Zhengwu Zhang

Gaussian Blurring-Invariant Comparison of Signals and Images

We present a Riemannian framework for analyzing signals and images in a manner that is invariant to their level of blurriness, under Gaussian blurring. Using a well known relation between Gaussian blurring and the heat equation, we establish an action of the blurring group on image space and define an orthogonal section of this action to represent and compare images at the same blur level. This comparison is based on geodesic distances on the section manifold which, in turn, are computed using a path-straightening algorithm. The actual implementations use coefficients of images under a truncated orthonormal basis and the blurring action corresponds to exponential decays of these coefficients. We demonstrate this framework using a number of experimental results, involving 1D signals and 2D images. As a specific application, we study the effect of blurring on the recognition performance when 2D facial images are used for recognizing people.

Blurring-invariant Riemannian metrics for comparing signals and images

We propose a novel Riemannian framework for comparing signals and images in a manner that is invariant to their levels of blur. This framework uses a log-Fourier representation of signals/images in which the set of all possible Gaussian blurs of a signal, i.e. its orbits under semigroup action of Gaussian blur functions, is a straight line. Using a set of Riemannian metrics under which the group actions are by isometries, the orbits are compared via distances between orbits. We demonstrate this framework using a number of experimental results involving 1D signals and 2D images.

Phase-Amplitude Separation and Modeling of Spherical Trajectories

ABSTRACT The problems of analysis and modeling of spherical trajectories, that is, continuous longitudinal data on , are important in several disciplines. These problems are challenging for two reasons: (1) nonlinear geometry of and (2) the presence of phase variability in given data. This article develops a geometric framework for separating phase variability from given trajectories, leaving only the shape or the amplitude variability. The key idea is to represent each trajectory with a pair of variables, a starting point, and a transported square-root velocity curve (TSRVC), a curve in the tangent (vector) space at the starting point. The space of all such curves forms a vector bundle and the norm, along with the standard Riemannian metric on , provides a natural, warping-invariant metric on this vector bundle. This leads to an efficient algorithm for registration of trajectories, that is, phase-amplitude separation, and computational tools, such as clustering, sample means, and principal component analysis (PCA) of the two components separately. It also helps derive simple statistical models of phase-amplitude components of spherical trajectories. This comprehensive framework is demonstrated using two datasets: a set of bird-migration trajectories and a set of hurricane paths in the Atlantic ocean. Supplementary material for this article is available online.

Discovering Change-Point Patterns in Dynamic Functional Brain Connectivity of a Population

This paper seeks to discover common change-point patterns, associated with functional connectivity (FC) in human brain, across multiple subjects. FC, represented as a covariance or a correlation matrix, relates to the similarity of fMRI responses across different brain regions, when a brain is simply resting or performing a task under an external stimulus. While the dynamical nature of FC is well accepted, this paper develops a formal statistical test for finding change-points in times series associated with FC observed over time. It represents instantaneous connectivity by a symmetric positive-definite matrix, and uses a Riemannian metric on this space to develop a graphical method for detecting change-points in a time series of such matrices. It also provides a graphical representation of estimated FC for stationary subintervals in between detected change-points. Furthermore, it uses a temporal alignment of the test statistic, viewed as a real-valued function over time, to remove temporal variability and to discover common change-point patterns across subjects, tasks, and regions. This method is illustrated using HCP database for multiple subjects and tasks.

Testing Stationarity of Brain Functional Connectivity Using Change-Point Detection in fMRI Data

This paper studies two questions: (1) Does the functional connectivity (FC) in a human brain remain stationary during performance of a task? (2) If it is non-stationary, how can one evaluate and estimate dynamic FC? The framework presented here relies on pre-segmented brain regions to represent instantaneous FC as symmetric, positive-definite matrices (SPDMs), with entries denoting covariances of fMRI signals across regions. The time series of such SPDMs is tested for change point detection using two important ideas: (1) a convenient Riemannian structure on the space of SPDMs for calculating geodesic distances and sample statistics, and (2) a graph-based approach, for testing similarity of distributions, that uses pairwise distances and a minimal spanning tree. This hypothesis test results in a temporal segmentation of observation interval into parts with stationary connectivity and an estimation of graph displaying FC during each such interval. We demonstrate these ideas using fMRI data from HCP database.

