Statistical Computing on Non-Linear Spaces for 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. 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 on objects like curves, surfaces and deformations that do not belong to standard Euclidean spaces. We explain in this chapter how the Riemannian structure can provide a powerful framework to build generic statistical computing tools. We show that few computational tools derive for each Riemannian metric can be used in practice as the basic atoms to build more complex generic algorithms such as interpolation, filtering and anisotropic diffusion on fields of geometric features. This computational framework is illustrated with the analysis of the shape of the scoliotic spine and the modeling of the brain variability from sulcal lines where the results suggest new anatomical findings

Image Registration and Predictive Modeling: Learning the Metric on the Space of Diffeomorphisms

We present a method for metric optimization in the Large Deformation Diffeomorphic Metric Mapping (LDDMM) framework, by treating the induced Riemannian metric on the space of diffeomorphisms as a kernel in a machine learning context. For simplicity, we choose the kernel Fischer Linear Discriminant Analysis (KLDA) as the framework. Optimizing the kernel parameters in an Expectation-Maximization framework, we define model fidelity via the hinge loss of the decision function. The resulting algorithm optimizes the parameters of the LDDMM norm-inducing differential operator as a solution to a group-wise registration and classification problem. In practice, this may lead to a biology-aware registration, focusing its attention on the predictive task at hand such as identifying the effects of disease. We first tested our algorithm on a synthetic dataset, showing that our parameter selection improves registration quality and classification accuracy. We then tested the algorithm on 3D subcortical shapes from the Schizophrenia cohort Schizconnect. Our Schizophrenia-Control predictive model showed significant improvement in ROC AUC compared to baseline parameters

Open Problem: Polynomial linearly-convergent method for geodesically convex optimization?

Let $f \colon \mathcal{M} \to \mathbb{R}$ be a Lipschitz and geodesically convex function defined on a $d$-dimensional Riemannian manifold $\mathcal{M}$. Does there exist a first-order deterministic algorithm which (a) uses at most $O(\mathrm{poly}(d) \log(\epsilon^{-1}))$ subgradient queries to find a point with target accuracy $\epsilon$, and (b) requires only $O(\mathrm{poly}(d))$ arithmetic operations per query? In convex optimization, the classical ellipsoid method achieves this. After detailing related work, we provide an ellipsoid-like algorithm with query complexity $O(d^2 \log^2(\epsilon^{-1}))$ and per-query complexity $O(d^2)$ for the limited case where $\mathcal{M}$ has constant curvature (hemisphere or hyperbolic space). We then detail possible approaches and corresponding obstacles for designing an ellipsoid-like method for general Riemannian manifolds

El problema de Weber sobre variedades Riemannianas: Algunas cotas Superiores para el mínimo de la función de Weber

In this paper we obtain some upper bounds for the minimum of the Weber function on a strongly convex ball in a Riemannian manifold with positive sectional curvature; where the minimum is reached on the weighted geometric median of “m” given points in the strongly convex.En este artículo se obtiene algunas cotas superiores para el mínimo de la función de Weber sobre una bola fuertemente convexa en una variedad Riemanniana con curvartura seccional positiva; dicho mínimo se alcanza sobre la mediana geométrica pesada de “m” puntos dados en la bola fuertemente convexa.