Covariance matrices play important roles in many areas of mathematics, statistics, and machine learning, as well as their applications. In computer vision and image processing, they give rise to a powerful data representation, namely the covariance descriptor, with numerous practical applications.
In this book, we begin by presenting an overview of the {\it finite-dimensional covariance matrix} representation approach of images, along with its statistical interpretation. In particular, we discuss the various distances and divergences that arise from the intrinsic geometrical structures of the set of Symmetric Positive Definite (SPD) matrices, namely Riemannian manifold and convex cone structures. Computationally, we focus on kernel methods on covariance matrices, especially using the Log-Euclidean distance.
We then show some of the latest developments in the generalization of the finite-dimensional covariance matrix representation to the {\it infinite-dimensional covariance operator} representation via positive definite kernels. We present the generalization of the affine-invariant Riemannian metric and the Log-Hilbert-Schmidt metric, which generalizes the Log Euclidean distance. Computationally, we focus on kernel methods on covariance operators, especially using the Log-Hilbert-Schmidt distance. Specifically, we present a two-layer kernel machine, using the Log-Hilbert-Schmidt distance and its finite-dimensional approximation, which reduces the computational complexity of the exact formulation while largely preserving its capability. Theoretical analysis shows that, mathematically, the approximate Log-Hilbert-Schmidt distance should be preferred over the approximate Log-Hilbert-Schmidt inner product and, computationally, it should be preferred over the approximate affine-invariant Riemannian distance.
Numerical experiments on image classification demonstrate significant improvements of the infinite-dimensional formulation over the finite-dimensional counterpart. Given the numerous applications of covariance matrices in many areas of mathematics, statistics, and machine learning, just to name a few, we expect that the infinite-dimensional covariance operator formulation presented here will have many more applications beyond those in computer vision.
Author Biography:
H� Quang Minh received his Ph.D. in mathematics from Brown University, Providence, RI, in May 2006, under the supervision of Steve Smale. He is currently a Researcher in the Department of Pattern Analysis and Computer Vision (PAVIS) with the Istituto Italiano di Tecnologia (IIT), Genova, Italy. Prior to joining IIT, he held research positions at the University of Chicago, the University of Vienna, Austria, and Humboldt University of Berlin, Germany. He was also a Junior Research Fellow at the Erwin Schrodinger International Institute for Mathematical Physics in Vienna and a Fellow at the Institute for Pure and Applied Mathematics (IPAM) at the University of California, Los Angeles (UCLA). His current research interests include applied and computational functional analysis, applied and computational differential geometry, machine learning, computer vision, and image and signal processing. His recent research contributions include the infinite-dimensional Log-Hilbert-Schmidt metric and Log-Determinant divergences between positive definite operators, along with their applications in machine learning and computer vision in the setting of kernel methods. He received the Microsoft Best Paper Award at the Conference on Uncertainty in Artificial Intelligence (UAI) in 2013 and the IBM Pat Goldberg Memorial Best Paper Award in Computer Science, Electrical Engineering, and Mathematics in 2013. Vittorio Murino is a full professor and head of the Pattern Analysis and Computer Vision (PAVIS) department at the Istituto Italiano di Tecnologia (IIT), Genoa, Italy. He received his Ph.D. in electronic engineering and computer science in 1993 at the University of Genoa, Italy. Then, he was first at the University of Udine and, since 1998, at the University of Verona where he was chairman of the Department of Computer Science from 2001 to 2007. Since 2009, he is leading the PAVIS department in IIT, which is involved in computer vision, pattern recognition and machine learning activities. His specific research interests are focused on statistical and probabilistic techniques for image and video processing, with application on (human) behavior analysis and related applications, such as video surveillance, biomedical imaging, and bioinformatics. Prof. Murino is co-author of more than 400 papers published in refereed journals and international conferences, member of the technical committees of important conference (CVPR, ICCV, ECCV, ICPR, ICIP, etc.), and guest co-editor of special issues in relevant scientific journals. He is currently a member of the editorial board of Computer Vision and Image Understanding, Pattern Analysis and Applications, and Machine Vision & Applications journals. Finally, he is a Senior Member of the IEEE and Fellow of the IAPR. G�rard Medioni received the Dipl�me d'Ingenieur in Information at The �cole Nationale Sup�rieure es T�l�communications, in 1977, and the M.S. and Ph.D. degrees in Computer Science from the University of Southern California, in 1980 and 1983, respectively. He has been at USC since then, and is currently Professor of Computer Science and Electrical Engineering, co-director of the Institute for Robotics and Intelligent Systems (IRIS), and co-director of the USC Games Institute. He served as Chairman of the Computer Science Department from 2001 to 2007. Prior to this, he was President and CEO of I.C. Vision, in Los Angeles, California, and held positions of Associate Professor, from 1992-1999, Assistant Professor, from 1987-1992, and Research Assistant Professor, from 1983-1987, at the Departments of Computer Science and Electrical Engineering, at the University of Southern California. From 1979-1983, he was a Research Assistant in the Intelligent Systems Group at the University of Southern California. Prior to his academic career, he was a research engineer at Underwater Signal Processing Division at Thomson-CSF, in Cagnes sur Mer, France. From 2000 to 2001, while on sabbatical leave, he was Chief Technical Officer at Geometrix, Inc. in San Jose, California.
