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In this chapter, we discuss applications of topological data analysis (TDA) to spatial systems. We briefly review the recently proposed level-set construction of filtered simplicial complexes, and we then examine persistent homology in two…
A scalable and robust 3D tissue transcriptomics profile can enable a holistic understanding of tissue organization and provide deeper insights into human biology and disease. Most predictive algorithms that infer ST directly from histology…
Spatial transcriptomic (ST) clustering employs spatial and transcription information to group spots spatially coherent and transcriptionally similar together into the same spatial domain. Graph convolution network (GCN) and graph attention…
Many datasets can be viewed as a noisy sampling of an underlying space, and tools from topological data analysis can characterize this structure for the purpose of knowledge discovery. One such tool is persistent homology, which provides a…
Topological Data Analysis (TDA) is an emergent field that aims to discover topological information hidden in a dataset. TDA tools have been commonly used to create filters and topological descriptors to improve Machine Learning (ML)…
Spatiotemporal (ST) image data are increasingly common and often high-dimensional (high-D). Modeling ST data can be a challenge due to the plethora of independent and interacting processes which may or may not contribute to the…
A fundamental tool in topological data analysis is persistent homology, which allows extraction of information from complex datasets in a robust way. Persistent homology assigns a module over a principal ideal domain to a one-parameter…
The rapid development of spatial transcriptomics(ST) enables the measurement of gene expression at spatial resolution, making it possible to simultaneously profile the gene expression, spatial locations of spots, and the matched…
Spatial transcriptomics provides an unprecedented perspective for deciphering tissue spatial heterogeneity. However, high-resolution spatial transcriptomic technology remains constrained by limited gene coverage, technical complexity, and…
Genome-wide association studies (GWAS) involving increasing sample sizes have identified hundreds of genetic variants associated with complex diseases, such as type 2 diabetes (T2D); however, it is unclear how GWAS hits form unique…
Persistent topological Laplacians constitute a new class of tools in topological data analysis (TDA). They are motivated by the necessity to address challenges encountered in persistent homology when handling complex data. These Laplacians…
We introduce PH-STAT, a comprehensive MATLAB toolbox designed for performing a wide range of statistical inferences and machine learning tasks on persistent homology, primarily for network and graph data, with an emphasis on brain network…
Recent advancements in Spatial Transcriptomics (ST) technology have facilitated detailed gene expression analysis within tissue contexts. However, the high costs and methodological limitations of ST necessitate a more robust predictive…
Persistence diagrams (PDs), often characterized as sets of death and birth of homology class, have been known for providing a topological representation of a graph structure, which is often useful in machine learning tasks. Prior works rely…
Developing reliable methods to discriminate different transient brain states that change over time is a key neuroscientific challenge in brain imaging studies. Topological data analysis (TDA), a novel framework based on algebraic topology,…
Topological data analysis (TDA) is a branch of computational mathematics, bridging algebraic topology and data science, that provides compact, noise-robust representations of complex structures. Deep neural networks (DNNs) learn millions of…
Topological correctness is critical for segmentation of tubular structures, which pervade in biomedical images. Existing topological segmentation loss functions are primarily based on the persistent homology of the image. They match the…
Spatial-temporal forecasting and imputation are important for real-world intelligent systems. Most existing methods are tailored for individual forecasting or imputation tasks but are not designed for both. Additionally, they are less…
This paper describes a localized algorithm for the topological simplification of scalar data, an essential pre-processing step of topological data analysis (TDA). Given a scalar field f and a selection of extrema to preserve, the proposed…
The inner representation of deep neural networks (DNNs) is indecipherable, which makes it difficult to tune DNN models, control their training process, and interpret their outputs. In this paper, we propose a novel approach to investigate…