Statistics on Lie groups : a need to go beyond the pseudo-Riemannian framework

Abstract. Lie groups appear in many fields from Medical Imaging to Robotics. In Medical Imaging and particularly in Computational Anatomy, an organ's shape is often modeled as the deformation of a reference shape, in other words: as an element of a Lie group. In this framework, if one wants to model the variability of the human anatomy, e.g. in order to help diagnosis of diseases, one needs to perform statistics on Lie groups. A Lie group G is a manifold that carries an additional group structure. Statistics on Riemannian manifolds have been well studied with the pioneer work of Fréchet, Karcher and Kendall [1, 2, 3, 4] followed by other

Statistics on Lie groups : a need to go beyond the pseudo-Riemannian framework

International audienceLie groups appear in many fields from Medical Imaging to Robotics. In Medical Imaging and particularly in Computational Anatomy, an organ's shape is often modeled as the deformation of a reference shape, in other words: as an element of a Lie group. In this framework, if one wants to model the variability of the human anatomy, e.g. in order to help diagnosis of diseases, one needs to perform statistics on Lie groups. A Lie group G is a manifold that carries an additional group structure. Statistics on Riemannian manifolds have been well studied with the pioneer work of Fréchet, Karcher and Kendall [1, 2, 3, 4] followed by others [5, 6, 7, 8, 9]. In order to use such a Riemannian structure for statistics on Lie groups, one needs to define a Riemannian metric that is compatible with the group structure, i.e a bi-invariant metric. However, it is well known that general Lie groups which cannot be decomposed into the direct product of compact and abelian groups do not admit a bi-invariant metric. One may wonder if removing the positivity of the metric, thus asking only for a bi-invariant pseudo-Riemannian metric, would be sufficient for most of the groups used in Computational Anatomy. In this paper, we provide an algorithmic procedure that constructs bi-invariant pseudo-metrics on a given Lie group G . The procedure relies on a classification theorem of Medina and Revoy. However in doing so, we prove that most Lie groups do not admit any bi-invariant (pseudo-) metric. We conclude that the (pseudo-) Riemannian setting is not the richest setting if one wants to perform statistics on Lie groups. One may have to rely on another framework, such as affine connection space

Invariant higher-order variational problems II

Motivated by applications in computational anatomy, we consider a second-order problem in the calculus of variations on object manifolds that are acted upon by Lie groups of smooth invertible transformations. This problem leads to solution curves known as Riemannian cubics on object manifolds that are endowed with normal metrics. The prime examples of such object manifolds are the symmetric spaces. We characterize the class of cubics on object manifolds that can be lifted horizontally to cubics on the group of transformations. Conversely, we show that certain types of non-horizontal geodesics on the group of transformations project to cubics. Finally, we apply second-order Lagrange--Poincar\'e reduction to the problem of Riemannian cubics on the group of transformations. This leads to a reduced form of the equations that reveals the obstruction for the projection of a cubic on a transformation group to again be a cubic on its object manifold.Comment: 40 pages, 1 figure. First version -- comments welcome

Singular solutions, momentum maps and computational anatomy

This paper describes the variational formulation of template matching problems of computational anatomy (CA); introduces the EPDiff evolution equation in the context of an analogy between CA and fluid dynamics; discusses the singular solutions for the EPDiff equation and explains why these singular solutions exist (singular momentum map). Then it draws the consequences of EPDiff for outline matching problem in CA and gives numerical examples

GaBoDS: The Garching-Bonn Deep Survey -- I. Anatomy of galaxy clusters in the background of NGC 300

The Garching-Bonn Deep Survey (GaBoDS) is a virtual 12 square degree cosmic shear and cluster lensing survey, conducted with the [email protected] MPG/ESO telescope at La Silla. It consists of shallow, medium and deep random fields taken in R-band in subarcsecond seeing conditions at high galactic latitude. A substantial amount of the data was taken from the ESO archive, by means of a dedicated ASTROVIRTEL program. In the present work we describe the main characteristics and scientific goals of GaBoDS. Our strategy for mining the ESO data archive is introduced, and we comment on the Wide Field Imager data reduction as well. In the second half of the paper we report on clusters of galaxies found in the background of NGC 300, a random archival field. We use weak gravitational lensing and the red cluster sequence method for the selection of these objects. Two of the clusters found were previously known and already confirmed by spectroscopy. Based on the available data we show that there is significant evidence for substructure in one of the clusters, and an increasing fraction of blue galaxies towards larger cluster radii. Two other mass peaks detected by our weak lensing technique coincide with red clumps of galaxies. We estimate their redshifts and masses.Comment: 20 pages, 16 figures, gzipped. An online postscript version with higher quality figures (3.3 MBytes) can be downloaded from http://www.mpa-garching.mpg.de/~mischa/ngc300/ngc300.ps.gz . Submitted to A&