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Graph-based representations are becoming increasingly popular for representing and analyzing video data, especially in object tracking and scene understanding applications. Accordingly, an essential tool in this approach is to generate statistical inferences for graphical time series associated with videos. This paper develops a Kalman-smoothing method for estimating graphs from noisy, cluttered, and incomplete data. The main challenge here is to find and preserve the registration of nodes (salient detected objects) across time frames when the data has noise and clutter due to false and missing nodes. First, we introduce a quotient-space representation of graphs that incorporates temporal registration of nodes, and we use that metric structure to impose a dynamical model on graph evolution. Then, we derive a Kalman smoother, adapted to the quotient space geometry, to estimate dense, smooth trajectories of graphs. We demonstrate this framework using simulated data and actual video graphs extracted from the Multiview Extended Video with Activities (MEVA) dataset. This framework successfully estimates graphs despite the noise, clutter, and missed detections. 

This paper addresses the problem of characterizing statistical distributions of cellular shape populations using shape samples from microscopy image data. This problem is challenging because of the nonlinearity and high-dimensionality of shape manifolds. The paper develops an efficient, nonparametric approach using ideas from k-modal mixtures and kernel estimators. It uses elastic shape analysis of cell boundaries to estimate statistical modes and clusters given shapes around those modes. (Notably, it uses a combination of modal distributions and ANOVA to determine k automatically.) A population is then characterized as k-modal mixture relative to this estimated clustering and a chosen kernel (e.g., a Gaussian or a flat kernel). One can compare and analyze populations using the Fisher-Rao metric between their estimated distributions. We demonstrate this approach for classifying shapes associated with migrations of entamoeba histolytica under different experimental conditions. This framework remarkably captures salient shape patterns and separates shape data for different experimental settings, even when it is difficult to discern class differences visually. 

We consider the problem of characterizing shape populations using highly frequent representative shapes. Framing such shapes as statistical modes – shapes that correspond to (significant) local maxima of the underlying pdfs – we develop a frequency-based, nonparametric approach for estimating sample modes. Using an elastic shape metric, we define ϵ-neighborhoods in the shape space and shortlist shapes that are central and have the most neighbors. A critical issue – How to automatically select the threshold ϵ? – is resolved using a combination of ANOVA and empirical mode distribution. The resulting modal set, in turn, helps characterize the shape population and performs better than the traditional cluster means. We demonstrate this framework using amoeba shapes from brightfield microscopy images and highlight its advantages over existing ideas. 

Elastic Riemannian metrics have been used successfully for statistical treatments of functional and curve shape data. However, this usage suffers from a significant restriction: the function boundaries are assumed to be fixed and matched. Functional data often comes with unmatched boundaries, {\it e.g.}, in dynamical systems with variable evolution rates, such as COVID-19 infection rate curves associated with different geographical regions. Here, we develop a Riemannian framework that allows for partial matching, comparing, and clustering functions under phase variability {\it and} uncertain boundaries. We extend past work by (1) Defining a new diffeomorphism group G over the positive reals that is the semidirect product of a time-warping group and a time-scaling group; (2) Introducing a metric that is invariant to the action of G; (3) Imposing a Riemannian Lie group structure on G to allow for an efficient gradient-based optimization for elastic partial matching; and (4) Presenting a modification that, while losing the metric property, allows one to control the amount of boundary disparity in the registration. We illustrate this framework by registering and clustering shapes of COVID-19 rate curves, identifying basic patterns, minimizing mismatch errors, and reducing variability within clusters compared to previous methods. 

Registering functions (curves) using time warpings (re-parameterizations) is central to many computer vision and shape analysis solutions. While traditional registration methods minimize penalized-L2 norm, the elastic Riemannian metric and square-root velocity functions (SRVFs) have resulted in significant improvements in terms of theory and practical performance. This solution uses the dynamic programming algorithm to minimize the L2 norm between SRVFs of given functions. However, the computational cost of this elastic dynamic programming framework – O(nT 2 k) – where T is the number of time samples along curves, n is the number of curves, and k < T is a parameter – limits its use in applications involving big data. This paper introduces a deep-learning approach, named SRVF Registration Net or SrvfRegNet to overcome these limitations. SrvfRegNet architecture trains by optimizing the elastic metric-based objective function on the training data and then applies this trained network to the test data to perform fast registration. In case the training and the test data are from different classes, it generalizes to the test data using transfer learning, i.e., retraining of only the last few layers of the network. It achieves the state-of-the-art alignment performance albeit at much reduced computational cost. We demonstrate the efficiency and efficacy of this framework using several standard curve datasets. 

Statistical analysis of shape evolution during cell migration is important for gaining insights into biological processes. This paper develops a time-series model for temporal evolution of cellular shapes during cell motility. It uses elastic shape analysis to represent and analyze shapes of cell boundaries (as planar closed curves), thus separating cell shape changes from cell kinematics. Specifically, it utilizes Transported Square-Root Velocity Field (TSRVF), to map non-Euclidean shape sequences into a Euclidean time series. It then uses PCA to reduce Euclidean dimensions and imposes a Vector Auto-Regression (VAR) model on the resulting low-dimensional time series. Finally, it presents some results from VAR-based statistical analysis: estimation of model parameters and diagnostics, synthesis of new shape sequences, and predictions of future shapes given past shapes.