Tangent bundle elastica and computer vision

Visual curve completion, an early visual process that completes the occluded parts between observed boundary fragments (a.k.a. inducers), is a major problem in perceptual organization and a critical step toward higher level visual tasks in both biological and machine vision. Most computational contributions to solving this problem suggest desired perceptual properties that the completed contour should satisfy in the image plane, and then seek the mathematical curves that provide them. Alternatively, few studies (including by the authors) have suggested to frame the problem not in the image plane but rather in the unit tangent bundle R² x S¹, the space that abstracts the primary visual cortex, where curve completion allegedly occurs. Combining both schools, here we propose and develop a biologically plausible theory of elastica in the tangent bundle that provides not only perceptually superior completion results but also a rigorous computational prediction that inducer curvatures greatly affects the shape of the completed curve, as indeed indicated by human perception.