Introduction to Riemannian Geometry and Geometric Statistics: from basic theory to implementation with Geomstats

International audienceAs data is a predominant resource in applications, Riemannian geometry is a natural framework to model and unify complex nonlinear sources of data.However, the development of computational tools from the basic theory of Riemannian geometry is laborious.The work presented here forms one of the main contributions to the open-source project geomstats, that consists in a Python package providing efficient implementations of the concepts of Riemannian geometry and geometric statistics, both for mathematicians and for applied scientists for whom most of the difficulties are hidden under high-level functions. The goal of this monograph is two-fold. First, we aim at giving a self-contained exposition of the basic concepts of Riemannian geometry, providing illustrations and examples at each step and adopting a computational point of view. The second goal is to demonstrate how these concepts are implemented in Geomstats, explaining the choices that were made and the conventions chosen. The general concepts are exposed and specific examples are detailed along the text.The culmination of this implementation is to be able to perform statistics and machine learning on manifolds, with as few lines of codes as in the wide-spread machine learning tool scikit-learn. We exemplify this with an introduction to geometric statistics

Facilitating Colorectal Cancer Diagnosis with Computed Tomographic Colonography

Computed tomographic colonography (CTC) is a diagnostic technique involving helical volume acquisition of the cleansed, distended colorectum to detect colorectal cancer or potentially premalignant polyps. This Thesis summarises the evidence base, identifies areas in need of further research, quantifies sources of bias and presents novel techniques to facilitate colorectal cancer diagnosis using CTC. CTC literature is reviewed to justify the rationale for current implementation and to identify fruitful areas for research. This confirms excellent diagnostic performance can be attained providing CTC is interpreted by trained, experienced observers employing state-of-the-art implementation. The technique is superior to barium enema and consequently, it has been embraced by radiologists, clinicians and health policy-makers. Factors influencing generalisability of CTC research are investigated, firstly with a survey of European educational workshop participants which revealed limited CTC experience and training, followed by a systematic review exploring bias in research studies of diagnostic test accuracy which established that studies focussing on these aspects were lacking. Experiments to address these sources of bias are presented, using novel methodology: Conjoint analysis is used to ascertain patients‘ and clinicians’ attitudes to false-positive screening diagnoses, showing that both groups overwhelmingly value sensitivity over specificity. The results inform a weighted statistical analysis for CAD which is applied to the results of two previous studies showing the incremental benefit is significantly higher for novices than experienced readers. We have employed eye-tracking technology to establish the visual search patterns of observers reading CTC, demonstrated feasibility and developed metrics for analysis. We also describe development and validation of computer software to register prone and supine endoluminal surface locations demonstrating accurate matching of corresponding points when applied to a phantom and a generalisable, publically available, CTC database. Finally, areas in need of future development are suggested