Elastic Shape Analysis Research Articles

Elastic Shape Analysis of Tree-Like 3D Objects Using Extended SRVF Representation.

How can one analyze detailed 3D biological objects, such as neuronal and botanical trees, that exhibit complex geometrical and topological variation? In this paper, we develop a novel mathematical framework for representing, comparing, and computing geodesic deformations between the shapes of such tree-like 3D objects. A hierarchical organization of subtrees characterizes these objects - each subtree has a main branch with some side branches attached - and one needs to match these structures across objects for meaningful comparisons. We propose a novel representation that extends the Square-Root Velocity Function (SRVF), initially developed for Euclidean curves, to tree-shaped 3D objects. We then define a new metric that quantifies the bending, stretching, and branch sliding needed to deform one tree-shaped object into the other. Compared to the current metrics such as the Quotient Euclidean Distance (QED) and the Tree Edit Distance (TED), the proposed representation and metric capture the full elasticity of the branches (i.e., bending and stretching) as well as the topological variations (i.e., branch death/birth and sliding). It completely avoids the shrinkage that results from the edge collapse and node split operations of the QED and TED metrics. We demonstrate the utility of this framework in comparing, matching, and computing geodesics between biological objects such as neuronal and botanical trees. We also demonstrate its application to various shape analysis tasks such as (i) symmetry analysis and symmetrization of tree-shaped 3D objects, (ii) computing summary statistics (means and modes of variations) of populations of tree-shaped 3D objects, (iii) fitting parametric probability distributions to such populations, and (iv) finally synthesizing novel tree-shaped 3D objects through random sampling from estimated probability distributions.

Longitudinal Elastic Shape Analysis of Brain Subcortical Structures

AbstractBackgroundOver the past 30 years, MRI has become a ubiquitous tool for accurately visualizing the development of the brain’s subcortical structures. However, the quantification of subcortical structures is still in its infancy due to challenges in shape extraction, representation, and modeling.MethodA simple and efficient framework of longitudinal elastic shape analysis (LESA) is developed for subcortical structures. By employing the elastic shape analysis of static surfaces using the square root normal field (SRNF) representations, we computed shape summaries of surfaces and represented the complex subcortical structures using a small number of basis functions. Further, we statistically modeled the sparse longitudinal data to predict the spatial‐temporal shape changes of subcortical structures. We also conducted the temporal registration using the square‐root velocity function (SRVF) framework to quantify the accelerated aging of Alzheimer’s disease (AD).ResultWe applied LESA to analyze longitudinal neuroimaging data sets and showcased its wide applications in estimating continuous shape trajectories, building life‐span growth patterns, and comparing shape differences among different groups. Using the Alzheimer’s Disease Neuroimaging Initiative (ADNI) data, we found that from 60 to 90 years old, the AD group always had the largest left ventricle (LV) surface area and the smallest left hippocampus (LHC) surface area, followed by the MCI and NC groups. The enlargement of LV and the atrophy of LHC had different speeds among different groups. Most of the shape‐changing differences between the AD and NC groups were visible in the anterior and posterior ends of LV. The atrophy of LHC happened mainly at the posterior end with aging. The AD group had the sharpest posterior end, and the NC group deformed the least with aging. Compared with the NC group, the AD group showed an average accelerated enlargement of 2.91 years and accelerated shape change of 2.76 years in LV; an average accelerated atrophy of 2.56 years and accelerated shape change of 2.59 years in LHC. By constructing the life‐span growth trajectories, we observed increases in shape changes’ speed in both LV and LHC after 60 years old.ConclusionAD can significantly speed up the shape change of ventricle and hippocampus compared with normal aging.

Archetypal Curves in the Shape and Size Space: Discovering the Salient Features of Curved Big Data by Representative Extremes

Curves are complex data. Tools for visualizing, exploring, and discovering the structure of a data set of curves are valuable. In this paper, we propose a scalable methodology to solve this challenge. On the one hand, we consider two distances in the shape and size space, one well-known distance and another recently proposed, which differentiate the contribution in shape and in size of the elements considered to compute the distance. On the other hand, we use archetypoid analysis (ADA) for the first time in elastic shape analysis. ADA is a recent technique in unsupervised statistical learning, whose objective is to find a set of archetypal observations (curves in this case), in such a way that we can describe the data set as convex combinations of these archetypal curves. This makes interpretation easy, even for non-experts. Archetypal curves or pure types are extreme cases, which also facilitates human understanding. The methodology is illustrated with a simulated data set and applied to a real problem. It is important to know the distribution of foot shapes to design suitable footwear that accommodates the population. For this purpose, we apply our proposed methodology to a real data set composed of foot contours from the adult Spanish population.

