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Problem of conformal invarincy in vision.

Dmitri Alekseevsky

Institute for Information Transmission Problems, Moscow, Russia and Masaryk University, Brno, Czech Republic

Paris, October 24, 2014

Dmitri Alekseevsky Problem of conformal invarincy in vision.

What we learn about vision during last 20 years Bruno A. Olshausen ”20 Years of Learning About Vision: Questions Answered, Questions Unanswered, and Questions Not Yet Asked” (in ”20 Years of Computational Neuroscience”,2013) ”We are still confronted with profound mysteries about how visual systems work. These are not just mysteries about biology, but also about the general principles that enable vision in any system whether it be biological or machine.” We learn that the visual system is much more complicated then we believe before and we do not understand even the basic principles. To understand the basic principles, we need synthesis of results and ideas from different sciences which deal with vision.

Dmitri Alekseevsky Problem of conformal invarincy in vision.

1.INTRODUCTION Aim of vision Visual system receives information from light (electro-magnetic radiation). The aim of vision is to obtain information about (Euclidean ) geometry of the external world from light which comes to retina and transform it into a finite set of ”invariants” (”gestalts”, words, emotions etc.) It must be objective, i.e. independent from position of observer ( that is invariant with respect to change of position of eyes, head, velocity etc.) Two questions arise : 1) Which information comes to eye (retina) and how does it change under movement of eye and head? 2) How do eyes and brain extract invariant information about the external geometry from the input subjective (dependent on position etc) information which comes to retina?

Dmitri Alekseevsky Problem of conformal invarincy in vision.

1 Light in approximation of geometric optics 2.Eye as an optical devise. Central projection 2.1. Energy function on retina 2.2. What is input function of retina? 2.3. Basic global and infinitesimal objects for early vision

• 3. Eye as a rotating rigid body.Saccades

3.1. Donder’s and Listing’s Laws 3.2.Fixation eye movements 3.3. Fixation eye movements as random walk in parabolic swamp. 3.4. Aim of eye movements

• 4. Transformation of retina image under eye’s rotation

4.1. Condition for conformality. 4.2. Problem of conformal invariancy as problem of conformal geometry of curves 5.Conformal geometry of sphere 5.1.Conformal M¨

• bius group and Tits model of conformal sphere.

5.2. Cartan connection associated with conformal sphere

• 6. Multiscale approximation of differential geometry ( following

Jan Koenderink and Luc Florack) and models of visual cells

Dmitri Alekseevsky Problem of conformal invarincy in vision.

6.1 Points in Differential Geometry 6.2. Gauss filters as an approximation of points 6.3. Sigma-approximation of tangent vectors 6.4. Visual neurons as functionals 7.Transformation of input function in retina and LGN 7.1. Scheme of processing and data conversion 7.2. Architecture of retina 7.3. Two pathway from receptors to ganglion cells 7.4. Retinotopic (topographic ) maps from retina to LGN. P-map and M-map

• 8. Primary visual cortex V1

8.1. Architecture of primary visual cortex V1. Columns and pinwheel field 8.2. Hypercolumns. Model of hypercolumn by Bressloff and Cowan 8.4. J.Petitots model of V1 cortex as S. Lee contact bundle PTV = PT ∗V 8.5. Generalized Petitot’s model by Sarti-Citti-Petitot: Parametrization of simple cells of a hypercolumn by (local) conformal group CO(R2) = R+ · SO2 · R2 and V1 as a principal

Dmitri Alekseevsky Problem of conformal invarincy in vision.

CO2-bundle 8.6. Parametrization of hypercolumns by the stability subgroup Gp and Tits model of the eye sphere. 8.7. Application to the problem of conformal invariance.

Dmitri Alekseevsky Problem of conformal invarincy in vision.

1.LIGHT IN APPROXIMATION OF GEOMETRIC OPTICS

Light is described in terms of the space L(E 3) = TS2 = T ∗S2 of straight lines. Light travels along a line ℓ ∈ L(E 3) with energy density I(ℓ) (the average value of the square norm of electric filed). We ignore the wave length (color) and polarization of light. Then all information for eye is coded in the energy function I : L(E 3) → R of light. Maxwell electrodynamics : light is a superposition of plane waves. Plane waves is associated with a shear-free congruence of isotropic lines in the Minkowski space (Robinson) and with a complex surface in the Penrose twistor space CP3 (Kerr). QED (quantum electrodynamics).

