The Bootstrap and Edgeworth Expansion

In 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 ?2 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.

/pdf/intrinsic-statistics-on-riemannian-manifolds-basic-tools-for-2uw8qmp8c5.pdf

Intrinsic Statistics on Riemannian Manifolds: Basic Tools for Geometric Measurements

Morphometric tools for landmark data: geometry and biology

For analyzing shapes of planar, closed curves, we propose differential geometric representations of curves using their direction functions and curvature functions. Shapes are represented as elements of infinite-dimensional spaces and their pairwise differences are quantified using the lengths of geodesics connecting them on these spaces. We use a Fourier basis to represent tangents to the shape spaces and then use a gradient-based shooting method to solve for the tangent that connects any two shapes via a geodesic. Using the Surrey fish database, we demonstrate some applications of this approach: 1) interpolation and extrapolations of shape changes, 2) clustering of objects according to their shapes, 3) statistics on shape spaces, and 4) Bayesian extraction of shapes in low-quality images.

Analysis of planar shapes using geodesic paths on shape spaces

Color science is a multidisciplinary field with broad applications in industries such as digital imaging, coatings and textiles, food, lighting, archiving, art, and fashion. Accurate definition and measurement of color appearance is a challenging task that directly affects color reproduction in such applications. Color Appearance Models addresses those challenges and offers insight into the preferred solutions. Extensive research on the human visual system (HVS) and color vision has been performed in the last century, and this book contains a good overview of the most important and relevant literature regarding color appearance models.

https://www.spiedigitallibrary.org/journals/Journal-of-Electronic-Imaging/volume-23/issue-2/029901/Color-Appearance-Models/10.1117/1.JEI.23.2.029901.pdf

Color Appearance Models

This article develops nonparametric inference procedures for estimation and testing problems for means on manifolds. A central limit theorem for Frechet sample means is derived leading to an asymptotic distribution theory of intrinsic sample means on Riemannian manifolds. Central limit theorems are also obtained for extrinsic sample means w.r.t. an arbitrary embedding of a differentiable manifold in a Euclidean space. Bootstrap methods particularly suitable for these problems are presented. Applications are given to distributions on the sphere Sd (directional spaces), real projective space ℝPN−1 (axial spaces), complex projective space ℂPk−2 (planar shape spaces) w.r.t. Veronese–Whitney embeddings and a three-dimensional shape space Σ34.

/pdf/large-sample-theory-of-intrinsic-and-extrinsic-sample-means-2a76fz6xmf.pdf

Large sample theory of intrinsic and extrinsic sample means on manifolds—II

This article develops nonparametric inference procedures for estimation and testing problems for means on manifolds. A central limit theorem for Frechet sample means is derived leading to an asymptotic distribution theory of intrinsic sample means on Riemannian manifolds. Central limit theorems are also obtained for extrinsic sample means w.r.t. an arbitrary embedding of a differentiable manifold in a Euclidean space. Bootstrap methods particularly suitable for these problems are presented. Applications are given to distributions on the sphere S^d (directional spaces), real projective space RP^{N-1} (axial spaces), complex projective space CP^{k-2} (planar shape spaces) w.r.t. Veronese-Whitney embeddings and a three-dimensional shape space \Sigma_3^4.

Large sample theory of intrinsic and extrinsic sample means on manifolds--II

This paper deals with projective shape analysis, which is a study of finite configurations of points modulo projective transformations. The topic has various applications in machine vision. We introduce a convenient projective shape space, as well as an appropriate coordinate system for this shape space. For generic configurations of k points in m dimensions, the resulting projective shape space is identified as a product of k - m -1 copies of axial spaces RP m . This identification leads to the need for developing multivariate directional and multivariate axial analysis and we propose parametric models, as well as nonparametric methods, for these areas. In particular, we investigate the Frechet extrinsic mean for the multivariate axial case. Asymptotic distributions of the appropriate parametric and nonparametric tests are derived. We illustrate our methodology with examples from machine vision.

/pdf/directions-and-projective-shapes-1lazcglumv.pdf

Directions and projective shapes

Given a probability distribution on an open book (a metric space obtained by gluing a disjoint union of copies of a half-space along their boundary hyperplanes), we define a precise concept of when the Fr\'{e}chet mean (barycenter) is sticky This nonclassical phenomenon is quantified by a law of large numbers (LLN) stating that the empirical mean eventually almost surely lies on the (codimension $1$ and hence measure $0$) spine that is the glued hyperplane, and a central limit theorem (CLT) stating that the limiting distribution is Gaussian and supported on the spine We also state versions of the LLN and CLT for the cases where the mean is nonsticky (ie, not lying on the spine) and partly sticky (ie, is, on the spine but not sticky)

Vic Patrangenaru

Papers

Large sample theory of intrinsic and extrinsic sample means on manifolds—II

Large sample theory of intrinsic and extrinsic sample means on manifolds--II

Directions and projective shapes

Sticky central limit theorems on open books

Nonparametic estimation of location and dispersion on Riemannian manifolds