Visual curve completion in the tangent bundle

Visual curve completion is a fundamental perceptual mechanism that completes the missing parts (e.g., due to occlusion) between observed contour fragments, and facilitates higher level vision. Recent computational, neurophysiological, and psychophysical studies suggest that completed curves emerge from activation patterns of orientation selective cells in the primary visual cortex, as if they were regular observable curves. In this thesis we suggest modeling these patterns as 3D curves in the mathematical continuous space R × S1, a.k.a. the unit tangent bundle associated with the image plane R , that abstracts the mammalian striate cortex. Then, we propose that the completed shape may follow physical/biological principles which are conveniently abstracted and analyzed in this space. First, we examine a principle of minimum energy consumption by seeking the pattern of the minimal number of active cells that links two visible boundary fragments. In the abstract, this pattern amounts to the (admissible) curve of minimum length in R 2 × S1. We review psychophysical findings for the geometrical characteristics of perceptually completed shapes, and match them to the geometrical properties that are derived by our proposed principle. Second, we show how this principle and theory can be implemented in a biologically plausible manner with locally connected parallel networks. We review old and recent neurophysiological evidence for visual completion, and match them to our proposed model.