Elastic Shape Analysis of Surfaces with Second-Order Sobolev Metrics: A Comprehensive Numerical Framework

This paper introduces a set of numerical methods for Riemannian shape analysis of 3D surfaces within the setting of invariant (elastic) second-order Sobolev metrics. More specifically, we address the computation of geodesics and geodesic distances between parametrized or unparametrized immersed surfaces represented as 3D meshes. Building on this, we develop tools for the statistical shape analysis of sets of surfaces, including methods for estimating Karcher means and performing tangent PCA on shape populations, and for computing parallel transport along paths of surfaces. Our proposed approach fundamentally relies on a relaxed variational formulation for the geodesic matching problem via the use of varifold fidelity terms, which enable us to enforce reparametrization independence when computing geodesics between unparametrized surfaces, while also yielding versatile algorithms that allow us to compare surfaces with varying sampling or mesh structures. Importantly, we demonstrate how our relaxed variational framework can be extended to tackle partially observed data. The different benefits of our numerical pipeline are illustrated over various examples, synthetic and real.

RV-ESA: A novel computer-aided elastic shape analysis system for retinal vessels in diabetic retinopathy

Diabetic retinopathy (DR), one of the most common and serious complications of diabetes, has become one of the main blindness diseases. The retinal vasculature is the only part of the human circulatory system that allows direct noninvasive visualization of the body’s microvasculature, which provides the opportunity to detect the structural and functional changes before DR becomes unable to intervene. For decades, as the fundamental step in computer-assisted analysis of retinopathy, retinal vascular extraction methods have been largely developed. However, further research focusing on retinal vascular analysis is still in its infancy. Meanwhile, due to the complexity of retinal vascular structure, the relationship between vascular geometry and DR has never been concluded. This paper aims to provide a novel computer-aided shape analysis system for retinal vessels. To perform retinal vascular shape analysis, a mathematical geometric representation is firstly generated by utilizing the proposed shape modeling method. Then, several useful statistical tools (e.g. Graph Mean, Graph PCA) are adopted to quantitatively analyze the vascular shape. Besides, in order to visualize the changes in vascular shape in the progression of DR, a geodesic tool is used to display the deformation process for ophthalmologists to observe. The efficacy of this analysis system is demonstrated in the EyePACS dataset and the subsequent visit records of 98 patients from the proprietary dataset. The experimental results show that there is a certain correlation between the variation of retinal vascular shape and DR progression, and the Graph PCA scores of retinal vessels are negatively correlated with DR grades. The code of our RV-ESA system can be publicly available at github.com/XiaolingLuo/RV-ESA.

Elastic shape analysis of brain structures for predictive modeling of PTSD.

It is well-known that morphological features in the brain undergo changes due to traumatic events and associated disorders such as post-traumatic stress disorder (PTSD). However, existing approaches typically offer group-level comparisons, and there are limited predictive approaches for modeling behavioral outcomes based on brain shape features that can account for heterogeneity in PTSD, which is of paramount interest. We propose a comprehensive shape analysis framework representing brain sub-structures, such as the hippocampus, amygdala, and putamen, as parameterized surfaces and quantifying their shape differences using an elastic shape metric. Under this metric, we compute shape summaries (mean, covariance, PCA) of brain sub-structures and represent individual brain shapes by their principal scores under a shape-PCA basis. These representations are rich enough to allow visualizations of full 3D structures and help understand localized changes. In order to validate the elastic shape analysis, we use the principal components (PCs) to reconstruct the brain structures and perform further evaluation by performing a regression analysis to model PTSD and trauma severity using the brain shapes represented via PCs and in conjunction with auxiliary exposure variables. We apply our method to data from the Grady Trauma Project (GTP), where the goal is to predict clinical measures of PTSD. The framework seamlessly integrates accurate morphological features and other clinical covariates to yield superior predictive performance when modeling PTSD outcomes. Compared to vertex-wise analysis and other widely applied shape analysis methods, the elastic shape analysis approach results in considerably higher reconstruction accuracy for the brain shape and reveals significantly greater predictive power. It also helps identify local deformations in brain shapes associated with PTSD severity.

