## List of abstracts

Dmitri ALEKSEEVSKY (Institute for Information Transmission Problems, Moscow, Russia)

Slides

Giovanni BELLETTINI (Università di Roma "Tor Vergata", Italy)

Slides

Daniel BENNEQUIN (Université Pierre et Marie Curie, France)

Ugo BOSCAIN (CMAP École Polytechnique, France)

Slides

Pascal CHOSSAT (Université de Nice Sophia-Antipolis, France)

Slides

Jack COWAN (University of Chicago, USA)

Slides

Remco DUITS (Eindhoven University of Technology, The Netherlands)

Slides

Olivier FAUGERAS (INRIA Sophia Antipolis, France)

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Yves FRÉGNAC (CNRS, France)

Jean LORENCEAU (CNRS, France)

Slides

Stéphane MALLAT (École Normale Supérieure, France)

Peter MICHOR (Universität Wien, Austria)

Slides

Xavier PENNEC (INRIA Sophia Antipolis, France)

Slides

Jean PETITOT (CAMS EHESS, France)

Alessandro SARTI (CAMS CNRS-EHESS, France)

Slides

Romain VELTZ (INRIA Sophia Antipolis, France)

Slides

Fred WOLF (Max-Planck-Institut für Dynamik und Selbstorganisation Theoretische Neurophysik, Germany)

Steve ZUCKER (Yale University, USA)

Problem of conformal invariance in early vision

Abstract
Within the framework of geometric optics, we analyze which input information about external world comes to the retina and how the input function
changes under eye movements. Under some assumptions, the change of the
input function (density of the light energy on retina) and contours (its level
curves with large gradient) is described by a conformal transformation of the
retina $\mathbb{R}\subset S^2$ which is a part of the eye sphere $S^2$.
Contours and their inﬁnitesimal versions - directions are main objects which are detected in early
vision (in primary visual cortex V1 and V2).
Since the eye is always participated in diﬀerent types of involuntary movements, there must be a universal neural mechanism of identiﬁcation of con-
formally equivalent contours, which represent for example the edge of a stationary object at diﬀerent positions of the eye.
We discuss two possible versions of such mechanism : ﬁrst, based on
the classical invariant description of curves on conformal sphere in terms of
differential invariants and signature, and second, based on a modiﬁcation of
Sarti-Citti-Petitot model of V1 cortex (2007), described in terms of the cotangent bundle $T^*S^2$ of the eye sphere,
and on proposed by Hubel and Wiesel concept of multicolumn in V1 cortex (recently found also in V2 area).
Also we shortly review the basic facts about ﬁxational eye movements, architecture of retina and primary visual cortex.

Slides

Giovanni BELLETTINI (Università di Roma "Tor Vergata", Italy)

Theory and algorithms for shape reconstruction from apparent contours.

Abstract
Motivated by the calculus of variations,
we shall discuss the problem of reconstructing the topology of
a three-dimensional smooth shape starting
from a plane graph (visible contour graph)
having as singularities only terminal points
and T-junctions. Manipulation of apparent contours via a complete
set of elementary moves will also be addressed, as well as
the problem of elimination of cusps.
A number of algorithmical applications will be shown.

Slides

Daniel BENNEQUIN (Université Pierre et Marie Curie, France)

Example of invariance for adaptation: spatial
frequency and orientation in cat's V1.

Abstract
Recent data of Jérome Ribot et al. on cat's V1 have shown an amazing structure of dipolar singularities for the spatial frequency map, situated at the pinwheels centers of the orientation map. The talk will explain in what sense this configuration is optimal for minimizing topological redundancy of the two maps. The pinwheel-dipole structure is scale invariant, and probably plays a role in V1 adaptation. We will also discuss more general principles linking
invariance and adaptation. Joint work with Jérome Ribot, Alberto Romagnoni, Chantal Milleret and Jonathan Touboul.

