reconstruction and pattern recognition via the petitot model

Reconstruction and pattern recognition via the Petitotmodel

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 SE2,NPattern recognition and texture discrimination

A few results


Papers

U. Boscain, †,J. Duplaix‡, J.P. Gauthier, F.Rossi, Antropomorphicimage reconstruction via hypoelliptic diffusion, SIAM J. on ControlSICON, 2012.¶U. Boscain, J.P. Gauthier, D. Prandi, A. Remizov, Hypoellipticdiffusion and human vision, a semi-discrete new twist, SIAM J. onImaging science, 2014.

J.P. Gauthier, J. Miteran, F. Smach, Generalized Fourier descriptorswith application to pattern recognition in SVM context, J. onmathematical imaging and vision, 30, 2008.

And the book by J. Petitot:"Vers une neurogéométrie de la vision", Ed de l’ecole Polytechnique, 2006.


The Petitot Model

In the visual cortex V1, groups of neurons are sensitive to both positionsanddirections.


Antropomorphic vision-1

the model is:

x = cos(θ)u, y = sin(θ)u, θ = v , J(u, v) =∫ T

0(u(t)2+ v(t)2)dt → min



To this model is associated a (hypoelliptic) diffusion equn:

dΨdt

= LΨ,

LΨ(z , θ) =12((cos(θ)

∂

∂x+ sin(θ)

∂

∂y)2 +

∂2

∂θ2)Ψ(x , y , θ),

That corresponds to go to a stochastic problem, exciting the systemby two independant Brownian motions: dxt = cos(θ)dut ,dyt = sin(θ)dut , dθt = dvt ,Geodesics can be computed using the PMP, they are given byclassical Jacobi elliptic functions,there are very close relations between SR-distance, geodesics andsmall-time asymptotics of the heat kernel (for instance,limt→0(t log(Pt (x)) = − 14d(0, x)2),Heat kernel (fundamental solution) can be computed usingnoncommutative harmonic analysis over the group SE (2). It isgiven 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


Pt (g) =

+∞∫0

(+∞

∑n=0ea

λn t < cen(θ,

λ2

4),κλ(X , θ)cen(θ,

λ2

4) > + (1)

+∞

∑n=0eb

λn t < sen(θ,

λ2

4),κλ(X , θ)sen(θ,

λ2

4) >)λ dλ.

Due to the small number of pinweels (≈ 20), the model is probablyin fact semi-discrete, with stochastic equation:

dzt =(cos(θt )sin(θt )

)dwt ,


Fokker-Planck with jumps-1

where θ is a jump process and z = (x , y). SetΛN = (λi ,j ), i , j = 0, ...,N − 1, whereλi ,j = limt→0 1t P [θt = ej |θ0 = ei ), with ej =

2jπN , and

λi ,j = −∑j 6=i λi ,j .

ΛN is the infinitesimal generator of the process θ.We assume Markov processes, where the law of the first jump time isexponential, with parameter λ (that will be specified later on). The jumphas probability 1

2 on each side.

Then we get a Poisson process, and the probability of k jumpsbetween 0 and t is:

P [k jumps)] =(λt)k

k !e−λt .


Foker-Planck with jumps-2

So that:

P [θt = ei+1|θ0 = ei ] =12[λt +

12

λ2t2 + ...]e−λt ,

P [θt = ei+2|θ0 = ei ] =14[12

λ2t2 + ...]e−λt ,

with the convention that ei is modulo N.

So that λi ,i+1 = λi ,i−1 =12λ, and λi ,i = −λ.

Then, the infinitesimal generator of the semi-group associated with (zt , θt )is of the form:

LNΨ(z , ei ) = (AΨ)i (z) + (ΛNΨ(z , ei ))i ,

where Ψj (z) = Ψ(z , ej ), and,



(AΨ)i (z) = AΨ(z , ei ) =12(cos(ei )

∂

∂x+ sin(ei )

∂

∂y)2Ψ(x , y , ei ),

(ΛNΨ(z , ei ))i =n−1∑j=0

λi ,jΨj (z) =λ

2(Ψi−1(z)− 2Ψi (z) +Ψi+1(z)).

