Efficient Recursive Algorithms for Statistical Analysis of Manifold-valued Data, (slides)
Presented by Prof. Baba C. Vemuri, University of Florida
Baba C. Vemuri received the PhD in Electrical and Computer Engineering from the University of Texas at Austin in 1987. He then joined the Department of Computer and Information Sciences at the University of Florida, Gainesville, and is currently a full professor of computer and information sciences and engineering. He also holds affiliate appointments in the Department of ECE and BME at the University of Florida. He served as a program chair for several conferences including the 11th IEEE International Conference on Computer Vision (ICCV 2007). He has been an area chair and a program committee member of several IEEE conferences. He was an associate editor for several journals, including the IEEE Transactions on Pattern Analysis and Machine Intelligence –TPAMI, (from 1992 to 1996), the IEEE Transactions on Medical Imaging — TMI, (from 1997 to 2003) and the journal of Computer Vision and Image Understanding (from 2000-2010). He is currently an associate editor for the Journal of Medical Image Analysis (MedIA) and the Intl. Journal of Computer Vision (IJCV). His research interests include Medical Image Computing, Computer Vision, Machine Learning, Information Geometry and Applied Mathematics. For the last several years, his research work has primarily focused on Information Geometric methods. Along this theme, he has been developing algorithms for the recursive computation of the statistics on Riemannian manifolds pertinent to manifold-valued data sets, analysis of diffusion weighted MRI and diffusion tensor MRI, 3D image segmentation, unimodal and multimodal image (rigid+nonrigid) registration, nonrigid registration of 3D point sets, metric learning, and large margin classifiers. He has published more than 180 refereed journal articles and conference proceedings on Medical Image Computing, Computer Vision, Graphics, and Applied Mathematics. He received the US National Science Foundation Research Initiation Award (NSF RIA) in 1988 and the Whitaker Foundation Award in 1994. He has received several best paper awards at various International Conferences and is a fellow of the IEEE and ACM.
With the advent of new sensing technologies and high powered computing resources, manifold-valued data sets have become ubiquitous in Science and Engineering. The most commonly encountered examples are matrix-valued fields e.g., diffusion tensor, structure tensor, covariance matrix and probability density fields respectively. Since these data do not live in a vector space, standard vector-space operations to process them are inappropriate and mathematical tools borrowed from the field of Differential Geometry are required. As is customary in conventional image analysis, it is useful to compute statistics from these data sets in order characterize the data quantitatively. Once again, it is important to respect the geometry of the space in which these data lie and hence one has to rely on manifold-valued statistics. In this talk, I will present algorithms for efficiently computing the most basic of all statistics, namely the intrinsic or Frechet mean of the manifold-valued data as well as the Principal Geodesic Analysis (PGA). Several applications of the aforementioned algorithms to Computer Vision and Medical Image. Computing will be presented interspersed during the talk.
Sampling, inference and clustering for data on graphs, (slides)
Pierre Vandergheynst, Prof. at Swiss Federal Institute of Technology
Pierre Vandergheynst received the M.S. degree in physics and the Ph.D. degree in mathematical physics from the Université catholique de Louvain, Louvain-la-Neuve, Belgium, in 1995 and 1998, respectively. From 1998 to 2001, he was a Postdoctoral Researcher with the Signal Processing Laboratory, Swiss Federal Institute of Technology (EPFL), Lausanne, Switzerland. He was Assistant Professor at EPFL (2002-2007), where he is now a Full Professor. His research focuses on harmonic analysis, sparse approximations and mathematical image processing with applications to higher dimensional, complex data processing. He was co-Editor-in-Chief of Signal Processing (2002-2006) and is Associate Editor of the IEEE Transactions on Signal Processing (2007-present). He has been on the Technical Committee of various conferences and was Co-General Chairman of the EUSIPCO 2008 conference. Pierre Vandergheynst is the author or co-author of more than 50 journal papers, one monograph and several book chapters. He’s a laureate of the Apple ARTS award and of the 2010-2011 De Boelpaepe prize from the Royal Academy of Sciences of Belgium. Pierre is strongly involved in technology transfer: he co-founded two start ups and holds numerous patents.
We study the problem of sampling k-bandlimited signals on graphs, as models for information over networks. We propose two sampling strategies that consist in selecting a small subset of nodes at random. The first strategy is non-adaptive, i.e., independent of the graph structure, and its performance depends on a parameter called the graph coherence. On the contrary, the second strategy is adaptive but yields optimal results. Indeed, no more than O(k log(k)) measurements are sufficient to ensure an accurate and stable recovery of all k-bandlimited signals. This second strategy is based on a careful choice of the sampling distribution, which can be estimated quickly. Then, we propose a computationally efficient inference algorithm to reconstruct k-bandlimited signals from their samples. We prove that it yields accurate reconstructions and that it is also stable to noise. Finally, we use these ideas to revisit spectral clustering and show that we can recover a k-partition from a k (logk)^2-dimensional sketch of the data, opening a new door to using these methods at scale. Joint work with Gilles Puy (now at Technicolor), Nicolas Tremblay and Rémi Gribonval (INRIA).