## Neuromathematics of VisionGiovanna Citti, Alessandro Sarti This book is devoted to the study of the functional architecture of the visual cortex. Its geometrical structure is the differential geometry of the connectivity between neural cells. This connectivity is building and shaping the hidden brain structures underlying visual perception. The story of the problem runs over the last 30 years, since the discovery of Hubel and Wiesel of the modular structure of the primary visual cortex, and slowly cams towards a theoretical understanding of the experimental data on what we now know as functional architecture of the primary visual cortex. Experimental data comes from several domains: neurophysiology, phenomenology of perception and neurocognitive imaging. Imaging techniques like functional MRI and diffusion tensor MRI allow to deepen the study of cortical structures. Due to this variety of experimental data, neuromathematematics deals with modelling both cortical structures and perceptual spaces. From the mathematical point of view, neuromathematical call for new instruments of pure mathematics: sub-Riemannian geometry models horizontal connectivity, harmonic analysis in non commutative groups allows to understand pinwheels structure, as well as non-linear dimensionality reduction is at the base of many neural morphologies and possibly of the emergence of perceptual units. But at the center of the neurogeometry is the problem of harmonizing contemporary mathematical instruments with neurophysiological findings and phenomenological experiments in an unitary science of vision. The contributions to this book come from the very founders of the discipline. |

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### Contents

1 | |

Shape Shading Brain and Awareness | 86 |

Why Shading Matters along Contours | 107 |

A Neuromathematical Approach | 131 |

Cuspless SubRiemannian Geodesics within theEuclidean Motion Group SEd | 172 |

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### Common terms and phrases

activity adaptation algorithm alignment ambiguity areas association field attractors Bednar brain cells Citti color computer vision consider contact structure corresponding cortical covariant curvature curves cusp cuspless sub-Riemannian geodesics David Mumford defined denotes derivative detection differential geometry diffusion direction dots dynamics elements equation Euclidean exponential map fibre bundle functional architecture Gabor Gaussian gestalt given global Hebbian learning Heisenberg group illusory contours input integral invariance isophote Journal kernel Koenderink Lie groups light source linear mathematical metric natural images neural neurogeometry neurons Neuroscience optical orientation maps parameters patterns Pcurve perception Petitot pinwheels plane primary visual problem projection psychophysical random receptive fields receptive profiles rectangle representation represents retina Sarti SE(d shading flow shape from shading singularities solution space spatial stimuli sub-Riemannian geometry sub-Riemannian manifold surface tangent theorem theory tion transformations variables vector field visual cortex visual system