Dmitri Alekseevsky Problem of conformal invarincy in vision.

Energy function associated with a surface Σ ⊂ E 3

A point A ∈ Σ is a source of (reflected) light which travels along rays in all directions outside of the surface. If the energy density I(AX) of a ray depends only on the source A ∈ Σ(diffuse reflection) and constant in time, we get a function I : Σ → R (energy function).

Dmitri Alekseevsky Problem of conformal invarincy in vision.

Eye

Dmitri Alekseevsky Problem of conformal invarincy in vision.

• 2. EYE AS AN OPTICAL DEVICE

Eye is a transparent ball B3 together with a lens L with center F ∈ B3 near the boundary sphere S2 which focuses light rays to the retina R ⊂ S2. We get a map π : L(F) → R ⊂ S2, ℓ → ℓ ∩ R from a surface Σ to retina R which depends on the position of the eye ball B(OF). The energy of light at a point ˆ A ∈ R is I R( ˆ A) :=

• Y ∈D

I(AY )dσ where D = {(AX) ∩ S2(A), X ∈ L} is the intersection of the cone

• ver lens L with vertex A with the unit sphere S2(A) with center A

and dσ is the standard measure of this sphere. The function I R : R → R is called the energy function.

Dmitri Alekseevsky Problem of conformal invarincy in vision.

What is the input function on the retina : the energy function I, 1-form dI or 1-distribution D = [dI] = kerdI? Basic global objects of early vision

Static and dynamics. Basic global objects of early vision are contours = curves on the retina R ⊂ S2 which are level set of the intensity function with ”big” gradient (w.r.t. which metric?) It is the image of edge (boundary of the object of external world.) For simplicity we consider only immovable objects. More elaborate answer is that the basic objects are piece-wise smooth surfaces in the 3-cylinder R × S2 where R is the

• time. Locally we may approximate R × S2 by R × R2 = R3.

Dmitri Alekseevsky Problem of conformal invarincy in vision.

Basic infinitesimal objects of early vision

First order infinitesimal approximation of a non parametrized curve (contour) is a tangent line. The space of such infinitesimal contours is the contact bundle PTS2 = PT ∗S2 = {(x, y, p = dy dx )} with the contact structure ker(dy − pdx). An infinitesimal contour of order k is a k-jet of a contour. The space of such objects can be identified with Jk(R, R) = {x, y, p = y′, · · · , y(k)}. A k-th order infinitesimal part of an input function I ∈ F(S2) is the k-jet jk

z (I) ∈ Jk(S2, R).

For k = 1, J1(S2, R) = R × T ∗S2. A better candidate for the space of first order infinitesimal functions is T ∗S2.

Dmitri Alekseevsky Problem of conformal invarincy in vision.

• 3. EYE AS A ROTATING RIGID BODY. Fixation eye

movements

Eye is a rigid ball B3

O which can rotate around the center O w.r.t.

three mutually orthogonal axes i, j, k. The center F0 ∈ B3

O of the eye crystal (lens) is near the boundary

sphere S2

O = ∂B3 O and the retina region R ⊂ S2 O is a big part of

S2

• O. For a fixed position of head, there is a privilege initial position

B(OF0) of the eye ball corresponding to the standard (frontal) direction (OF0) of the gaze.

Dmitri Alekseevsky Problem of conformal invarincy in vision.

Donder’s and Listing’s laws

Donder’s law (1846)(No twist). If the head is fixed, the result of a movement from position B(OF0) to a new position B(OF) is uniquely defined by the gaze OF and do not depend on previous movement. Mathematically, it defines a section s : S2 → SO3 of the frame bundle SO3 → S2 = SO3/SO2 such that a curve γ(t) in S2 has lift sγ(t) to the group of rotations SO3. Due to this law, a movement

• f the eye is determined by a curve on the eye sphere.

