SIAM CSE21 Minisymposium

Interaction of Topology and Geometry in Biomedical Image Analysis

An unprecedented amount of imaging data such as brain scans, cell membranes, and bone structures, are currently being generated for analysis, thus bringing exciting opportunities for new scientific discoveries in biology and medicine. Yet, valuable information in the sheer amount of complex biomedical imaging data maybe hidden in patterns that cannot be decoded easily with standard analytical tools. Geometric approaches have been very effective in quantifying and characterizing complex anatomical shape differences and changes in biomedical images, while various topological approaches such as level sets, graph cuts and fuzzy connectedness have been effective in image segmentation. It is, however, highly challenging to separate topology from geometry in practice. Recently, topological and geometric data analysis (TGDA) has emerged as an effective computational tool for decoding multiscale features in data that are simply inaccessible or require considerable modifications of standard methods. TGDA often employs geometric methods to quantify topological changes in imaging data, such as enforcing topological constraints to achieve consistent shape-preserving image deformation. This has set up TGDA as an effective companion for statistical and machine learning of heterogeneous biomedical images. The main aim of this minisymposium is to promote the awareness of TGDA in the mathematical sciences community and accelerate the development of this emerging area of research.

Organizer: Yuan Wang

University of South Carolina, U.S.

- Interaction of Topology and Geometry in Biomedical Image Analysis: An Introduction

Yuan Wang, University of South Carolina, U.S.

Abstract. This talk will provide an overview of the interaction of topology and geometry in biomedical image analysis, as well as some recent methodological advances in topological learning of neuroimaging data motivated by Morse Theory and their applications in post-stroke aphasia. The topological methods identify robust neuroimaging markers for speech and language impairment post-stroke aphasia despite the heterogeneity of stroke conditions associated with the disorder.

- Topological Loss for Deep Generative Models

Chao Chen, Stony Brook University, U.S.

Abstract. Existing generative models (GAN or VAE) focus on generating realistic images based on CNN-derived image features, but fail to preserve the structural properties of real images. This can be fatal in applications where the underlying structure (e.g., neurons, vessels, membranes, and road networks) of the image carries crucial semantic meaning. We propose a novel GAN model that learns the topology of real images, i.e., connectedness and loopy-ness. In particular, we introduce a topological GAN loss that bridges the gap between synthetic image distribution and real image distribution in the topological feature space. By optimizing this loss, the generator produces images with the same structural topology as real images. We also propose new GAN evaluation metrics that measure the topological realism of the synthetic images. We show in experiments that our method generates synthetic images with realistic topology. We also highlight the increased performance that our method brings to downstream tasks such as segmentation.

- Integrating Topologically Different Structural and Functional Brain Networks

Moo Chung, University of Wisconsin, Madison, U.S.

Abstract. Structural connectivity between brain regions is often measured by the number of white matter fiber tracts connecting them. Such fiber tracts are obtained through diffusion tensor imaging (DTI), which is a noninvasive imaging modality that can characterize the microstructure of biological tissues by using magnitude, anisotropy, and anisotropic orientation associated with the diffusion of water molecules in the fibers. The structural brain networks are mainly characterized by 0-dimensional topological features such as the 0-th Betti number. On the other hand, functional activity in brain regions can be measured by functional magnetic resonance imaging (fMRI) that measures blood-oxygen-level dependent (BOLD) signals in the brain. Functional connectivity between brain regions is then measured as correlation between BOLD signals. The functional brain networks are mainly characterized by 1-dimenstional topological features such as the 1-th Betti number. Thus, at the structural level, the brain networks are tree-like while at the functional level, brain networks have a lot of cycles. So far it is unclear how to integrate and correlate networks of different topology of the same subject. In this talk, we present a new integrative framework that enables us to relate topologically varying networks of different sizes and shapes together. In doing so, we are able to integrate both structural and functional brain networks tougher in a coherent statistical modeling framework.

