Completing visual contours: The relationship between relatability and minimizing inflections
Manish Singh and Donald D. Hoffman
Perception & Psychophysics, 61: 943-951 (1999).

Abstract

 Visual completion is a ubiquitous phenomenon: Human vision often constructs contours and surfaces in regions that have no sharp gradients in any image property. When does human vision interpolate a contour between a given pair of luminance-defined edges? Two different answers have been proposed: relatability and minimizing inflections. We state and prove a proposition that links these two proposals by showing that, under appropriate conditions, relatability is mathematically equivalent to the existence of a smooth curve with no inflection points that interpolates between the two edges. The proposition thus provides a set of necessary and sufficient conditions for two edges to be relatable.
  Based on these conditions, we suggest a way to extend the definition of relatability:

• to include the role of genericity, and
• to extend the current all-or-none character of relatability to a graded measure that can track the gradedness in psychophysical data.