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We show how a scale-invariant measure of three-dimensional shape can be derived from the velocity field generated by a rigid curved surface patch under perspective projection. We use invariance under rotation of the image plane [the Lie group SO(2)] to decompose the second-order velocity field in differential invariants. From a combination of these invariants we construct an approximation of the absolute value of Koenderink’s shape index [ Image Vis. Comput.10557 ( 1992)]. We show that the effect of these approximations on the shape index is small, especially under parallel projection. Furthermore, we provide an explanation for the psychophysical finding that elliptical shapes are more readily detected than parabolic or hyperbolic shapes. From the invariants we can also derive approximations of the principal directions, the curvedness, the slant, and the tilt.

Extraction of three-dimensional shape from optic flow: a geometric approach