- Visual contour completion is a fundamental task in perceptual organization. Previous computational studies, which pursue a mathematical description for the shape of the completed contour (the “shape problem”), are typically motivated by formal description of perceptual characteristics such as minimum total curvature, extensibility, roundedness, and scale-invariance (e.g., Ullman 1976; Kimia etal. 2003). However, it is difficult to examine and determine these characteristics psychophysically, and there is no consensus in the perceptual literature for what they should be. (e.g., see Ullman 1976 vs Guttman & Kellman 2004 regarding the axiom of roundedness, or Kimia etal. 2003 vs. Gerbino & Fantoni 2006 regarding scale invariance). Instead, we suggest to leverage the fact that contour completion occurs in low level vision in order to formalize the problem in a space that explicitly abstracts the primary visual cortex. Such a suitable abstraction is the (unit) tangent bundle R2xS1, where curves represent the activation pattern or orientation selective cells due to real or completed image contours. We show that a basic principle of “minimum energy consumption”, namely completion according to minimum total length in the unit tangent bundle, entails desired perceptual properties for the completion in the image plane, minimizing both total curvature and total length in the image plane and thus expressing the two Gestalt principles of good continuation and proximity in a single elegant framework. Moreover, we show that our model does not support the roundedness and scale-invariance axioms, as has been promoted by recent psychophsyical findings. We present our modal and amodal completion result on both natural and synthetic images and discuss further implications and connections to perceptual findings and theories.