Alexandre AFGOUSTIDIS (Institut de Mathématiques de Jussieu, Paris, France)
Orientation maps in non-Euclidean geometries

Abstract
Existing models for the geometry and development of orientation preference maps in the primary visual cortex of higher mammals make a crucial use of symmetry considerations. We start from the fact that the dominant features of probabilistic models for V1 maps can be described with the words of Group theory, the mathematical common ground for symmetry arguments ; with this in mind we review the probabilistic arguments that allow to estimate pinwheel densities and predict the observed value of \pi. Then, in order to test the relevance of general symmetry considerations and to introduce methods which could be of use in modelling curved regions, we use representation theory to build orientation maps adapted to the most famous non- Euclidean geometries, viz. spherical and hyperbolic geometry. We find that the famous features of V1 maps are preserved in these ; the graphic layout is very telling of course, but there are also precise counterparts to the results on Euclidean pinwheel densities.

Poster
Davide BARBIERI (Universidad Autonoma de Madrid, Spain)
Spatio-temporal spectral clustering with cortical kernels.

Abstract
The geometric structure of V1 spatio-temporal receptive fields defines a 5 dimensional fiber bundle $M=(R^2_x \times R^+_t) \times (S^1_\theta \times R^+_v) of positions and activation times together with the locally detected features of orientation and apparent velocity. Subelliptic diffusions over this manifold allow to implement affinity kernels that measure the distance between different regions of spatio-temporal visual stimuli according to the collinearity of level lines and their dynamic evolution patterns. The spectral analysis of such kernels, computed over synthetic moving images, provides decays of eigenvalues that are sharp enough to obtain neat clustering results, which are compatible with the psychophysical behaviors of visual grouping.
* Joint work with G. Citti,
G. Cocci,
M. Favali and A. Sarti. *

Poster
Alexey MASHTAKOV (Eindhoven University of Technology, The Netherlands)
Solutions to an association field model on the
retinal sphere: sub-Riemannian geodesics in
SO(3) with cuspless spherical projections.

Abstract
Joint work with R. Duits, Yu. Sachkovz and I. Beschastnyix.

Poster
Nina MIOLANE (INRIA Sophia Antipolis, France)
Statistics on Lie groups : a need to go beyond the pseudo-Riemanian framework.

Abstract
Lie groups appear in many geometrical models related to vision, from the field of Medical Imaging to the field of Robotics. In Medical Imaging and particularly in Computational Anatomy, an organ's shape is often modeled as the deformation of a reference shape, in other words : as an element of a Lie group. In this framework, if one wants to model the variability of the human anatomy observable in images, e.g. in order to help diagnosis of diseases, one needs to perform statistics on Lie groups. A Lie group G is a manifold that carries an additional group structure. Statistics on Riemannian manifolds have been well studied with the pioneer work of Fréchet, Karcher and Kendall followed by others. In order to use such a Riemannian structure for statistics on Lie groups, one needs to define a Riemannian metric that is compatible with the group structure, i.e a bi-invariant metric. However, it is well known that general Lie groups which cannot be decomposed into the direct product of compact and abelian groups do not admit a bi-invariant metric. One may wonder if removing the positivity of the metric, thus asking only for a bi-invariant pseudo-Riemannian metric, would be sufficient for most of the groups used in Computational Anatomy. In this paper, we provide an algorithmic procedure that constructs bi-invariant pseudo-metrics on a given Lie group G. The procedure relies on a classification theorem of Medina and Revoy. However in doing so, we prove that most Lie groups do not admit any bi-invariant (pseudo-) metric. We conclude that the (pseudo-) Riemannian setting is not the richest setting if one wants to perform statistics on Lie groups. One may have to rely on another framework, such as affine connection space.

*Joint work with X. Pennec. *

Poster
Dario PRANDI (Université de Toulon, France)
Highly corrupted image inpainting through hypoelliptic diffusion.

Abstract
We present an image inpainting algorithm based on hypoelliptic diffusion for the Citti-Petitot- Sarti model of the primary visual cortex V1 [1,2,3]. In particular, we focus on the inpainting problem with complete knowledge of the location of the corruption: the damaged image consists of pixels, and we know whether each pixel is damaged or not. Our techniques are an improvement on the semi-discrete approach introduced in [4] for the Citti-Petitot-Sarti mode, obtained by introducing heuristic methods that allows for very good reconstructions of highly corrupted images (i.e., with more than 80\% of corrupted pixels).
*Joint work with U. Boscain, J.-P. Gauthier and A. Remizov. *

Poster
Gonzalo SANGUINETTI (Department of Mathematics and Computer Science, TUE Eindhoven, The Netherlands)
Existence of sub-Riemannian mean curvature flow and amodal completion.

Abstract
We reconsider the sub-riemannian cortical model of completion introduced by Citti and Sarti in 2003. This model combines two mechanisms: sub-riemannian diffusion and concentration, giving rise to a diffusion driven motion by curvature. In this study we give a formal proof of the existence of viscosity solutions of sub-riemannian motion by curvature, extending the results of Evans and Spruck. We also show that the proposed sub-riemannian finite difference scheme of OS type is convergent. Results of completion and enhancement on a number of natural images are shown, and compared with other techniques.

*Joint work with G. Citti, B. Franceschiello and A. Sarti. *

Poster
Jonathan VACHER (Université Paris Dauphine, France)
Dynamic Textures for Probing Visual Perception

Abstract
Perception is often described as a predictive process based on an optimal inference with respect to a generative model. Under such an assumption, we study here the principled construction of generative models specifically parametrized for probing visual perception. In that context, we first provide an axiomatic derivation of the generative model along with biologically relevant probability distributions. This model synthesizes dynamic random textures which are defined by stationary Gaussian distributions obtained by the random aggregation of small warped patterns. Herein, we show that this model can equivalently be described as a stochastic partial differential equation. Firstly, these findings have direct applications to enable a real time generative numerical scheme. Moreover, it allows to recast the motion-energy model into a principled Bayesian inference framework.

*Joint work with G. Peyré, L. Perrinet and A.I. Meso. *

Poster