Listing’s law (1845) The movement from B(OF0) to B(OF) is

• btained by rotation with respect to the axe
• OF0 ×

OF. The curve in SO3 is the parallel lift of the initial frame along the arc F0F ⊂ S2. Question Is Donder’s and Listing’s law valid for involuntary fixation movements ?

Dmitri Alekseevsky Problem of conformal invarincy in vision.

Fixation eye movements

Fixation eye movements include: tremor, drifts and microsaccades. Tremor is an aperiodic, wave-like motion of the eyes of high frequency but very small amplitude. Drifts occur simultaneously with tremor and are slow motions of eyes, in which the image of the fixation point for each eye remains within the fovea! Drifts occurs between the fast, jerk-like, linear microsaccades.

Dmitri Alekseevsky Problem of conformal invarincy in vision.

Characteristics of fixation eye movements

Amplitude Duration Frequency Speed Tremor 20-40 sec

• 30-100 Hz

Max 20 min/s Drift 1-9 min 0.2-0.8 s 95-97% of time 1-30 min/s Micsac 1-50 min 0.01-0.02 s 0.1-5 Hz 10 − 50◦/s Per 1 s tremor moves on 1-1.5 diameters of the fovea cone drift moves on 10-15 diameters microsaccads moves on 15-300 diameters. Under tremor the axis of eye draws a cone for 0.1 s. In 2-3 sec after compensation of fixation eye movement, a human lost ability to see an immobile object. (Yarbus)

Dmitri Alekseevsky Problem of conformal invarincy in vision.

Microdaccades, drift and tremor Drift and microsaccades

Dmitri Alekseevsky Problem of conformal invarincy in vision.

Model of eye movements by R.Engbert, K. Mergenthaler,

• P. Sinn, A. Pikovsky ”Self-avoiding Random Walk”

Involuntary eye movement described as a self-avoiding random walk on the square lattice Z2 with quadratic potential (”Random walk in a swamp on a paraboloid”). Physiological aim of such movement (when gaze fix a point A): the images ( ¯ A)(t) of A on retina must be homogeneously distributed between all receptors of the fovea.

Dmitri Alekseevsky Problem of conformal invarincy in vision.

Why we need fixation eye movements

1.(Geometry) For a fix gaze OF, the retina gets information only about the 2-dimensional Lagrangian submanifold L(F) = RP2 of the 4-dimensional space of lines L(E 3). When eye moves in a neighborhood of a fixed point OF, it gets information from a neighborhood of L(F) in 4-manifold L(E 3).

• 2. To see immobile objects (Yarbus)
• 3. To determine direction of moving external objects (Roords et

al., 2013) 4(Neuroscience) For better identification of contours in V1 cortex.

Dmitri Alekseevsky Problem of conformal invarincy in vision.

• 4. TRANSFORMATION OF AN IMAGE UNDER EYE’S
• ROTATION. Central projection

Let S2 = {A, OA

2 = x2 + y2 + z2 = r2 0 }

be the eye sphere in coordinates wit origin at the center O and and first ort i ∼ OF s.t. F = (r.0.0). The image of a point A = (x, y, z) ∈ Σ on retina R ⊂ S2is given by the central projection πF : Σ ∋ A → ¯ A = F − f (A)(A − F) = f (A)A + (1 − f (A))F where f (A) is the positive solution of the quadratic equation (A − F)2f 2 − 2F · (A − F)f + F 2 − r2

0 = 0.

If F ∈ S2, i.e. r0 = r, the equation becomes linear and ¯ A = (r, 0, 0) − 2r(x−r)

R2

(x − r, y, z) =

r R2 (−(x − r)2 + y2 + z2, 2(x − r)y, 2(x − r)z),

where R2 := (A − F)2 = (x − r)2 + y2 + z2.

Dmitri Alekseevsky Problem of conformal invarincy in vision.