LESA: Longitudinal Elastic Shape Analysis of Brain Subcortical Structures

Over the past 30 years, magnetic resonance imaging has become a ubiquitous tool for accurately visualizing the change and development of the brain’s subcortical structures (e.g., hippocampus). Although subcortical structures act as information hubs of the nervous system, their quantification is still in its infancy due to many challenges in shape extraction, representation, and modeling. Here, we develop a simple and efficient framework of longitudinal elastic shape analysis (LESA) for subcortical structures. Integrating ideas from elastic shape analysis of static surfaces and statistical modeling of sparse longitudinal data, LESA provides a set of tools for systematically quantifying changes of longitudinal subcortical surface shapes from raw structure MRI data. The key novelties of LESA include: (i) it can efficiently represent complex subcortical structures using a small number of basis functions and (ii) it can accurately delineate the spatiotemporal shape changes of the human subcortical structures. We applied LESA to analyze three longitudinal neuroimaging datasets and showcase its wide applications in estimating continuous shape trajectories, building life-span growth patterns, and comparing shape differences among different groups. In particular, with the Alzheimer’s Disease Neuroimaging Initiative (ADNI) data, we found that Alzheimer’s Disease (AD) can significantly speed the shape change of the lateral ventricle and the hippocampus from 60 to 75 years olds compared with normal aging. Supplementary materials for this article are available online.

Elastic statistical analysis of interval-valued time series

We investigate the problem of statistical analysis of interval-valued time series data – two nonintersecting real-valued functions, representing lower and upper limits, over a period of time. Specifically, we pay attention to the two concepts of phase (or horizontal) variability and amplitude (or vertical) variability, and propose a phase-amplitude separation method. We view interval-valued time series as elements of a function (Hilbert) space and impose a Riemannian structure on it. We separate phase and amplitude variability in observed interval functions using a metric-based alignment solution. The key idea is to map an interval to a point in , view interval-valued time series as parameterized curves in , and borrow ideas from elastic shape analysis of planar curves, including PCA, to perform registration, summarization, analysis, and modeling of multiple series. The proposed phase-amplitude separation provides a new way of PCA and modeling for interval-valued time series, and enables shape clustering of interval-valued time series. We apply this framework to three different applications, including finance, meteorology and physiology, proves the effectiveness of proposed methods, and discovers some underlying patterns in the data. Experimental results on simulated data show that our method applies to the point-valued time series.

Elastic Shape Analysis of Planar Objects Using Tensor Field Representations

Elastic Shape Analysis of Planar Objects Using Tensor Field Representations

Application of Elastic Shape Analysis to User Authentication and Identification

Application of Elastic Shape Analysis to User Authentication and Identification

Elastic Shape Analysis for Anomaly Detection in Fabric Images

Elastic Shape Analysis for Anomaly Detection in Fabric Images

Analysis of shape data: From landmarks to elastic curves.

Proliferation of high-resolution imaging data in recent years has led to sub-stantial improvements in the two popular approaches for analyzing shapes of data objects based on landmarks and/or continuous curves. We provide an expository account of elastic shape analysis of parametric planar curves representing shapes of two-dimensional (2D) objects by discussing its differences, and its commonalities, to the landmark-based approach. Particular attention is accorded to the role of reparameterization of a curve, which in addition to rotation, scaling and translation, represents an important shape-preserving transformation of a curve. The transition to the curve-based approach moves the mathematical setting of shape analysis from finite-dimensional non-Euclidean spaces to infinite-dimensional ones. We discuss some of the challenges associated with the infinite-dimensionality of the shape space, and illustrate the use of geometry-based methods in the computation of intrinsic statistical summaries and in the definition of statistical models on a 2D imaging dataset consisting of mouse vertebrae. We conclude with an overview of the current state-of-the-art in the field.

