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Kernel methods on Riemannian manifolds Richard Hartley, Sadeep Jayasumana, Mehrtash Harandi, Mathieu Salzmann Hongdong Li, Khurrum Aftab Oxford, March 2014

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Page 1: Kernel methods on Riemannian manifolds - robots.ox.ac.ukvgg/rg/slides/Oxford-Mar-2014.pdf · Kernel methods on Riemannian manifolds Richard Hartley, Sadeep Jayasumana, Mehrtash Harandi,

Kernel methods on Riemannian manifolds

Richard Hartley, Sadeep Jayasumana, Mehrtash Harandi, Mathieu Salzmann

Hongdong Li, Khurrum Aftab

Oxford, March 2014

Page 2: Kernel methods on Riemannian manifolds - robots.ox.ac.ukvgg/rg/slides/Oxford-Mar-2014.pdf · Kernel methods on Riemannian manifolds Richard Hartley, Sadeep Jayasumana, Mehrtash Harandi,
Page 3: Kernel methods on Riemannian manifolds - robots.ox.ac.ukvgg/rg/slides/Oxford-Mar-2014.pdf · Kernel methods on Riemannian manifolds Richard Hartley, Sadeep Jayasumana, Mehrtash Harandi,
Page 4: Kernel methods on Riemannian manifolds - robots.ox.ac.ukvgg/rg/slides/Oxford-Mar-2014.pdf · Kernel methods on Riemannian manifolds Richard Hartley, Sadeep Jayasumana, Mehrtash Harandi,
Page 5: Kernel methods on Riemannian manifolds - robots.ox.ac.ukvgg/rg/slides/Oxford-Mar-2014.pdf · Kernel methods on Riemannian manifolds Richard Hartley, Sadeep Jayasumana, Mehrtash Harandi,
Page 6: Kernel methods on Riemannian manifolds - robots.ox.ac.ukvgg/rg/slides/Oxford-Mar-2014.pdf · Kernel methods on Riemannian manifolds Richard Hartley, Sadeep Jayasumana, Mehrtash Harandi,
Page 7: Kernel methods on Riemannian manifolds - robots.ox.ac.ukvgg/rg/slides/Oxford-Mar-2014.pdf · Kernel methods on Riemannian manifolds Richard Hartley, Sadeep Jayasumana, Mehrtash Harandi,
Page 8: Kernel methods on Riemannian manifolds - robots.ox.ac.ukvgg/rg/slides/Oxford-Mar-2014.pdf · Kernel methods on Riemannian manifolds Richard Hartley, Sadeep Jayasumana, Mehrtash Harandi,

Image from http://en.wikipedia.org/wiki/File:Tangentialvektor.svg

Page 9: Kernel methods on Riemannian manifolds - robots.ox.ac.ukvgg/rg/slides/Oxford-Mar-2014.pdf · Kernel methods on Riemannian manifolds Richard Hartley, Sadeep Jayasumana, Mehrtash Harandi,

Exponential map.  Set out in a given direction and see where you get to.

Page 10: Kernel methods on Riemannian manifolds - robots.ox.ac.ukvgg/rg/slides/Oxford-Mar-2014.pdf · Kernel methods on Riemannian manifolds Richard Hartley, Sadeep Jayasumana, Mehrtash Harandi,

Logarithm map.  What direction do I need to go to get from X to Y ?

Page 11: Kernel methods on Riemannian manifolds - robots.ox.ac.ukvgg/rg/slides/Oxford-Mar-2014.pdf · Kernel methods on Riemannian manifolds Richard Hartley, Sadeep Jayasumana, Mehrtash Harandi,
Page 12: Kernel methods on Riemannian manifolds - robots.ox.ac.ukvgg/rg/slides/Oxford-Mar-2014.pdf · Kernel methods on Riemannian manifolds Richard Hartley, Sadeep Jayasumana, Mehrtash Harandi,
Page 13: Kernel methods on Riemannian manifolds - robots.ox.ac.ukvgg/rg/slides/Oxford-Mar-2014.pdf · Kernel methods on Riemannian manifolds Richard Hartley, Sadeep Jayasumana, Mehrtash Harandi,
Page 14: Kernel methods on Riemannian manifolds - robots.ox.ac.ukvgg/rg/slides/Oxford-Mar-2014.pdf · Kernel methods on Riemannian manifolds Richard Hartley, Sadeep Jayasumana, Mehrtash Harandi,
Page 15: Kernel methods on Riemannian manifolds - robots.ox.ac.ukvgg/rg/slides/Oxford-Mar-2014.pdf · Kernel methods on Riemannian manifolds Richard Hartley, Sadeep Jayasumana, Mehrtash Harandi,
Page 16: Kernel methods on Riemannian manifolds - robots.ox.ac.ukvgg/rg/slides/Oxford-Mar-2014.pdf · Kernel methods on Riemannian manifolds Richard Hartley, Sadeep Jayasumana, Mehrtash Harandi,

