A Biologically-Inspired Theory for Non-axiomatic Parametric Curve Completion

Visual curve completion is typically handled in an axiomatic fashion where the shape of the sought-after completed curve follows formal descriptions of desired, image-based perceptual properties (e.g, minimum curvature, roundedness, etc...). Unfortunately, however, these desired properties are still a matter of debate in the perceptual literature. Instead of the image plane, here we study the problem in the mathematical space

${\mathbf R}^{2}\times {\mathcal S}^{1}$

that abstracts the cortical areas where curve completion occurs. In this space one can apply basic principles from which perceptual properties in the image plane are

derived

rather than

imposed

. In particular, we show how a “least action” principle in

${\mathbf R}^{2}\times {\mathcal S}^{1}$

entails many perceptual properties which have support in the perceptual curve completion literature. We formalize this principle in a variational framework for general parametric curves, we derive its differential properties, we present numerical solutions, and we show results on a variety of images.