Intrinsic shape analysis in archaeology: A case study on ancient sundials

This paper explores a novel mathematical approach to extract archaeological insights from ensembles of similar artifact shapes. We show that by considering all the shape information in a find collection, it is possible to identify shape patterns that would be difficult to discern by considering the artifacts individually or by classifying shapes into predefined archaeological types and analyzing the associated distinguishing characteristics. Recently, series of high-resolution digital representations of artifacts have become available, and we explore their potential on a set of 3D models of ancient Greek and Roman sundials, with the aim of providing alternatives to the traditional archaeological method of ``trend extraction by ordination'' (typology). In the proposed approach, each 3D shape is represented as a point in a shape space -- a high-dimensional, curved, non-Euclidean space. By performing regression in shape space, we find that for Roman sundials, the bend of the sundials' shadow-receiving surface changes with the location's latitude. This suggests that, apart from the inscribed hour lines, also a sundial's shape was adjusted to the place of installation. As an example of more advanced inference, we use the identified trend to infer the latitude at which a sundial, whose installation location is unknown, was placed. We also derive a novel method for differentiated morphological trend assertion, building upon and extending the theory of geometric statistics and shape analysis. Specifically, we present a regression-based method for statistical normalization of shapes that serves as a means of disentangling parameter-dependent effects (trends) and unexplained variability.Comment: accepted for publication from the ACM Journal on Computing and Cultural Heritag

Geometric Data Analysis: Advancements of the Statistical Methodology and Applications

Data analysis has become fundamental to our society and comes in multiple facets and approaches. Nevertheless, in research and applications, the focus was primarily on data from Euclidean vector spaces. Consequently, the majority of methods that are applied today are not suited for more general data types. Driven by needs from fields like image processing, (medical) shape analysis, and network analysis, more and more attention has recently been given to data from non-Euclidean spaces–particularly (curved) manifolds. It has led to the field of geometric data analysis whose methods explicitly take the structure (for example, the topology and geometry) of the underlying space into account. This thesis contributes to the methodology of geometric data analysis by generalizing several fundamental notions from multivariate statistics to manifolds. We thereby focus on two different viewpoints. First, we use Riemannian structures to derive a novel regression scheme for general manifolds that relies on splines of generalized Bézier curves. It can accurately model non-geodesic relationships, for example, time-dependent trends with saturation effects or cyclic trends. Since Bézier curves can be evaluated with the constructive de Casteljau algorithm, working with data from manifolds of high dimensions (for example, a hundred thousand or more) is feasible. Relying on the regression, we further develop a hierarchical statistical model for an adequate analysis of longitudinal data in manifolds, and a method to control for confounding variables. We secondly focus on data that is not only manifold- but even Lie group-valued, which is frequently the case in applications. We can only achieve this by endowing the group with an affine connection structure that is generally not Riemannian. Utilizing it, we derive generalizations of several well-known dissimilarity measures between data distributions that can be used for various tasks, including hypothesis testing. Invariance under data translations is proven, and a connection to continuous distributions is given for one measure. A further central contribution of this thesis is that it shows use cases for all notions in real-world applications, particularly in problems from shape analysis in medical imaging and archaeology. We can replicate or further quantify several known findings for shape changes of the femur and the right hippocampus under osteoarthritis and Alzheimer's, respectively. Furthermore, in an archaeological application, we obtain new insights into the construction principles of ancient sundials. Last but not least, we use the geometric structure underlying human brain connectomes to predict cognitive scores. Utilizing a sample selection procedure, we obtain state-of-the-art results

The Role of Riemannian Manifolds in Computer Vision: From Coding to Deep Metric Learning