Central projection of a plane to sphere

Let Σ = Π = Πρ

n = {A, n · A = ρ} is the plane with normal vector

n = (cos ϕ, sin ϕ, 0) where ρ = dist(Π, O) and coordinates (y, z). Then Πρ

n = {A = ρn + (sin ϕy, − cos ϕy, z) =

(ρ cos ϕ + sin ϕy, ρ sin ϕ − cos ϕy, z)} The central projection is πF : A → ¯ A = F − f (A)(A − F) = F − 2F · (A − F) (A − F)2 (A − F).

Dmitri Alekseevsky Problem of conformal invarincy in vision.

When the central projection of a plane is a conformal map?

The induces metric gS2 of the sphere S2 w.r.t. the local coordinates y, z s.t. ¯ A(y, z) = πFA(y, z) is gS2 = d ¯ A2 = f 2dA2 − 2r sin ϕdydf where dA2 = dy2 + dz2 is the metric of the plane Πn

ρ

f = − 2F·(A−F)

(A−F)2

= − 2r(sin ϕy+β)

R2

, R2 = |A − F|2 := (y − sin ϕ)2 + z2 + (ρ − sin ϕ)2, β = ρ cos ϕ − r and df = 2r R4 [{ρ2−2r2+y2+z2+2y(r −2ρ cos ϕ}dy −2z(β+sin ϕy)dz] It is a conformal map iff the plane is frontal ( i.e. orthogonal to the frontal direction , i.e. ϕ = 0. A small rotation R = Rα

O of eye (which is equivalent to a rotation

R−1 of the external space in opposite direction) produces (approximately) a conformal transformation of the eye sphere.

Dmitri Alekseevsky Problem of conformal invarincy in vision.

Problem of conformal invariant perception of contours (problem of stability) as the main problem of conformal geometry of curves

Importance of conformal group in vision (Hoffman, 1989 ) The main problem of differential geometry of curves in a homogeneous manifold M = G/H is to construct the full system of G-invariants of a curve C ⊂ M which determines it up to a transformation from G. For Euclidean plane E 2 = SE(2)/SO2, a solution given by Frenet associates with a curve γ the natural equation K = K(s) where s is the natural parameter (arc-length) and K(s) = |¨ z(s)| (the curvature = acceleration of the path z(s)). For conformal geometry S2 = SO1,3/Conf (E 2) of sphere similar solution is known ( A. Fialkov, J. Haantjes, R. Sharp, F.Brustall and D. Calderbank.) The natural equation of a curve γ ⊂ S2 is K = K(s) where s is a ”conformal parameter ” along a curve γ (defined up to a fractional linear transformations) and K is the conformal curvature which depends on 5-jet of γ.

Dmitri Alekseevsky Problem of conformal invarincy in vision.

5 CONFORMAL GEOMETRY OF SPHERE

Let (V = R1,3, g) be the Minkowski vector space, V 0 the light cone of isotropic vectors and S2 := PV 0 = {[p] := Rp, p ∈ V0} ≃ S2 the celestial sphere. The metric g induces a conformal structure [g0] in S2 and the connected Lorentz group G = SO(V ) ≃ SO1,3 acts transitively on S2 as the group of conformal transformations (the M¨

• bius group).

Dmitri Alekseevsky Problem of conformal invarincy in vision.

Gauss decomposition of conformal Lie algebra and conformal group

The gradation V = V−1 + V0 + V1 = Rp + E 2 + Rq, g(p, p) = g(q, q) = g(Rp + Rq, E 2) = 0, g(p, q) = 1 defines a gradation of the Lie algebra g = so(V ) = g−1 + g0 + g1 = p ∧ E + so(E) + q ∧ E. It defines the Gauss decomposition of the conformal group G = SO(V ) = G− · G0 · G+. S2 = G/B = G/G0 · G+, G± ≃ Sim(E 2) = R+ · SO2 · R2.

Dmitri Alekseevsky Problem of conformal invarincy in vision.

Sphere S2 as the Riemannian sphere C ∪ {∞}

In terms of the holomorphic coordinate z ∈ S2 ≃ C ∪ {∞}, SO(V ) = SL2(C)/(±1) ∋ ±A = ± a b c d

• : z → az + b

cz + d . The Lie algebra sl2(C) = {(b + az + cz2)∂z}. The gradation is sl2(C) = g−1 + g0 + g1 = {b∂z} + {az∂z} + {cz2∂z}. The (local) Gauss decomposition is G = SL2(C) = G− · G0 · G+ = 1 b 1

• ,

a d 1 c 1

• Note that S2 = G/B = SL2(C)/G 0 · G + where B = G 0 · G 1 is the

Borel subgroup of upper triangular matrices.