Elastic Shape Analysis of Functions, Curves and Trajectories

We present a Riemannian framework for geometric shape analysis of curves, functions, and trajectories on nonlinear manifolds. Since scalar functions and trajectories can also have important geometric features, we use shape as an all-encompassing term for the descriptors of curves, scalar functions and trajectories. Our framework relies on functional representation and analysis of curves and scalar functions, by square-root velocity fields (SRVF) under the Fisher–Rao metric, and of trajectories by transported square-root vector fields (TSRVF). SRVFs are general functional representations that jointly capture both the shape (geometry) and the reparameterization (sampling speed) of curves, whereas TSRVFs also capture temporal reparameterizations of time-indexed shapes. The space of SRVFs for shapes of curves becomes a subset of a spherical Riemannian manifold under certain special constraints. A fundamental tool in shape analysis is the construction and implementation of geodesic paths between shapes. This is used to accomplish a variety of tasks, including the definition of a metric to compare shapes, the computation of intrinsic statistics for a set of shapes, and the definition of probability models on shape spaces. We demonstrate our approach using several applications from computer vision and medical imaging including the analysis of (1) curves, (2) human growth, (3) bird migration patterns, and (4) human actions from video surveillance images and skeletons from depth images.

Bayesian Shape Clustering

Curve clustering is an important fundamental problem in biomedical applications involving clustering protein sequences or cell shapes in microscopy images. Existing model-based clustering techniques rely on simple probability models that are not generally valid for analyzing shapes of curves. In this chapter, we talk about an efficient Bayesian method to cluster curve data using a carefully chosen metric on the shape space. Rather than modeling the infinite-dimensional curves, we focus on modeling a summary statistic which is the inner product matrix obtained from the data. The inner-product matrix is modeled using a Wishart with parameters with carefully chosen hyperparameters which induce clustering and allow for automatic inference on the number of clusters. Posterior is sampled through an efficient Markov chain Monte Carlo procedure based on the Chinese restaurant process. This method is demonstrated on a variety of synthetic data and real data examples on protein structure analysis.

3D object search through semantic component

In this paper, we present a novel concept named semantic component for 3D object search which describes a key component that semantically defines a 3D object. In most cases, the semantic component is intra-category stable and therefore can be used to construct an efficient 3D object retrieval scheme. By segmenting an object into segments and learning the similar segments shared by all the objects in the same category, we can summarise what human uses for object recognition, from the analysis of which we develop a method to find the semantic component of an object. In our experiments, the proposed method is justified and the effectiveness of our algorithm is also demonstrated.

Relationships between Human Brain Structural Connectomes and Traits

Advanced brain imaging techniques make it possible to measure individuals’ structural connectomes in large cohort studies non-invasively. However, due to limitations in image resolution and pre-processing, questions remain about whether reconstructed connectomes are measured accurately enough to detect relationships with human traits and behaviors. Using a state-of-the-art structural connectome processing pipeline and a novel dimensionality reduction technique applied to data from the Human Connectome Project (HCP), we show strong relationships between connectome structure and various human traits. Our dimensionality reduction approach uses a tensor characterization of the connectomes and relies on a generalization of principal components analysis. We analyze over 1100 scans for 1076 subjects from the HCP and the Sherbrooke test-retest data set as well as 175 human traits that measure domains including cognition, substance use, motor, sensory and emotion. We find that brain connectomes are associated with many traits. Specifically, fluid intelligence, language comprehension, and motor skills are associated with increased cortical-cortical brain connectivity, while the use of alcohol, tobacco, and marijuana are associated with decreased cortical-cortical connectivity.

Rate-invariant analysis of covariance trajectories

Statistical analysis of dynamic systems, such as videos and dynamic functional connectivity, is often translated into a problem of analyzing trajectories of relevant features, particularly covariance matrices. As an example, in video-based action recognition, a natural mathematical representation of activity videos is as parameterized trajectories on the set of symmetric, positive-definite matrices (SPDMs). The execution rates of actions, implying arbitrary parameterizations of trajectories, complicate their analysis. To handle this challenge, we represent covariance trajectories using transported square-root vector fields, constructed by parallel translating scaled-velocity vectors of trajectories to their starting points. The space of such representations forms a vector bundle on the SPDM manifold. Using a natural Riemannian metric on this vector bundle, we approximate geodesic paths and geodesic distances between trajectories in the space of this vector bundle. This metric is invariant to the action of the re-parameterization group, and leads to a rate-invariant analysis of trajectories. In the process, we remove the parameterization variability and temporally register trajectories. We demonstrate this framework in multiple contexts, using both generative statistical models and discriminative data analysis. The latter is illustrated using several applications involving video-based action recognition and dynamic functional connectivity analysis.