Ugo BOSCAIN (CMAP École Polytechnique, France)

Hypoelliptic diffusion and human vision: a semi-discrete new twist

Abstract
We present a semi-discrete alternative to the theory of neurogeometry
of vision, due to Citti, Petitot and Sarti. We propose a new
ingredient, namely working on the group of translations and discrete
rotations SE(2,N). The theoretical side of our study relates the
stochastic nature of the problem with the Moore group structure of
SE(2,N). Harmonic analysis over this group leads to very simple finite
dimensional reductions. We then apply these ideas to the inpainting
problem which is reduced to the integration of a completely
parallelizable finite set of Mathieu-type diffusions (indexed by the
dual of SE(2,N) in place of the points of the Fourier plane, which is
a drastic reduction). The integration of the the Mathieu equations can
be performed by standard numerical methods for elliptic diffusions and
leads to a very simple and efficient class of inpainting algorithms.
We illustrate the performances of the method on a series of deeply
corrupted images. Joint work with R. Chertovskih, J.P. Gauthier, A. Remizov.

Slides

Pascal CHOSSAT (Université de Nice Sophia-Antipolis, France)

Texture perception, one step forward beyond the ring model

Abstract
The "ring" model was a successful attempt to account for experimental observations about the processing of orientation detection by the visual cortex V1 area, using the mean field equations of Wilson and Cowan.
More recently, extensions of the ring model to other features such as spatial frequency or contrast have been proposed. I present such a model where the orientation is replaced by the "structure tensor", an object which has long been known from engineers in image processing. This concept was applied to texture perception by O. Faugeras, G. Faye and PC. I introduce the model and its analysis and discuss its relevance to texture perception theory.

Slides

Jack COWAN (University of Chicago, USA)

Geometric Visual Hallucinations:
what they tell us about the
architecture of the brain

Abstract
In 1979 Ermentrout and Cowan showed how to analyze
spatial pattern formation in the form of perceived geometric
hallucinations in a simplified model of visual cortex, using
equivariant bifurcation theory applied to the mean-field
Wilson-Cowan equations. In doing so they introduced the
idea that the visual cortex possessed the Euclidean
symmetry of the plane E(2), and had the Cartesian
coordinates $\{x,y\}$ at every point of the plane $\mathbb{R}^2$.
They then
provided the first account of how geometric visual
hallucinations might be generated in the brain. In 2001
Bressloff, Cowan, Golubitsky, Thomas, and Wiener
introduced a more elaborate model of visual cortex, in which
the coordinates of any cortical location were now $\{x,y,\phi\}$
where $\phi$ is the orientation preference at the point $\{x,y\}$.
The
space is now $\mathbb{R}^2 \times S^1$. In order to preserve Euclidean
symmetry on mapping $\mathbb{R}^2 \times S^1$ onto a discrete lattice, the
rotation operator $\theta$ maps $\{x,y,\phi\}$ into $\{R_{\theta}\{x,y\},\phi+\theta\}$. In
computer vision this is called a shift-twist operator. They
then used the equivariant branching lemma of Golubitsky,
Stewart, and Schaeffer (1988), to extend the classes of
hallucinatory images generated by such models. In 2002
Bressloff and Cowan showed that this model could in turn be
extended to include spatial frequency preference by
representing the cortical space as $\mathbb{R}^2 \times S^2$ in which orientation and spatial frequency preferences are represented by the
angular coordinates of the sphere $S^2$.
In this lecture I hope to show that the remaining
features of early vision: color, depth and motion can also be
represented in similar fashion, and that visual hallucinations
involving such features can be incorporated into the
geometric patterns seen as visual hallucinations.

Slides

Remco DUITS (Eindhoven University of Technology, The Netherlands)

Lie Group Analysis for Medical Image Processing.