Dmitri Alekseevsky Problem of conformal invarincy in vision.

Tits model of the conformal sphere

Since NG(B) = B, the conformal sphere S2 can be identified with the set of all Borel subgroups in G = SO(V ) = SL2(C) In terms of real coordinates x, y of the sphere,where z = x + iy, the generators of G consist of generators of G 0: (dilatation)E = r∂r = x∂x + y∂y, (rotation) R = x∂y − y∂x, generators of G − : (translations) ∂x, ∂y, generators of G +: (special fields) Y 1 = (x2 − y2)∂x + 2xy∂y = 2xE − r2∂x = 2xr∂r − r2∂x, Y 2 = 2xy∂x − (x2 − y2)∂y = 2yE − r2∂y = 2xr∂r − r2∂y.

Dmitri Alekseevsky Problem of conformal invarincy in vision.

Cartan connection associated with a conformal structure

Differential geometry teaches that the most important object associated with a conformal geometry is the (normal) Cartan connection (a principal bundle with an absolute parallelism ). In the case of conformal sphere, the Cartan connection is the principal bundle π : SO1,3 → S2 = SO1,3/Sim(E 2) = G/G0 · G+

• f second order frames with the absolute parallelism , defined by

the Maurer-Cartan form µ : TSO1,3 → so1,3.

Dmitri Alekseevsky Problem of conformal invarincy in vision.

• 6. MULTISCALE APPROXIMATION OF DIFFERENTIAL

GEOMETRY AND MODELS OF VISUAL CELLS. Points in Differential Geometry (DG)

From quantum point of view, the main object of DG is the algebra C ∞(M) of function on a manifold. Point z ∈ M is a special linear functional (called ”Dirac delta function”) δz0 : f → δz0(f ) = f (z0). Tangent vector at z0 is a linear functional V : C ∞(M) → R which satisfies the Leibnitz rule V (fg) = f (z0)V (g) + g(z0)V (f ). Moreover, such functional can be consider as a partial derivative of the delta function due to the formula (∂xδz0)(f ) = −(∂xf ) where (x, y) are local coordinates of a point z ∈ M2. Approximating the delta functional by functionals associated with smooth functions (e.g. Gauss functions G σ) we get a ”sigma” -approximation of DG.

Dmitri Alekseevsky Problem of conformal invarincy in vision.

Gauss filter as sigma-approximation of a point

We assume that M = R2 = {z = (x, y)}. The delta functional δz0 associated with a point z0 = (x0, y0) is approximated by Gauss functionals (”Gauss filter”) TG σ : I →

• G σ(z)I(z)dxdy

where G = G σ

z0(z) =

1 √ 2πσ exp(−|z − z0|2 2σ2 ) is the Gauss function. Note that Gauss function and functional are isotropic,i.e. invariant under rotation w.r.t. z0

Dmitri Alekseevsky Problem of conformal invarincy in vision.

Sigma-approximation of tangent vectors as functionals associated with derivative of Gauss function

If X is a divergent free vector field, then the functional TX·G associated with X · G is a σ-approximation of the vector −Xz0 etc.

Dmitri Alekseevsky Problem of conformal invarincy in vision.

Visual neurons as functionals (”filters”)

Many visual cells of early vision (in retina, LGN and V1 cortex) can be considered as functionals I → F(I) on the space of input functions, which associate with an input function I on retina the number, the ”degree of excitation” of the cell. At some approximation, this functional can be considered as a linear functional ( generalized function or ”linear filter”) which measure the integral of the input function I in some small domain D with appropriate weight F (receptive profile (RP)): TF : I →

• D

F(z)I(z)dxdy, where F is a smooth function with support D (”receptive field” (RF) of the cell). RP is ordinary constructed from the Gauss function.