Shape Analysis of Framed Space Curves

In the elastic shape analysis approach to shape matching and object classification, plane curves are represented as points in an infinite-dimensional Riemannian manifold, wherein shape dissimilarity is measured by geodesic distance. A remarkable result of Younes, Michor, Shah and Mumford says that the space of closed planar shapes, endowed with a natural metric, is isometric to an infinite-dimensional Grassmann manifold via the so-called square root transform. This result facilitates efficient shape comparison by virtue of explicit descriptions of Grassmannian geodesics. In this paper, we extend this shape analysis framework to treat shapes of framed space curves. By considering framed curves, we are able to generalize the square root transform by using quaternionic arithmetic and properties of the Hopf fibration. Under our coordinate transformation, the space of closed framed curves corresponds to an infinite-dimensional complex Grassmannian. This allows us to describe geodesics in framed curve space explicitly. We are also able to produce explicit geodesics between closed, unframed space curves by studying the action of the loop group of the circle on the Grassmann manifold. We apply our results to compute means for collections of space curves and to perform statistical analysis of circular DNA molecule shapes.

A robust tangent PCA via shape restoration for shape variability analysis

This paper presents a novel method for handling the effects of shape outliers in statistical shape analysis. Usually performed by a variant of classical principal component analysis (PCA), variability analysis may be highly affected by erroneous shapes. Principal components may thus imply aberrant modes, while eigenshapes may not accurately describe variability in a given set of shapes. Our robust analysis is performed using an elastic metric associated with the square-root velocity representation of shapes. This elastic shape analysis allows shape variability to be described with natural and intuitive deformations. The proposed method based on shape outlier detection applies the shape restoration procedure to rectify aberrant shapes. The resultant components are thus obtained from a tangent PCA on the restored database. By performing experiments based on MPEG-7 and HAND databases, we demonstrate that the proposed scheme is effective for shape variability analysis in the presence of outlying shapes. Our method is then compared with two existing schemes for robust data variability analysis: minimum covariance determinant-based PCA and projection pursuit-based PCA.

Elastic Statistical Shape Analysis of Biological Structures with Case Studies: A Tutorial.

We describe a recent framework for statistical shape analysis of curves and show its applicability to various biological datasets. The presented methods are based on a functional representation of shape called the square-root velocity function and a closely related elastic metric. The main benefit of this approach is its invariance to reparameterization (in addition to the standard shape-preserving transformations of translation, rotation and scale), and ability to compute optimal registrations (point correspondences) across objects. Building upon the defined distance between shapes, we additionally describe tools for computing sample statistics including the mean and covariance. Based on the covariance structure, one can also explore variability in shape samples via principal component analysis. Finally, the estimated mean and covariance can be used to define Wrapped Gaussian models on the shape space, which are easy to sample from. We present multiple case studies on various biological datasets including (1) leaf outlines, (2) internal carotid arteries, (3) Diffusion Tensor Magnetic Resonance Imaging fiber tracts, (4) Glioblastoma Multiforme tumors, and (5) vertebrae in mice. We additionally provide a MATLAB package that can be used to produce the results given in this manuscript.

Elastic Shape Analysis of Three-Dimensional Objects

Abstract Statistical analysis of shapes of 3D objects is an important problem with a wide range of applications. This analysis is difficult for many reasons, including the fact that objects differ in both geometry and topology. In this manuscript, we narrow the problem by focusing on objects with fixed topology, say objects that are diffeomorphic to unit spheres, and develop tools for analyzing their geometries. The main challenges in this problem are to register points across objects and to perform analysis while being invariant to certain shape-preserving transformations. We develop a comprehensive framework for analyzing shapes of spherical objects, i.e., objects that are embeddings of a unit sphere in ℝ, including tools for: quantifying shape differences, optimally deforming shapes into each other, summarizing shape samples, extracting principal modes of shape variability, and modeling shape variability associated with populations. An important strength of this framework is that it is elastic: it perf...

