Color and Grassmann-Cayley coordinates of shape

摘要

Abstract: A new concept of surface color is developed and the variety of all perceived colors is proved to be a 9-D set of 3 $MUL 3 matrices corresponding to different surface colors. We consider the Grassmann manifold Q of orbits Q $EQ $LFBC@B $DOT h $PLU c; detB$DNEQ 0$RTBC where c is an arbitrary vector of colorimetric space, B is a 3 $MUL 3 matrix, and h(x,y) is a color image of a Lambertian surface assumed to be a linear vector-function of the normal vectors. Different orbits Q(n) correspond to different shapes but they are invariant under color and illuminant transformation. Coordinates of an orbit Q in Q can be computed as 3 $MUL 3 (2 $MUL 2, sometimes) determinants the elements of which are values of some linear functionals (receptive fields) of h(x,y). Based on the approach, a shape-from-shading algorithm was developed and successfully tested on the three-color images of various real objects (an egg, cylinders and cones made of paper, etc.).!