Non-Euclidean Example: The Unit Sphere

Differential Geometry

• Formal mathematical theory

• Work with small ‘patches’– the ‘patches’ look Euclidean

• Do calculus over the patches

Manifolds

• Open Sets

• Coordinate neighbourhood

• Compatible neighbourhoods

Tangents

• Tangent Vectors

• Tangent Space,

• Inner Product– Norm:

– depends on – varies smoothly

Geodesics and Metrics

• The shortest path between two points is the geodesic

• The length of the geodesic is the distance between the points

Exponential and Logarithm Maps

• : Maps tangents to the manifold

• : Maps points on the manifold to

•

• Both maps are locally well defined

Gradient

• In Euclidean space: direction of fastest increase

• On a manifold: tangent of fastest increase

• Definition: is a real valued function. The gradient at is satisfying

directional derivative along delta

The Conversion Table

1. X. Pennec, P. Fillard and N. Ayache , “A Riemannian Framework for Tensor Computing,” International. Journal of Computer Vision., 66(1), 41–66, 2006.

Matrix Lie Groups

• Sets of matrices which– form a group under matrix multiplication– are Riemannian manifolds

• Examples– Rigid body transformations SE(n)– Rotations SO(n)– Affine motions A(n)

1. W. Rossman, “Lie Groups: An Introduction through Linear Groups,” Oxford University Press, 2003.

Grassmann Manifolds, .

• Each point on the Grassmann manifold, , represents a dimensional subspace of .

• Numerically, represented by an orthonormal basis – matrix such that

• Representation is not unique– computation should account for this

1. A. Edelman, T. A. Arias and S. T. Smith, “The Geometry of Algorithms with Orthogonality Constraints,” SIAM Journal on Matrix Analysis and Applications, 20(2), 303–353, 1998.

The Essential Manifold

• Set of matrices with– two equal and one zero singular value– let the two equal singular values be 1

• Equivalent to SO(3)xSO(3)– two-time covering of the essential manifold

• Can also be expressed as a homogeneous space

1. S. Soatto, R. Frezza and P. Perona , “Recursive Estimation on the Essential Manifold,” 3rd Europan Conference on Computer Vision, Stockholm, Sweden, May 1994, vol.II, p.61-72.

Twisted Pairs

The Symmetric Manifold

• contains symmetric positive definite matrices. e.g. diffusion tensor MRI– it has two different metrics

• The Affine Invariant metric• The Log-Euclidean Metric

– practically similar to the affine invariant metric– computationally easier to work with

1. V. Arsigny , P. Fillard, X. Pennec and N. Ayache , “Geometric Means in a Novel Vector Space Structure on Symmetric Positive-Definite Matrices,” SIAM Journal of Matrix Analysis and Applications, 29(1), 328–347, 2006.

Mean Shift for Euclidean Spaces

• The kernel density estimate

• Mean shift as normalized gradient of

where

• The iteration

Mean Shift

• Gradient ascent on kernel density– but, no line search

• Equivalent, to expectation-maximization

• Nonparametric Clustering

1. D. Comaniciu and P. Meer , “Mean Shift: A Robust Approach Towards Feature Space Analysis ,” IEEE Transactions on Pattern Analysis and Machine Intelligence, vol.24, 603–619, 2002.

2. D. Comaniciu, V. Ramesh and P. Meer , “Kernel-based Object Tracking,” IEEE Transactions on Pattern Analysis and Machine Intelligence, vol.25, 564–577, 2003.

Mean Shift for Manifolds

• The kernel density estimate

• Mean shift as normalized gradient of

• The iteration

Mean Shift for Riemannian Manifolds

• Map points to tangent space

• Get weighted average of tangent vectors– this is the mean shift vector

• Map the mean shift vector back to the manifold

Theoretical Properties

• Gradient ascent on kernel density

• Nonlinear mean shift is provably convergent– upper limit on allowed bandwidth

• nonlinear mean shift is equivalent to EM– for homogeneous spaces

Motion Segmentation

• Hypothesis Generation Lie groups– Randomly pick elemental subset– Generate parameter hypothesis

• Clustering– Cluster parameters on the manifold

• Return– Number of dominant modes– Positions of dominant modes

Affine Motion: Results A(2)

Camera Pose Estimation SE(3)

Camera Pose Estimation: Results

Factorization: Results G10,3

Essential Matrix: Results SO(3)xSO(3)

Discontinuity Preserving Filtering

• An image is a mapping from a lattice in to data lying on a manifold

• Filtering: Run mean shift in the space

Iterations update spatial and parameter values.

• If the iteration from converges to , set in the filtered image

Chromatic Noise Filtering M = G3,1

Chromatic Noise Filtering

DTI Images R3xSym+3