Comparative Analysis of Kernel Methods for Statistical Shape Learning

Rathi, Yogesh; Dambreville, Samuel; Tannenbaum, Allen

doi:10.1007/11889762_9

Cited by 9 publications

(7 citation statements)

References 14 publications

(27 reference statements)

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“…Later, Twining and Taylor (2001) proposed the use of Kernel PCA instead, which they claim to be more general than other methods. Recently, there is a growing interest in using Kernel PCA for implicit shape analysis (Cremers et al, 2003;Dambreville et al, 2008;Rathi et al, 2006). Major reason for this development is that non-linear decompositions can cope with the level-set specific problem mentioned in Section 1.1.2, i.e.…”

Section: Dimensionality Reductionmentioning

confidence: 99%

Statistical shape models for 3D medical image segmentation: A review

Heimann

Meinzer

2009

Medical Image Analysis

1,245

752

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Section: Dimensionality Reductionmentioning

confidence: 99%

Statistical shape models for 3D medical image segmentation: A review

Heimann

Meinzer

2009

Medical Image Analysis

1,245

752

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“…This implies that there is no hope in obtaining a dense training set, even with a loose grid (say N bins for each degree of freedom, which makes N d bins), even with billions of examples. Consequently, methods involving only distances [12] or nearest neighbors [6,13,16] are not likely to be successful for high d, whereas our approach based on transport of …”

Section: Propagating Informationmentioning

confidence: 93%

“…Distance-based algorithms, such as kernel methods, were proposed [12,6] to handle high variability, but at the price of considering only distances between shapes, instead of deformations, thus losing the crucial information they carry. These methods consider training sets as graphs, whose nodes are shapes and whose edges are distances (for a particular metric chosen).…”

Section: Introductionmentioning

confidence: 99%

Learning shape metrics based on deformations and transport

Charpiat

2009

2009 IEEE 12th International Conference on Computer Vision Workshops, ICCV Workshops

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Shape evolutions, as well as shape matchings or image segmentation with shape prior, involve the preliminary choice of a suitable metric in the space of shapes. Instead of choosing a particular one, we propose a framework to learn shape metrics from a set of examples of shapes, designed to be able to handle sparse sets of highly varying shapes, since typical shape datasets, like human silhouettes, are intrinsically high-dimensional and non-dense. We formulate the task of finding the optimal metrics on an empirical manifold of shapes as a classical minimization problem ensuring smoothness, and compute its global optimum fast.First, we design a criterion to compute point-to-point matching between shapes which deals with topological changes. Then, given a training set of shapes, we use these matchings to transport deformations observed on any shape to any other one. Finally, we estimate the metric in the tangent space of any shape, based on transported deformations, weighted by their reliability. Experiments on difficult sets are shown, and applications are proposed.

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“…Ker-nel density estimation in feature space was introduced by Cremers et al [15] to incorporate the probability of 2D silhouettes of 3D objects in 2D image segmentation. An overview on related kernel density methods is given by Rathi et al [37]. An abstract distance measure between objects in (different) metric spaces is the Gromov-Hausdorff distance, which allows to compute an isometrically invariant distance measure between shapes.…”

Section: Review Of Related Workmentioning

confidence: 99%

An Elasticity-Based Covariance Analysis of Shapes

Rumpf

Wirth

2010

Int J Comput Vis

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We introduce the covariance of a number of given shapes if they are interpreted as boundary contours of elastic objects. Based on the notion of nonlinear elastic deformations from one shape to another, a suitable linearization of geometric shape variations is introduced. Once such a linearization is available, a principal component analysis can be investigated. This requires the definition of a covariance metric-an inner product on linearized shape variations. The resulting covariance operator robustly captures strongly nonlinear geometric variations in a physically meaningful way and allows to extract the dominant modes of shape variation. The underlying elasticity concept represents an alternative to Riemannian shape statistics. In this paper we compare a standard L 2 -type covariance metric with a metric based on the Hessian of the nonlinear elastic energy. Furthermore, we explore the dependence of the principal component analysis on the type of the underlying nonlinear elasticity. For the built-in pairwise elastic registration, a relaxed model formulation is employed which allows for a non-exact matching. Shape contours are approximated by single well phase fields, which enables an extension of the method to a covariance analysis of image morphologies. The model is implemented with multilinear finite elements embedded in a multi-scale approach. The characteristics of the approach are demonstrated on a number of illustrative and real world examples in 2D and 3D.

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Comparative Analysis of Kernel Methods for Statistical Shape Learning

Cited by 9 publications

References 14 publications

Statistical shape models for 3D medical image segmentation: A review

Statistical shape models for 3D medical image segmentation: A review

Learning shape metrics based on deformations and transport

An Elasticity-Based Covariance Analysis of Shapes

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