Topological and variational properties of a model for the reconstruction of three-dimensional transparent images with self-occlusions

We introduce and study a two-dimensional variational model for the reconstruction of a smooth generic solid shape E, which may handle the self-occlusions and that can be considered as an improvement of the 2.1D sketch of Nitzberg and Mumford (Proceedings of the Third International Conference on Computer Vision, Osaka, 1990). We characterize from the topological viewpoint the apparent contour of E, namely, we characterize those planar graphs that are apparent contours of some shape E. This is the classical problem of recovering a three-dimensional layered shape from its apparent contour, which is of interest in theoretical computer vision. We make use of the so-called Huffman labeling (Machine Intelligence, vol. 6, Am. Elsevier, New York, 1971), see also the papers of Williams (Ph.D. Dissertation, 1994 and Int. J. Comput. Vis. 23:93-108, 1997) and the paper of Karpenko and Hughes (Preprint, 2006) for related results. Moreover, we show that if E and F are two shapes having the same apparent contour, then E and F differ by a global homeomorphism which is strictly increasing on each fiber along the direction of the eye of the observer. These two topological theorems allow to find the domain of the functional ℱ describing the model. Compactness, semicontinuity and relaxation properties of ℱ are then studied, as well as connections of our model with the problem of completion of hidden contours