Dmitri Alekseevsky Problem of conformal invarincy in vision.

Marr and Gabor filters

Sometimes, more realistic assumption is that a cell with RP F acts as the convolution operator I → F ∗ I, where the new function (F ∗ I)(z0) := TFIz0 is obtained by integration of the shifted function Iz0(z) := I(z − z0) which is produced by movements of eye. A first model of cell with isotropic RP was proposed by D. Marr. It based on Kuffler description of ganglion cells (especially, P-cells) and is applied also to many types of cells in LGN and even in V1

• cortex. Marr cell works as Marr filter (linear isotropic functional

with RP ∆G σ

z0).

Important anisotropic model is the model of simple cells. It is described as Gabor filters : linear functional with Gabor function as RP: Gab = kG σ

0 (cos 2y + i sin 2y) w.r.t.some Cartesian

coordinates (x, y) with center at 0. It depends on position, variance σ and orientation.

Dmitri Alekseevsky Problem of conformal invarincy in vision.

A differential model of visual cells (M.Hansard and R. Horaud, 2011)

Since the radius r of an eye S2 is very big with respect to the diameter of RF of V1 cells near fovea and the curvature K = 1

r2 of

the induces physical metric g on S2 is small, we may assume that the metric is flat. We can choose a holomorphic coordinate z = ρeiϕ centered at any point such that g = dzd ¯

• z. Then ∂z is

the covariant derivative w.r.t. Levi-Civita connection and we can represent the k-jet of the Gauss function G = G σ

0 = 1 √ 2πσe− z ¯

z 2σ2 at

the origin as sum of symmetric forms G j of degree j = 0, · · · , k as follows jk

0 G = G 0 + G 1 + G 2 + · · · + G k

G 0 = G(0), G 1 = dG|0 = Gzdz + G¯

zd ¯

z = − G 2σ2 (¯ zdz + zd ¯ z), G 2 = Gzzdz2 + 2Gz ¯

zdzd ¯

z + G¯

z ¯ zd ¯

zd ¯ z = G 4σ4 (¯ z2dz2 + z2d ¯ z2) + G σ2 ( 1 2σ2 r2 − 1)dzd ¯ z, etc

Dmitri Alekseevsky Problem of conformal invarincy in vision.

Differential model

Evaluated symmetric forms G j on a parallel unit vector field Vθ := cos θ∂x + sin θ∂y = 1 2(eiθ∂z + e−iθ∂¯

z)

we get functions Gj := Gj(Vθ, · · · , Vθ). M.Hansard and R. Horaud proposed to consider them as RP of simple cells. In particular, G1 = (G1)σ,θ(z, ¯ z) = − G 4σ2 (¯ zeiθ +ze−iθ) = − G 2σ2 (x cos θ+y sin θ) Up to higher order in |z|, this function coincides with the odd Gabor function. Gabσ,θ = − G 2σ2 sin(x cos θ + y sin θ).

Dmitri Alekseevsky Problem of conformal invarincy in vision.

Differential model

Similar, we get G2 = G σ,θ

2

(z, ¯ z) = G 2σ4 g(Y 1, V2θ) + G σ2 ( 1 8σ2 z ¯ z − 1). where Y 1 = (x2 − y2)∂x + 2xy∂y = Re(z2∂z) is the special conformal vector field. The second term ∂z∂¯

zG(Vθ, Vθ) = 1/4∆G

coincides (up to a factor) with the Marr function Mσ := ∆G σ.

Dmitri Alekseevsky Problem of conformal invarincy in vision.

Model of complex cell by Hansard and Horand

Let S1(z, σ, θ) = G σθ

1

∗ I(z) be the response by the first order simple cell with parameters σ, θ. It will be large if there is a contour through z in direction perpendicular to Vθ = (cos θ, sin θ). A model of a complex cell , constructed from simple cells is defined by the formula C(z, σ, θ) := maxt,|t|<t0|S1(z, σ, θ)|.

Dmitri Alekseevsky Problem of conformal invarincy in vision.

• 7. TRANSFORMATION OF INPUT FUNCTION IN

RETINA AND LGN. Processing and data conversion in retina, LGN and V1 cortex

• 1. Retina produces a smoothing and contourization of the input

function by Marr filters.

