Statistical Shape Analysis & Modeling Group

Publications

Journals

2014

2D Affine and Projective Shape

IEEE Transactions on Pattern Analysis and Machine Intelligence
36
(5)
,
May
2014

2013

Computing Equilibrium Wealth Distributions in Models with Heterogeneous-Agents, Incomplete Markets, and Idiosyncratic Risk

Journal of Computational Economics
41
(2)
,
January
2013

Estimating Summary Statistics in Spike Train Space

Journal
34
(3)
,
June
2013

Segmentation and Statistical Analysis of Biosignals with Application to Disease Classification

Journal of Applied Statistics
49
(40)
,
June
2013

RNA Alignment in the Joint Sequence-Structure Space Using Elastic Shape Analysis

Journal of Nucleic Acids Research
41
(11)
,
June
2013

2012

Elastic Geodesic Paths in Shape Space of Parameterized Surfaces

IEEE Transactions on Pattern Analysis and Machine Intelligence
2012

[PDF]

Generative Models for Functional Data Using Phase and Amplitude Separation

Computational Statistics and Data Analysis
2012

Constructing generative models for functional observations is an important task in statistical functional analysis. In general, functional data contains both phase (or x or horizontal) and amplitude (or y or vertical) variability. Traditional methods often ignore the phase variability and focus solely on the amplitude variation, using cross-sectional techniques such as fPCA for dimensional reduction and data modeling. Ignoring phase variability leads to a loss of structure in the data and ineciency in data models. This paper presents an approach that relies on separating the phase (x-axis) and amplitude (y-axis), then modeling these components using joint distributions. This separation, in turn, is performed using a technique called elastic shape analysis of curves that involves a new mathematical representation of functional data. Then, using individual fPCAs, one each for phase and amplitude components, while respecting the nonlinear geometry of the phase representation space; impose joint probability models on principal coecients of these components. These ideas are demonstrated using random sampling, for models estimated from simulated and real datasets, and show their superiority over models that ignore phase-amplitude separation. Furthermore, the generative models are applied to classification of functional data and achieve high performance in applications involving SONAR signals of underwater objects, handwritten signatures, and periodic body movements recorded by smart phones.

On Advances in Geometric Approaches for 2D and 3D Shape Analysis and Activity Recognition

Journal of Image and Vision Computing
30
(6)
,
June
2012

In this paper we summarize recent advances in shape analysis and shape-based activity recognition problems with a focus on techniques that use tools from differential geometry and statistics. We start with general goals and challenges faced in shape analysis, followed by a summary of the basic ideas, strengths and limitations, and applications of different mathematical representations used in shape analyses of 2D and 3D objects. These representations include point sets, curves, surfaces, level sets, deformable templates, medial representations, and other feature-basedmethods.We discuss some common choices of Riemannianmetrics and computational tools used for evaluating geodesic paths and geodesic distances for several of these shape representations. Then, we study the use of Riemannian frameworks in statistical modeling of variability within shape classes. Next, we turn to models and algorithms for activity analysis from various perspectives. We discuss how mathematical representations for human shape and its temporal evolutions in videos lead to analyses over certain special manifolds.We discuss the various choices of shape features, and parametric and non-parametricmodels for shape evolution, and how these choices lead to appropriate manifold-valued constraints. We discuss applications of these methods in gait-based biometrics, action recognition, and video summarization and indexing. For reader convenience, we also provide a short overview of the relevant tools from geometry and statistics on manifolds in the Appendix.

Statistical Modeling of Curves Using Shapes and Related Features

Journal of American Statistical Association
107
(499)
,
October
2012

2011

A Gradient-Descent Method for Curve Fitting on Riemannian Manifolds

Foundations of Computational Mathematics
2011

Given data points p0, . . . , pN on a closed submanifold M of Rn and time instants 0 = t0 < t1 < · · · < tN = 1, we consider the problem of finding a curve γ on M that best approximates the data points at the given instants while being as “regular” as possible. Specifically, γ is expressed as the curve that minimizes the weighted sum of a sum-of-squares term penalizing the lack of fitting to the data points and a regularity term defined, in the first case as the mean squared velocity of the curve, and in the second case as the mean squared acceleration of the curve. In both cases, the optimization task is carried out by means of a steepest-descent algorithm on a set of curves on M. The steepest-descent direction, defined in the sense of the first-order and second-order Palais metric, respectively, is shown to admit analytical expressions involving parallel transport and covariant integral along curves. Illustrations are given in Rn and on the unit sphere.

