Calculus of Variations and Geometric Measure Theory

G. Bellettini - V. Beorchia - M. Paolini

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

created by belletti on 28 Jul 2006
modified on 16 May 2007


Submitted Paper

Inserted: 28 jul 2006
Last Updated: 16 may 2007

Year: 2006


We introduce and study a two-dimensional model for the reconstruction of a smooth generic three-dimensional scene $E$, which may handle the self-occlusions and that can be considered as an improvement of the 2.1D sketch of Nitzberg and Mumford \cite{NiMu:90}. We characterize from the topological viewpoint the apparent contour of $E$, namely, we characterize those planar graphs $G$ that are apparent contours of some scene $E$. We make use of the so-called Huffman labelling \cite{Hu:71}, see also the paper of Williams and \cite{Wi:97} and the paper of Karpenko-Hughes \cite{KaHu:06} for related results. Moreover, we show that if $E$ and $F$ are two of these scenes, 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 $\mathcal F$ describing the model. Compactness, semicontinuity and relaxation properties of $\mathcal F$ are then studied, as well as connections of our model with the problem of completion of hidden contours.