xy

Page 17: Kernel methods on Riemannian manifolds - robots.ox.ac.ukvgg/rg/slides/Oxford-Mar-2014.pdf · Kernel methods on Riemannian manifolds Richard Hartley, Sadeep Jayasumana, Mehrtash Harandi,
Page 18: Kernel methods on Riemannian manifolds - robots.ox.ac.ukvgg/rg/slides/Oxford-Mar-2014.pdf · Kernel methods on Riemannian manifolds Richard Hartley, Sadeep Jayasumana, Mehrtash Harandi,

Kernels on Positive Definite Matrices

Page 19: Kernel methods on Riemannian manifolds - robots.ox.ac.ukvgg/rg/slides/Oxford-Mar-2014.pdf · Kernel methods on Riemannian manifolds Richard Hartley, Sadeep Jayasumana, Mehrtash Harandi,
Page 20: Kernel methods on Riemannian manifolds - robots.ox.ac.ukvgg/rg/slides/Oxford-Mar-2014.pdf · Kernel methods on Riemannian manifolds Richard Hartley, Sadeep Jayasumana, Mehrtash Harandi,
Page 21: Kernel methods on Riemannian manifolds - robots.ox.ac.ukvgg/rg/slides/Oxford-Mar-2014.pdf · Kernel methods on Riemannian manifolds Richard Hartley, Sadeep Jayasumana, Mehrtash Harandi,
Page 22: Kernel methods on Riemannian manifolds - robots.ox.ac.ukvgg/rg/slides/Oxford-Mar-2014.pdf · Kernel methods on Riemannian manifolds Richard Hartley, Sadeep Jayasumana, Mehrtash Harandi,
Page 23: Kernel methods on Riemannian manifolds - robots.ox.ac.ukvgg/rg/slides/Oxford-Mar-2014.pdf · Kernel methods on Riemannian manifolds Richard Hartley, Sadeep Jayasumana, Mehrtash Harandi,
Page 24: Kernel methods on Riemannian manifolds - robots.ox.ac.ukvgg/rg/slides/Oxford-Mar-2014.pdf · Kernel methods on Riemannian manifolds Richard Hartley, Sadeep Jayasumana, Mehrtash Harandi,
Page 25: Kernel methods on Riemannian manifolds - robots.ox.ac.ukvgg/rg/slides/Oxford-Mar-2014.pdf · Kernel methods on Riemannian manifolds Richard Hartley, Sadeep Jayasumana, Mehrtash Harandi,
Page 26: Kernel methods on Riemannian manifolds - robots.ox.ac.ukvgg/rg/slides/Oxford-Mar-2014.pdf · Kernel methods on Riemannian manifolds Richard Hartley, Sadeep Jayasumana, Mehrtash Harandi,
Page 27: Kernel methods on Riemannian manifolds - robots.ox.ac.ukvgg/rg/slides/Oxford-Mar-2014.pdf · Kernel methods on Riemannian manifolds Richard Hartley, Sadeep Jayasumana, Mehrtash Harandi,

Example: Kernel Dictionary Learning on Manifolds

Harandi et al (ICCV 2013)

Page 28: Kernel methods on Riemannian manifolds - robots.ox.ac.ukvgg/rg/slides/Oxford-Mar-2014.pdf · Kernel methods on Riemannian manifolds Richard Hartley, Sadeep Jayasumana, Mehrtash Harandi,
Page 29: Kernel methods on Riemannian manifolds - robots.ox.ac.ukvgg/rg/slides/Oxford-Mar-2014.pdf · Kernel methods on Riemannian manifolds Richard Hartley, Sadeep Jayasumana, Mehrtash Harandi,
Page 30: Kernel methods on Riemannian manifolds - robots.ox.ac.ukvgg/rg/slides/Oxford-Mar-2014.pdf · Kernel methods on Riemannian manifolds Richard Hartley, Sadeep Jayasumana, Mehrtash Harandi,
Page 31: Kernel methods on Riemannian manifolds - robots.ox.ac.ukvgg/rg/slides/Oxford-Mar-2014.pdf · Kernel methods on Riemannian manifolds Richard Hartley, Sadeep Jayasumana, Mehrtash Harandi,