Abstract
The human visual system outperforms state-of-the art computer algorithms due to smart encoding of early cognitive features that allow for subsequent perceptional organization. We model this by constructing scores from images, that extend the image domain to a larger non-commutative Lie group G allowing us to deal with multiple features that are locally present. These scores are obtained by probing the image with a family of group coherent wavelets, and coincide e.g. with continuous wavelet transforms on G=SIM(d), coherent state transforms on G=SE(d), or Gabor Transforms on G=H(2d+1), d=2,3. The key challenge is to exploit these scores, their underlying group structure, and their intrinsic invertibility in order to perform contextual operators for robust enhancement and detection of basic patterns in medical images. This is done via left-invariant PDE's such as adaptive diffusions and Hamilton-Jacobi-Bellman equations and differential invariants, using the score as initial condition, and subsequent tracking via left-invariant ODE's arising from optimal control problems in (sub-Riemannian manifolds within) G.
Via the Cartan connection we briefly categorize both the (horizontal) exp-curves along which the PDE-flow take place, and
the (sub)-Riemannian geodesics for tracking. Then for applications both curves need score adaptation, and this is done
by optimal exponential curve fits and induced gauge frames in the score, and by including external costs in
new wave front propagation methods for optimal geometric control within the score.
We present many analytic optimal solutions to left-invariant PDE's and geometric control problems with uniform cost,
and use them to check our numerical PDE and ODE approaches.

We show that the score-adaptive PDE's, ODE's and differential invariants on Lie groups are beneficial in the following applications:

- G=SE(2) & G=SIM(2): fully automated detection of the vascular tree in retinal imaging, important for diagnosis of glaucoma/diabetic retinopathy.

- G=H(5): quantification of cardiac wall deformation from frequency fields obtained via left-invariant processing of Gabor transforms, important for identification of infarcted tissue.

- G=SE(3): detection of blood-vessels in 3D MRI-angiography, important for diagnosis and surgery planning.

- G=SE(3): fiber enhancement and tracking in brain white matter in DW-MRI neuroimaging, important for epilepsy surgery planning.

We show that the score-adaptive PDE's, ODE's and differential invariants on Lie groups are beneficial in the following applications:

- G=SE(2) & G=SIM(2): fully automated detection of the vascular tree in retinal imaging, important for diagnosis of glaucoma/diabetic retinopathy.

- G=H(5): quantification of cardiac wall deformation from frequency fields obtained via left-invariant processing of Gabor transforms, important for identification of infarcted tissue.

- G=SE(3): detection of blood-vessels in 3D MRI-angiography, important for diagnosis and surgery planning.

- G=SE(3): fiber enhancement and tracking in brain white matter in DW-MRI neuroimaging, important for epilepsy surgery planning.

Slides

Olivier FAUGERAS (INRIA Sophia Antipolis, France)

Neural fields and Riemannian geometry as models for how colors and color edges may be represented in visual area V1.

Abstract
Neural fields and Riemannian geometry as models for how colors and color edges may be represented in visual area V1.
I first go through some basic facts about the neurophysiological bases of color representation and processing in early visual areas, then continue with the psychophysics of color perception and discrimination, covering Hering's theory of color opponency and the Riemannian structure of color space in humans. Building on this I propose to describe color related activity in V1 with the formalism of neural fields theory and I relate it to previous work with Pascal Chossat on the representation in V1 of visual edges with the structure tensor. Based on this I speculate on the possibility of observing certain spontaneous patterns of activity in V1.

Slides

Yves FRÉGNAC (CNRS, France)

Perceptual association waves and collective belief in Visual Cortex

Abstract
Combination of intracellular recordings and network imaging (voltage sensitive dye) allows to explore
binding mechanisms operating beyond the classical receptive field, in the “silent” periphery of visual
cortical neurons. Using apparent motion noise at saccadic speed, we have inferred from the synaptic
echoes (recorded intracellularly) the existence of long-distance propagation of visually evoked activity
through lateral (and possibly feedback) connectivity outside the classical receptive field. VSD imaging has
been used to visualize, at the mesoscopic level, the propagation patterns travelling at the speed inferred
from our microscopic recordings. Results obtained at UNIC in collaboration with F. Gerard-Mercier,
P. Carelli, M. Pananceau and more recently X. Troncoso demonstrate the propagation of intracortical
depolarizing waves at the V1 map level. These waves are interpreted as broadcasting an elementary form
of collective “belief” to distant parts of the network. Their functional features support the hypothesis of a
dynamic perceptual association field, facilitating synaptic modulation in space and time during oculomotor
exploration. They may serve as a substrate for implementing the psychological Gestalt principles of
common fate and axial collinearity.