Then, if we set: λ = N 24π2, we get:

(ΛNΨ(z , ei ))i =12

Ψi−1(z)− 2Ψi (z) +Ψi+1(z)( 2πN )

2,

=12

∂2

∂θ2Ψ(z , ei ) +O(

1N).



At the limit, we get:

LΨ(z , θ) =12((cos(θ)

∂

∂x+ sin(θ)

∂

∂y)2 +

∂2

∂θ2)Ψ(x , y , θ),

which is our diffusion equation, while the exact Foker-Planck equationwith small number of angles is:

dpjdt(t, z) =

12(cos(ej )

∂

∂x+ sin(ej )

∂

∂y)2pj (t, z)+

λ

2(pj−1(t, z)− 2pj (t, z) + pj+1(t, z))


Heat kernel via representations-1.

The group law of SE (2,N) is:

(z , ei ) ∗ (w , ej ) = (z + R iw , ei+j ),where R is the rotation of angle 2π

N .It is a Moore group!!

Unitary irreducible reoresentations are given by the Mackey’s imprimitivitytheorem. They work on Mackey’s orbits that are all Z/NZ.They are parametrized by the orbit of the action of the discrete rotationson the plane, i.e. the dual is the "slice of camembert" SN : (With thetopology of the dual, you have to fold it in order to get the "french friescone" FN ). Let λ, ν parametrize SN . Then the unitary irreduciblerepresentations are given by:

(χλ,ν(z , er )) = diagk (ei<Vλ,ν,R k z>)S r ,

where S is the shift modN of the components in CN . Also,Vλ,ν = (λ cos(ν),λ sin(ν)). The Plancherel measure is λdλdν.


Heat kernel via representations-2.

The GFT transforms our hypoelliptic equation into a continuous sum of(**Elliptic**) ones. In the following, Mλ,µ is a N ×N matrix:

dMλ,ν

dt= ΛNMλ,ν − diagk [λ2 cos(ek − ν)]Mλ,ν

= Aλ,νMλ,ν.

This is a matrix Matthieu-type diffusion.And via the inverse GFT, we get:

pt (z , er ) =∫SNtrace[eAλ,νt .diagk (e

i<Vλ,ν,R k z>)S r ]λdλdν.

This is the Jump heat Kernel. A much simpler formula than in the case ofSE (2).


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 :

dUdt= ΛNU − diagk [λ2 cos(ek − θ)2]U.

Here, we write too many Matthieu equation. But this step can beimproved on.2. Integrate w.r.t. t3. Take ordinary inverse Fourier transform.If you do this, you get the exact solution of the discrete diffusion.


The algorithm-2.

In fact, first, start from the discrete diffusion:dpkdt(t, z) =

12(cos(ek )

∂

∂x+ sin(ek )

∂

∂y)2pj (t, z)+

λ

2(pk−1(t, z)− 2pki (t, z) + pk+1(t, z)).

1. Take a space discretization, and the ordinary finite differencesapproximation, to get:

dpkdt(t, zi ,j ) =

12(cos(ek )A ∂

∂x+ sin(ek )A ∂

∂y)2pk (t, z)+

λ

2(pk−1(t, z)− 2pk (t, z) + pk+1(t, z)), (D)

This is a very big linear differential system: (512× 512×N). 2. Take thedouble FFT, to get a completely parallel systemd Pi ,j

dt (t) = Di ,j Pi ,j .3. Integrate with respect to time the 512×512 matthieu-like equations. 4.Take the inverse double FFT.

TheoremThis algorithm provides the exact solution to the space-discretized system(D).


The algorithm-3.

I assume that the initial image is infinite, twice periodic.

TheoremThis algorithm provides the exact solution to the space-discretized system(D).

dpkdt(t, zi ,j ) =

12(cos(ek )A ∂

∂x+ sin(ek )A ∂

∂y)2pk (t, z)+

λ

2(pk−1(t, z)− 2pk (t, z) + pk+1(t, z)), (D)

Using the Heat Kernel, one could integrate much less Matthieu-like eqs:only one for each point of the slice SN .


