Poster - Geometrical Models in Vision

Orientation maps on non-Euclidean spaces
defined with group representations
Alexandre Afgoustidis
Institut de mathématiques de Jussieu - Paris Rive Gauche
Euclidean symmetry in gaussian field models for V1 orientation maps
I In
this poster, we discuss a symmetry-based model for the layout of orientation
preferences (in V1) in species with continuous orientation maps, and introduce
generalizations to homogeneous non-Euclidean geometries.
I
Niebur and Worgötter, Wolf and Geisel, Barbieri-Citti-Sanguinetti-Sarti and many
others : three ingredients are enough for getting realistic V1-like orientation maps
(ask me for defintions !),
From Bosking et al. (1997).
V1 orientation map obtained by
optical imaging in a tree shrew.
Details in the upper right corner
show pinwheels, points in the
map where all orientations are
represented.
Gaussian random field on the plane
+
Euclidean invariance of the probability distribution
+
Dominance of a single wavelength in the correlation spectrum
From Niebur and Worgötter (1994).
Correlation spectrum in an
experimental V1 map.
From Schnabel’s thesis (2008).
Correlation spectrum of a real tree
shrew map.
To simulate a monochromatic
SE(2)-invariant field ΦΛ, I drew
this map from a superposition of
30 plane waves with random
propagation directions, gaussian
random complex amplitudes and
the same wavelength Λ.
Pinwheel density and variance of hypercolumn size in invariant Gaussian fields
The spectral thinness condition must correspond to
having a well-defined hypercolumn size in the map.
I
Can we prove it ?
Kaschube, Schnabel et al. (2010) : in many species, the correlation spectrum of experimental V1
maps does concentrate on a typical wavelength Λ, and
Fig. below shows how, in SE(2)-invariant gaussian
fields, the variance of typical spacing between
iso-orientation domains varies with the spectral
width (numerical evaluation from an exact formula).
2
Mean number of pinwheels in an area Λ = 3.14 ± 2%
I In
a centered gaussian field with SE(2)-invariant probability distribution,
E (Nb. of points on a line segment of length ` with a given orientation preference) ∝ `.
I Let
I
us define the quasiperiod Λth as the only number ` for which it is equal to one.
For any invariant gaussian field on the Euclidean plane, with thin spectrum or not,
E Number of of pinwheels in an area
2
Λth
= π (for details, ask me !)
Group representation theory : a symmetry-based toolbox
Representation theory is a symmetry-based generalization
of harmonic analysis to non-Euclidean spaces with a group
action.
I Weyl and Wigner proved that it can be used as an
engineering tool in physics, to guess at models from little
more than symmetry assumptions.
I The set of maps which can be sampled from Φλ
carries an irreducible unitary representation of SE(2).
I
I Yaglom
: zero-mean Gaussian invariant random fields on a homogeneous
space G/K correspond (via their correlation functions) to K -bi-invariant
(deterministic) functions G → C.
I Élie
Cartan, Harish-Chandra, Helgason and others : for some pairs
(G, K ), studying K -bi-invariant functions on G is the same as studying
some unitary irreducible representations of G which can be realized as
spaces of smooth functions on G/K .
Using group representations to define orientation maps on non-Euclidean spaces
I
If G is a group and X a G-homogeneous
riemannian space, we can look for
gaussian fields defined on X whose
probability distribution is G-invariant, and
whose samples probe an irreducible
representation of G contained in L2(X ).
I
We call these gaussian fields
monochromatic ; when X is one of Élie
Cartan’s riemannian symmetric spaces,
they can be described explicitly.
I
There are generalizations to fields having
shift-twist symmetry, but they are technically
more difficult to study.
Negative curvature
Positive curvature
When X is a riemannian symmetric space of the
noncompact type, we can use Helgason waves
to mimic the Euclidean constructions. Here the
analogies with Euclidean space are strongest.
When X is a compact symmetric space, the
irreducible constituents of L2(X ) are finitedimensional and generated by (generalized)
spherical harmonics ; there is but a countable set
of monochromatic fields.
Example with G = SL2(R) :
Example with G = SO(3) :
Pinwheel densities in non-Euclidean maps
I
I
In riemannian symmetric spaces, let us first define the quasiperiod in an invariant
gaussian field as the only real number Λ such that, on a geodesic segment of length
Λ, the expectation for the number of points with a given orientation is equal to one.
For invariant gaussian fields defined on symmetric spaces, Λ can be computed
from an eigenvalue of the Laplace-Beltrami operator.
Many thanks to :
I My advisor Daniel Bennequin, for many suggestions and constant support ;
I Alessandro Sarti, Michel Duflo, Alain Berthoz, for helpful discussion ;
I Christine and Sophie Cachot, for help with computer software.
We can then use recent generalizations of the Kac-Rice formula to evaluate the
pinwheel density δ (mean number of zeroes in a region of invariant volume Λdim X ) of
a monochromatic invariant field :
δ = π dim(X )/2.
Further reading (ask me for detailed references !)
I Kaschube et al. (2010) Science Vol. 330 no. 6007, pp. 1113-1116
I F. Wolf and T. Geisel (2003) Journal of Physiology - Paris 97:253-264.
I A. M. Yaglom (1961), 2d Order Homogeneous Random fields.
I R. Adler, J. Taylor (2005). The geometry of Random Fields (Springer).
I S. Helgason (1984) Groups and Geometric Analysis