Shape Analysis of 3D Objects, S.Kurtek, E.Klassen, A.Srivastava
Capturing Growth Deformation Using GRID Model, Q.Xie, A.Srivastava
Structure-based RNA Function Prediction using Elastic Shape Analysis, J.Laborde, J.Zhang, A.Srivastava
Blurring-Invariant Riemannian Metrics for Comparing Signals and Images, Z.Zhang, E.Klassen, A.Srivastava
Affine and Projective Shape Analysis, D.Bryner, E.Klassen, A.Srivastava
Shape Analysis of Annotated Curves, W.Liu, J.Zheng, A.Srivastava
Riemannian Metrics for Analyzing Orientations and Shapes in HARDI Data, S.Ncube, Q.Xie, A.Srivastava
Partial Shape Matching, D.Robinson, E.Klassen
Alignment, Modeling, and Classification of Elastic Functional Data, J.D.Tucker, W.Wu, A.Srivastava
Registration and Alignment of Functional Data , A. Srivastava, W. Wu, S. Kurtek, E. Klassen
Statistical Inferences in the Spike Train Space, W.Wu, A.Srivastava
Detection and Estimation of Shapes from Point Clouds in Euclidean spaces , J.Su, Z.Zhu, F.Huffer, A.Srivastava
Fitting Optimal Curves to Time-Indexed, Noisy Observations on Nonlinear Manifolds ,J.Su, I.L.Dryden, E.Klassen, H.Le and A.Srivastava
A Novel Framework for Joint Registration, Comparison and Averaging of Paths on Nonlinear Manifolds , J.Su, S.Kurtek, E.Klassen, A.Srivastava

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Shape Analysis of 3D Objects
S.Kurtek, E.Klassen, A.Srivastava

The goal of this work is to develop techniques that can be used for comprehensive shape analysis of 3D objects (as continuous, parameterized surfaces) under a Riemannian framework. This requires a Riemannian metric that allows: (1) re-parameterizations of surfaces by isometries, and (2) efficient computations of geodesic paths between surfaces. These tools allow for computing Karcher means and covariances (using tangent PCA) for shape classes, and a probabilistic classification of surfaces. The process of computing geodesics requires optimal re- parameterizations (deformations of grids) of surfaces. This results in a superior alignment of geometric features. The resulting means and covariances are better representatives of the original data and lead to parsimonious shape models.
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The mathematical and statistical modeling of biological growth over time is an important problem with a variety of applications ranging from medical diagnostics to evolutionary biology. Here we use 2D and/or 3D images taken across time, species, or specimens to compare or to extract salient differences in anatomical structures, and to analyze and model their variations both within and across biological classes. There is a large body of work on representing differences in imaged objects using deformations of background space. Motivated by Grenander's Growth as Random Iterated Diffeomorphisms (GRID) model, our goal is to solve its inverse problem where we want to decompose large biological growth into smaller biologically-interpretable units.
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When one considers the atomic coordinates of RNA structures utilizing their backbone we can view them as 3D open curves. We are able to perform RNA structure alignment based on the geometry of the backbones and we further attempt structure-based function prediction using the SCOR functional classification as reference. We use Elastic Shape Analysis of these backbone RNA structures to functionally classify them. This task is accomplished by means of computing distance matrices and performing leave-one-out classification and computing the accuracy to compare our method with some benchmark method. We show that Elastic Shape Analysis have significant classification rate.
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Blurring is omnipresent in image data. Standard metrics for image comparisons provide results that are affected by the amount of blurring present in images. A common solution to deal with this problem is to deblur the images or extract features that invariant to the blurring, and then compare them. 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- Euclidean representation of signals/images in which the set of all possible blurrings 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 using distances between orbits.
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Affine and Projective Shape Analysis
D.Bryner, E.Klassen, A.Srivastava

The most commonly used type of shape analysis is designed to compare shapes under the invariants of scale, translation, rotation, and in certain methods re-parameterization. These methods are not invariant to affine or projective transformations, which apply certain types of perspective distortion to the shape, skewing it in a manner that differs greatly from rigid motion and global scaling. Recent research efforts have focused on developing a framework to compare discrete point sets or pre-determined landmarks in 2-dimensional affine or projective shape spaces, but we treat shapes as fully elastic, continuous curves, allowing for a more accurate and robust shape matching procedure. The goal of this research is to properly define affine shape space and projective shape space in the context of continuous curves and formulate an elastic distance measure on each space. From these distance measures, we can then build geodesics and shape statistics in these spaces.
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Shape Analysis of Annotated Curves
W.Liu, J.Zheng, A.Srivastava

In object analysis of digital world , besides the geometric shape properties, there always exist some additional attributes along the interested curve. These attributes also represent some important properties of objects, and can be labeled along the curve to form so-called annotated curve. Our goal is to develop a framework for analyzing the annotated curve with combining both geometry and extra labeled information jointly to improve the result of shape only analysis.
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Interpolation between PDFs in the joint shape-orientation space.