*(Work supported by CNRS, the French ANR (NatStats and V1-complex) and the European Community (FE-Bio-I3 integrated programs (IP FP6: FACETS (015879), IP FP7: BRAINSCALES(269921) and Brain-i-nets (243914)).)*Jean LORENCEAU (CNRS, France)

Dynamics of propagation through long range horizontal connections:
spatial facilitation and spike time alignment

Abstract
The architecture of long range horizontal connections in primary visual cortex suggests it implements geometrical
constraints akin to the statistical distribution of local orientations in natural scenes. Propagating activity within this network is slow,
owing to the thin unmyelinated axons running along the cortical surface. Such spatio-temporal organization results in delayed neuronal
facilitation which can induce perceptual biases correlated to MEG evoked activity in primary visual cortex. I shall present psychophysical
and MEG experimental data showing that response latencies to flashed aligned and non-aligned sequences of Gabor patches probe the
cortical dynamics within this network and shall present a simple spike-time-alignment model (STAM) derived from these and other results
where spatial facilitation induces a temporal alignment of the neuronal responses to elongated contours of varying contrast. STAM predicts that the
variance of the population response latencies to elongated contours is reduced; the consequences of this prediction on further processing stages will be discussed.

Slides

Stéphane MALLAT (École Normale Supérieure, France)

Invariant Image Classification with Deep Neural Networks

Abstract
Image classification and understanding requires to circumvent the curse of
dimensionality which affects all high-dimensional classification problems.
The use of invariants over Lie Groups provides powerful techniques to reduce
dimensionality for classification. Invariants to actions of diffeomorphisms
are obtained with scale separation techniques implemented with wavelet
transforms. It leads to deep neural network algorithms, called
scattering transforms.
For complex image classification, invariants are learned and
adapted to the image databasis. We shall explain how invariants can be
learned and constructed with hierarchical grouping algorithms, which build wavelets
over unknow groups. Classification
applications are shown over several image data bases.

Peter MICHOR (Universität Wien, Austria)

Overview on geometries of shape spaces, diffeomorphism groups, and
spaces of Riemannian metrics

Abstract
1. A short introduction to convenient calculus in infinite dimensions.
2. Manifolds of mappings (with compact source) and diffeomorphism
groups as convenient manifolds
3. A diagram of actions of diffeomorphism groups
4. Riemannian geometries of spaces of immersions, diffeomorphism groups, and
shape spaces, their geodesic equations with well posedness results and vanishing geodesic
distance.
5. Riemannian geometries on spaces of Riemannian metrics and pulling them
back to diffeomorphism groups.
6. Robust Infinite Dimensional Riemannian manifolds,
and Riemannian homogeneous spaces of diffeomorphism groups.
We will discuss geodesic equations of many different metrics on these spaces
and make contact to many well known equations
(Cammassa-Holm, KdV, Hunter-Saxton, Euler for ideal fluids), if time permits.

Slides

Xavier PENNEC (INRIA Sophia Antipolis, France)

Geometric Structures for Statistics on Shapes and Deformations in Computational Anatomy

Abstract
Computational anatomy is an emerging discipline at the interface of geometry, statistics, image analysis and medicine that aims at analysing and modelling the
biological variability of the organs shapes at the population level. The goal is to model the mean anatomy and its normal variation among a population and to discover
morphological differences between normal and pathological populations. For instance, the analysis of population-wise structural brain changes with ageing in Alzheimer's
disease requires first the analysis of longitudinal morphological changes for a specific subject. This can be evaluated through the non-rigid registration. Second, To perform
a longitudinal group-wise analysis, the subject-specific longitudinal trajectories need to be transported in a common reference (using some parallel transport).
To reach this goal, one needs to design a consistent statistical framework on manifolds and Lie groups. The geometric structure considered so far was Riemannian geometry.
The main steps are to redefine the mean as the minimizer of an intrinsic quantity: the Riemannian squared distance to the data points. When the Fréchet mean is determined,
one can pull back the distribution on the tangent space at the mean to define higher order moments like the covariance matrix. In the context of medical shape analysis, the
powerful framework of Riemannian (right) invariant metric on groups of diffeomorphisms (aka LDDMM) has often been investigated for such analyses in computational
anatomy. In parallel, efficient image registration methods and discrete parallel transport methods based on diffeomorphisms parametrized by stationary velocity fields (SVF)
(DARTEL, log-demons, Schild's ladder etc) have been developed with a great success from the practical point of view but with less theoretical support.
In this talk, I will detail the Riemannian framework for geometric statistics and partially extend if to affine connection spaces and more particularly to Lie groups provided
with the canonical Cartan-Schouten connection (a non-metric connection). In finite dimension, this provides strong theoretical bases for the use of one-parameter subgroups.
The generalization to infinite dimensions would ground the SVF-framework. Other useful but challenging extensions of geometric statistics would be needed in the future to
include sub-Riemannian and nonholonomic geometry for statistics on curves, surfaces and complex tissue types.