More precisely-1.

x , y : 0→√M, ,x = k−1√

M, y = l−1√

M.

FFTM (u)k ;l = 1M ∑M

r ,s=1 ur ,se−2πi [ (k−1)(r−1)+(l−1)(s−1)M ];

and conversely uk ;l = 1M ∑M

r ,s=1 FFTM (u)r ,se2πi [ (k−1)(r−1)+(l−1)(s−1)M ]. Then

the discretized operator ∂∂x is mapped to:

(∂u∂x)ˆk =

1√M

M

∑r=1

(ur+1 − ur−1)2√M

e−2πi [ (k−1)(r−1)M ]

=√M

1

2√M[(M

∑r=1

ur+1e−2πi [ (k−1)(r+1−1)M ])e2πi k−1M −

(M

∑r=1

ur−1e−2πi [ (k−1)(r−1−1)M ])e−2πi k−1M ]

=

√M2uk (e

2πi k−1M − e−2πi k−1M ) = i√Muk sin(

2π(k − 1)M

).


More precisely-2.

Then, cos(θ) ∂∂x + sin(θ)

∂∂y →k ,l

iM(sin( 2π(k−1)M ) cos(θ) + sin(θ) sin( 2π(l−1)

M )),

and:[ ∂2

∂θ2+ α(cos(θ) ∂

∂x + sin(θ)∂

∂y )2]u→k ,l

∂2

∂θ2uk ,l − αM2(sin( 2π(k−1)

M ) cos(θ) + sin(θ) sin( 2π(l−1)M ))2uk ,l .

Finally, the diffusion becomes:

2∂urk ,l

∂t= α

ur−1k ,l − 2urk ,l + ur+1k ,l

( πM )

2 −

βM2(sin(2π(k − 1)

M) cos(θ) + sin(θ) sin(

2π(l − 1)M

))2urk ,l .

One can see the natural projectivisation (due to the square).


Relation to kernel and to dual-1.

The (limit) diffusion is, after space-Fourier transform:

dudt=

∂2u∂θ2− ρ2 cos(ω− θ)2u, or for each fixed ρ,ω :

duρ,ω(θ, t)dt

=∂2uρ,ω

∂θ2− ρ2 cos(ω− θ)2uρ,ω.

Set uρ,ω(ω− θ, t) = uρ,ω(θ, t), ω− θ = θ, to obtain:

duρ,ω(θ, t)dt

=∂2uρ,ω(θ, t)

∂θ2− ρ2 cos(θ)2uρ,ω(θ, t).

It means that we need to compute only resolvants along the dual half-line.In the discrete case, we need to compute resolvants at each point of theslice of camembert only.


Based upon this diffusion and certain heuristic complements, we get nicealgorithms for image completion:


An other point of view.

In relation with what follows (for pattern recognition), there is a differentpoint of view . There are finite dimensional subspaces [with arbitrarilylarge dimension] of the space of almost periodic functions over SE (2,N)that are invariant under the diffusion operator: any finite direct sum ofN-dimensional spaces of irreducible representations of SE (2,N).In restriction to these spaces, the diffusion (not the discretization, but theexact diffusion) can be EXACTLY integrated by the previous algorithm.


The lifts 1

These considerations here work in the general context of a semi-directproduct G = N nH of a compact group N by an abelian locally compactgroup H, and the Haar measure on H is invariant under the action R of N.In that case, all linear left invariant lifts L from L2(H) to L2(G ), such thatf → Lf (0, e) is densely defined and bounded in L2 norm are of the form:

Lf (n,X ) = [(RnΨ) ∗ f ](X ),

where (RnΨ)(X ) = Ψ(Rn−1X ), and Ψ ∈ L2(H) .Moreover such a map L is injective iff∫

N

|Ψ(Rnχ)|2dn > 0 for a.e. χ ∈ H.

Example: we can take for Ψ a standard orientation filter (Gabor forinstance).


The lifts 2

Define, for f ∈ L2(H) and χ ∈ H, ωf (χ) ∈ L2(H) byωf (χ)(n) = f (Rn−1χ).Problem: The Fourier transform Lf (Tχ) of a left invariant lift is always arank1 operator:

Lf (Tχ) = ωf (χ)∗⊗ωΨ(χ).

