An Elasticity-Based Covariance Analysis of Shapes

Rumpf, Martin; Wirth, Benedikt

doi:10.1007/s11263-010-0358-2

Cited by 23 publications

(17 citation statements)

References 46 publications

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“…Two of the most prominent ideas in this research area have been the use of fluid flow [28,29,108] and elasticity notions [98,113,114]. One domain of application of these methods is computational anatomy, where finding precise correspondence between homologous anatomical structures is often the goal.…”

Section: Related Workmentioning

confidence: 99%

Gromov–Wasserstein Distances and the Metric Approach to Object Matching

2011

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This paper discusses certain modifications of the ideas concerning the Gromov-Hausdorff distance which have the goal of modeling and tackling the practical problems of object matching and comparison. Objects are viewed as metric measure spaces, and based on ideas from mass transportation, a Gromov-Wasserstein type of distance between objects is defined. This reformulation yields a distance between objects which is more amenable to practical computations but retains all the desirable theoretical underpinnings. The theoretical properties of this new notion of distance are studied, and it is established that it provides a strict metric on the collection of isomorphism classes of metric measure spaces. Furthermore, the topology generated by this metric is studied, and sufficient conditions for the pre-compactness of families of metric measure spaces are identified. A second goal of this paper is to establish links to several other practical methods proposed in the literature for comparing/matching shapes in precise terms. This is done by proving explicit lower bounds for the proposed distance that involve many of the invariants previously reported by researchers. These lower bounds can be computed in polynomial time. The numerical implementations of the ideas are discussed and computational examples are presented.

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Section: Related Workmentioning

confidence: 99%

Gromov–Wasserstein Distances and the Metric Approach to Object Matching

2011

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“…This constitutes a major difference with the work of [56] and as demonstrated in Section 3 of the supplementary material, it entails substantial adaptations in the mathematical developments. The first two methods rely on fundamental notions of elastic behaviour and the following observation made in [56] : "the classical covariance tensor can be identified with the covariance tensor of the displacements obtained by adding a small fraction of the i-th spring force under the Hooke's law". Whilst the first method is based on the linearisation of the stored energy function around the identity, which might result in the loss of the initial nonlinear nature of the deformations but has the advantage of being fast, the second approach is more intricate.…”

Section: 7mentioning

confidence: 99%

“…It allows for the derivation of image statistics, the retrieval of the inherent dynamics of a single individual's organ, the estimation of the probability that a particular spatial location takes on a certain label, the detection and quantisation of abnormalities, that is, more generally, it allows to characterise and understand how geometrical and structural changes influence health. A large body of papers feeds the field of atlas generation and shape statistics among which [36] (atlas generation problem phrased in the Large Deformation Diffeomorphic Metric Mapping (LDDMM) framework [8]), [16] (the shapes to be analysed are modeled as random histograms and in order to learn principal modes of variation from such data, the Wasserstein distance between probability measures is introduced), [70] (dedicated to elastic shape analysis; a unified registration/parameterised object statistical analysis framework is tackled, based on square-root transformations and able to process data as diverse as curves, functions, surfaces and images), [3] (statistics performed on the space of diffeomorphisms), [35] (the use of a kernel descriptor that characterises local shape properties ensures geometrically meaningful correspondence between shapes with statistical studies of the deformations), [55,56] (the shapes are viewed as closed contours approximated by phase fields, and shape averaging and covariance analysis are carried out in a nonlinear elasticity setting), to name a few.…”

Section: Introductionmentioning

confidence: 99%

“…More precisely, the novelty of the paper rests upon: (i) an original modeling involving the stored energy function of an Ogden material complemented by two new hard constraints on the Jacobian and its inverse (in addition to the theoretical utility of these constraints, it also allows to control changes of length), the Potts model for segmentation, and an original discrepancy measure ensuring edge mapping; (ii) the derivation of theoretical results encompassing non straightforward mathematical tools; (iii) the analysis and comparison of three different methods to perform statistical analysis on the obtained deformation: the first one, based on linearised elasticity principles largely inspired by [56], the second one using the Cauchy-stress tensors motivated by [56], and the last one, more novel and on which the paper focuses, relying on tensor-based smoothing D m splines, influenced by [41]. Section 2 is devoted to the analysis of the mathematical model including a theorem of existence of minimisers, while Section 3 is dedicated to the theoretical analysis of a numerical method of resolution based on a splitting approach and implying the Alternating Direction Method of Multipliers (ADMM) techniques and proximal gradient descent algorithms.…”