Planes through the origin

Projective space – lines through the origin

Page 32: Kernel methods on Riemannian manifolds - robots.ox.ac.ukvgg/rg/slides/Oxford-Mar-2014.pdf · Kernel methods on Riemannian manifolds Richard Hartley, Sadeep Jayasumana, Mehrtash Harandi,
Page 33: Kernel methods on Riemannian manifolds - robots.ox.ac.ukvgg/rg/slides/Oxford-Mar-2014.pdf · Kernel methods on Riemannian manifolds Richard Hartley, Sadeep Jayasumana, Mehrtash Harandi,
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Page 35: Kernel methods on Riemannian manifolds - robots.ox.ac.ukvgg/rg/slides/Oxford-Mar-2014.pdf · Kernel methods on Riemannian manifolds Richard Hartley, Sadeep Jayasumana, Mehrtash Harandi,
Page 36: Kernel methods on Riemannian manifolds - robots.ox.ac.ukvgg/rg/slides/Oxford-Mar-2014.pdf · Kernel methods on Riemannian manifolds Richard Hartley, Sadeep Jayasumana, Mehrtash Harandi,
Page 37: Kernel methods on Riemannian manifolds - robots.ox.ac.ukvgg/rg/slides/Oxford-Mar-2014.pdf · Kernel methods on Riemannian manifolds Richard Hartley, Sadeep Jayasumana, Mehrtash Harandi,
Page 38: Kernel methods on Riemannian manifolds - robots.ox.ac.ukvgg/rg/slides/Oxford-Mar-2014.pdf · Kernel methods on Riemannian manifolds Richard Hartley, Sadeep Jayasumana, Mehrtash Harandi,

Atul Kanaujia

Guillaume Charpiat

Shape Manifold.Captures the configuration of a set of points, allowing for rotation, translation and scaling.

Page 39: Kernel methods on Riemannian manifolds - robots.ox.ac.ukvgg/rg/slides/Oxford-Mar-2014.pdf · Kernel methods on Riemannian manifolds Richard Hartley, Sadeep Jayasumana, Mehrtash Harandi,
Page 40: Kernel methods on Riemannian manifolds - robots.ox.ac.ukvgg/rg/slides/Oxford-Mar-2014.pdf · Kernel methods on Riemannian manifolds Richard Hartley, Sadeep Jayasumana, Mehrtash Harandi,

The Hopf Fibration – the realization of CP1 as a fibration

Page 41: Kernel methods on Riemannian manifolds - robots.ox.ac.ukvgg/rg/slides/Oxford-Mar-2014.pdf · Kernel methods on Riemannian manifolds Richard Hartley, Sadeep Jayasumana, Mehrtash Harandi,
Page 42: Kernel methods on Riemannian manifolds - robots.ox.ac.ukvgg/rg/slides/Oxford-Mar-2014.pdf · Kernel methods on Riemannian manifolds Richard Hartley, Sadeep Jayasumana, Mehrtash Harandi,
Page 43: Kernel methods on Riemannian manifolds - robots.ox.ac.ukvgg/rg/slides/Oxford-Mar-2014.pdf · Kernel methods on Riemannian manifolds Richard Hartley, Sadeep Jayasumana, Mehrtash Harandi,

Leaf Database

Page 44: Kernel methods on Riemannian manifolds - robots.ox.ac.ukvgg/rg/slides/Oxford-Mar-2014.pdf · Kernel methods on Riemannian manifolds Richard Hartley, Sadeep Jayasumana, Mehrtash Harandi,
Page 45: Kernel methods on Riemannian manifolds - robots.ox.ac.ukvgg/rg/slides/Oxford-Mar-2014.pdf · Kernel methods on Riemannian manifolds Richard Hartley, Sadeep Jayasumana, Mehrtash Harandi,

Acknowledgements.

This talk deals with work done by myself and my collaborators, particularly

1. Mehrtash Harandi2. Sadeep Jayasumana3. Mathieu Salzmann4. Hongdong Li5. Brian Lovell

Page 46: Kernel methods on Riemannian manifolds - robots.ox.ac.ukvgg/rg/slides/Oxford-Mar-2014.pdf · Kernel methods on Riemannian manifolds Richard Hartley, Sadeep Jayasumana, Mehrtash Harandi,

The End


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