• 2. LGN organizes an additional isotropic preparation of the input
• data. Using feed back from the higher levels of the visual system ,

it modifies and correct parameters of visual cells (e.g. the scale ). 3.Recognition of local pieces of contours is started in V1 cortex.

• 4. There are two independent channel of data conversion:

P-channel from P-cells of retina to 4 upper parvocellular layers of LGN ( which consists of parvocellular small cells) and then to 4β layer of V1 cortex.It is responsible for stable contours. M-channel from M-cells of retina to lower two layers of LGN (containing magnocellular big cells) and then to layer 4α of V1

• cortex. It is responsible for detection and analysis of moving
• bjects.

Dmitri Alekseevsky Problem of conformal invarincy in vision.

Retina

Retina

Dmitri Alekseevsky Problem of conformal invarincy in vision.

Architecture of retina

Retina consists of 5 layers. In rabbit’s retina, there are 55 different types of cells and in human approx. 80. The bottom layer consists of receptors, photoelements which transform light energy into electric signals. They measure the intensity function I R : R → R≥0 and send information to ganglion cells. In fovea one cone is connected with 1 ganglion. In peripheria, one rode is connected with 102 − 103 ganglions. There are 1 million of ganglions and 125 − 150 millions of receptors. RF of ganglion cells are rotationally invariant and contain central disc and surround ring.

Dmitri Alekseevsky Problem of conformal invarincy in vision.

On and Off Marr cells

Dmitri Alekseevsky Problem of conformal invarincy in vision.

Two pathways from receptors to ganglions

Direct path: receptor - bipolar - ganglion activates the center of ganglion cells, which work as a linear filter. Antagonistic surround is activated by (linear) negative feed back from horizontal cells ( via indirect path : receptor-horizontal cell- (amacril)- bipolar - ganglion). A nonlinear rectifying mechanism (associated with contrast gain control) is related with amacril cells. For sufficiently small contrast, ganglion P-cells is working as linear Marr filter. M-cells, responsible for perception of moving objects, are working as essentially non-linear filters. Response depends on stimulus contrast and temporal frequency. (E. Kaplan, E. Benardette, Dynamics of ganglion cells, 2001)

Dmitri Alekseevsky Problem of conformal invarincy in vision.

Retinotopic map from retina to LGN and V1 cortex

P cells project to upper 4 parvocellular layers of LGN and then to the layer 4β of V 1 M cells project to lower two magnocellular layers of LGN and then to the layer 4α of V1. Schwartz meromorphic dipole formula for retinotopic maps (E. Schwartz, 2002) z → w = k z + a z + b Conformality.

Dmitri Alekseevsky Problem of conformal invarincy in vision.

Retinotopic map from LGN to V1 cortex

Dmitri Alekseevsky Problem of conformal invarincy in vision.

• 8. PRIMARY VISUAL CORTEX V1. Columnar structure

and pinwheel field

Local quantities in V1 ( RF, orientation, spatial frequency,temporal frequency, ocular dominance etc.) Columnar structure of V Cells with approximately the same RF are organized in vertical columns. Simple cells of a column at a point acts as Gabor filter with some

• rientation.

For a regular point p ∈ V all simple cells of the column have the same orientation Γp = RXp ∈ PTpV . Simple cells of a column at a singular point (called pinwheels) have all possible orientations and are parametrized by a circle. We get a fundamental 1-dimensional distribution with singularities p → Γp (Pinwheel field)

Dmitri Alekseevsky Problem of conformal invarincy in vision.

Pinwheel field of directions

Dmitri Alekseevsky Problem of conformal invarincy in vision.

• Hypercolumn. Model by Paul Bressloff and Jack Cowan

(2002)

Hubel and Wiezel define notion of hypercolumn as minimal collection of columns (module), which contain all possible

• rientations and are sufficient to reconstruct the local structure of

a contour.

