Polynomial Regression on Riemannian Manifolds

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Special Case: Geodesic (k = 1) Adjoint system is N∑ ∇ ˙γ λ 0 = δ(t − t i )(grad g i (γ(t))) + R(λ 1 , v 1 )v 1 i=1 ∇ ˙γ λ 1 = −λ 0 Between data points this is (∇ ˙γ ) 2 λ 1 = −R(λ 1 , ˙γ) ˙γ This is the Jacobi equation, λ 1 is a Jacobi field. <strong>Polynomial</strong> <strong>Regression</strong> on Riemannian Manifolds 17
Kendall Shape Space Space of N landmarks in d dimensions, R Nd , modulo translation, scale, rotation Prevents skewed statistics due to similarity transformed data d = 2, complex projective space C P N−2 <strong>Polynomial</strong> <strong>Regression</strong> on Riemannian Manifolds 18
Page 1 and 2: Polynomial <strong
Page 3 and 4: Parametric Regression</stro
Page 5 and 6: Riemannian Polynomial</stro
Page 15 and 16: Rolling maps Leite 2008 Unroll curv
Page 17 and 18: Polynomial <strong
Page 19 and 20: Lagrange multiplier (adjoint) vecto
Page 21 and 22: Rewrite using integration by parts
Page 23 and 24: For any smooth family of curves γ
Page 25 and 26: Now do the same with another term T
Page 27 and 28: Combine all terms to get adjoint eq
Page 29: Polynomial <strong
Page 33 and 34: Covariant derivative in shape space
Page 35 and 36: First fundamental form is (O’Neil
Page 37 and 38: Corpus Collosum Aging (www.oasis-br
Page 39 and 40: Landmark Space Space L of N points
Page 41 and 42: Have simple formula for cometric g
Page 43 and 44: For curvature, need Christoffel sym
Page 45 and 46: Landmark Regression</strong
Page 47: Thank You!

Polynomial Regression on Riemannian Manifolds

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