Reconstructing 3D Compact Sets

Reconstructing a 3D shape from sample points is a central problem faced in medical applications, reverse engineering, natural sciences, cultural heritage projects, etc. While these applications motivated intense research on 3D surface reconstruction, the problem of reconstructing more general shapes hardly received any attention. This paper develops a reconstruction algorithm changing the 3D reconstruction paradigm as follows. First, the algorithm handles general shapes, i.e. compact sets, as opposed to surfaces. Under mild assumptions on the sampling of the compact set, the reconstruction is proved to be correct in terms of homotopy type. Second, the algorithm does not output a single reconstruction but a nested sequence of plausible reconstructions. Third, the algorithm accommodates topological persistence so as to select the most stable features only. Finally, in case of reconstruction failure, it allows the identification of under-sampled areas, so as to possibly fix the sampling. These key features are illustrated by experimental results on challenging datasets, and should prove instrumental in enhancing the processing of such datasets in the aforementioned applications.