Reconstruction and pattern recognition via the Petitot model J.P. Gauthier, U. Boscain, Dario Prandi University of Toulon and Ecole Polytechnique, Paris [01/15]January 2015 J.P. Gauthier, U. Boscain, Dario Prandi (University of Toulon and Ecole Polytechnique, Paris) Reconstruction and pattern recognition via the Petitot model [01/15]January 2015 1 / 36
Plan The Petitot Model The Hypoelliptic diffusion and the semi-discrete diffusion The lifts Chu categories and Moore groups The case of compact groups The case of SE 2 , N Pattern recognition and texture discrimination A few results J.P. Gauthier, U. Boscain, Dario Prandi (University of Toulon and Ecole Polytechnique, Paris) Reconstruction and pattern recognition via the Petitot model [01/15]January 2015 2 / 36
Papers U. Boscain, † ,J. Duplaix ‡ , J.P. Gauthier, F.Rossi, Antropomorphic image reconstruction via hypoelliptic diffusion, SIAM J. on Control SICON, 2012. ¶ U. Boscain, J.P. Gauthier, D. Prandi, A. Remizov, Hypoelliptic diffusion and human vision, a semi-discrete new twist, SIAM J. on Imaging science, 2014. J.P. Gauthier, J. Miteran, F. Smach, Generalized Fourier descriptors with application to pattern recognition in SVM context, J. on mathematical imaging and vision, 30, 2008. And the book by J. Petitot: "Vers une neurogéométrie de la vision", Ed de l’ecole Polytechnique, 2006. J.P. Gauthier, U. Boscain, Dario Prandi (University of Toulon and Ecole Polytechnique, Paris) Reconstruction and pattern recognition via the Petitot model [01/15]January 2015 3 / 36
The Petitot Model In the visual cortex V1, groups of neurons are sensitive to both positions anddirections. J.P. Gauthier, U. Boscain, Dario Prandi (University of Toulon and Ecole Polytechnique, Paris) Reconstruction and pattern recognition via the Petitot model [01/15]January 2015 4 / 36
Antropomorphic vision-1 the model is: � T 0 ( u ( t ) 2 + v ( t ) 2 ) dt → min y = sin ( θ ) u , ˙ x = cos ( θ ) u , ˙ ˙ θ = v , J ( u , v ) = J.P. Gauthier, U. Boscain, Dario Prandi (University of Toulon and Ecole Polytechnique, Paris) Reconstruction and pattern recognition via the Petitot model [01/15]January 2015 5 / 36
Antropomorphic vision-2 To this model is associated a (hypoelliptic) diffusion equn: d Ψ dt = L Ψ , ∂ y ) 2 + ∂ 2 2 (( cos ( θ ) ∂ ∂ x + sin ( θ ) ∂ L Ψ ( z , θ ) = 1 ∂θ 2 ) Ψ ( x , y , θ ) , That corresponds to go to a stochastic problem, exciting the system by two independant Brownian motions: dx t = cos ( θ ) du t , dy t = sin ( θ ) du t , d θ t = dv t , Geodesics can be computed using the PMP, they are given by classical Jacobi elliptic functions, there are very close relations between SR-distance, geodesics and small-time asymptotics of the heat kernel (for instance, lim t → 0 ( t log ( P t ( x )) = − 1 4 d ( 0 , x ) 2 ) , Heat kernel (fundamental solution) can be computed using noncommutative harmonic analysis over the group SE ( 2 ) . It is given as a series of Mathieu functions: J.P. Gauthier, U. Boscain, Dario Prandi (University of Toulon and Ecole Polytechnique, Paris) Reconstruction and pattern recognition via the Petitot model [01/15]January 2015 6 / 36
Antropomorphic vision-3 + ∞ � + ∞ n t < ce n ( θ , λ 2 4 ) , κ λ ( X , θ ) ce n ( θ , λ 2 e a λ ∑ P t ( g ) = ( 4 ) > + (1) n = 0 0 + ∞ n t < se n ( θ , λ 2 4 ) , κ λ ( X , θ ) se n ( θ , λ 2 e b λ ∑ 4 ) > ) λ d λ . n = 0 Due to the small number of pinweels ( ≈ 20 ) , the model is probably in fact semi-discrete, with stochastic equation: � cos ( θ t ) � dz t = dw t , sin ( θ t ) J.P. Gauthier, U. Boscain, Dario Prandi (University of Toulon and Ecole Polytechnique, Paris) Reconstruction and pattern recognition via the Petitot model [01/15]January 2015 7 / 36
Fokker-Planck with jumps-1 where θ is a jump process and z = ( x , y ) . Set Λ N = ( λ i , j ) , i , j = 0 , ..., N − 1, where λ i , j = lim t → 0 1 t P [ θ t = e j | θ 0 = e i ) , with e j = 2 j π N , and λ i , j = − ∑ j � = i λ i , j . Λ N is the infinitesimal generator of the process θ . We assume Markov processes, where the law of the first jump time is exponential, with parameter λ (that will be specified later on). The jump has probability 1 2 on each side. Then we get a Poisson process, and the probability of k jumps between 0 and t is: P [ k jumps) ] = ( λ t ) k e − λ t . k ! J.P. Gauthier, U. Boscain, Dario Prandi (University of Toulon and Ecole Polytechnique, Paris) Reconstruction and pattern recognition via the Petitot model [01/15]January 2015 8 / 36
