@phdthesis{Freifeld:Thesis:2013,
  title = {Statistics on Manifolds with Applications to Modeling Shape Deformations},
  abstract = {Statistical models of non-rigid deformable shape have wide application in many fields,
  including computer vision, computer graphics, and biometry. We show that shape deformations
  are well represented through nonlinear manifolds that are also matrix Lie groups.
  These pattern-theoretic representations lead to several advantages over other alternatives,
  including a principled measure of shape dissimilarity and a natural way to compose deformations.
  Moreover, they enable building models using statistics on manifolds. Consequently,
  such models are superior to those based on Euclidean representations. We
  demonstrate this by modeling 2D and 3D human body shape. Shape deformations are
  only one example of manifold-valued data. More generally, in many computer-vision and
  machine-learning problems, nonlinear manifold representations arise naturally and provide
  a powerful alternative to Euclidean representations. Statistics is traditionally concerned
  with data in a Euclidean space, relying on the linear structure and the distances associated
  with such a space; this renders it inappropriate for nonlinear spaces. Statistics can,
  however, be generalized to nonlinear manifolds. Moreover, by respecting the underlying
  geometry, the statistical models result in not only more effective analysis but also consistent
  synthesis. We go beyond previous work on statistics on manifolds by showing how,
  even on these curved spaces, problems related to modeling a class from scarce data can be
  dealt with by leveraging information from related classes residing in different regions of the
  space. We show the usefulness of our approach with 3D shape deformations. To summarize
  our main contributions: 1) We define a new 2D articulated model -- more expressive than
  traditional ones -- of deformable human shape that factors body-shape, pose, and camera
  variations. Its high realism is obtained from training data generated from a detailed 3D
  model. 2) We define a new manifold-based representation of 3D shape deformations that
  yields statistical deformable-template models that are better than the current state-of-the-
  art. 3) We generalize a transfer learning idea from Euclidean spaces to Riemannian
  manifolds. This work demonstrates the value of modeling manifold-valued data and their
  statistics explicitly on the manifold. Specifically, the methods here provide new tools for
  shape analysis.},
  school = {Brown University},
  month = aug,
  year = {2013},
  slug = {freifeld-thesis-2013},
  author = {Freifeld, Oren},
  month_numeric = {8}
}