A GEOMETRICAL DESCRIPTION OF VISUAL PERCEPTIONThe Leuven Café Erasmus model and the Bristol Café Wall illusionLeopold VerstraelenK.U. Leuven - K.U. Brussel Center PADGE,
(Pure and Applied Differential Geometry),
Departement Wiskunde, Celestijnenlaan 200B, B 3001 Leuven, Belgium Abstract. Intimite relations between the functioning
of the natural world in general and of the human intelligence in
particular have resulted in some mathematical models which give
reasonable formal descriptions of some fundamental objects and
processes occuring in nature as observed by human beings. These
models constitute the essence of our general scientific knowledge
and are illustrations of Feynman's saying that ''Nature
speaks to us in the language of mathematics''.
And, in the words of Chern: ''While algebra and analysis
provide the foundations of mathematics, geometry is at the
core''. Geometry is the field of mathematics whose main source of
intuition is human visual perception. So, it seems appropriate
that geometry would contribute somewhat to a better understanding
of visual perception. Paraphrasing Feynman, what follows may
illustrate that ''Nature likes to be looked at with
geometer's eyes and brains''.
Basically, a visual observation amounts to the recording
of light-energy (further on called ''luminosity''). In
mathematical terms this is well described by a surface
(further on called ''visual-stimulus-surface''). Based on this
visual information, our visual system (in the way this has been
developed in our ancestors and in ourselves via their and our
wider contacts with the observed realities of the surrounding
world, and which evolutions indeed also have had and have
influence on this recording of light-energy itself) makes us aware
of a corresponding image which is our actual registration of this
visual observation. And this image can essentially only be
determined by the geometrical properties of this surface.
My purpose here is to present this natural, and therefore simple,
geometrical model in some more detail and to discuss a bit its
application to some so-called visual illusions.
