A spatialized model of visual texture perception using the structure tensor formalism

Grégory Faye; Pascal Chossat; Grégory Faye; Pascal Chossat

doi:10.3934/nhm.2013.8.211

Networks and Heterogeneous Media

2013, Volume 8, Issue 1: 211-260. doi: 10.3934/nhm.2013.8.211

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A spatialized model of visual texture perception using the structure tensor formalism

Grégory Faye ^{1
,},
Pascal Chossat ^{2
,}

1.
School of Mathematics, University of Minnesota, 206 Church Street S.E., Minneapolis, MN 55455
2.
J-A Dieudonné Laboratory, CNRS and University of Nice Sophia-Antipolis, Parc Valrose, 06108 Nice Cedex 02

Received: 01 January 2012 Revised: 01 December 2012
Primary: 37G40, 34C23; Secondary: 92B20.

The primary visual cortex (V1) can be partitioned into fundamental domains or hypercolumns consisting of one set of orientation columns arranged around a singularity or ``pinwheel'' in the orientation preference map. A recent study on the specific problem of visual textures perception suggested that textures may be represented at the population level in the cortex as a second-order tensor, the structure tensor, within a hypercolumn. In this paper, we present a mathematical analysis of such interacting hypercolumns that takes into account the functional geometry of local and lateral connections. The geometry of the hypercolumn is identified with that of the Poincaré disk $\mathbb{D}$. Using the symmetry properties of the connections, we investigate the spontaneous formation of cortical activity patterns. These states are characterized by tuned responses in the feature space, which are doubly-periodically distributed across the cortex.
- Neural field equations,
- symmetry,
- equivariant bifurcation,
- pattern formation,
- Poincaré disk
Citation: Grégory Faye, Pascal Chossat. A spatialized model of visual texture perception using the structure tensor formalism[J]. Networks and Heterogeneous Media, 2013, 8(1): 211-260. doi: 10.3934/nhm.2013.8.211

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Abstract

The primary visual cortex (V1) can be partitioned into fundamental domains or hypercolumns consisting of one set of orientation columns arranged around a singularity or ``pinwheel'' in the orientation preference map. A recent study on the specific problem of visual textures perception suggested that textures may be represented at the population level in the cortex as a second-order tensor, the structure tensor, within a hypercolumn. In this paper, we present a mathematical analysis of such interacting hypercolumns that takes into account the functional geometry of local and lateral connections. The geometry of the hypercolumn is identified with that of the Poincaré disk $\mathbb{D}$. Using the symmetry properties of the connections, we investigate the spontaneous formation of cortical activity patterns. These states are characterized by tuned responses in the feature space, which are doubly-periodically distributed across the cortex.

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