Shape Analysis of Elastic Curves in Euclidean Spaces

IEEE Transactions on Pattern Analysis and Machine Intelligence
July
2011
, accepted for publication

This paper introduces a square-root velocity (SRV) representation for analyzing shapes of curves in Euclidean spaces under an elastic metric. Due to this SRV representation the elastic metric simplifies to the $\ltwo$ metric, the re-parameterization group acts by isometries, and the space of unit length curves becomes the unit sphere. The shape space of closed curves is quotient space of (a submanifold of) the unit sphere, modulo rotation and re-parameterization groups, and we find geodesics in that space using a path-straightening approach. These geodesics and geodesic distances provide a framework for optimally matching, deforming and comparing shapes. These ideas are demonstrated using: (i) Shape analysis of cylindrical helices for studying protein backbones, (ii) Shape analysis of facial curves for recognizing faces, (iii) A wrapped probability distribution for capturing shapes of planar closed curves, and (iv) Parallel transport of deformations for predicting shapes from novel poses.

Gesture and Action Recognition via Modeling Trajectories on Shape Manifolds

Computer Vision and Image Understanding Journal
115
(3)
,
March
2011

This paper addresses the problem of recognizing human gestures from videos using models that are built from the Riemannian geometry of shape spaces. We represent a human gesture as a temporal sequence of human poses, each characterized by a contour of the associated human silhouette. The shape of a contour is viewed as a point on the shape space of closed curves and, hence, each gesture is characterized and modeled as a trajectory on this shape space. We propose two approaches for modeling these trajectories. In the first template-based approach, we use dynamic time warping (DTW) to align the different trajectories using elastic geodesic distances on the shape space. The gesture templates are then calculated by averaging the aligned trajectories. In the second approach, we use a graphical model approach similar to an exemplar-based hidden Markov model, where we cluster the gesture shapes on the shape space, and build non-parametric statistical models to capture the variations within each cluster. We model each gesture as a Markov model of transitions between these clusters. To evaluate the proposed approaches, an extensive set of experiments was performed using two different data sets representing gesture and action recognition applications. The proposed approaches not only are successfully able to represent the shape and dynamics of the different classes for recognition, but are also robust against some errors resulting from segmentation and background subtraction.

Towards Statistical Summaries of Spike Train Data

Journal of Neuroscience Methods
195
(1)
,
January
2011

Statistical inference has an important role in analysis of neural spike trains. While current approaches are mostly model-based, and designed for capturing the temporal evolution of the underlying stochastic processes, we focus on a data-driven approach where statistics are defined and computed in function spaces where individual spike trains are viewed as points. The first contribution of this paper is to endow spike train space with a parameterized family of metrics that takes into account different time warpings and generalizes several currently used metrics. These metrics are essentially penalized Lp norms, involving appropriate functions of spike trains, with penalties associated with time-warpings. The second contribution of this paper is to derive a notion of a mean spike train in the case when p = 2. We present an efficient recursive algorithm, termed Matching-Minimization algorithm, to compute the sample mean of a set of spike trains. The proposed metrics as well as the mean computations are demonstrated using an experimental recording from the motor cortex.

An Information-Geometric Framework for Statistical Inferences in the Neural Spike Train Space

Journal of Computational Neuroscience
May
2011

[Abstract]

Statistical inferences are essentially important in analyzing neural spike trains in computational neuroscience. Current approaches have followed a general in- ference paradigm where a parametric probability model is often used to characterize the temporal evolution of the underlying stochastic processes. To directly capture the overall variability and distribution in the space of the spike trains, we focus on a data- driven approach where statistics are defined and computed in the function space in which spike trains are viewed as individual points. To this end, we at first develop a parametrized family of metrics that takes into account different warpings in the time domain and generalizes several currently used spike train distances. These new met- rics are essentially penalized Lp norms, involving appropriate functions of spike trains, with penalties associated with time-warping. The notions of means and variances of spike trains are then defined based on the new metrics when p = 2 (corresponding to the “Euclidean distance”). Using some restrictive conditions, we present an efficient recursive algorithm, termed Matching-Minimization algorithm, to compute the sample mean of a set of spike trains with arbitrary numbers of spikes. The proposed metrics as well as the mean spike trains are demonstrated using simulations as well as an ex- perimental recording from the motor cortex. It is found that all these methods achieve desirable performance and the results support the success of this novel framework.

