Intrinsic Statistics on Riemannian Manifolds: Basic Tools for Geometric Measurements

A preliminary appeared as INRIA RR-5093, January 2004.International audienceIn medical image analysis and high level computer vision, there is an intensive use of geometric features like orientations, lines, and geometric transformations ranging from simple ones (orientations, lines, rigid body or affine transformations, etc.) to very complex ones like curves, surfaces, or general diffeomorphic transformations. The measurement of such geometric primitives is generally noisy in real applications and we need to use statistics either to reduce the uncertainty (estimation), to compare observations, or to test hypotheses. Unfortunately, even simple geometric primitives often belong to manifolds that are not vector spaces. In previous works [1, 2], we investigated invariance requirements to build some statistical tools on transformation groups and homogeneous manifolds that avoids paradoxes. In this paper, we consider finite dimensional manifolds with a Riemannian metric as the basic structure. Based on this metric, we develop the notions of mean value and covariance matrix of a random element, normal law, Mahalanobis distance and X² law. We provide a new proof of the characterization of Riemannian centers of mass and an original gradient descent algorithm to efficiently compute them. The notion of Normal law we propose is based on the maximization of the entropy knowing the mean and covariance of the distribution. The resulting family of pdfs spans the whole range from uniform (on compact manifolds) to the point mass distribution. Moreover, we were able to provide tractable approximations (with their limits) for small variances which show that we can effectively implement and work with these definitions

Interrater and intrarater reliability of four different classification methods for evaluating acromial morphology on standardized radiographs

Background: Acromial morphology is an important pathophysiological factor for the development of subacromial impingement syndrome. There are 3 radiological methods to evaluate acromial morphology: Bigliani, modified Epstein, and acromial angle. However, their reliability have not been compared in a single study, nor using standardized radiographs. Consequently, the evaluation of acromial morphology is currently not validated though its widespread use across the world. The objective of this study was to investigate reliability of the 3 known classifications and the novel Copenhagen Acromial Curve classification. Methods: Three experienced clinicians rated 102 standardized supraspinatus outlet view radiographs with the 4 classification methods in 2 separate sessions a month apart. All measurements were blinded. With an expected kappa (κ) and intraclass correlation coefficient (ICC) > 0.7 (+/−0.15), the target sample size was 87 radiographs. Results: The Bigliani classification had interrater and intrarater reliability ranging from fair to good (κ 0.32-0.41 and 0.26-0.62). The modified Epstein classification had fair to good interrater and intrarater reliability (κ 0.24-0.69 and 0.57-0.63). The acromial angle classification had moderate to good interrater and intrarater reliability (κ 0.53-0.60 and 0.59-0.72). The novel Copenhagen Acromial Curve classification showed moderate to good interrater and intrarater reliability (ICC 0.66-0.71 and 0.75-0.78, respectively). Conclusion: The Copenhagen Acromial Curve was the only classification method with an ICC value > 0.7. The popular Bigliani classification had the worst reliability. The Copenhagen Acromial Curve classification produces numerical data, as opposed to the other 3 classification methods. This could potentially be utilized in future research to establishing cut-off values for treatment stratification