Shape Analysis Using the Fisher-Rao Riemannian Metric: Unifying Shape Representation and Deformation

Peter, Adrian M.; Rangarajan, Anand

doi:10.1109/isbi.2006.1625130

Cited by 34 publications

(34 citation statements)

References 16 publications

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“…The geodesic and the corresponding geodesic distance seem to be a natural thing to study and has, for example, found applications in image analysis (see [19,20], for example).…”

Section: Applicationsmentioning

confidence: 99%

On the Limiting Behaviour of the Fundamental Geodesics of Information Geometry

Critchley

Marriott

2017

Entropy

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Abstract:The Information Geometry of extended exponential families has received much recent attention in a variety of important applications, notably categorical data analysis, graphical modelling and, more specifically, log-linear modelling. The essential geometry here comes from the closure of an exponential family in a high-dimensional simplex. In parallel, there has been a great deal of interest in the purely Fisher Riemannian structure of (extended) exponential families, most especially in the Markov chain Monte Carlo literature. These parallel developments raise challenges, addressed here, at a variety of levels: both theoretical and practical-relatedly, conceptual and methodological. Centrally to this endeavour, this paper makes explicit the underlying geometry of these two areas via an analysis of the limiting behaviour of the fundamental geodesics of Information Geometry, these being Amari's (+1) and (0)-geodesics, respectively. Overall, a substantially more complete account of the Information Geometry of extended exponential families is provided than has hitherto been the case. We illustrate the importance and benefits of this novel formulation through applications.

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“…The geodesic and the corresponding geodesic distance seem to be a natural thing to study and has, for example, found applications in image analysis (see [19,20], for example).…”

Section: Applicationsmentioning

confidence: 99%

On the Limiting Behaviour of the Fundamental Geodesics of Information Geometry

Critchley

Marriott

2017

Entropy

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“…In order to compare them in a manner that is invariant to their parameterizations, one forms a quotient space, called the shape space, that is defined as the space of closed curves modulo all possible re-parameterizations [7]. A related problem is to perform non-rigid registration of points across curves [10,11].…”

Section: Introductionmentioning

confidence: 99%

“…This task is accomplished by imposing Riemannian structures on appropriate manifolds formed by these functions. The most natural Riemannian metric in this context is the so-called Fisher-Rao metric, which has been used extensively in computer vision [6,5,10,11].Čencov [2] showed that this is the only metric that is invariant to re-parametrizations of those functions. This metric has also played an important role in information geometry due to the pioneering efforts of Amari [1].…”

Section: Introductionmentioning

confidence: 99%

Riemannian Analysis of Probability Density Functions with Applications in Vision

Srivastava

Jermyn

Joshi

2007

2007 IEEE Conference on Computer Vision and Pattern Recognition

164

158

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“…Another interesting approach proposes an entropy based criterion to find shape correspondences, but requires implicit surface representations (10). Other works combine the search for correspondences with shape based classification (11,12) or with shape analysis (13), however, these methods are not easily adaptable to multiple observations of unstructured point sets. The approach in (14) for unstructured point sets focuses only on the mean shape.…”

Section: Introductionmentioning

confidence: 99%

Generation of a statistical shape model with probabilistic point correspondences and the expectation maximization- iterative closest point algorithm

et al. 2008

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Objective: Identification of point correspondences between shapes is required for statistical analysis of organ shapes differences. Since manual identification of landmarks is not a feasible option in 3D, several methods were developed to automatically find one-to-one correspondences on shape surfaces. For unstructured point sets, however, one-to-one correspondences do not exist but correspondence probabilities can be determined. Materials and methods: A method was developed to compute a statistical shape model based on shapes which are represented by unstructured point sets with arbitrary point numbers. A fundamental problem when computing statistical shape models is the determination of correspondences between the points of the shape observations of the training data set. In the absence of landmarks, exact correspondences can only be determined between continuous surfaces, not between unstructured point sets. To overcome this problem, we introduce correspondence probabilities instead of exact correspondences. The correspondence probabilities are found by aligning the observation shapes with the affine expectation maximization-iterative closest points (EM-ICP) registration algorithm. In a second step, the correspondence probabilities are used as input to compute a mean shape (represented once again by an unstructured point set). Both steps are unified in a single optimization criterion which depe nds on the two parameters 'registration transformation' and 'mean shape'. In a last step, a variability model which best represents the variability in the training data set is computed. Experiments on synthetic data sets and in vivo brain structure data sets (MRI) are then designed to evaluate the performance of our algorithm. Results: The new method was applied to brain MRI data sets, and the estimated point correspondences were compared to a statistical shape model built on exact correspondences. Based on established measures of "generalization ability" and "specificity", the estimates were very satisfactory. Conclusion: The novel algorithm for building a generative statistical shape model (gSSM) does not need one-to-one point correspondences but relies solely on point correspondence probabilities for the computation of mean shape and eigenmodes. It is well-suited for shape analysis on unstructured point sets.

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Shape Analysis Using the Fisher-Rao Riemannian Metric: Unifying Shape Representation and Deformation

Cited by 34 publications

References 16 publications

On the Limiting Behaviour of the Fundamental Geodesics of Information Geometry

On the Limiting Behaviour of the Fundamental Geodesics of Information Geometry

Riemannian Analysis of Probability Density Functions with Applications in Vision

Generation of a statistical shape model with probabilistic point correspondences and the expectation maximization- iterative closest point algorithm

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