Polynomial Regression on Riemannian Manifolds

More documents

Recommendations

Info

Corpus Collosum Aging 0.08 0.06 0.04 0.02 0 −0.02 −0.04 ˙γ(0) ∇ ˙γ ˙γ(0) (∇ ˙γ ) 2 ˙γ(0) −0.06 −0.08 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 Initial conditions are collinear, implying time reparametrization <strong>Polynomial</strong> <strong>Regression</strong> on Riemannian Manifolds 25
Landmark Space Space L of N points in R d . Geodesic equations: d dt x i = N∑ γ(|x i − x j | 2 )α j j=1 d dt α i = −2 N∑ (x i − x j )γ ′ (|x i − x j |) 2 αi T α j j=1 Usually use Gaussian kernel γ(r) = e −r/(2σ2 ) x ∈ L and α ∈ T ∗ x L is a covector (momentum) <strong>Polynomial</strong> <strong>Regression</strong> on Riemannian Manifolds 26
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 and 32: Kendall Shape Space Space of N land
Page 33 and 34: Covariant derivative in shape space
Page 35 and 36: First fundamental form is (O’Neil
Page 37: Corpus Collosum Aging (www.oasis-br
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

Create successful ePaper yourself

Delete template?

Save as template?