Topological evaluation of volume reconstructions by voxel carving

Homology is topologically invariant, meaning it is a property of an object that does not change under continuous (elastic) transformations of the object. Persistent homology studies homology classes and their life-times (persistence) in the belief that significant topological attributes must have a long life-time in a filtration (an increasing nested sequence of subcomplexes). In this demo, we work to apply persistence theory for the evaluation of a 3D reconstruction process: voxel carving. Voxel carving is a technique for creating a three-dimensional reconstruction of an object from a series of twodimensional images captured from cameras placed around the object at different viewing angles. The technique involves capturing a series of synchronised images of an object, and, by analysis of these images and with prior knowledge of the exact three-dimensional location of the cameras, deriving an approximation of the shape of the object. The reconstruction process, along increasing number of cameras, can be modelled as a suitable input for persistent homology computation, giving the possibility of a topological understanding of the whole process. More specifically, we analyse the persistence barcodes which are graphical representations of pairs of birth and death times of homology classes as a collection of horizontal line segments (intervals) in a plane, in order to extract information about the topological evolution of the reconstructions.