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Publications about Metric
Result of the query in the list of publications :
Article |
1 - Invariant Bayesian estimation on manifolds.
I. H. Jermyn. Annals of Statistics, 33(2): pages 583--605, April 2005.
Keywords : Bayesian estimation, MAP, MMSE, Invariant, Metric, Jeffrey's.
@ARTICLE{jermyn_annstat05,
|
| author |
= |
{Jermyn, I. H.}, |
| title |
= |
{Invariant Bayesian estimation on manifolds}, |
| year |
= |
{2005}, |
| month |
= |
{April}, |
| journal |
= |
{Annals of Statistics}, |
| volume |
= |
{33}, |
| number |
= |
{2}, |
| pages |
= |
{583--605}, |
| url |
= |
{http://dx.doi.org/10.1214/009053604000001273}, |
| pdf |
= |
{ftp://ftp-sop.inria.fr/ariana/Articles/jermyn_annstat05.pdf}, |
| keyword |
= |
{Bayesian estimation, MAP, MMSE, Invariant, Metric, Jeffrey's} |
| } |
Abstract :
A frequent and well-founded criticism of the maximum em a posteriori (MAP) and minimum mean squared error (MMSE) estimates of a continuous parameter param taking values in a differentiable manifold paramspace is that they are not invariant to arbitrary `reparametrizations' of paramspace. This paper clarifies the issues surrounding this problem, by pointing out the difference between coordinate invariance, which is a em sine qua non for a mathematically well-defined problem, and diffeomorphism invariance, which is a substantial issue, and then provides a solution. We first show that the presence of a metric structure on paramspace can be used to define coordinate-invariant MAP and MMSE estimates, and we argue that this is the natural way to proceed. We then discuss the choice of a metric structure on paramspace. By imposing an invariance criterion natural within a Bayesian framework, we show that this choice is essentially unique. It does not necessarily correspond to a choice of coordinates. In cases of complete prior ignorance, when Jeffreys' prior is used, the invariant MAP estimate reduces to the maximum likelihood estimate. The invariant MAP estimate coincides with the minimum message length (MML) estimate, but no discretization or approximation is used in its derivation. |
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3 Conference articles |
1 - Removing Shape-Preserving Transformations in Square-Root Elastic (SRE) Framework for Shape Analysis of Curves.
S. Joshi and E. Klassen and A. Srivastava and I. H. Jermyn. In Proc. Energy Minimization Methods in Computer Vision and Pattern Recognition (EMMCVPR), Ezhou, China, August 2007.
Keywords : Shape, Reparameterization, Metric, Geodesic.
Copyright : The original publication is available at www.springerlink.com.
@INPROCEEDINGS{Joshi07b,
|
| author |
= |
{Joshi, S. and Klassen, E. and Srivastava, A. and Jermyn, I. H.}, |
| title |
= |
{Removing Shape-Preserving Transformations in Square-Root Elastic (SRE) Framework for Shape Analysis of Curves}, |
| year |
= |
{2007}, |
| month |
= |
{August}, |
| booktitle |
= |
{Proc. Energy Minimization Methods in Computer Vision and Pattern Recognition (EMMCVPR)}, |
| address |
= |
{Ezhou, China}, |
| pdf |
= |
{ftp://ftp-sop.inria.fr/ariana/Articles/2007_Joshi07b.pdf}, |
| keyword |
= |
{Shape, Reparameterization, Metric, Geodesic} |
| } |
Abstract :
This paper illustrates and extends an efficient framework, called the square-root-elastic (SRE) framework, for studying shapes of closed curves, that was first introduced in [2]. This framework combines the strengths of two important ideas - elastic shape metric and path-straightening methods - for finding geodesics in shape spaces of curves. The elastic metric allows for optimal matching of features between curves while path-straightening ensures that the algorithm results in geodesic paths. This paper extends this framework by removing two important shape preserving transformations: rotations and re-parameterizations, by forming quotient spaces and constructing geodesics on these quotient spaces. These ideas are demonstrated using experiments involving 2D and 3D curves. |
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2 - A Novel Representation for Riemannian Analysis of Elastic Curves in R^n.
S. Joshi and E. Klassen and A. Srivastava and I. H. Jermyn. In Proc. IEEE Computer Vision and Pattern Recognition (CVPR), Minneapolis, USA, June 2007.
Keywords : Shape, Metric, Geodesic, Prior.
@INPROCEEDINGS{Joshi07a,
|
| author |
= |
{Joshi, S. and Klassen, E. and Srivastava, A. and Jermyn, I. H.}, |
| title |
= |
{A Novel Representation for Riemannian Analysis of Elastic Curves in R^n}, |
| year |
= |
{2007}, |
| month |
= |
{June}, |
| booktitle |
= |
{Proc. IEEE Computer Vision and Pattern Recognition (CVPR)}, |
| address |
= |
{Minneapolis, USA}, |
| url |
= |
{http://dx.doi.org/10.1109/CVPR.2007.383185}, |
| pdf |
= |
{ftp://ftp-sop.inria.fr/ariana/Articles/2007_Joshi07a.pdf}, |
| keyword |
= |
{Shape, Metric, Geodesic, Prior} |
| } |
Abstract :
We propose an efficient representation for studying shapes of closed curves in R^n. This paper combines the strengths of two important ideas---elastic shape metric and path-straightening methods---and results in a very fast algorithm for finding geodesics in shape spaces. The elastic metric allows for optimal matching of features between the two curves while path-straightening ensures that the algorithm results in geodesic paths. For the novel representation proposed here, the elastic metric becomes the simple L^2 metric, in contrast to the past usage where more complex forms were used. We present the step-by-step algorithms for computing geodesics and demonstrate them with 2-D as well as 3-D examples. |
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3 - Riemannian Analysis of Probability Density Functions with Applications in Vision.
S. Joshi and A. Srivastava and I. H. Jermyn. In Proc. IEEE Computer Vision and Pattern Recognition (CVPR), Minneapolis, USA, June 2007.
Keywords : Probability density function, Metric, Geodesic, Reparameterization.
@INPROCEEDINGS{Joshi07,
|
| author |
= |
{Joshi, S. and Srivastava, A. and Jermyn, I. H.}, |
| title |
= |
{Riemannian Analysis of Probability Density Functions with Applications in Vision}, |
| year |
= |
{2007}, |
| month |
= |
{June}, |
| booktitle |
= |
{Proc. IEEE Computer Vision and Pattern Recognition (CVPR)}, |
| address |
= |
{Minneapolis, USA}, |
| url |
= |
{http://dx.doi.org/10.1109/CVPR.2007.383188 }, |
| pdf |
= |
{ftp://ftp-sop.inria.fr/ariana/Articles/2007_Joshi07.pdf}, |
| keyword |
= |
{Probability density function, Metric, Geodesic, Reparameterization} |
| } |
Abstract :
Applications in computer vision involve statistically analyzing an important class of constrained, non- negative functions, including probability density functions (in texture analysis), dynamic time-warping functions (in activity analysis), and re-parametrization or non-rigid registration functions (in shape analysis of curves). For this one needs to impose a Riemannian structure on the spaces formed by these functions. We propose a em spherical version of the Fisher-Rao metric that provides closed form expressions for geodesics and distances, and allows an efficient computation of statistics. We compare this metric with some previously used metrics and present an application in planar shape classification. |
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