Foker-Planck with jumps-2 So that: P [ θ t = e i + 1 | θ 0 = e i ] = 1 2 [ λ t + 1 2 λ 2 t 2 + ... ] e − λ t , P [ θ t = e i + 2 | θ 0 = e i ] = 1 4 [ 1 2 λ 2 t 2 + ... ] e − λ t , with the convention that e i is modulo N . So that λ i , i + 1 = λ i , i − 1 = 1 2 λ , and λ i , i = − λ . Then, the infinitesimal generator of the semi-group associated with ( z t , θ t ) is of the form: L N Ψ ( z , e i ) = ( A Ψ ) i ( z ) + ( Λ N Ψ ( z , e i )) i , where Ψ j ( z ) = Ψ ( z , e j ) , and, J.P. Gauthier, U. Boscain, Dario Prandi (University of Toulon and Ecole Polytechnique, Paris) Reconstruction and pattern recognition via the Petitot model [01/15]January 2015 9 / 36
Foker-Planck with jumps-3 ( A Ψ ) i ( z ) = A Ψ ( z , e i ) = 1 2 ( cos ( e i ) ∂ ∂ x + sin ( e i ) ∂ ∂ y ) 2 Ψ ( x , y , e i ) , n − 1 λ i , j Ψ j ( z ) = λ ∑ ( Λ N Ψ ( z , e i )) i = 2 ( Ψ i − 1 ( z ) − 2 Ψ i ( z ) + Ψ i + 1 ( z )) . j = 0 Then, if we set: λ = N 2 4 π 2 , we get: Ψ i − 1 ( z ) − 2 Ψ i ( z ) + Ψ i + 1 ( z ) ( Λ N Ψ ( z , e i )) i = 1 , ( 2 π 2 N ) 2 ∂ 2 = 1 ∂θ 2 Ψ ( z , e i ) + O ( 1 N ) . 2 J.P. Gauthier, U. Boscain, Dario Prandi (University of Toulon and Ecole Polytechnique, Paris) Reconstruction and pattern recognition via the Petitot model [01/15]January 2015 10 / 36
Foker-Planck with jumps-4 At the limit, we get: ∂ y ) 2 + ∂ 2 L Ψ ( z , θ ) = 1 2 (( cos ( θ ) ∂ ∂ x + sin ( θ ) ∂ ∂θ 2 ) Ψ ( x , y , θ ) , which is our diffusion equation, while the exact Foker-Planck equation with small number of angles is: dp j dt ( t , z ) = 1 2 ( cos ( e j ) ∂ ∂ x + sin ( e j ) ∂ ∂ y ) 2 p j ( t , z )+ λ 2 ( p j − 1 ( t , z ) − 2 p j ( t , z ) + p j + 1 ( t , z )) J.P. Gauthier, U. Boscain, Dario Prandi (University of Toulon and Ecole Polytechnique, Paris) Reconstruction and pattern recognition via the Petitot model [01/15]January 2015 11 / 36
Heat kernel via representations-1. The group law of SE ( 2 , N ) is: ( z , e i ) ∗ ( w , e j ) = ( z + R i w , e i + j ) , where R is the rotation of angle 2 π N . It is a Moore group !! Unitary irreducible reoresentations are given by the Mackey’s imprimitivity theorem. They work on Mackey’s orbits that are all Z / N Z . They are parametrized by the orbit of the action of the discrete rotations on the plane, i.e. the dual is the "slice of camembert" S N : (With the topology of the dual, you have to fold it in order to get the "french fries cone" F N ). Let λ , ν parametrize S N . Then the unitary irreducible representations are given by: ( χ λ , ν ( z , e r )) = diag k ( e i < V λ , ν , R k z > ) S r , where S is the shift mod N of the components in C N . Also, V λ , ν = ( λ cos ( ν ) , λ sin ( ν )) . The Plancherel measure is λ d λ d ν . J.P. Gauthier, U. Boscain, Dario Prandi (University of Toulon and Ecole Polytechnique, Paris) Reconstruction and pattern recognition via the Petitot model [01/15]January 2015 12 / 36
Heat kernel via representations-2. The GFT transforms our hypoelliptic equation into a continuous sum of (**Elliptic**) ones. In the following, M λ , µ is a N × N matrix: dM λ , ν = Λ N M λ , ν − diag k [ λ 2 cos ( e k − ν )] M λ , ν dt = ˜ A λ , ν M λ , ν . This is a matrix Matthieu-type diffusion. And via the inverse GFT, we get: � trace [ e ˜ A λ , ν t . diag k ( e i < V λ , ν , R k z > ) S r ] λ d λ d ν . p t ( z , e r ) = S N This is the Jump heat Kernel. A much simpler formula than in the case of SE ( 2 ) . J.P. Gauthier, U. Boscain, Dario Prandi (University of Toulon and Ecole Polytechnique, Paris) Reconstruction and pattern recognition via the Petitot model [01/15]January 2015 13 / 36
The algorithm-1. We could start with the Kernel. What we do now is a bit less economic, but more understandable. 1. First, take ordinary Fourier transform with respect to space variable z . Write w for the dual variable to z . Set also w = ( λ cos ( θ ) , λ sin ( θ )) . Diffusion becomes, at w : dU dt = Λ N U − diag k [ λ 2 cos ( e k − θ ) 2 ] U . Here, we write too many Matthieu equation. But this step can be improved on. 2. Integrate w.r.t. t 3. Take ordinary inverse Fourier transform. If you do this, you get the exact solution of the discrete diffusion. J.P. Gauthier, U. Boscain, Dario Prandi (University of Toulon and Ecole Polytechnique, Paris) Reconstruction and pattern recognition via the Petitot model [01/15]January 2015 14 / 36
Recommend
More recommend