A Mathematical Framework for Protein Structure Comparison

PloS Computational Biology
7
(2)
,
February
2011
, doi:10.1371/journal.pcbi.1001075

Comparison of protein structures is important for revealing the evolutionary relationship among proteins, predicting protein functions and predicting protein structures. Many methods have been developed in the past to align two or multiple protein structures. Despite the importance of this problem, rigorous mathematical or statistical frameworks have seldom been pursued for general protein structure comparison. One notable issue in this field is that with many different distances used to measure the similarity between protein structures, none of them are proper distances when protein structures of different sequences are compared. Statistical approaches based on those non-proper distances or similarity scores as random variables are thus not mathematically rigorous. In this work, we develop a mathematical framework for protein structure comparison by treating protein structures as three-dimensional curves. Using an elastic Riemannian metric on spaces of curves, geodesic distance, a proper distance on spaces of curves, can be computed for any two protein structures. In this framework, protein structures can be treated as random variables on the shape manifold, and means and covariance can be computed for populations of protein structures. Furthermore, these moments can be used to build Gaussian-type probability distributions of protein structures for use in hypothesis testing. The covariance of a population of protein structures can reveal the population-specific variations and be helpful in improving structure classification. With curves representing protein structures, the matching is performed using elastic shape analysis of curves, which can effectively model conformational changes and insertions/deletions. We show that our method performs comparably with commonly used methods in protein structure classification on a large manually annotated data set.

Parameterization-Invariant Shape Comparisons of Anatomical Surfaces

IEEE Transactions on Medical Imaging
30
(3)
,
March
2011

We consider 3D brain structures as continuous parameterized surfaces and present a metric for their comparisons that is invariant to the way they are parameterized. Past comparisons of such surfaces involve either volume deformations or non-rigid matching under fixed parameterizations of surfaces. We propose a new mathematical representation of surfaces, called q-maps, such that L2 distances between such maps are invariant to re-parameterizations. This property allows for removing the parameterization variability by optimizing over the reparameterization group, resulting in a proper parameterizationinvariant distance between shapes of surfaces. We demonstrate this method in shape analysis of multiple brain structures, for 34 subjects in the Detroit Fetal Alcohol and Drug Exposure Cohort study, which results in a 91%classification rate for ADHD (Attention Deficit Hyperactivity Disorder) cases and controls. This method outperforms some existing techniques such as SPHARM-PDM (spherical harmonic point distribution model) or ICP (iterative closest point).

Shape Analysis of Local Patches for 3D Facial Expression Recognition

Pattern Recognition
44
(8)
,
August
2011

[Abstract]

We propose an approach based on shape analysis of local facial patches. We apply a Riemannian framework to study and compare the shapes of patches taken from different faces under various expressions. A computation of the length of the geodesic path between corresponding patches in a shape space provides us with a quantitative information about their similarities (or dis- similarities). These measures are then used as inputs to several classifica- tion methods. The experimental results demonstrate the effectiveness of our method. Using Multiboost and Support Vector Machines (SVM) classifiers, we achieved 98.81% and 97.75% recognition average rates, respectively, for recognition of the six prototypical facial expressions on the publicly avail- able 3D facial expression database from Binghamton University (BU-3DFE). Comparisons with past approaches in the same experimental setting show that the results obtained by our approach outperform the state of the art results.

Fitting Smoothing Splines to Time-Indexed, Noisy Points on Nonlinear Manifolds

Journal of Image and Vision Computing,
2011
, http://dx.doi.org/10.1016/j.imavis.2011.09.006

[Abstract]

We address the problem of estimating full curves/paths on certain nonlinear manifolds using only a set of time-indexed points, for use in interpolation, smoothing, and prediction of dynamic systems. These curves are analogous to smooth splines on Euclidean spaces as they are optimal under a similar objective function, which is a weighted sum of a fitting-related (data term) and a regularity-related (smoothing term) cost functions. The search for smoothing splines on manifolds is based on a Palais-based steepest-decent algorithm developed in Samir et al. (2010). Using three representative manifolds: the rotation group for pose tracking, the space of symmetric positive-definite matrices for DTI image analysis, and Kendall

2010

Statistical Computations on Special Manifolds for Image and Video-Based Recognition

IEEE Transactions on Pattern Analysis and Machine Intelligence
2010

[Abstract]

