Title: Global tools for separating noise from data

Abstract:
Separating noise from data is a universal problem in science and much has been written about it. It is often difficult or impossible to distinguish fine structure in data from experimental or other noise. In this talk i'll present two vignettes, one theoretical and one experimental, showing how a global understanding of the structure of the problem can help with this task.

A. Topological mode analysis

We present a clustering algorithm that combines a mode-seeking phase with a cluster merging phase. While mode detection is performed by a standard graph-based hill-climbing scheme, the novelty of our approach resides in its use of topological persistence theory to guide the merging of clusters. Our algorithm provides additional feedback in the form of a set of points in the plane called a persistence diagram. This provably reflects the prominence of the modes of the density. In practice, this feedback enables the user to determine a set of parameter values to compute a relevant clustering. Meanwhile, its complexity remains practical: although the size of the input distance matrix may be up to quadratic in the number of data points, a careful implementation only uses a linear amount of memory and takes barely more time to run than to read through the input.

Joint work with Primoz Skraba, Steve Y. Oudot, and Frédéric Chazal (INRIA).

B. Robust single-view geometry and motion reconstruction

We present a framework and algorithms for robust geometry and motion reconstruction of complex deforming shapes. Our method makes use of a smooth template that provides a crude approximation of the scanned object and serves as a geometric and topological prior for reconstruction. Large-scale motion of the acquired object is recovered using a novel space-time adaptive, non-rigid registration method. Fine-scale details such as wrinkles and folds are synthesized with an efficient linear mesh deformation algorithm. Subsequent spatial and temporal filtering of detail coefficients allows transfer of persistent geometric detail to regions not observed by the scanner. We show how this two-scale process allows faithful recovery of small-scale shape and motion features leading to a high-quality reconstruction. We illustrate the robustness and generality of our algorithm on a variety of examples composed of different materials and exhibiting a large range of dynamic deformations.

Joint work with Hao Li, Mark Pauly (ETH Zürich) and Bart Adams (KU Leuven).