Stephan Huckemann

Principal component analysis for Riemannian manifolds, with an application to triangular shape spaces

Classical principal component analysis on manifolds, for example on Kendalls shape spaces, is carried out in the tangent space of a Euclidean mean equipped with a Euclidean metric. We propose a method of principal component analysis for Riemannian manifolds based on geodesics of the intrinsic metric, and provide a numerical implementation in the case of spheres. This method allows us, for example, to compare principal component geodesics of different data samples. In order to determine principal component geodesics, we show that in general, owing to curvature, the principal component geodesics do not pass through the intrinsic mean. As a consequence, means other than the intrinsic mean are considered, allowing for several choices of definition of geodesic variance. In conclusion we apply our method to the space of planar triangular shapes and compare our findings with those of standard Euclidean principal component analysis.

Sticky central limit theorems on open books

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 Frechet 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 (i.e., not lying on the spine) and partly sticky (i.e., is, on the spine but not sticky).

Intrinsic MANOVA for Riemannian Manifolds with an Application to Kendall's Space of Planar Shapes

We propose an intrinsic multifactorial model for data on Riemannian manifolds that typically occur in the statistical analysis of shape. Due to the lack of a linear structure, linear models cannot be defined in general; to date only one-way MANOVA is available. For a general multifactorial model, we assume that variation not explained by the model is concentrated near elements defining the effects. By determining the asymptotic distributions of respective sample covariances under parallel transport, we show that they can be compared by standard MANOVA. Often in applications manifolds are only implicitly given as quotients, where the bottom space parallel transport can be expressed through a differential equation. For Kendalls space of planar shapes, we provide an explicit solution. We illustrate our method by an intrinsic two-way MANOVA for a set of leaf shapes. While biologists can identify genotype effects by sight, we can detect height effects that are otherwise not identifiable.

Nested Sphere Statistics of Skeletal Models

We seek a form of object model that exactly and completely captures the interior of most non-branching anatomic objects and simultaneously is well suited for probabilistic analysis on populations of such objects. We show that certain nearly medial, skeletal models satisfy these requirements. These models are first mathematically defined in continuous three-space, and then discrete representations formed by a tuple of spoke vectors are derived. We describe means of fitting these skeletal models into manual or automatic segmentations of objects in a way stable enough to support statistical analysis, and we sketch means of modifying these fits to provide good correspondences of spoke vectors across a training population of objects. Understanding will be developed that these discrete skeletal models live in an abstract space made of a Cartesian product of a Euclidean space and a collection of spherical spaces. Based on this understanding and the way objects change under various rigid and nonrigid transformations, a method analogous to principal component analysis called composite principal nested spheres will be seen to apply to learning a more efficient collection of modes of object variation about a new and more representative mean object than those provided by other representations and other statistical analysis methods. The methods are illustrated by application to hippocampi.

INTRINSIC INFERENCE ON THE MEAN GEODESIC OF PLANAR SHAPES AND TREE DISCRIMINATION BY LEAF GROWTH

For planar landmark based shapes, taking into account the non-Euclidean geometry of the shape space, a statistical test for a common mean first geodesic principal component (GPC) is devised which rests on one of two asymptotic scenarios. For both scenarios, strong consistency and central limit theorems are established, along with an algorithm for the computation of a Ziezold mean geodesic. In application, this allows to verify the geodesic hypothesis for leaf growth of Canadian black poplars and to discriminate genetically different trees by observations of leaf shape growth over brief time intervals. With a test based on Procrustes tangent space coordinates, not involving the shape spaces curvature, neither can be achieved.

Filter Design and Performance Evaluation for Fingerprint Image Segmentation.