- Recent Advances in Geometric Analysis of Topologically-Varying Shapes

Anuj Srivastava, Florida State University, U.S.

Abstract. Structures of brain arterial networks (BANs) -- that are complex arrangements of individual arteries, their branching patterns, and inter-connectivities -- play an important role in characterizing and understanding brain physiology. One would like tools for statistically analyzing the shapes of BANs, i.e. quantify shape differences, compare population of subjects, and study the effects of covariates on these shapes. This paper mathematically represents and statistically analyzes BAN shapes as elastic shape graphs. Each elastic shape graph is made up of nodes that are connected by a number of 3D curves, and edges, with arbitrary shapes. We develop a mathematical representation, a Riemannian metric and other geometrical tools, such as computations of geodesics, means and covariances, and PCA for analyzing elastic graphs and BANs. This analysis is applied to BANs after separating them into four components -- top, bottom, left, and right. This framework is then used to generate shape summaries of BANs from 92 subjects, and to study the effects of age and gender on shapes of BAN components. We conclude that while gender effects require further investigation, the age has a clear, quantifiable effect on BAN shapes. Specifically, we find an increased variance in BAN shapes as age increases.

- Fiber Bundles in Probabilistic Models

Sayan Mukherjee and Henry Kirveslahti, Duke University, U.S.

Abstract. We will present how fiber bundles can be used in probabilistic modeling. Applications in geometric morphometrics will be used as examples, this means we will model shapes and surfaces for evolutionary biology and biomedical applications. We will consider three problems: (1) regression using shapes as covariates, (2) sub-shape or variable selection, and (3) Gaussian process models indexed via fiber bundles. For the first two problems we will use ideas based in integral geometry and for the third we will extend approaches based on diffusions on fiber bundles to a hierarchical Bayesian Gaussian process model.

UofSC BDHSC Big Data Analytics Workshop

Statistical and Machine Learning in Challenging Neuroimaging Problems

The increasing volume and variety of neuroimaging datasets demand advanced modeling techniques to answer scientifically relevant questions such as causality and treatment effect. This workshop features novel statistical and machine learning approaches for analyzing high-dimensional brain signals, tree-shaped sulcal and gyral structures, and neurons.


Using Brain Imaging for Computer Aided Prognosis of Stroke

Chris Rorden

University of South Carolina

Abstract: Stroke is the leading cause of major disability in developed countries. Speech and language impairment (aphasia) is a common consequence of stroke. South Carolina has consistently had one of the highest rates of stroke in the world. Stroke inicidence is increasing dramatically for those less than 65 years old. Medical advances have increased stroke survival rates. These two factors are leading to a dramatic increase in stroke survivors. The clinicians I partner with have developed innovative treatments that help some, but not all patients. We believe that brain imaging can be combined with other measures (acute impairment, age, genetics) to predict outcome. I outline three ways that this work can change the standard of care. First, we use brain imaging to ensure brain stimulation targets the residual eloquent cortex. Second, we be modeling expected outcome we can match individuals based on prognosis, minimizing variability in clinical trials. Third, we can identify which patients are likely to benefit from a particular form of treatment. Like all forms of deep learning, we need to develop large training datasets. Our Center for the Study of Aphasia Recovery ( is not only acquiring this data, but also preparing to disseminate curated data to allow other teams to tackle this critical issue.

Machine Learning Applied to Neuroimaging Research in Neurology

Leonardo Bonilha

Medical University of South Carolina

Abstract: In this presentation, we will discuss how structural and functional neuroimaging can be integrated with multiple forms of machine learning to better understand the relationship between brain anatomy and function with neurological disorders. Focusing on stroke related language disorders or epilepsy, we will present some recent approaches to use machine learning as well as deep learning to investigate pathophysiology and predict outcomes. We will also approach translational uses of these methodologies.