In this paper, we examine image and video based recognition applications where the underlying models have a special structure – the linear subspace structure. We discuss how commonly used parametric models for videos and image-sets can be described using the unified framework of Grassmann and Stiefel manifolds. We first show that the parameters of linear dynamic models are finite dimensional linear subspaces of appropriate dimensions. Unordered image-sets as samples from a finite-dimensional linear subspace naturally fall under this framework. We show that the study of inference over subspaces can be naturally cast as an inference problem on the Grassmann manifold. To perform recognition using subspace-based models, we need tools from the Riemannian geometry of the Grassmann manifold. This involves a study of the geometric properties of the space, appropriate definitions of Riemannian metrics, and definition of geodesics. Further, we derive statistical modeling of inter- and intra-class variations that respect the geometry of the space. We apply techniques such as intrinsic and extrinsic statistics, to enable maximum-likelihood classification. We also provide algorithms for unsupervised clustering derived from the geometry of the manifold. Finally, we demonstrate the improved performance of these methods in a wide variety of vision applications such as activity recognition, video-based face recognition, object recognition from image-sets, and activity-based video clustering.

Joint Gait-Cadence Analysis for Human Identification Using An Elastic Shape Framework

Communications in Statistics – Theory and Methods
2010

[Abstract]

We perform human identification by gait recognition where subjects’ gait is represented by silhouettes which are elements of a manifold of Square-Root Velocity functions. Gait cycles become stochastic processes on this manifold; cadence its rate of execution. Using geometry of this manifold, we compute mean gait cycle templates for subjects. An observation model, where test sequences are random perturbations of templates, produces likelihood functions for classification. We perform temporal registration—linear and nonlinear—of cycles with templates, removing cadence effects. In an experiment on 26 individuals, linear registration, preserving cadence, performs better than nonlinear registration, which removes cadence.

Optimal Linear Projections for Enhancing Desired Data Statistics

Journal of Statistics and Computing
2010

[Abstract]

Problems involving high-dimensional data, such as pattern recognition, image analysis, and gene clustering, often require a preliminary step of dimension reduction before or during statistical analysis. If one restricts to a linear technique for dimension reduction, the remaining issue is the choice of the projection. This choice can be dictated by desire to maximize certain statistical criteria, including variance, kurtosis, sparseness, and entropy, of the projected data. Motivations for such criteria comes from past empirical studies of statistics of natural and urban images. We present a geometric framework for finding projections that are optimal for obtaining certain desired statistical properties. Our approach is to define an objective function on spaces of orthogonal linear projections—Stiefel and Grassmann manifolds, and to use gradient techniques to optimize that function. This construction uses the geometries of these manifolds to perform the optimization. Experimental results are presented to demonstrate these ideas for natural and facial images

2009

Looking for Shapes in Two-Dimensional, Cluttered Point Cloud

IEEE Transactions on Pattern Analysis and Machine Intelligence
2009

[Abstract]

We study the problem of identifying shape classes in point clouds. These clouds contain sampled points along contours and are corrupted by clutter and observation noise. Taking an analysis-by-synthesis approach, we simulate high-probability configurations of sampled contours using models learned from training data to evaluate the given test data. To facilitate simulations, we develop statistical models for sources of (nuisance) variability: 1) shape variations within classes, 2) variability in sampling continuous curves, 3) pose and scale variability, 4) observation noise, and 5) points introduced by clutter. The variability in sampling closed curves into finite points is represented by positive diffeomorphisms of a unit circle. We derive probability models on these functions using their square-root forms and the Fisher-Rao metric. Using a Monte Carlo approach, we simulate configurations from a joint prior on the shape-sample space and compare them to the data using a likelihood function. Average likelihoods of simulated configurations lead to estimates of posterior probabilities of different classes and, hence, Bayesian classification.