Fingerprint recognition plays an important role in many commercial applications and is used by millions of people every day, e.g. for unlocking mobile phones. Fingerprint image segmentation is typically the first processing step of most fingerprint algorithms and it divides an image into foreground, the region of interest, and background. Two types of error can occur during this step which both have a negative impact on the recognition performance: ‘true’ foreground can be labeled as background and features like minutiae can be lost, or conversely ‘true’ background can be misclassified as foreground and spurious features can be introduced. The contribution of this paper is threefold: firstly, we propose a novel factorized directional bandpass (FDB) segmentation method for texture extraction based on the directional Hilbert transform of a Butterworth bandpass (DHBB) filter interwoven with soft-thresholding. Secondly, we provide a manually marked ground truth segmentation for 10560 images as an evaluation benchmark. Thirdly, we conduct a systematic performance comparison between the FDB method and four of the most often cited fingerprint segmentation algorithms showing that the FDB segmentation method clearly outperforms these four widely used methods. The benchmark and the implementation of the FDB method are made publicly available.

Principal component geodesics for planar shape spaces

In this paper a numerical method to compute principal component geodesics for Kendalls planar shape spaces-which are essentially complex projective spaces-is presented. Underlying is the notion of principal component analysis based on geodesics for non-Euclidean manifolds as proposed in an earlier paper by Huckemann and Ziezold [S. Huckemann, H. Ziezold, Principal component analysis for Riemannian manifolds with an application to triangular shape spaces, Adv. Appl. Prob. (SGSA) 38 (2) (2006) 299-319]. Currently, principal component analysis for shape spaces is done on the basis of a Euclidean approximation. In this paper, using well-studied datasets and numerical simulations, these approximation errors are discussed. Overall, the error distribution is rather dispersed. The numerical findings back the notion that the Euclidean approximation is good for highly concentrated data. For low concentration, however, the error can be strongly notable. This is in particular the case for a small number of landmarks. For highly concentrated data, stronger anisotropicity and a larger number of landmarks may also increase the error.

Tree-Oriented Analysis of Brain Artery Structure

Statistical analysis of magnetic resonance angiography (MRA) brain artery trees is performed using two methods for mapping brain artery trees to points in phylogenetic treespace: cortical landmark correspondence and descendant correspondence. The differences in end-results based on these mappings are highlighted to emphasize the importance of correspondence in tree-oriented data analysis. Representation of brain artery systems as points in phylogenetic treespace, a mathematical space developed in (Billera et al. Adv. Appl. Math 27:733–767, 2001), facilitates this analysis. The phylogenetic treespace is a rich setting for tree-oriented data analysis. The Fréchet sample mean or an approximation is reported. Multidimensional scaling is used to explore structure in the data set based on pairwise distances between data points. This analysis of MRA data shows a statistically significant effect of age and sex on brain artery structure. Variation in the proximity of brain arteries to the cortical surface results in strong statistical difference between sexes and statistically significant age effect. That particular observation is possible with cortical correspondence but did not show up in the descendant correspondence.

Analysis of Rotational Deformations From Directional Data

This article discusses a novel framework to analyze rotational deformations of real three-dimensional objects. The rotational deformations such as twisting or bending have been observed as the major variation in some medical applications, where the features of the deformed three-dimensional objects are directional data. We propose modeling and estimation of the global deformations in terms of generalized rotations of directions. The proposed method can be cast as a generalized small circle fitting on the unit sphere. We also discuss the estimation of descriptors for more complex deformations composed of two simple deformations. The proposed method can be used for a number of different three-dimensional object models. Two analyses of three-dimensional object data are presented in detail: one using skeletal representations in medical image analysis and the other from biomechanical gait analysis of the knee joint. Supplementary materials for this article are available online.

Möbius deconvolution on the hyperbolic plane with application to impedance density estimation

In this paper we consider a novel statistical inverse problem on the Poincare, or Lobachevsky, upper (complex) half plane. Here the Riemannian structure is hyperbolic and a transitive group action comes from the space of 2 x 2 real matrices of determinant one via Mobius transformations. Our approach is based on a deconvolution technique which relies on the Helgason― Fourier calculus adapted to this hyperbolic space. This gives a minimax nonparametric density estimator of a hyperbolic density that is corrupted by a random Mobius transform. A motivation for this work comes from the reconstruction of impedances of capacitors where the above scenario on the Poincare plane exactly describes the physical system that is of statistical interest.