Pierre-Antoine Absil (UCL)

Title: Averaging diffusion tensors

Abstract:

This talk concerns the problem of averaging a collection of symmetric positive-definite (SPD) matrices. Generally speaking, averaging methods are required notably to aggregate several noisy measurements of the same object, or to compute the mean of clusters in k-means clustering algorithms, or as a subtask in higher-level tasks such as curve fitting. The problem of averaging SPD matrices arises for example in medical imaging (denoising and segmentation tasks in Diffusion Tensor Imaging), mechanics (elasticity tensor computation), and in video tracking and radar detection tasks. Among several possible definitions for the mean of SPD matrices (including the straightforward arithmetic mean), the “Karcher mean” (specifically the least-squares mean in the sense of the so-called affine-invariant metric) is of widespread interest in the research literature, since it possesses several pleasant properties while being challenging to compute. In this talk, we will review recent advances in iterative methods that converge to the Karcher mean, and in methods that approach it using limited resources.
This work is in collaboration with Xinru Yuan and Kyle Gallivan (Florida State University), Wen Huang (Rice University), and Estelle Massart and Julien Hendrickx (UCLouvain).