Rate-invariant recognition of humans and their activities

IEEE Transactions on Image Processing
8
(6)
,
June
2009

[Abstract]

Pattern Recognition in video is a challenging task because of the multitude of spatio-temporal variations that occur in different videos capturing the exact same event. While traditional pattern theoretic approaches account for the spatial changes that occur due to lighting and pose very little has been done to address the effect of temporal rate changes in the executions of an event. In this paper, we provide a systematic model based approach to learn the nature of such temporal variations (time warps) while simultaneously allowing for the spatial variations in the descriptors. We discuss the model in the context of action recognition and provide experimental justification for the importance of accounting for rate variations in action recognition. The model is composed of a nominal activity trajectory and a function space capturing the probability distribution of activity-specific time warping transformations. We use the square-root parameterization of time warps to derive geodesics, probability distributions and distance measures on the space of time warping functions. We then design a Bayesian algorithm for execution rate-invariant classification of activities. This approach allows us to learn the space of time warps for each activity while simultaneously allowing for other intra- and inter-class variations. Next we discuss a special case of this approach which leads to a uniform distribution on the space of time warping functions and show how computationally efficient inference algorithms may be derived for this special case. We discuss the relative advantages and disadvantages of both approaches and show their efficacy using experiments on gait based person identification and activity recognition.

Intrinsic Bayesian Active Contours for Extraction of Object Contours in Images

International Journal of Computer Vision
81
(3)
,
March
2009

[Abstract]

We present a framework for incorporating prior information about high-probability shapes in the process of contour extraction and object recognition in images. Here one studies shapes as elements of an infinite-dimensional, non-linear quotient space, and statistics of shapes are defined and computed intrinsically using differential geometry of this shape space. Prior models on shapes are constructed using probability distributions on tangent bundles of shape spaces. Similar to the past work on active contours, where curves are driven by vector fields based on image gradients and roughness penalties, we incorporate the prior shape knowledge in the form of vector fields on curves. Through experimental results, we demonstrate the use of prior shape models in the estimation of object boundaries, and their success in handling partial obscuration and missing data. Furthermore, we describe the use of this framework in shape-based object recognition or classification.

An Intrinsic Framework for Analysis of Facial Surfaces

International Journal of Computer Vision
82
(1)
,
April
2009

[Abstract]

A statistical analysis of shapes of facial surfaces can play an important role in biometric authentication and other face-related applications. The main difficulty in developing such an analysis comes from the lack of a canonical system to represent and compare all facial surfaces. This paper suggests a specific, yet natural, coordinate system on facial surfaces, that enables comparisons of their shapes. Here a facial surface is represented as an indexed collection of closed curves, called facial curves, that are level curves of a surface distance function from the tip of the nose. Defining the space of all such representations of face, this paper studies its differential geometry and endows it with a Riemannian metric. It presents numerical techniques for computing geodesic paths between facial surfaces in that space. This Riemannian framework is then used to: (i) compute distances between faces to quantify differences in their shapes, (ii) find optimal deformations between faces, and (iii) define and compute average of a given set of faces. Experimental results generated using laser-scanned faces are presented to demonstrate these ideas.

Elastic Shape Models for Face Analysis Using Curvilinear Coordinates

Journal of Mathematical Imaging and Vision
33
(2)
,
February
2009

[Abstract]

This paper studies the problem of analyzing variability in shapes of facial surfaces using a Riemannian framework, a fundamental approach that allows for joint matchings, comparisons, and deformations of faces under a chosen metric. The starting point is to impose a curvilinear coordinate system, named the Darcyan coordinate system, on facial surfaces; it is based on the level curves of the surface distance function measured from the tip of the nose. Each facial surface is now represented as an indexed collection of these level curves. The task of finding optimal deformations, or geodesic paths, between facial surfaces reduces to that of finding geodesics between level curves, which is accomplished using the theory of elastic shape analysis of 3D curves. The elastic framework allows for nonlinear matching between curves and between points across curves. The resulting geodesics between facial surfaces provide optimal elastic deformations between faces and an elastic metric for comparing facial shapes. We demonstrate this idea using examples from FSU face database.

s

2014

Statistical Analysis of Trajectories On Riemannian Manifolds: Bird MIgration, Hurricane Tracking, and Video Surveillance

Annals of Applied Statistics
8
(1)
,
April
2014

Elastic Shape Analysis of Cylindrical Surfaces for 3D/2D Registration in Endometrial Tissue Chracterization

IEEE Transactions on Medical Image Analysis
33
(5)
,
May
2014

Differential Geometric Representations and Algorithms for Some Pattern Recognition and Computer Vision Problems

Pattern Recognition Letters
43
(1)
,
July
2014

2013

Landmark-Guided Elastic Shape Analysis of Spherically-Parameterized Surfaces

Computer Graphics Forum
32
(2)
,
May
2013

3D Face Recognition Under Expressions, Occlusions, and Pose Variations

IEEE Transactions on Pattern Analysis and Machine Intelligence
35
(9)
,
September
2013

