Noise robustness of persistent homology on greyscale images, across filtrations and signatures

Three-dimensional topology-based analysis segments volumetric and spatiotemporal fluorescence microscopy

Image analysis techniques provide objective and reproducible statistics for interpreting microscopy data. At higher dimensions, three-dimensional (3D) volumetric and spatiotemporal data highlight additional properties and behaviors beyond the static 2D focal plane. However, increased dimensionality carries increased complexity, and existing techniques for general segmentation of 3D data are either primitive, or highly specialized to specific biological structures. Borrowing from the principles of 2D topological data analysis (TDA), we formulate a 3D segmentation algorithm that implements persistent homology to identify variations in image intensity. From this, we derive two separate variants applicable to spatial and spatiotemporal data, respectively. We demonstrate that this analysis yields both sensitive and specific results on simulated data and can distinguish prominent biological structures in fluorescence microscopy images, regardless of their shape. Furthermore, we highlight the efficacy of temporal TDA in tracking cell lineage and the frequency of cell and organelle replication.

Constrained Adaptive Distillation Based on Topological Persistence for Wearable Sensor Data

Wearable sensor data analysis with persistence features generated by topological data analysis (TDA) has achieved great successes in various applications, however, it suffers from large computational and time resources for extracting topological features. In this paper, our approach utilizes knowledge distillation (KD) that involves the use of multiple teacher networks trained with the raw time-series and persistence images generated by TDA, respectively. However, direct transfer of knowledge from the teacher models utilizing different characteristics as inputs to the student model results in a knowledge gap and limited performance. To address this problem, we introduce a robust framework that integrates multimodal features from two different teachers and enables a student to learn desirable knowledge effectively. To account for statistical differences in multimodalities, entropy based constrained adaptive weighting mechanism is leveraged to automatically balance the effects of teachers and encourage the student model to adequately adopt the knowledge from two teachers. To assimilate dissimilar structural information generated by different style models for distillation, batch and channel similarities within a mini-batch are used. We demonstrate the effectiveness of the proposed method on wearable sensor data.

Persistent Homology-Based Machine Learning Method for Filtering and Classifying Mammographic Microcalcification Images in Early Cancer Detection

Microcalcifications in mammogram images are primary indicators for detecting the early stages of breast cancer. However, dense tissues and noise in the images make it challenging to classify the microcalcifications. Currently, preprocessing procedures such as noise removal techniques are applied directly on the images, which may produce a blurry effect and loss of image details. Further, most of the features used in classification models focus on local information of the images and are often burdened with details, resulting in data complexity. This research proposed a filtering and feature extraction technique using persistent homology (PH), a powerful mathematical tool used to study the structure of complex datasets and patterns. The filtering process is not performed directly on the image matrix but through the diagrams arising from PH. These diagrams will enable us to distinguish prominent characteristics of the image from noise. The filtered diagrams are then vectorised using PH features. Supervised machine learning models are trained on the MIAS and DDSM datasets to evaluate the extracted features' efficacy in discriminating between benign and malignant classes and to obtain the optimal filtering level. This study reveals that appropriate PH filtering levels and features can improve classification accuracy in early cancer detection.

Topology-based fluorescence image analysis for automated cell identification and segmentation

Cell segmentation refers to the body of techniques used to identify cells in images and extract biologically relevant information from them; however, manual segmentation is laborious and subjective. We present Topological Boundary Line Estimation using Recurrence Of Neighbouring Emissions (TOBLERONE), a topological image analysis tool which identifies persistent homological image features as opposed to the geometric analysis commonly employed. We demonstrate that topological data analysis can provide accurate segmentation of arbitrarily-shaped cells, offering a means for automatic and objective data extraction. One cellular feature of particular interest in biology is the plasma membrane, which has been shown to present varying degrees of lipid packing, or membrane order, depending on the function and morphology of the cell type. With the use of environmentally-sensitive dyes, images derived from confocal microscopy can be used to quantify the degree of membrane order. We demonstrate that TOBLERONE is capable of automating this task.

Selecting embedding delays: An overview of embedding techniques and a new method using persistent homology

Delay embedding methods are a staple tool in the field of time series analysis and prediction. However, the selection of embedding parameters can have a big impact on the resulting analysis. This has led to the creation of a large number of methods to optimize the selection of parameters such as embedding lag. This paper aims to provide a comprehensive overview of the fundamentals of embedding theory for readers who are new to the subject. We outline a collection of existing methods for selecting embedding lag in both uniform and non-uniform delay embedding cases. Highlighting the poor dynamical explainability of existing methods of selecting non-uniform lags, we provide an alternative method of selecting embedding lags that includes a mixture of both dynamical and topological arguments. The proposed method, Significant Times on Persistent Strands (SToPS), uses persistent homology to construct a characteristic time spectrum that quantifies the relative dynamical significance of each time lag. We test our method on periodic, chaotic, and fast-slow time series and find that our method performs similar to existing automated non-uniform embedding methods. Additionally, n-step predictors trained on embeddings constructed with SToPS were found to outperform other embedding methods when predicting fast-slow time series.