Mathematical Models of Visual Perception for the Analysis of Geometrical Optical Illusions

In this chapter a cortical-based mathematical models for Geometrical Optical illusions (GOIs) is provided. GOIs have been of great interest due to the possibility to understand, through the effect they produce on neural connections, the behaviour of low-level visual processing. They have been defined in the XIX century by German psycologists (Oppel, Ueber geometrisch-optische Tauschungen. Jahresbericht des physikalischen Vereins zu Frankfurt am Main, pp. 37–47, 1854–1855; Hering, Beitrage zur Physiologie, von Wilhelm Engelmann, Leipzig, pp. 1–5, 1861) in terms of phenomenology of vision, as situations in which there is an awareness of a mismatch of geometrical properties between an item in object space and its associated percept (Westheimer, Vis. Res. 48(20):2128–2142, 2008). As pointed out by Eagleman (2001) the study of these systematic misperceptions combined with recent techniques for brain’s activity recording provides a brilliant insight to lead new experiments on receptive fields of V1, as well as new hypothesis about the behaviour of perception. In this framework, starting from the geometrical model for the primary visual cortex introduced by Citti and Sarti in 2003, we provide an efficient mathematical model which allows to interpret these phenomena and to measure the perceived misperception based on the simulated response of simple cells in V1. The model involves image-processing techniques and allows to recover the perceived displacement by means of partial differential equations.