Geometric probability, psychophysics and invariance

Abstract

The perception of the noisy visual world around us naturally combines geometry and probability with psychophysics. So how do we perceive geometric objects from a probabilistic perspective, i.e. infer randomness in a spatial setting ? To test this psychophysically, we use a set of simple experiments to distinguish between probability distributions of planar line images connected with Buffon’s needle and Bertrand’s paradox, two classic exemplars of geometric probability. We find that participants associate greater randomness with images that are invariant under the sub-groups of translation, rotation, and scale. An information theoretic framework centered around the Radon (Hough) transform captures the observed behavioral results, and suggests that symmetry and chance are embedded in human visual perception.