Statistical Analysis of Manual Segmentations of Structures in Medical Images

Computer Vision and IMage Understanding
117
(9)
,
2013

Elastic Shape Models for Improving Segmentation of Object Boundaries in Synthetic Aperture Sonar Images

Computer Vision and Image Understanding
117
(12)
,
December
2013

Gaussian-Blurring Invariant Comparisons of Signals and Images

IEEE Transactions on Image Processing
22
(8)
,
August
2013

2012

Fitting Optimal Curves to Time-Indexed, Noisy Observations on Nonlinear Manifolds

Journal of Image and Vision Computing
30
,
June
2012

Boosting 3D-Geometric Features for Efficient 3D Face Recognition and Gender Classification

IEEE Transactions on Information Forensics and Security
7
(6)
,
2012

Journalss

2013

Detection, Classification and Estimation of Shapes in 2D and 3D Point Clouds

Computational Statistics and Data Analysis
58
(58)
,
February
2013

[PDF]

Conferences

2012

A Geometric Analysis of ODFs As Oriented Surfaces for Interpolation, Averaging and Denoising in HARDI Data

IEEE Workshop of Mathematical Methods in Biomedical Image Aanalysis
2012

[Abstract]

We propose a Riemannian framework for analyzing orientation distribution functions (ODFs), or corresponding probability density functions (PDFs), in HARDI for use in comparing, interpolating, averaging, and denoising. Recent approaches based on the Fisher-Rao Riemannian metric result in geodesic paths that have limited biological interpretations. As an alternative, we develop a framework where we separate the shape and orientation features of PDFs, compute geodesics under their respective Riemannian metrics and then combine them to form pseudo-geodesics on the product space. These pseudo-geodesic paths have better biological interpretation (in terms of interpolating points between given PDFs by preserving shape diffusivity and anisotropy) and provide tools for pairwise comparison and averaging of a collection of PDFs. The latter tools, in turn, are useful for interpolation, denoising, and improved tractography in HARDI data. We demonstrate these ideas using both synthetic and real HARDI data.

Elastic Symmetry Analysis of Anatomical Structures

IEEE Workshop on Mathematical Methods in Biomedical Image Analysis (MMBIA)
2012
, Won Best Paper Award at the Conference

[PDF]

Affine-Invariant, Elastic Shape Analysis of Planar Contours

IEEE Conference on Computer Vision and Pattern Recognition (CVPR)
2012

[PDF]

A Novel Framework for Metric-Based Image Registration

Workshop on Biomedical Image Registration
2012

[PDF]

Analysis of Signals Under Compositional Noise With Applications to SONAR Data

Proc. of MTS/IEEE Oceans
2012

[PDF]

Which Geometric Features Give Up Your Identity?

International Conference on Biometrics
June
2012

2011

Parmeterization-Invariant Shape Statistics and Probabilistic Classification of Anatomical Surfaces

Information Processing in Medical Imaging (IPMI), Monastree Irsay, Germany
July
2011

[Abstract]

We consider the task of computing shape statistics and classification of 3D anatomical structures (as continuous, parameterized surfaces) under a Riemannian framework. This task requires a Riemannian metric that allows: (1) reparameterizations 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 into disease and control classes. In a separate paper [13], we introduced a mathematical representation of surfaces, called q-maps, and we used the L2 metric on the space of q-maps to induce a Riemannian metric on the space of parameterized surfaces. We also developed a path-straightening algorithm for computing geodesic paths [14]. This process requires optimal reparameterizations (deformations of grids) of surfaces and achieves a superior alignment of geometric features across surfaces. The resulting means and covariances are better representatives of the original data and lead to parsimonious shape models. These two moments specify a normal probability model on shape classes, which are then used for classifying test shapes. Through improved random sampling and a higher classification performance, we demonstrate the success of this model over some past methods. In addition to toy objects, we use the Detroit Fetal Alcohol and Drug Exposure Cohort data to study brain structures and present classification results for the Attention Deficit Hyperactivity Disorder cases and controls in this study.We find that using the mean and covariance structure of the given data, we are able to attain a 88% classification rate, which is an improvement over a previously reported result of 82% on the same data.

A Novel Riemannian Metric for Analyzing HARDI Data

SPIE Medical Imaging Conference on Image Processing, Orlando, FL
February
2011

[Abstract]

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 modi¯cation 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.