Slides

Jean PETITOT (CAMS EHESS, France)

Introduction

Slides
Alessandro SARTI (CAMS CNRS-EHESS, France)

Phenomenological Gestalten and figural completion: a neurogeometrical approach

Abstract
First I will recall neurogeometrical models of the functional architecture of V1 due to J.Petitot and G.Citti-A.Sarti. In this sub-Riemannian geometrical structure 3 main problems of visual perception will be faced: constitution of Gestalten, modal and amodal figural completion.
Joint work with Giovanna Citti.

Slides

Romain VELTZ (INRIA Sophia Antipolis, France)

On the effects of the pinwheel network symmetries on cortical response

Abstract
The goal of this work is to study how long-range connections in the primary visual cortex of mammals can influence local cortical activity as seen in optical imaging experiments. It is known that the representation of contours in the primary visual cortex is locally organised around “pinwheels” and possesses an approximate Euclidean invariance when only local connections are considered. Moreover recent experimental evidences support that the local circuitry operates at the edge of an instability where the network shows self-sustained stationary/oscillatory activity. Assuming (to simplify the analysis) that the pinwheels are organised in a discrete lattice, we consider the effects of long-range (non local) connections as modelled in Bressloff:03. It produces a forced symmetry-breaking in the equations which, as we have shown, can lead to oscillatory dynamics around the static square/hexagonal pattern produced by the local connections. The tools are equivariant dynamical systems theory and degree theory. Numerical experiments support the theoretical analysis.

Slides

Fred WOLF (Max-Planck-Institut für Dynamik und Selbstorganisation Theoretische Neurophysik, Germany)

The neural field theory of visual cortical architecture. Symmetries, Optimization, and Phase space structure

Abstract
Over the past 65 million years, the evolution of mammals led - in several lineages - to a dramatic increase in brain size. During this process, some neocortical areas, including the primary sensory ones, expanded by many orders of magnitude. The primary visual cortex, for instance, measured about a square millimeter in late cretaceous stem eutherians but in homo sapiens comprises more than 2000 mm2. If we could rewind time and restart the evolution of large and large brained mammals, would the network architecture of neocortical circuits take the same shape or would the random tinkering process of biological evolution generate different or even fundamentally distinct designs? In this talk, I will argue that, based on the consolidated mammalian phylogenies available now, this seemingly speculative question can be rigorously approached using a combination of quantitative brain imaging, computational, and dynamical systems techniques. Our studies on visual cortical circuit layout in a broad range of eutherian species indicate that neuronal plasticity and developmental network self-organization have restricted the evolution of neuronal circuitry underlying orientation columns to a few discrete design alternatives. Our theoretical analyses predict that different evolutionary lineages adopt virtually identical circuit designs when using only qualitatively similar mechanisms of developmental plasticity

Steve ZUCKER (Yale University, USA)

Color, orientation and shape

Abstract
The inference of shape from shading information is confounded when
there are material changes. We show how the shape-from-shading inference
can be structured geometrically on shading flows, and derive a new (iso-hue)
flow which, when parallel to the shading flow, suppresses the shape percept.
An appropriate version of frequency further indicates lighting (rather
than material)
effects.
Joint research with D. Holtmann-Rice, B. Kunsberg, E. Alexander and R.
Fleming.