A diverse number of tasks in computer vision and machine learning enjoy from representations of data that are compact yet discriminative, informative and robust to critical measurements. Two notable representations are offered by Region Covariance Descriptors (RCovD) and linear subspaces which are naturally analyzed through the manifold of Symmetric Positive Definite (SPD) matrices and the Grassmann manifold, respectively, two widely used types of Riemannian manifolds in computer vision. As our first objective, we examine image and video-based recognition applications where the local descriptors have the aforementioned Riemannian structures, namely the SPD or linear subspace structure. Initially, we provide a solution to compute Riemannian version of the conventional Vector of Locally aggregated Descriptors (VLAD), using geodesic distance of the underlying manifold as the nearness measure. Next, by having a closer look at the resulting codes, we formulate a new concept which we name Local Difference Vectors (LDV). LDVs enable us to elegantly expand our Riemannian coding techniques to any arbitrary metric as well as provide intrinsic solutions to Riemannian sparse coding and its variants when local structured descriptors are considered. We then turn our attention to two special types of covariance descriptors namely infinite-dimensional RCovDs and rank-deficient covariance matrices for which the underlying Riemannian structure, i.e. the manifold of SPD matrices is out of reach to great extent. %Generally speaking, infinite-dimensional RCovDs offer better discriminatory power over their low-dimensional counterparts. To overcome this difficulty, we propose to approximate the infinite-dimensional RCovDs by making use of two feature mappings, namely random Fourier features and the Nystrom method. As for the rank-deficient covariance matrices, unlike most existing approaches that employ inference tools by predefined regularizers, we derive positive definite kernels that can be decomposed into the kernels on the cone of SPD matrices and kernels on the Grassmann manifolds and show their effectiveness for image set classification task. Furthermore, inspired by attractive properties of Riemannian optimization techniques, we extend the recently introduced Keep It Simple and Straightforward MEtric learning (KISSME) method to the scenarios where input data is non-linearly distributed. To this end, we make use of the infinite dimensional covariance matrices and propose techniques towards projecting on the positive cone in a Reproducing Kernel Hilbert Space (RKHS). We also address the sensitivity issue of the KISSME to the input dimensionality. The KISSME algorithm is greatly dependent on Principal Component Analysis (PCA) as a preprocessing step which can lead to difficulties, especially when the dimensionality is not meticulously set. To address this issue, based on the KISSME algorithm, we develop a Riemannian framework to jointly learn a mapping performing dimensionality reduction and a metric in the induced space. Lastly, in line with the recent trend in metric learning, we devise end-to-end learning of a generic deep network for metric learning using our derivation

Topological and geometric inference of data

The overarching problem under consideration is to determine the structure of the subspace on which a distribution is supported, given only a finite noisy sample thereof. The special case in which the subspace is an embedded manifold is given particular attention owing to its conceptual elegance, and asymptotic bounds are obtained on the admissible level of noise such that the manifold can be recovered up to homotopy equivalence. Attention is turned on how to accomplish this in practice. Following ideas from topological data analysis, simplicial complexes are used as discrete analogues of spaces suitable for computation. By utilising the prior assumption that the data lie on a manifold, topologically inspired techniques are proposed for refining the simplicial complex to better approximate this manifold. This is applied to the problem of nonlinear dimensionality reduction and found to improve accuracy of reconstructing several synthetic and real-world datasets. The second chapter focuses on extending this work to the case where the ambient space is non-Euclidean. The interfaces between topological data analysis, functional data analysis, and shape analysis are thoroughly explored. Lipschitz bounds are proved which relate several metrics on the space of positive semidefinite matrices; they are then interpreted in the context of topological data analysis. This is applied to diffusion tensor imaging and phonology. The final chapter explores the case where the points are non-uniformly distributed over the embedded subspace. In particular, a method is proposed to overcome the shortcomings of witness complex construction when there are large deviations in the density. The theory of multidimensional persistence is leveraged to provide a succinct setting in which the structure of the data can be interpreted as a generalised stratified space.EPSR

