Riemannian geometries on spaces of plane curves

Michor, Peter W.; Mumford, David

doi:10.4171/jems/37

Cited by 317 publications

(364 citation statements)

References 10 publications

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“…This metric was first studied in the context of shape analysis in [95]. The geodesic equation for the…”

Section: The Space Of Riemannian Metricsmentioning

confidence: 99%

“…We have the following result. For the space B i,f (S 1 , R 2 ) an explicit construction of the path with arbitrarily short length was given in [95]. Heuristically, if the curve is made to zig-zag wildly, then the normal component of the motion will be inversely proportional to the length of the curve.…”

Section: The Space Of Riemannian Metricsmentioning

confidence: 99%

“…with A > 0 in [95] and the conformal rescalings of the [115,135], both on the space of plane curves. More general almost local metrics on the space of plane curves were considered in [96] and they have been generalized to hypersurfaces in higher dimensions in [3,12,13].…”

Section: Almost Local Metrics On Shape Spacementioning

confidence: 99%

See 2 more Smart Citations

Overview of the Geometries of Shape Spaces and Diffeomorphism Groups

2014

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This article provides an overview of various notions of shape spaces, including the space of parametrized and unparametrized curves, the space of immersions, the diffeomorphism group and the space of Riemannian metrics. We discuss the Riemannian metrics that can be defined thereon, and what is known about the properties of these metrics. We put particular emphasis on the induced geodesic distance, the geodesic equation and its well-posedness, geodesic and metric completeness and properties of the curvature.

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“…This metric was first studied in the context of shape analysis in [95]. The geodesic equation for the…”

Section: The Space Of Riemannian Metricsmentioning

confidence: 99%

Section: The Space Of Riemannian Metricsmentioning

confidence: 99%

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Overview of the Geometries of Shape Spaces and Diffeomorphism Groups

2014

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“…There is some previous work on computing geodesic distances in the space of curves [7,8], but there is little work when the shape is represented by signed distance functions.…”

Section: Euclidean Distance Between Signed Distance Functionsmentioning

confidence: 99%

Nonparametric shape priors for active contour-based image segmentation

2007

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When segmenting images of low quality or with missing data, statistical prior information about the shapes of the objects to be segmented can significantly aid the segmentation process. However, defining probability densities in the space of shapes is an open and challenging problem. In this paper, we propose a nonparametric shape prior model for image segmentation problems. In particular, given example training shapes, we estimate the underlying shape distribution by extending a Parzen density estimator to the space of shapes. Such density estimates are expressed in terms of distances between shapes, and we propose two distance metrics that could be used in this framework. We then incorporate the learned shape prior distribution into a maximum a posteriori estimation framework for segmentation. This results in an optimization problem, which we solve using active contours. We demonstrate the effectiveness of the resulting algorithm in segmenting images that involve low-quality data and occlusions. The proposed framework is especially powerful in handling "multimodal" shape densities, involving multiple classes of objects.

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“…Once again, the k nearest neighbors can be computed using the distance relation (9) or any other metric on the space of shapes [22,11,7,[23][24][25].…”

Section: Kernel Llementioning

confidence: 99%

Comparative Analysis of Kernel Methods for Statistical Shape Learning

Rathi

Dambreville

Tannenbaum

2006

Computer Vision Approaches to Medical Image Analysis

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Abstract. Prior knowledge about shape may be quite important for image segmentation. In particular, a number of different methods have been proposed to compute the statistics on a set of training shapes, which are then used for a given image segmentation task to provide the shape prior. In this work, we perform a comparative analysis of shape learning techniques such as linear PCA, kernel PCA, locally linear embedding and propose a new method, kernelized locally linear embedding for doing shape analysis. The surfaces are represented as the zero level set of a signed distance function and shape learning is performed on the embeddings of these shapes. We carry out some experiments to see how well each of these methods can represent a shape, given the training set.

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Riemannian geometries on spaces of plane curves

Cited by 317 publications

References 10 publications

Overview of the Geometries of Shape Spaces and Diffeomorphism Groups

Overview of the Geometries of Shape Spaces and Diffeomorphism Groups

Nonparametric shape priors for active contour-based image segmentation

Comparative Analysis of Kernel Methods for Statistical Shape Learning

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