• Conjecture. RF of hypercolumn cells cover area of retina, which

contains images ¯ A(t) of an immobile external point A under fixation eye movements. Hypercolumns as spheres. Considering parametrisation of simple cells by two parameters: orientation θ and spatial frequency p ∈ [pL, pH], Bresloff and Cowan proposed a model of hypercolumn as a sphere associated with two pinwheels, which corresponds to north and south poles, with spherical coordinates ϕ = π log(p/pL)

log(pH/pL), θ.

Dmitri Alekseevsky Problem of conformal invarincy in vision.

Jean Petitots model of V1 cortex. Parametrization of simple cells

J.Petitot considers primary visual cortex VI as a surface V with a field of directions Γ. He observes that if we consider parametrization of simple cells according to their function (as Gabor filters), they will be parametrized by a surface ˜ V which is the blowing up of V at all centers of pinwheels. All simple cells of the column at a regular point acts as the same Gabor filter and define one point in ˜ V . Simple cells of columns at a singular point z (pinwheel) measure contours of any direction and parametrized by a circle (preimage of z in ˜ V ).

Dmitri Alekseevsky Problem of conformal invarincy in vision.

Petitot’s model: Primary cortex as a contact bundle

Under approximation that all points are centers of pinwheels, J.Petitot concludes that the space of simple cells can be identified with the contact bundle PTV = PT ∗V → V of directions with the natural contact structure.(Before it was guessed by Hoffman). Simple cells of V detect not only points z of a contour CV , but also its direction TzCT . So they determine the horizontal lift of the contour to the horizontal curve ¯ CT ⊂ PT(V ). If (x, y) are coordinates in V such that contours are described as y = y(x), then the contact manifold PTV can be locally identified with the manifold J1(R) of 1-jets of functions with coordinates (x, y, p = dy

dx ) and the contact form η = dy − pdx. The contact

manifold J1(R) is identified with the Heisenberg group Heis3 with a left invariant contact structure.

Dmitri Alekseevsky Problem of conformal invarincy in vision.

Generalized Petitot’s model by Sarti-Citti-Petitot

The simple cells of a singular column z0(i.e. Gabor filter with center at z0) are parametrized by an angle θ ∈ PTz0V = S1 = SO2 and a scale σ that is with points of the group CO2 = R+ × SO2 or with the set of conformal frames in

• Tz0S2. The set of simple cells locally, can be parametrized by the

Borel group B− = G0 · G− = Sim(E 2) = CO2 · R2 and the space of simple cells in V1 (in Petitot’s approximation) can be considered as the principal CO2-bundle R+ × PT ∗V = T ∗V of conformal frames on S2 with natural symplectic structure. This is the basic assumption of the generalized Petitot model.

Dmitri Alekseevsky Problem of conformal invarincy in vision.

Parametrization of a hypercolumn by stability group G+ = G0 · G+ = Sim(E 2) and V1 as the principal bundle of conformal frames of second order (Cartan connection)

We conjecture that simple cells of a hypercolumn are parametrized locally by points of the stability subgroup B = G0 · G+ ≃ CO2 · R2 (or, equivalently) its Lie algebra. Two new parameters (a, b) ∈ R2

+

(the coordinates of an element from G+ ≃ R2 with respect to the basis Y1, Y2) corresponds to the differential dσ of the scale σ. They are second order objects, i.e. are defined by the second jet. Then in Petitot approximation (when we consider all points as center of pinwheels), we get the Tits model of the eye sphere where points are identified with the corresponding stability subgroups ( or subalgebras). One of the advantage of this generalized Petitot model is that it allows to explain partially the invariance of perception w.r.t. fixation eyes movements.

Dmitri Alekseevsky Problem of conformal invarincy in vision.

A neurophysiological approach to problem of conformal invariancy

Let G be a transformation group of a manifold V (e.g the group SO2 of rotation in V = R2) and S is a G orbit. If observers are distributed along S and send information (say, about a curve in V ) to some center O anonymously, the information obtained by O will be G invariant. If simple cells of a hypercoloumn send information to a complex cell C it become invariant w.r.t. the stability subgroup. If C get information from other neighborhood hypercolumns, it become invariant w.r. t. the conformal group G.

Dmitri Alekseevsky Problem of conformal invarincy in vision.