Problem of conformal invarincy in vision. Dmitri Alekseevsky - PowerPoint PPT Presentation

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¨ obius 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 ( R 2 ) = R + · SO 2 · R 2 and V1 as a principal Dmitri Alekseevsky Problem of conformal invarincy in vision.

CO 2 -bundle 8.6. Parametrization of hypercolumns by the stability subgroup G p 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 ) = TS 2 = T ∗ S 2 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 C P 3 (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 B 3 together with a lens L with center F ∈ B 3 near the boundary sphere S 2 which focuses light rays to the retina R ⊂ S 2 . We get a map π : L ( F ) → R ⊂ S 2 , 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 ) := I ( AY ) d σ Y ∈ D where D = { ( AX ) ∩ S 2 ( A ) , X ∈ L } is the intersection of the cone over lens L with vertex A with the unit sphere S 2 ( A ) with center A and d σ 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 ] = ker dI ? Basic global objects of early vision Static and dynamics. Basic global objects of early vision are contours = curves on the retina R ⊂ S 2 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 × S 2 where R is the time. Locally we may approximate R × S 2 by R × R 2 = R 3 . 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 PTS 2 = PT ∗ S 2 = { ( 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 J k ( R , R ) = { x , y , p = y ′ , · · · , y ( k ) } . A k -th order infinitesimal part of an input function I ∈ F ( S 2 ) is z ( I ) ∈ J k ( S 2 , R ). the k -jet j k For k = 1, J 1 ( S 2 , R ) = R × T ∗ S 2 . A better candidate for the space of first order infinitesimal functions is T ∗ S 2 . Dmitri Alekseevsky Problem of conformal invarincy in vision.

3. EYE AS A ROTATING RIGID BODY. Fixation eye movements Eye is a rigid ball B 3 O which can rotate around the center O w.r.t. three mutually orthogonal axes i , j , k . The center F 0 ∈ B 3 O of the eye crystal (lens) is near the boundary sphere S 2 O = ∂ B 3 O and the retina region R ⊂ S 2 O is a big part of S 2 O . For a fixed position of head, there is a privilege initial position B ( OF 0 ) of the eye ball corresponding to the standard (frontal) direction ( OF 0 ) 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 ( OF 0 ) 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 : S 2 → SO 3 of the frame bundle SO 3 → S 2 = SO 3 / SO 2 such that a curve γ ( t ) in S 2 has lift s γ ( t ) to the group of rotations SO 3 . Due to this law, a movement of the eye is determined by a curve on the eye sphere. Listing’s law (1845) The movement from B ( OF 0 ) to B ( OF ) is OF 0 × � � obtained by rotation with respect to the axe OF . The curve in SO 3 is the parallel lift of the initial frame along the arc F 0 F ⊂ S 2 . 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.