Section: Introductionmentioning

confidence: 99%

“…As already mentioned, our motivation is to investigate how the linear elasticity based approach and the Cauchy-stress based method compare to the D m splines approximation based procedure. The first two are non-straightforward adaptations of the techniques envisioned in [56], but the view we take to model the shapes -phase field rather than closed contours -implies substantial changes in the physics of the problem, while the emphasis is put on the last one for its novelty. Section 5 focuses on numerical simulations with a thorough comparison between sequential treatments and the proposed joint model, demonstrating the ability of our model to handle large deformations and to produce in the end, an atlas with sharp edges, high contrast and reflecting a realistic shape.…”

Section: Introductionmentioning

confidence: 99%

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A Variational Model Dedicated to Joint Segmentation, Registration, and Atlas Generation for Shape Analysis

Debroux¹,

Aston²,

Bonardi³

et al. 2020

SIAM J. Imaging Sci.

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In medical image analysis, constructing an atlas, i.e. a mean representative of an ensemble of images, is a critical task for practitioners to estimate variability of shapes inside a population, and to characterise and understand how structural shape changes have an impact on health. This involves identifying significant shape constituents of a set of images, a process called segmentation, and mapping this group of images to an unknown mean image, a task called registration, making a statistical analysis of the image population possible. To achieve this goal, we propose treating these operations jointly to leverage their positive mutual influence, in a hyperelasticity setting, by viewing the shapes to be matched as Ogden materials. The approach is complemented by novel hard constraints on the L ∞ norm of both the Jacobian and its inverse, ensuring that the deformation is a bi-Lipschitz homeomorphism. Segmentation is based on the Potts model, which allows for a partition into more than two regions, i.e. more than one shape. The connection to the registration problem is ensured by the dissimilarity measure that aims to align the segmented shapes. A representation of the deformation field in a linear space equipped with a scalar product is then computed in order to perform a geometry-driven Principal Component Analysis (PCA) and to extract the main modes of variations inside the image population. Theoretical results emphasizing the mathematical soundness of the model are provided, among which existence of minimisers, analysis of a numerical method of resolution, asymptotic results and a PCA analysis, as well as numerical simulations demonstrating the ability of the modeling to produce an atlas exhibiting sharp edges, high contrast and a consistent shape.

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Variational Methods in Shape Analysis

Rumpf¹,

Wirth²

2011

Handbook of Mathematical Methods in Imaging

Self Cite

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The analysis of shapes as elements in a frequently infinite-dimensional space of shapes has attracted increasing attention over the last decade. There are pioneering contributions in the theoretical foundation of shape space as a Riemannian manifold as well as path-breaking applications to quantitative shape comparison, shape recognition, and shape statistics. The aim of this chapter is to adopt a primarily physical perspective on the space of shapes and to relate this to the prevailing geometric perspective. Indeed, we here consider shapes given as boundary contours of volumetric objects, which consist either of a viscous fluid or an elastic solid.In the first case, shapes are transformed into each other via viscous transport of fluid material, and the flow naturally generates a connecting path in the space of shapes. The viscous dissipation rate-the rate at which energy is converted into heat due to friction-can be defined as a metric on an associated Riemannian manifold. Hence, via the computation of shortest transport paths one defines a distance measure between shapes.In the second case, shapes are transformed via elastic deformations, where the associated elastic energy only depends on the final state of the deformation and not on the path along which the deformation is generated. The minimal elastic energy required to deform an object into another one can be considered as a dissimilarity measure between the corresponding shapes.In what follows we discuss and extensively compare the path-based and the state-based approach. As applications of the elastic shape model we consider shape averages and a principal component analysis of shapes. The viscous flow model is used to exemplarily cluster 2D and 3D shapes and to construct a flow type nonlinear interpolation scheme. Furthermore, we show how to approximate the viscous, path-based approach with a time-discrete sequence of state-based variational problems. A review of different shape space conceptsThe structure of shape spaces and statistical analyses of shapes have been examined in various settings, and applications range from the computation of priors for segmentation [1,2,3] and shape classification [4,5,6,7] to the construction of standardized anatomical atlases [8,9,10]. Among all existing approaches, a number of different concepts of a shape are employed, including landmark vectors [11,1], planar curves [12,13,14], surfaces in R 3 [4,15,16], boundary contours of

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An Elasticity-Based Covariance Analysis of Shapes

Cited by 23 publications

References 46 publications

Gromov–Wasserstein Distances and the Metric Approach to Object Matching

Gromov–Wasserstein Distances and the Metric Approach to Object Matching

A Variational Model Dedicated to Joint Segmentation, Registration, and Atlas Generation for Shape Analysis

Variational Methods in Shape Analysis

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