3IA PhD/Postdoc Seminar #8

Towards Efficient Algorithms in Computational Topology

Abstract: At the turn of the 19th and 20th centuries, Henri Poincaré published a series of six papers that revolutionized topology, the branch of mathematics concerned with robust properties of shapes (in a very general sense) that remain unchanged by continuous deformation.

In these papers Poincaré laid the foundations of homology theory, an algebraic framework for quantifying the components, holes and voids of shapes, which has become a cornerstone of topological data analysis (TDA) hundred years later. He also posed his famous conjecture, which was resolved in a celebrated effort by Perelman only in 2003, providing a major driving force behind many advances in modern 3-dimensional topology.

In the first part of my talk I outline the basic ideas and premises of topological data analysis, and invite the audience to GUDHI, an open-source C++ library with a Python interface actively developed at INRIA, that provides state-of-the-art algorithms and data structures for TDA.

Then, for the second part, we switch gears and restrict our focus to 3-manifolds: shapes that locally look like the 3-dimensional Euclidean space, but globally can be highly complex and non-linear. 3-manifolds naturally appear in various contexts, such as the theory of knots and links, astronomy, or theoretical physics. Here a key challenge concerns the possibility of finding "nice" combinatorial descriptions of these shapes that allow their efficient algorithmic study.

Joint work in progress with Clément Maria.

Diffusion MRI based Brain Tissue Microstructure Characterization Using Autoencoder Neural-Networks

Abstract: Diffusion Magnetic Resonance Imaging (dMRI) is the only non-invasive imaging technique that is able to probe brain tissue microstructure in-vivo.

The dMRI signal in each voxel can be viewed as a 3D function parametrized by the gradient strength and direction. Mathematical models are often used to give a continuous and interpretable representation of the signal and extract brain tissue features from each voxel. Among the most common models used in dMRI are the Multi-Compartment (MC) models which try to identify how much of the water molecules in the voxels are inside or outside a given compartment. These compartments are modeled as cell structures (e.g. neuron bodies and axons). Therefore we can for example estimate the fraction of axons in a voxel by looking at the weight of the intra-axonal compartment in that voxel.

In recent years, MC models have been widely used to try to characterize brain tissue microstructure from dMRI data. One of the main drawbacks of this approach is that the number of microstructural features needs to be decided a priori and it is embedded in the model definition (e.g. number and type of compartments). However, the number of microstructural features which is possible to obtain from dMRI data given the acquisition scheme is still not clear.

In this work, we aim at characterizing brain tissue using autoencoder neural networks in combination with rotation-invariant features. By changing the number of neurons in the autoencoder latent space, we can effectively control the number of microstructural features that we obtained from the data. By plotting the autoencoder reconstruction error to the number of features we were able to find the optimal trade-off between data fidelity and the number of microstructural features. Our results show how this number is impacted by the number of shells and the gradient strength used to sample the dMRI signal. We also show how our technique paves the way to a richer characterization of the brain tissue microstructure in-vivo.
 

Open discussion on the two contributions