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 Neuromathématiques A neurogeometrical approach

Modal completion

Amodal completion

Amodal completion

The hypercolumnar module

The pinwheel structure

The Cortex as a fiber bundle C = ( G, π , B ) π : G → B W.Hoffman, J.Koenderink, S.Zucker, Bressloff Cowan, J. Petitot, Citti-Sarti, R.Duits, Boscain-Gauthier

The Cortex as a fiber bundle on the Lie group SE(2) C = ( G, π , B ) = ( E (2) , π , R 2 )

Infinitesimal transformation and the Lie algebra

The stratified Lie algebra of SE(2) and the sub-Riemannian structure X 1 = cos ( θ ) ∂ x + sin ( θ ) ∂ y X 2 = ∂ θ X 3 = [ X 2 , X 1 ] = − sin ( θ ) ∂ x + cos ( θ ) ∂ y are left invariant for E(2) X 1 , X 2 , X 3 The Hormander condition holds Sarti , Citti 2003 Citti, Sarti 2006

The integral curves of the algebra

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Horizontal Connectivity

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 •

Amodal completion

The amodal completion flow Horizontal mean c urvature flow u t = X 11 u ( X 2 u ) 2 − 2 X 1 uX 2 uX 12 u + X 22 u ( X 1 u ) 2 in R 2 × S 1 \ Σ 0 ( X 1 u ) 2 + ( X 2 u ) 2 Horizontal Laplace-Beltrami flow v t = X 11 v ( X 2 u ) 2 − 2 X 1 uX 2 uX 12 v + X 22 v ( X 1 u ) 2 in R 2 × S 1 \ Σ 0 ( X 1 u ) 2 + ( X 2 u ) 2 Sarti, Citti 2003 Citti, Sarti 2006 �

The amodal completion flow Horizontal mean c urvature flow u t = X 11 u ( X 2 u ) 2 − 2 X 1 uX 2 uX 12 u + X 22 u ( X 1 u ) 2 + ✏ 2 ∆ u ( X 1 u ) 2 + ( X 2 u ) 2 + ✏ 1 Horizontal Laplace-Beltrami flow v t = X 11 v ( X 2 u ) 2 − 2 X 1 uX 2 uX 12 v + X 22 v ( X 1 u ) 2 + ✏ 2 ∆ v ( X 1 u ) 2 + ( X 2 u ) 2 + ✏ 1

Inpainting See the poster for the proof of existence of the sub-Riemannian mean curvature flow of Citti-Sarti (2003,2006) and convergence of numerics.

Modal completion

Retinex model I ( x, y ) ∆ log I ∆ log f = ∆ log I Land et al 1974, Kimmel et al 2003, Morel et al. 2010

Retinex model I ( x, y ) ∆ h h = log I φ = log f ∆ φ = ∆ h Z | r φ � r h | 2 dxdy L 1 =

The modal completion sub-Riemannian Lagrangian Z Z Z | r � � ~ | X 1 ~ | r � � r h | 2 dxdy + A | 2 dxdy + A | 2 dxdy The Euler-Lagrange Equation ∆ � = 1 2( ∆ h + div ( ~ A )) X 11 ~ A = �r � + ~ A G.Citti, A.Sarti 2014

The field term X 11 ~ A = �r �

The field term X 11 ~ A = �r �

The particle term ∆ � = 1 2( ∆ h + div ( ~ A ))

The particle term ∆ � = 1 2( ∆ h + div ( ~ A )) 1 50 100 150 200 250 1 50 100 150 200 250 Citti, Sarti 2014

Inverted contrast

Different apertures 37

Alternate polarity

Fragmentation

Koffka cross: narrow

Koffka cross: wide

The field term X 11 ~ A = �r �

Constitution of perceptual units

Ermentraut-Cowan mean field equation of neural activity

Horizontal c onnectivity kernel ω ( ξ , 0) δ = X 1 ω ( ξ , 0) + X 22 ω ( ξ , 0) δ = X 11 ω ( ξ , 0) + X 22 ω ( ξ , 0) ω ( ξ , 0) ≈ e − d 2 c ( ξ , 0) 46

The E-C equation in the domain of the input

The eigenvalue problem Z ω ( ξ , ξ 0 ) u ( ξ 0 ) d ξ 0 = ˜ λ k u ω ( ξ i , ξ j ) u i = ˜ λ k u i sub-Riemannian kernel PCA A.S., G.Citti, 2010,2014

Spectral decomposition: 1st eigenvector

Spectral decomposition: 2nd eigenvector

M.Favali, G.Citti, A.Sarti preprint 2014

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Organizers: G.Citti, A.Destexhe, O.Faugeras, J.P. Nadal, J.Petitot, A.Sarti Seminar of European Institute of Neuromathematics of Theoretical Neuroscience � Paris Vision