Modeling Spectral Causality in a Brain Network

Hernando Ombao

King Abdullah University of Science and Technology

Abstract: The first part of the talk will be background information on the various spectral measures of dependence in a network of brain signals. There is no one measure that can completely characterize dependence between signals in a network. Here, our focus will be on characterizing dependence through their various oscillatory components. We give an overview of coherence, partial coherence, dual-frequency coherence and the time-evolving dual-frequency coherence. One limitation of these measures is that they do not give information about causality, that is, it they do not quantify the amount of variation in a channel that is explained by past activity of another channel. One way to approach this is by fitting a vector autoregressive (VAR) model. Under the VAR model, one can derive frequency-based directed measures such as partial directed coherence. One limitation of examining directionality through the classical VAR model is that it constrains the lead-lag dependence across different frequency bands to be the same. In this talk we will demonstrate why this can be problematic and then propose a spectral vector autoregressive model that allows us to examine directed dependence between oscillatory activity at different frequency bands. The model will be illustrated on electroencephalograms and the analysis will highlight some of the interesting lead-lag dependence between oscillations. This is joint work with Chee-Ming Ting (KAUST) and Marco Pinto (City University of Oslo).

Heat Kernel Smoothing on Riemannian manifolds and Its Application to Brain Images

Moo K. Chung

University of Wisconsin-Madison

Abstract: We will introduce the basic concept and properties of heat kernel smoothing, a Hilbert space framework for solving the diffusion equation on Riemannian manifolds. We illustrate the method with two recent applications. In the first application, we show how to compute correlations dynamically without sliding windows in resting-state fMRI (arXiv:1911.02731). Due to the discrete nature of the sliding windows, spurious high-frequency fluctuations are often observed in the estimated dynamic correlations. The problem can be easily remedied if we avoid using the sliding windows. We address the problem using heat kernel smoothing. The method is shown to improve the state space estimation. In the second application, we show how to analyze sulcal and gyral curves of the brain (arXiv:1911.02721). The sulcal and gyral curves are tree shaped patterns obtained from MRI. The sulcal and gyral curves do not align well across subjects even after image registration. Building coherent statistical models on such complex tree shapes at the vertex level has been a challenge. We address the problem by introducing the algebraic representation for trees via heat kernel smoothing.

Single-Trial Identification in fMRI Data: Applications to Representation of Affective States

Svetlana Shinkareva

University of South Carolina

Multivariate pattern analyses methods are sensitive to subtle changes of neural activity under different conditions being investigated and can be applied at individual or group levels of analysis. This talk will overview the use of multivariate pattern classification and representational similarity analyses methods in examining the representation and identification of cognitive states. Specific examples will be drawn from examining the representation of affective states.

Learning with Topological Priors and Constraints

Chao Chen

Stony Brook University

Abstract: In various contexts, it is challenging to incorporate global topological prior or constraints into an end-to-end training system. In this talk, we explain how topological information, e.g., the number of connected components and handles, can indeed be formulated as a differentiable penalty function through the theory of persistent homology. We show how the topological information can be effectively leveraged in different contexts. In biomedical image analysis, it helps training a topology-aware image segmentation network to segment fine-scale structures such as neurons with correct topology. In machine learning, topological information helps improving the generalization power and robustness of classifiers.

Topological Signal Processing in Neuroimaging Studies

Yuan Wang

University of South Carolina

Abstract: Topological data analysis (TDA) can decode multi-scale patterns in electroencephalographic (EEG) signals not captured by standard temporal and spectral features. A challenge for applying TDA to groups of long EEG recordings is the ambiguity of performing computationally efficient statistical inference. To address this problem, we advance a new formulation of topological signal processing and apply a unified permutation-based inference framework to test statistical difference in the topological feature persistence landscape of multi-trial EEG signals. The present study applies the topological inference framework to investigate the EEG correlates of speech sensorimotor impairment in post-stroke aphasia patients under a speech altered auditory feedback paradigm.