This will be a big problem later on.From this point of view, there are better lifts than the left invariant ones.For instance if H = R2, the "cyclic lift"

f c (n,X ) = f (RnX + Xc ),

Xc =1

f (0)

∫H

Xf (X )dX


Chu Duality 1

Chu duality is an extension of Tannaka duality, working for certainnoncompact groups. Groups (locally compact) that have Chu Duality:Abelian, compact, Moore.Remark: not all MAP groups have Chu duality (Roeders’s example).The group M2,N is Moore and then has Chu duality.

Chu dualG a topological group, RPN (G ) denotes the set of all N-dimensionalcontinuous unitary representations R of G in CN , with thecompact-open topology, and RP(G ) is the topological summ of theRPN (G ) over N ≥ 1. RPN (G ) is second countable provided that G isso.RP(G ) is called the Chu dual of G .


Quasi representations

Quasi representationsA quasi representation of G is a continuous map Q from RP(G ) tothe topological sum U = ∪n1U(n) of all unitary groups, with thefollowing properties:(Q1) Q(R) ∈ U(n(R)), (Q2) Q(R ⊕ R ′) = Q(R)⊕Q(R ′), (Q3)Q(R ⊗ R ′) = Q(R)⊗Q(R ′), (Q4) Q(U−1RU) = U−1Q(R)U for allR,R ′ ∈ RP(G ) and U ∈ U(n(R)).Denote by RP(G ) the union of all quasi representations of Gembedded with the compact open topology. RP(G ) is called the Chuquasi dual of G .Set E (R) = Idn(R ), Q

−1(R) = Q(R−1). Then, RP(G ) is a Hausdorftopological group, with E as its identity.

for g ∈ G set g(R) = R(g). consider Ω : G → RP(G ), Ω(g) = g .Ω is a continuous homomorphism, injective provided that G is MAP.


Chu duality again

Def: the group G has the Chu duality property if Ω is a topologicalisomorphism.

TheoremMoore groups have the Chu duality property.

Not all MAP groups have Chu duality property

SE2,N has the Chu duality property while SE2 has not.

Chu duality is a (topological) generalization of Tannaka duality forcompact groups (which is itself an analog of Pontryagin’s duality forabelian groups).


Pattern recognition: the bispectral principle

Let G be a locally compact group. Let G denote as usual the dual ofG , i.e. the set of (equivalence classes of) unitary irreduciblerepresentations of G . For f ∈ L2(G ), Define If : G × G → C,

If (λ1,λ2) = f (λ1)⊗ f (λ2) f (λ1 ⊗ λ2)∗, in which f (λ) is the Fouriertransform of f , and λ1 ⊗ λ2 is the tensor product representation.By the properties of the Fourier transform, If is invariant under the leftaction of G on L2(G ).

Bispectral principle:

TheoremLet G be separable, abelian or compact, or Moore (with certainrestriction). Then, there is a residual Subset R of L2(G ) such thatIf = Ih implies that h = g0.f for some g0 ∈ G .

That is, over the very big subset R of L2(G ), functions are separatedmodulo translations by the invariants If .


Proof of the bispectral principle (compact case, sketch)

The generic set R is the subset of L2(G ) such that the Fourier transformf (λ) is invertible for all λ ∈ G .Assume If = Ih. Apply it to λ1 = λ,λ2 = T , the trivial representation,to get that f (λ) f (λ)∗ = h(λ) h(λ)∗. It follows thath(λ) = f (λ)U(λ), for some unitary operator U(λ).The map U extends uniquely to a map RP(G )→ U by requiring Q2(commutation with ⊕). Due to the definition of the Fourier transform, themap U meets Q4 (commutation with unitary equivalences). PropertyQ3 (commutation with tensor product) is obtained from the equalityIf = Ih and the definition of the generic set R.It follows that U is a quasi representation, and by Chu (Tannaka) duality,there is g ∈ G such that U = g . Then, h(λ) = f (λ) λ(g), and by theelementary property of the Fourier transform, h = g .f .


