Statistics Explains Geometrical Optical Illusions

Azriel Rosenfeld has been our mentor for the last decade. In everyday conversation he stressed the importance of intuition, the unreasonable effectiveness of mathematics in understanding the world, and the power of simplicity inherent in deep ideas. Because we mostly worked on 3D vision, he often argued that 2D vision has been, is, and will continue to be a large source of problems. Strangely enough, we arrived at this study, which is our first on 2D vision, through problems we encountered in our work in 3D motion and shape. Azriel Rosenfeld was also one of the first to apply statistics [ 27 , 28 ] to image analysis and understanding, and he always reminded us of the uncertainties involved in visual computations. This paper shows that statistics cannot be ignored, not even in the interpretation of two simple straight intersecting lines. It demonstrates that uncertainty in the visual data causes problems for the early visual processes. Because of noise, the estimation of features, such as lines, intersections of lines, and local image motion, is biased. The inevitability of this bias provides an explanation for many well-known geometrical optical illusions.