Rotation Averaging and Optimization on Manifolds

June 30, 2011
Richard Hartley | Australian National University

I will discuss averaging on manifolds, mainly the manifold SO3 of 3D rotations. A number of theoretical results will be given, in particular conditions for convexity of distance measures on SO3, and basins of convergence of averaging. Different averaging problems, such as single rotation and multiplerelative rotation averaging will be discussed along with their applications in
different Computer Vision problems, such as structure from motion and hand-eye
coordination.

Recent work in L1 averaging on SO3 will be presented, based on the classical Weiszfeld algorithm (1937), which gives a solution to the so-called Fermat or Fermat-Weber problem, concerning L1-averaging in Rⁿ. Using this algorithm, we can compute the orientation of all 595 cameras in the Notredame data set with accuracy of 1 degree, in about 3 minutes.

Extension of the Weiszfeld algorithm to averaging on the essential manifold will be discussed, including a new (I think more natural) metric on this manifold.Key is the computation of the geodesics on this manifold, and hence the exponential and logarithm maps

Speaker Details

EDUCATION:
University of Toronto, Canada PhD Mathematics, 1976, MSc 1972
Stanford University, MSc Computer Science, 1985
Australian National University, BSc, 1971

EXPERIENCE:
Professor Richard Hartley is a member of the Vision Science, Technology and Applications Program in National ICT Australia; from 2003 until 2006 he was the leader of this research group. This program seeks to apply method of Computer Vision and Sensor Technology in a range of real-world problems, ranging from motor-vehicle safety to improved methods of health care. To this end, the research program supports research projects in Intelligent Vehicles, Surveillance, Mobile Robotics and Medical Imaging. In 2001, Professor Hartley returned from the USA to a position in the Department of Information Engineering at the Australian National University. Before that, he worked at the General Electric Research and Development Center in Schenectady New York from 1985 to 2001. He began work in Image Understanding and Scene Reconstruction for GE’s Simulation and Control Systems Division. This division built large-scale flight-simulators. His projects in this area were in the construction of terrain models and texture mosaics from aerial and satellite imagery. In 1991, he began an extended research effort in the area of applying geometric techniques to the analysis of video. This research led to advances in machine-understanding of video, and opened up one of the most popular areas of Computer Vision research in the 1990s. The most visible outcome of this research was in automating the creation of special effects in the film entertainment industry. In 2000, he co-authored a book “Multiple View Geometry in Computer Vision” for Cambridge University Press, summarizing the previous decade’s research in this area. This has become one of the most popular research reference texts in Computer Vision