## A sub-Riemannian model with frequency-phase and its application to orientation map construction

Our objective is to develop a geometrical model of the primary visual cortex in accordance with the neural characteristics of the cortex and construct orientation maps by using the relevant model geometry. Our departure point is the visual cortex model of the orientation selective cortical neurons, which was presented in [1] by Citti and Sarti. We spatially extend this model to a five dimensional sub-Riemannian geometry and provide a novel geometric model of the primary visual cortex which models orientation-frequency selective, phase shifted cortical cell behavior and the associated neural connectivity. This model extracts orientation, frequency and phase information of any given two dimensional input image. We employ in particular an input image with uniformly distributed white noise as the mathematical interpretation of internal stimulation on the retinal plane. Then, we start from the very first step mechanisms of visual perception and by using our sub-Riemannian model in order to extract visual features from the noise image, we provide a neurally inspired geometric procedure for multi-feature orientation map construction.

Bibliography

[1] G. Citti and A. Sarti, “A cortical based model of perceptual completion in the roto-translation space,” Journal of Mathematical Imaging and Vision, vol. 24, no. 3, pp. 307–326, 2006.