In our work, we propose a novel Riemannian framework for analyzing orientation distribution functions (ODFs) in HARDI data sets, for use in comparing, interpolating, averaging, and denoising ODFs. A recently used Fisher-Rao metric does not provide physically feasible solutions, and we suggest a modification that removes orientations from ODFs and treats them as separate variables. This way a comparison of any two ODFs is based on separate comparisons of their shapes and orientations. Furthermore, this provides an explicit orientation at each voxel for use in tractography. We demonstrate these ideas by computing geodesics between ODFs and Karcher means of ODFs, for both the original Fisher-Rao and the proposed framework.
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Partial Shape Matching
D.Robinson, E.Klassen
Partial shape matching on a key.

Shapes that are very different when considered in their entirety can nonetheless share significant features. This situation arises in practice when parts of an observed shape represent noise, occluding objects, or uninteresting features rather than the features of interest. Our goal is to develop efficient partial matching algorithms which will provide meaningful shape comparisons in these cases.
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The task of clustering and modeling underwater objects using acoustic spectrum is complicated by the uncertainties in aspect angles at different data collections. Small changes in the aspect angles introduce compositional noise in the signals. The traditional alignment techniques are based on energy functions that are not proper distances, and necessitate a separate (and thus suboptimal) choice of distance to compare the aligned functions. We present a comprehensive technique, for removing compositional noise and aligning functions
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Registration and Alignment of Functional Data
A. Srivastava, W. Wu, S. Kurtek, E. Klassen

We introduce a novel geometric framework for separating the phase and the amplitude variability in functional data of the type frequently studied in growth curve analysis. This framework uses the Fisher-Rao Riemannian metric to derive a proper distance on the quotient space of functions modulo the time-warping group. A convenient square-root velocity function (SRVF) representation transforms the Fisher-Rao metric into the standard L2 metric, simplifying the computations. This distance is then used to define a Karcher mean template and warp the individual functions to align them with the Karcher mean template. The strength of this framework is demonstrated by deriving a consistent estimator of a signal observed under random warping, scaling, and vertical translation. The new method is empirically shown to be superior in performance to several recently published methods for registration and alignment of functional data.
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The research goal of this project is to develop a framework for a novel, data-driven statistical inference in the space of neural spike trains. Statistical inference methods have played an essential role in the analysis of neural spike trains. However, current approaches are mainly based on probabilistic representations at each time and therefore cannot address the statistical nature in the space of spike trains directly. Motivated by the abundance of spike train data from experimental recordings, we propose a data-driven framework to address this important issue by treating each spike train as one point in an infinite dimensional function space. The new framework will be based on novel metric (distance) systems on spike trains. We will construct new tools for: 1) quantifying differences in spike trains, 2) computing summary statistics such as means and covariance of spike trains, and 3) performing statistical inferences in the spike train space.
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A variety of practical algorithms can be used to extract a set of feature points associated with objects containing in image data and an important task is to detect shape classes of interest in such point cloud data. This problem is challenging because the data is typically noisy, cluttered, and unordered. In addition, once a shape is detected in a point cloud, we want to reconstruct the shape from the cloud.
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Fitting Optimal Curves to Time-Indexed, Noisy Observations on Nonlinear Manifolds
J. Su, I.L. Dryden, E. Klassen, H. Le and A. Srivastava

This project studies the interaction of two disparate but inherent aspects of computer vision systems: the temporal evolution of a dynamic scene and the nonlinear representations of features of interest. Specifically, we are interested in dynamical systems where a process of interest evolves over time on a nonlinear manifold and one observes this process only at limited times. The goal is to estimate/predict the remaining process using the observed values under some predetermined criterion. The motivation of such a problem comes from many applications. Consider the evolution of the shape of a human silhouette in a video, in a situation where one has an unobstructed view of the person in only a few frames. Given these observed shapes, along with their observation times, one would like to estimate shapes at some intermediate times and perhaps even predict future shape evolution.
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We address the problem of registration, comparison and averaging of temporal paths on nonlinear Riemannian manifolds such as spheres and shape manifolds. Past methods for registration of trajectories on manifolds, such as those used in temporal alignment of human activity data, used quantities that are not proper distances (they are not even symmetric). Without a proper distance, it is difficult to define average paths (or templates) or setup a classification solution. An important property needed here is that the chosen quantity should be a distance and it should be invariant to identical time-wrappings (or re-parameterizations) of the paths. [ more ]

Figure: Given two paths (in red and blue), we obtain an average without temporal alignment (left, in black) and with temporal alignment (right, in black).