Models of the Visual Cortex in Lie Groups

Abstract

The most classical and exhaustive theory which states and studies the phenomenological laws of visual reconstruction is Gestalt theory [73, 74]. It formalizes visual perceptual phenomena in terms of geometric concepts, such as good continuation, orientation, or vicinity. Consequently, phenomenological models of vision have been expressed in terms of geometrical instruments and minima of calculus of variations [5, 51, 96]. On the other hand, the recent progress of medical imaging and integrative neuroscience allows to study neurological structures related to perception of space and motion. The first results that used instruments of differential geometry to model the cortex and justify macroscopical visual phenomena in terms of the microscopical behavior of the cortex were due to Hoffmann [70], Koenderink [77], and Petitot–Tondut [100, 102]. More recently, in [37] and [109], the visual cortex was modeled as a Lie group with a sub-Riemannian metric. Other models in Lie groups are proposed in [12, 22, 39, 49, 62, 63, 112, 117, 118]. We refer to these papers for a complete description of this type of problems.