as biology, neuroscience, and materials science, where understanding the shape and connectivity of data points is crucial. When compared to other topological data analysis techniques, the core and alpha-core bifiltrations offer a unique perspective by focusing on core distances and Voronoi cells, which can reveal important geometric information about the dataset. However, the performance of these techniques may vary depending on the specific characteristics of the dataset, and it is essential to consider the nature of the data and the research question at hand when choosing the appropriate method for analysis.