Classification of mathematics deficiency using shape and scale analysis of 3D brain structures

SPIE Medical Imaging Conference on Image Processing
February
2011

[Abstract]

We investigate the use of a recent technique for shape analysis of brain substructures in identifying learning disabilities in third-grade children. This Riemannian technique provides a quantification of differences in shapes of parameterized surfaces, using a distance that is invariant to rigid motions and re-parameterizations. Additionally, it provides an optimal registration of points across surfaces for improved matching and comparisons. We utilize an efficient gradient based method to obtain the optimal re-parameterizations of surfaces. In this study we consider 20 different substructures in human brain and correlate the differences in their shapes with abnormalities manifested in deficiency of mathematical skills in 106 subjects. The selection of these structures is motivated in part by the past links between their shapes and cognitive skills, albeit in broader contexts. We have studied the use of both individual substructures and multiple structures jointly for disease classification. Using a leave-one-out classifier, we obtained a 62.3% classification rate based on the shape of the left hippocampus. The use of multiple structures resulted in an improved classification rate of 71.4%.

Blurring-Invariant Riemannian Metrics for Comparing Images and Signals

International Conference on Computer Vision (ICCV)
November
2011

[Abstract]

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 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. We demonstrate this framework using a number of experimental results involving 1D signals and 2D images.

A GRID-based Parametric Representation of Large Diffeomorphic Deformations

3rd Workshop on Mathematical Foundations of Computational Anatomy (MFCA)
September
2011

[Abstract]

The growth by random iterated diffeomorphisms (GRID) model seeks to decompose large deformations, caused by growth, anomaly, or other reasons, into smaller, biologically-meaningful components. These components are spatially local and parametric, and are characterized by radial deformation patterns around randomly-placed seeds. A sequential composition of these components, using the group structure of diffeo- morphism group, models cumulative deformation. The actual decom- position requires estimation of GRID parameters from observations of large growth, typically from 2D or 3D images. While past papers have estimated parameters under certain simplifying assumptions, including that different components are spatially separated and non-interacting, we address the problem of parameter estimation under the original GRID model that advocates sequential composition of arbitrarily interacting components. Using a gradient-based approach, we present an algorithm for estimation of GRID parameters by minimizing an energy function and demonstrate its superiority over the past additive methods.

Structure-based RNA Function Prediction using Elastic Shape Analysis

2011 IEEE International Conference on Bioinformatics and Biomedicine
2011

[Abstract]

In recent years, RNAs have been found to have diverse functions beyond being a messenger in gene transcription. The functions of non-coding RNAs are determined by their structures. Structure comparison/alignment of RNAs provides an effective means to predict their functions. Despite many previous studies on RNA structure alignment, it is still a challenging problem to predict the function of RNA molecules based on their structure information. In this study, we developed a new RNA structure alignment method based on elastic shape analysis (ESA). ESA treats RNA structures as three dimensional curves and performs flexible alignment between two RNA molecules by bending and stretching one of the molecules to match the other. The amount of bending and stretching is quantified by a formal distance, geodesic distance. Based on ESA, a rigorous mathematical framework can be built for RNA structure comparison. Means and covariances can be computed and probability distributions can be constructed for a group of RNA structures. We further applied the method to predict functions of RNA molecules. Our method achieved good performance when tested on benchmark datasets.

Statistical analysis and classification of acoustic color functions

Proc SPIE
8017
,
April
2011

[Abstract]

In this paper we present a method for clustering and classification of acoustic color data based on statistical analysis of functions using square-root velocity functions (SVRF). The convenience of the SVRF is that it trans- forms the Fisher-Rao metric into the standard L2 metric. As a result, a formal distance can be calculated using geodesic paths. Moreover, this method allows optimal deformations between acoustic color data to be computed for any two targets allowing for robustness to measurement error. Using the SVRF formulation statistical models can then be constructed using principal component analysis to model the functional variation of acoustic color data. Empirical results demonstrate the utility of functional data analysis for improving performance results in pattern recognition using acoustic color data.