Landmark-Constrained Elastic Shape Analysis of Planar Curves

ABSTRACT Various approaches to statistical shape analysis exist in current literature. They mainly differ in the representations, metrics, and/or methods for alignment of shapes. One such approach is based on landmarks, that is, mathematically or structurally meaningful points, which ignores the remaining outline information. Elastic shape analysis, a more recent approach, attempts to fix this by using a special functional representation of the parametrically defined outline to perform shape registration, and subsequent statistical analyses. However, the lack of landmark identification can lead to unnatural alignment, particularly in biological and medical applications, where certain features are crucial to shape structure, comparison, and modeling. The main contribution of this work is the definition of a joint landmark-constrained elastic statistical shape analysis framework. We treat landmark points as constraints in the full shape analysis process. Thus, we inherit benefits of both methods: the landmarks help disambiguate shape alignment when the fully automatic elastic shape analysis framework produces unsatisfactory solutions. We provide standard statistical tools on the landmark-constrained shape space including mean and covariance calculation, classification, clustering, and tangent principal component analysis (PCA). We demonstrate the benefits of the proposed framework on complex shapes from the MPEG-7 dataset and two real data examples: mice T2 vertebrae and Hawaiian Drosophila fly wings. Supplementary materials for this article are available online.

Numerical Inversion of SRNF Maps for Elastic Shape Analysis of Genus-Zero Surfaces.

Recent developments in elastic shape analysis (ESA) are motivated by the fact that it provides a comprehensive framework for simultaneous registration, deformation, and comparison of shapes. These methods achieve computational efficiency using certain square-root representations that transform invariant elastic metrics into euclidean metrics, allowing for the application of standard algorithms and statistical tools. For analyzing shapes of embeddings of in , Jermyn etal.[1] introduced square-root normal fields (SRNFs), which transform an elastic metric, with desirable invariant properties, into the metric. These SRNFs are essentially surface normals scaled by square-roots of infinitesimal area elements. A critical need in shape analysis is a method for inverting solutions (deformations, averages, modes of variations, etc.) computed in SRNF space, back to the original surface space for visualizations and inferences. Due to the lack of theory for understanding SRNF maps and their inverses, we take a numerical approach, and derive an efficient multiresolution algorithm, based on solving an optimization problem in the surface space, that estimates surfaces corresponding to given SRNFs. This solution is found to be effective even for complex shapes that undergo significant deformations including bending and stretching, e.g., human bodies and animals. We use this inversion for computing elastic shape deformations, transferring deformations, summarizing shapes, and for finding modes of variability in a given collection, while simultaneously registering the surfaces. We demonstrate the proposed algorithms using a statistical analysis of human body shapes, classification of generic surfaces, and analysis of brain structures.

Landmark-guided elastic shape analysis of human character motions

Motions of virtual characters in movies or video games are typically generated by recording actors using motion capturing methods. Animations generated this way often need postprocessing, such as improving the periodicity of cyclic animations or generating entirely new motions by interpolation of existing ones. Furthermore, search and classification of recorded motions becomes more and more important as the amount of recorded motion data grows. In this paper, we will apply methods from shape analysis to the processing of animations. More precisely, we will use the by now classical elastic metric model used in shape matching, and extend it by incorporating additional inexact feature point information, which leads to an improved temporal alignment of different animations.

An efficient algorithm for protein structure comparison using elastic shape analysis.

BackgroundProtein structure comparison play important role in in silico functional prediction of a new protein. It is also used for understanding the evolutionary relationships among proteins. A variety of methods have been proposed in literature for comparing protein structures but they have their own limitations in terms of accuracy and complexity with respect to computational time and space. There is a need to improve the computational complexity in comparison/alignment of proteins through incorporation of important biological and structural properties in the existing techniques.ResultsAn efficient algorithm has been developed for comparing protein structures using elastic shape analysis in which the sequence of 3D coordinates atoms of protein structures supplemented by additional auxiliary information from side-chain properties are incorporated. The protein structure is represented by a special function called square-root velocity function. Furthermore, singular value decomposition and dynamic programming have been employed for optimal rotation and optimal matching of the proteins, respectively. Also, geodesic distance has been calculated and used as the dissimilarity score between two protein structures. The performance of the developed algorithm is tested and found to be more efficient, i.e., running time reduced by 80–90 % without compromising accuracy of comparison when compared with the existing methods. Source codes for different functions have been developed in R. Also, user friendly web-based application called ProtSComp has been developed using above algorithm for comparing protein 3D structures and is accessible free.ConclusionsThe methodology and algorithm developed in this study is taking considerably less computational time without loss of accuracy (Table 2). The proposed algorithm is considering different criteria of representing protein structures using 3D coordinates of atoms and inclusion of residue wise molecular properties as auxiliary information.