There are many more references in the literature related
to this topic. We are listing only a handful to
give some ideas.
Some Text Books on Introductory Differential Geometry,
and Applications in Vision.
W. M. Boothby.
An Introduction to Differential Manifolds and Riemannian
Geometry.
Academic Press, Inc., 1986.
F. Warner.
Foundations of Differential Geometry and Lie Groups.
Graduate Texts in Mathematics, Springer-Verlag, 1983.
J. Lee. Introduction to Smooth Manifolds.
Springer-Verlag, 2002.
Michael Spivak.
A Comprehensive Introduction to Differential Geometry, Vol I &
II.
Publish or Perish, Inc., Berkeley, 1979.
K. Kanatani.
Geometric Computation for Machine Vision.
Clarendon Press, Oxford, 1993.
K. V. Mardia.
Statistics of Directional Data.
Academic Press, 1972.
I. L. Dryden and K. V. Mardia.
Statistical Shape Analysis.
John Wiley & Son, 1998.
C. G. Small.
The Statistical Theory of Shape.
Springer, 1996.
Some Papers on Analysis on Nonlinear Manifolds.
J. M. Oller and J. M. Corcuera.
Intrinsic analysis of statistical estimation.
Annals of Statistics, 23(5):1562--1581, 1995.
H. Hendricks.
A Cramer-Rao type lower bound for estimators with values in a
manifold.
Journal of Multivariate Analysis, 38:245--261, 1991.
A. Edelman, T. Arias, and S. T. Smith.
The geometry of algorithms with orthogonality constraints.
SIAM Journal of Matrix Analysis and Applications,
20(2):303--353, 1998.
A. Srivastava and E. Klassen.
Monte Carlo extrinsic estimators for manifold-valued parameters.
IEEE Trans. on Signal Processing, 50(2):299--308, February
2001.
Some Papers on Applications in Image and Signal
Processing
U. Grenander, M. I. Miller, and A. Srivastava.
Hilbert-Schmidt lower bounds for estimators on matrix Lie
groups for {ATR}.
IEEE Transactions on PAMI, 20(8):790--802, 1998.
A. Srivastava and E. Klassen.
Bayesian, geometric subspace tracking.
Journal for Advances in Applied Probability, 36(1):to appear,
March 2004.
E. Klassen, A. Srivastava, W. Mio, and S. Joshi.
Analysis of planar shapes using geodesic paths on shape spaces.
IEEE Pattern Analysis and Machiner Intelligence,
26(3):372--383, March, 2004.
X. Liu, A. Srivastava, and K. Gallivan.
Optimal linear representations of images for object recognition.
IEEE Transactions on Pattern Analysis and Machine Intelligence,
26(5):662--666, 2004.
Some papers on Learning Image Manifolds
S. T. Roweis and L. K. Saul.
Nonlinear dimensionality reduction by locally linear embedding.
Science, 290:2323--2326, 2000.
J. B. Tenenbaum, V. Silva, , and J. C. Langford.
A global geometric framework for nonlinear dimensionality reduction.
Science, 290:2319--2323, 2000.
H. S. Seung and D. D. Lee.
The manifold ways of perception.
Science, 290:2268--2269, 2000.
Papers in CVPR 2004 related to nonlinear manifolds
Manifolds Learning
Y. Chang, C. Hu and M. Turk,
"Probabilistic Expression Analysis on Manifolds."
K. Weinberger and L. Saul.
"Unsupervised Learning of Image Manifolds by Semidefinite
Programming."
A. Elgammal and C.S. Lee.
"Separating Style and Content on a Nonlinear Manifold."
L. Younes, J. Glaunes and A. Trouve,
"Diffeomorphic Matching of Distributions: A New Algorithm for Point
Sets and Sub-manifold Matching."
Shape Analysis
W. Mio and A. Srivastava,
"Elastic-String Models for Representation and Analysis of Planar Shapes."
A. Willis and D. Cooper,
"Bayesian Assembly of 3D Axially Symmetric Shapes from Fragments."
X. Huang, D. Metaxas and T. Chen,
"MetaMorphs: Deformable Shape and Texture Models."
J. Xiao and T. Kanade,
"Non-Rigid Shape and Motion Recovery: Degenerate Deformations."
Lie Groups
V. M. Govindu, "Lie-Algebraic Averaging For Globally
Consistent Motion Estimation."
A. Veeraraghavan, A. Roy Chowdhury and R. Chellappa,
"Role of Shape and Kinematics in Human Movement Analysis."
Geometry
K. Dana and J. Wang, " Hybrid Textons: Integrating Appearance and
Geometry for Surface Modeling."