Professor Medioni has made significant contributions to the field of computer vision. His research covers a broad spectrum of the field, such as edge detection, stereo and motion analysis, shape inference and description, and system integration. He has published 3 books, over 50 journal papers and 150 conference articles, and is the recipient of 8 international patents.
Prof. Medioni is associate editor of the Image and Vision Computing Journal, associate editor of the Pattern Recognition and Image Analysis Journal, and associate editor of the International Journal of Image and Video Processing.
Prof. Medioni served as program co-chair of the 1991 IEEE CVPR Conference in Hawaii, of the 1995 IEEE Symposium on Computer Vision in Miami, general co-chair of the1997 IEEE CVPR Conference in Puerto Rico, conference co-chair of the 1998 ICPR Conference in Australia, general co-chair of the 2001 IEEE CVPR Conference in Kauai, general co-chair of the 2007 IEEE CVPR Conference in Minneapolis, and general co-chair of the upcoming 2009 IEEE CVPR Conference in Miami. He is a Fellow of IAPR, a Fellow of the IEEE, and a Fellow of AAAI.
Sven Dickinson received the B.A.Sc. degree in Systems Design Engineering from the University of Waterloo in 1983, and the M.S. and Ph.D. degrees in Computer Science from the University of Maryland, in 1988 and 1991, respectively. He is currently Professor of Computer Science at the University of Toronto, where he serves as Acting Chair. Prior to that, he served as Departmental Vice Chair, from 2003-2006, and as Associate Professor, from 2000-2007. From 1995-2000, he was an Assistant Professor of Computer Science at Rutgers University, where he also held a joint appointment in the Rutgers Center for Cognitive Science (RuCCS) and membership in the Center for Discrete Mathematics and Theoretical Computer Science (DIMACS). From 1994-1995, he was a Research Assistant Professor in the Rutgers Center for Cognitive Science, and from 1991-1994, a Research Associate at the Artificial Intelligence Laboratory, University of Toronto. He has held affiliations with the MIT Media Laboratory (Visiting Scientist, 1992-1994), the University of Toronto (Visiting Assistant Professor, 1994 1997), and the Computer Vision Laboratory of the Center for Automation Research at the University of Maryland (Assistant Research Scientist, 1993-1994, Visiting Assistant Professor, 1994 1997). Prior to his academic career, he worked in the computer vision industry, designing image processing systems for Grinnell Systems Inc., San Jose, CA, 1983-1984, and optical character recognition systems for DEST, Inc., Milpitas, CA, 1984-1985.
His research interests revolve around the problem of object recognition, in general, and generic object recognition, in particular. He has explored a multitude of generic shape representations, and their common representation as hierarchical graphs has led to his interest in inexact graph indexing and matching. His interest in shape representation and matching has also led to his research in object tracking, vision-based navigation, content based image retrieval, and the use of language to guide perceptual grouping, object recognition, and motion analysis. One of the focal points of his research is the problem of image abstraction, which he believes is critical in bridging the representational gap between exemplar-based and generic object recognition. He has published over 100 papers on these topics in refereed journals, conferences, and edited collections. In 1996, he received the NSF CAREER award for his work in generic object recognition, and in 2002, received the Government of Ontario Premiere's Research Excellence Award (PREA), also for his work in generic object recognition. He was co-chair of the 1997, 1999, 2004, and 2007 IEEE International Workshops on Generic Object Recognition (or Object Categorization), co chaired the DIMACS Workshop on Graph Theoretic Methods in Computer Vision in 1999, and co-chaired the First International Workshop on Shape Perception in Human and Computer Vision in 2008. From 1998-2002, he served as Associate Editor of the IEEE Transactions on Pattern Analysis and Machine Intelligence, in which he also co-edited a special issue on graph algorithms and computer vision, which appeared in 2001. He currently serves as Associate Editor for the journals: International Journal of Computer Vision; Image and Vision Computing; Pattern Recognition Letters; IET Computer Vision; and the Journal of Electronic Imaging.