Combining shape and color. A bottom-up approach to evaluate object similarities

The objective of the present work is to develop a bottom-up approach to estimate the similarity between two unknown objects. Given a set of digital images, we want to identify the main objects and to determine whether they are similar or not. In the last decades many object recognition and classification strategies, driven by higher-level activities, have been successfully developed. The peculiarity of this work, instead, is the attempt to work without any training phase nor a priori knowledge about the objects or their context. Indeed, if we suppose to be in an unstructured and completely unknown environment, usually we have to deal with novel objects never seen before; under these hypothesis, it would be very useful to define some kind of similarity among the instances under analysis (even if we do not know which category they belong to). To obtain this result, we start observing that human beings use a lot of information and analyze very different aspects to achieve object recognition: shape, position, color and so on. Hence we try to reproduce part of this process, combining different methodologies (each working on a specific characteristic) to obtain a more meaningful idea of similarity. Mainly inspired by the human conception of representation, we identify two main characteristics and we called them the implicit and explicit models. The term "explicit" is used to account for the main traits of what, in the human representation, connotes a principal source of information regarding a category, a sort of a visual synecdoche (corresponding to the shape); the term "implicit", on the other hand, accounts for the object rendered by shadows and lights, colors and volumetric impression, a sort of a visual metonymy (corresponding to the chromatic characteristics). During the work, we had to face several problems and we tried to define specific solutions. In particular, our contributions are about: - defining a bottom-up approach for image segmentation (which does not rely on any a priori knowledge); - combining different features to evaluate objects similarity (particularly focusiing on shape and color); - defining a generic distance (similarity) measure between objects (without any attempt to identify the possible category they belong to); - analyzing the consequences of using the number of modes as an estimation of the number of mixture’s components (in the Expectation-Maximization algorithm)

Statistical computing on manifolds: from Riemannian geometry to computational anatomy

International audienceComputational anatomy is an emerging discipline that aims at analyzing and modeling the individual anatomy of organs and their biological variability across a population. The goal is not only to model the normal variations among a population, but also discover morphological differences between normal and pathological populations, and possibly to detect, model and classify the pathologies from structural abnormalities. Applications are very important both in neuroscience, to minimize the influence of the anatomical variability in functional group analysis, and in medical imaging, to better drive the adaptation of generic models of the anatomy (atlas) into patient-specific data (personalization).However, understanding and modeling the shape of organs is made difficult by the absence of physical models for comparing different subjects, the complexity of shapes, and the high number of degrees of freedom implied. Moreover, the geometric nature of the anatomical features usually extracted raises the need for statistics and computational methods on objects that do not belong to standard Euclidean spaces. We investigate in this chapter the Riemannian metric as a basis for developing generic algorithms to compute on manifolds. We show that few computational tools derived from this structure can be used in practice as the atoms to build more complex generic algorithms such as mean computation, Mahalanobis distance, interpolation, filtering and anisotropic diffusion on fields of geometric features. This computational framework is illustrated with the joint estimation and anisotropic smoothing of diffusion tensor images and with the modeling of the brain variability from sulcal lines

Proceedings of the Fourth International Workshop on Mathematical Foundations of Computational Anatomy - Geometrical and Statistical Methods for Biological Shape Variability Modeling (MFCA 2013), Nagoya, Japan

International audienceComputational anatomy is an emerging discipline at the interface of geometry, statistics and image analysis which aims at modeling and analyzing the biological shape of tissues and organs. The goal is to estimate representative organ anatomies across diseases, populations, species or ages, to model the organ development across time (growth or aging), to establish their variability, and to correlate this variability information with other functional, genetic or structural information. The Mathematical Foundations of Computational Anatomy (MFCA) workshop aims at fostering the interactions between the mathematical community around shapes and the MICCAI community in view of computational anatomy applications. It targets more particularly researchers investigating the combination of statistical and geometrical aspects in the modeling of the variability of biological shapes. The workshop is a forum for the exchange of the theoretical ideas and aims at being a source of inspiration for new methodological developments in computational anatomy. A special emphasis is put on theoretical developments, applications and results being welcomed as illustrations. Following the first edition of this workshop in 2006, second edition in New-York in 2008, the third edition in Toronto in 2011, the forth edition was held in Nagoya Japan on September 22 2013