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Alessandro Sarti joint work with Giovanna Citti
Phenomenological Gestalten and figural completion: A neurogeometrical approach
Center of Mathematics CNRS-EHESS, Paris Equipe Neuromathématiques
Phenomenological Gestalten and Center of Mathematics figural - - PowerPoint PPT Presentation
Phenomenological Gestalten and Center of Mathematics figural - - PowerPoint PPT Presentation
Alessandro Sarti joint work with Giovanna Citti Phenomenological Gestalten and Center of Mathematics figural completion: CNRS-EHESS, Paris Equipe Neuromathmatiques A neurogeometrical approach Modal completion Amodal completion
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Amodal completion
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Amodal completion
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The hypercolumnar module
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The pinwheel structure
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The Cortex as a fiber bundle
W.Hoffman, J.Koenderink, S.Zucker, Bressloff Cowan,
- J. Petitot, Citti-Sarti, R.Duits, Boscain-Gauthier
π : G → B C = (G, π, B)
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The Cortex as a fiber bundle on the Lie group SE(2)
C = (G, π, B) = (E(2), π, R2)
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Infinitesimal transformation and the Lie algebra
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The stratified Lie algebra of SE(2) and the sub-Riemannian structure
X1 = cos(θ)∂x + sin(θ)∂y X3 = [X2, X1] = −sin(θ)∂x + cos(θ)∂y X2 = ∂θ are left invariant for E(2) X1, X2, X3 The Hormander condition holds
Sarti , Citti 2003 Citti, Sarti 2006
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The integral curves of the algebra
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≈
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Horizontal Connectivity
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The neurogeometrical model
- The cortex is a continuous-differentiable manifold
- Fiber bundle
- Lie symmetries of SE2 and sub-Riemannian structure
- Neural activity is constrained by the structure
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Amodal completion
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in R2 × S1\Σ0 in R2 × S1\Σ0
Horizontal mean curvature flow Horizontal Laplace-Beltrami flow The amodal completion flow
Sarti, Citti 2003 Citti, Sarti 2006
- ut = X11u(X2u)2 − 2X1uX2uX12u + X22u(X1u)2
(X1u)2 + (X2u)2 vt = X11v(X2u)2 − 2X1uX2uX12v + X22v(X1u)2 (X1u)2 + (X2u)2
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Horizontal mean curvature flow Horizontal Laplace-Beltrami flow The amodal completion flow
vt = X11v(X2u)2 − 2X1uX2uX12v + X22v(X1u)2 (X1u)2 + (X2u)2 + ✏1 + ✏2∆v ut = X11u(X2u)2 − 2X1uX2uX12u + X22u(X1u)2 (X1u)2 + (X2u)2 + ✏1 + ✏2∆u
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See the poster for the proof of existence of the sub-Riemannian mean curvature flow
- f Citti-Sarti (2003,2006) and convergence of numerics.
Inpainting
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Modal completion
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Retinex model Land et al 1974, Kimmel et al 2003, Morel et al. 2010
I(x, y) ∆ log I ∆ log f = ∆ log I
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Retinex model
I(x, y) h = log I ∆h ∆φ = ∆h φ = log f L1 = Z |rφ rh|2dxdy
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The modal completion sub-Riemannian Lagrangian
Z |r rh|2dxdy + Z |r ~ A|2dxdy + Z |X1 ~ A|2dxdy ∆ = 1 2(∆h + div( ~ A)) X11 ~ A = r + ~ A
The Euler-Lagrange Equation
G.Citti, A.Sarti 2014
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The field term
X11 ~ A = r
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The field term
X11 ~ A = r
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∆ = 1 2(∆h + div( ~ A))
The particle term
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∆ = 1 2(∆h + div( ~ A))
The particle term
Citti, Sarti 2014
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Inverted contrast
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Different apertures
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Alternate polarity
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Fragmentation
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Koffka cross: narrow
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Koffka cross: wide
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The field term
X11 ~ A = r
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Constitution of perceptual units
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Ermentraut-Cowan mean field equation of neural activity
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Horizontal connectivity kernel
δ = X11ω(ξ, 0) + X22ω(ξ, 0) δ = X1ω(ξ, 0) + X22ω(ξ, 0) ω(ξ, 0) ω(ξ, 0) ≈ e−d2
c(ξ,0)
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The E-C equation in the domain of the input
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The eigenvalue problem sub-Riemannian kernel PCA
Z ω(ξ, ξ0)u(ξ0)dξ0 = ˜ λku ω(ξi, ξj)ui = ˜ λkui
A.S., G.Citti, 2010,2014
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Spectral decomposition: 1st eigenvector
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Spectral decomposition: 2nd eigenvector
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M.Favali, G.Citti, A.Sarti preprint 2014
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Seminar of Neuromathematics of Vision
European Institute of Theoretical Neuroscience Paris Organizers: G.Citti, A.Destexhe, O.Faugeras, J.P. Nadal, J.Petitot, A.Sarti