Synthetic Data Generation and Deep Learning for the Topological Analysis of 3D Data

This research uses deep learning to estimate the topology of manifolds represented by sparse, unordered point cloud scenes in 3D. A new labelled dataset was synthesised to train neural networks and evaluate their ability to estimate the genus of these manifolds. This data used random homeomorphic deformations to provoke the learning of visual topological features. We demonstrate that deep learning models could extract these features and discuss some advantages over existing topological data analysis tools that are based on persistent homology. Semantic segmentation was used to provide additional geometric information in conjunction with topological labels. Common point cloud multi-layer perceptron and transformer networks were both used to compare the viability of these methods. The experimental results of this pilot study support the hypothesis that, with the aid of sophisticated synthetic data generation, neural networks can perform segmentation-based topological data analysis. While our study focused on simulated data, the accuracy achieved suggests a potential for future applications using real data.

Topology Estimation of Simulated 4D Image Data by Combining Downscaling and Convolutional Neural Networks

Four-dimensional image-type data can quickly become prohibitively large, and it may not be feasible to directly apply methods, such as persistent homology or convolutional neural networks, to determine the topological characteristics of these data because they can encounter complexity issues. This study aims to determine the Betti numbers of large four-dimensional image-type data. The experiments use synthetic data, and demonstrate that it is possible to circumvent these issues by applying downscaling methods to the data prior to training a convolutional neural network, even when persistent homology software indicates that downscaling can significantly alter the homology of the training data. When provided with downscaled test data, the neural network can estimate the Betti numbers of the original samples with reasonable accuracy.