Mathematical Biology Seminar|
Tom Fletcher, SCI, University of Utah
Wednesday March 28, 2012
3:05pm in LCB 323
Statistics on Riemannian Manifolds and its Application to Imaging
Manifold representations are useful for many different types of data,
including directional data, transformation matrices, tensors, and shape. Statistical analysis of these data is an important problem in a wide range of image analysis and computer vision applications. However, defining statistics on a manifold is not a straightforward process. Even the simplest statistics, such as the mean, depend on the vector space structure of Euclidean space. This structure is not available for a general manifold. In this talk I will discuss how many common statistics can be defined for manifold-valued data by utilizing the geodesic distance on the manifold. The first example is the mean value of data on a manifold, which can be defined as the point that minimizes the sum-of-squared geodesic distances to the data. Following this example, we can also define variance, principal components analysis, regression, and the median. Computation of all of these statistics is achieved using only two tools, the Riemannian exponential and log mappings. After explaining the algorithms for computing statistics on manifolds, I will demonstrate their application in several image analysis problems.