Signal Estimation Under Random Time-Warpings and Nonlinear Signal Alignment

Neural Information Processing Systems (NIPS)
2011

[PDF]

2010

Protein Structure Alignment Using Elastic Shape Analysis

ACM Conference on Bioinformatics and Computational Biology (ACM-BCB), Niagara Falls, NY
August
2010

[Abstract]

In this paper we present a method for flexible protein structure alignment based on elastic shape analysis of backbones, which can incorporate different characteristics of the backbones. In particular, it can include the backbone geometry, the secondary structures, and the amino-acid sequences in the matching process. As a result, a formal distance can be calculated and geodesic paths, showing optimal deformations between conformations/structures, can be computed for any two backbone structures. It can also be used to average shapes of conformations associated with similar proteins. Using examples of protein backbones we demonstrate the matching and clustering of proteins using the backbone geometries, the secondary labels and the primary sequences.

Elastic Radial Curves to Model Facial Deformations

British Machine Vision Conference (BMVC), Aberystwth, UK
September
2010

[Abstract]

In this paper we explore the use of shapes of elastic radial curves to model 3D facial deformations, caused by changes in facial expressions. We represent facial surfaces by indexed collections of radial curves on them, emanating from the nose tips, and compare the facial shapes by comparing the shapes of their corresponding curves. Using a past approach on elastic shape analysis of curves, we obtain an algorithm for comparing facial surfaces. We also introduce a quality control module which allows our approach to be robust to pose variation and missing data. Comparative evaluation using a common experimental setup on GAVAB dataset, considered as the most expression-rich and noise-prone 3D face dataset, shows that our approach outperforms other state-of-the-art approaches.

A Novel Riemannian Framework for Shape Analysis of 3D Objects

IEEE Conference on computer Vision and Pattern Recognition (CVPR), San Francisco, CA,
June
2010

[Abstract]

In this paper we introduce a novel Riemannian frame- work for shape analysis of parameterized surfaces. We derive a distance function between any two surfaces that is invariant to rigid motion, global scaling, and re- parametrization. It is the last part that presents the main difficulty. Our solution to this problem is twofold: (1) we define a special representation, called a q-map, to repre- sent each surface, and (2) we develop a gradient-based al- gorithm to optimize over different re-parameterizations of a surface. The second step is akin to deforming the mesh on a fixed surface to optimize its placement. (This is differ- ent from the current methods that treat the given meshes as fixed.) Under the chosen representation, with the L2 met- ric, the action of the re-parametrization group is by isome- tries. This results in, to our knowledge, the first Rieman- nian distance between parameterized surfaces to have all the desired invariances. We demonstrate this framework with several examples using some toy shapes, and real data with anatomical structures, and cropped facial surfaces. We also successfully demonstrate clustering and classification of these objects under the proposed metric.

Detection of Shapes in 2D Point Clouds Generated from Images, (ICPR)

International Conference on Pattern Recognition, Istanbul, Turkey
August
2010

We present a novel statistical framework for detecting pre-determined shape classes in 2D cluttered point clouds, which are in turn extracted from images. In this model based approach, we use a 1D Poisson process for sampling points on shapes, a 2D Poisson process for points from background clutter, and an additive Gaussian model for noise. Combining these with a past stochastic model on shapes of continuous 2D contours, and optimization over unknown pose and scale, we develop a generalized likelihood ratio test for shape detection. We demonstrate the efficiency of this method and its robustness to clutter using both simulated and real data.

A fully statistical framework for shape detection in image primitives

The Seventh Workshop on Perceptual Organization in Computer Vision (POCV) in conjunction with CVPR, San Francisco, CA
June
2010

[Abstract]

We present a fully statistical framework for detecting pre-determined shape classes in 2D clouds of primitives (points, edges, and arcs), which are in turn extracted from images. An important goal is to provide a likelihood, and thus a confidence, of finding a shape class in a given data. This requires a model-based approach. We use a composite Poisson process: 1D Poisson process for primitives belonging to shapes and a 2D Poisson process for primitives belonging to clutter. An additive Gaussian model is assumed for noise in shape primitives. Combining these with a past stochastic model on shapes of continuous 2D contours, and optimization over unknown pose and scale, we develop a generalized likelihood ratio test for shape detection. We demonstrate the efficiency of this method and its robustness to clutter using both simulated and real data.

Statistical Shape Detection in Over-Segmented Images

Workshop on Applications of Discrete Geometry and Mathematical Morphology in conjunction with ICPR, Istanbul, Turkey
August
2010

[Abstract]

We summarize a fully statistical method for finding a shape of interest in an over-segmented image. This method is based on sequentially curve growing where the main idea is to avoid local solutions by adding a viability test that evaluates the likelihood of the full target shape being present in the data. And an elastic comparison of the overall estimated shape to the target shape provides a stopping criterion.