In the case of 1-dimensional signals f (t), the bispectral invariants are just

B(λ1,λ2) = f (λ1)f (λ2)f (λ1 + λ2)

Note that B (λ1,0)f (0)

is just the "power spectral density" of the signal, and

B(λ1,λ2) contains the missing phase informations.

The B(λ1,λ2) are used in several areas of signal processing. For instance:Dubnov S, Tishby N and Cohen D. (1997). "Polyspectra as Measures ofSound Texture and Timbre". Journal of New Music Research 26: 277—314.

The If (λ1,λ2) (the Bispectral Invariants) are the generalization of these,to 2-D signals (more precisely to their lifts). It is expected that theycontain all information modulo motions.


The case of SE(2,N) 1

Define I λ1,λ2,k2 (f ) =< ωf (λ1 + k.λ2),ωf (λ1)ωf (k.λ2) >L2(N ), where

is the pointwise product.Then, we have the following relation:

[Lf (λ1 ⊗ λ2) Lf (λ1)∗ ⊗ Lf (λ1)∗].F (u, uk−1) =I λ1,λ2,k2 (f )Ψ(u−1(λ1 + k.λ2)) (ωΨ(λ1)

∗ ⊗ωΨ(λ2)∗.F )

From the bispectral principle, we could expect that the I λ1,λ2,k2 form a

(weakly) complete set of invariants of the action of SE (2,N) on L2(H),but it doesn’t work since the Fourier transform operators have all rank 1.This is still a conjecture. We have arguments to conjecture that this couldbe false.


The case of SE(2,N) 2

Define I λ1,λ2,k2 (f ) = [Lf (λ1 ⊗ λ2) Lf (λ1)∗ ⊗ Lf (R−kλ2)∗].

We say that ωf (λ) is cyclic if Snωf (λ) is a basis of L2(N). A functionf ∈ L2(H) is said weakly cyclic if ωf (λ) is cyclic for almost all λ ∈ HNote that I λ1,λ2,k

2 are not invariant under translations (work for centeredfunctions only). We can prove:

Theorem

(T) Assume that Ψ is weakly cyclic and Ψ(λ) 6= 0 for almost all λ ∈ H.Let f , g , be weakly cyclic functions with compact support having the sameinvariants. Then, there is n ∈ N such that f = Rng .

The sketch of the proof is the same as the general proof for compactgroups: we construct a quasi-representation of G . But a lot of complicateddetails come. A key point is the induction-reduction theorem (an analogof the Clebsch-Gordan decomposition for compact groups).


Texture discrimination 1

Let K be a (finite or countable) subset of R2, which is stable under theaction of ZN .It is natural to consider images whose lift are now almost-periodicfunctions f over SE (2,N). Those are called "texture images".

f (n, x) = ∑n

h∈K

a(n, h)e i<Rnh,x>

and that are in the B2 Besikovich class. The theory above can be adaptedto these spaces of functions, and an analog of theorem (T) above can beproved.Moreover,

If K is finite, this space can be used (as was pointed out) to solveexactly the Diffusion equation (first part of the talk). It is a finitedimensional invariant subspace of the action of SE (2,N), andnaturally a sum of spaces of irreducible representations.The (analog of) conjecture of completeness of the invariants I λ1,λ2,k

2is false (we have a counterexample)


A few results 1

Bispectral invariants have good properties w.r.t. scale: letfα(x) := f (αx), then:B(λ1,λ2)(fα) = 1

α6B(λ1

α ,λ2α )(f ). We can use this

relation to eliminate the scale effects.At this point, we use a standard strategyBispectral invariants are used together with a learning machine (SVM), torealize "pseudo 3D" pattern recognition.Vapnik, Vladimir N.; The Nature of Statistical Learning Theory,Springer-Verlag, 1995Also, for texture discrimination, wit is very natural to use thealmost-periodic context.


A few results 2

A long time ago, we got a series of very nice results on standard academicdata bases. We get nice results w.r.t. standard strategies.


A few results 3

Face detection on the ORL data base


Work on texture discrimation is going on

We thank you for your attention


reconstruction and pattern recognition via the petitot model

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