Poster - Geometrical Models in Vision

Orientation maps on non-Euclidean spaces
defined with group representations
Alexandre Afgoustidis
Institut de math´ ematiques de Jussieu - Paris Rive Gauche
Euclidean symmetry in gaussian field models for V1 orientation maps
IIn 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.
INiebur and Worg¨
otter, 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 !),
Gaussian random field on the plane
+
Euclidean invariance of the probability distribution
+
Dominance of a single wavelength in the correlation spectrum
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.
From Niebur and Worg ¨ otter (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.
Can we prove it ?
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).
IKaschube, Schnabel et al. (2010) : in many species, the correlation spectrum of experimental V1
maps does concentrate on a typical wavelength Λ, and
Mean number of pinwheels in an area Λ2=3.14 ±2%
IIn 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)∝`.
ILet us define the quasiperiod Λth as the only number `for which it is equal to one.
IFor any invariant gaussian field on the Euclidean plane, with thin spectrum or not,
ENumber of of pinwheels in an area Λ2
th=π(for details, ask me !)
Group representation theory : a symmetry-based toolbox
IRepresentation theory is a symmetry-based generalization
of harmonic analysis to non-Euclidean spaces with a group
action.
IWeyl and Wigner proved that it can be used as an
engineering tool in physics, to guess at models from little
more than symmetry assumptions.
IThe set of maps which can be sampled from Φλ
carries an irreducible unitary representation of SE(2).
IYaglom : zero-mean Gaussian invariant random fields on a homogeneous
space G/Kcorrespond (via their correlation functions) to K-bi-invariant
(deterministic) functions G→C.
I´
Elie Cartan, Harish-Chandra, Helgason and others : for some pairs
(G,K), studying K-bi-invariant functions on Gis the same as studying
some unitary irreducible representations of Gwhich can be realized as
spaces of smooth functions on G/K.
Using group representations to define orientation maps on non-Euclidean spaces
IIf Gis a group and XaG-homogeneous
riemannian space, we can look for
gaussian fields defined on Xwhose
probability distribution is G-invariant, and
whose samples probe an irreducible
representation of Gcontained in L2(X).
IWe call these gaussian fields
monochromatic ; when Xis one of ´ Elie
Cartan’s riemannian symmetric spaces,
they can be described explicitly.
IThere are generalizations to fields having
shift-twist symmetry, but they are technically
more difficult to study.
Negative curvature
When Xis 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.
Example with G=SL2(R):
Positive curvature
When Xis a compact symmetric space, the
irreducible constituents of L2(X)are finite-
dimensional and generated by (generalized)
spherical harmonics ; there is but a countable set
of monochromatic fields.
Example with G=SO(3):
Pinwheel densities in non-Euclidean maps
IIn riemannian symmetric spaces, let us first define the quasiperiod in an invariant
gaussian field as the only real number Λsuch that, on a geodesic segment of length
Λ, the expectation for the number of points with a given orientation is equal to one.
IFor invariant gaussian fields defined on symmetric spaces, Λcan be computed
from an eigenvalue of the Laplace-Beltrami operator.
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=πdim(X)/2.
Many thanks to :
IMy advisor Daniel Bennequin, for many suggestions and constant support ;
IAlessandro Sarti, Michel Duflo, Alain Berthoz, for helpful discussion ;
IChristine and Sophie Cachot, for help with computer software.
Further reading (ask me for detailed references !)
IKaschube et al. (2010) Science Vol. 330 no. 6007, pp. 1113-1116
IF. Wolf and T. Geisel (2003) Journal of Physiology - Paris 97:253-264.
IA. M. Yaglom (1961), 2dOrder Homogeneous Random fields.
IR. Adler, J. Taylor (2005). The geometry of Random Fields (Springer).
IS. Helgason (1984) Groups and Geometric Analysis