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.
2.
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.
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
– Randomly pick elemental subset
– Generate parameter hypothesis
• Clustering
– Cluster parameters on the manifold
• Return
– Number of dominant modes
– Positions of dominant modes
Lie groups
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
lying on a manifold
to data
• 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