Polynomial Regression on Riemannian Manifolds

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Kendall Shape Space Geometry (d = 2) Center point-set and scale so that ∑ N i=1 |x i| 2 = 1 (resulting object is called a preshape) Preshapes lie on sphere S 2N−1 , represented as vectors in (R 2 ) N = C N Riemannian submersion from preshape to shape space: vertical direction holds rotations of R 2 Exponential and log map available in closed form (for d = 2) <strong>Polynomial</strong> <strong>Regression</strong> on Riemannian Manifolds 19
Covariant derivative in shape space in terms of preshape (O’Neill1966): ∇ ∗ X ∗ Y ∗ = H∇ X Y Vertical direction is JN, where N is outward unit normal at the preshape, J is almost complex structure on C N . So parallel transport in small steps in upstairs space then do horizontal projection. <strong>Polynomial</strong> <strong>Regression</strong> on Riemannian Manifolds 20
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 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 31: Kendall Shape Space Space of N land
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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