Applications of Multiview Tensors in Higher Dimensions

This chapter is devoted to applications of multiview tensors, in higher dimension, to projective recostruction of segmented or dynamic scenes. Particular emphasis is placed on the analysis of critical configurations and their loci in this context, i.e. configurations of chosen scene-points and cameras that turn out to prevent successful reconstruction or allow for multiple possible solutions giving rise to ambiguities. A general geometric set up for higher dimensional spaces ad projections is firstly recalled. Examples of segmented and dynamic scenes, interpreted as static scenes in higher dimensional projective spaces, are then considered, following Shashua and Wolf. A theoretical approach to multiview tensors in higher dimension is presented, according to Hartley and Schaffalitzky. Using techniques of multilinear algebra and proper formalized language of algebraic geometry, a complete description of the geometric structure of the loci of critical configurations in any dimension is given. Supporting examples are supplied, both for reconstruction from one view and from multiple views. In an experimental context, the following two cases are realized as static scenes in P

4:

3

D

points lying on two bodies moving relatively to each other by pure translation and 3

D

points moving independently along parallel straight lines with constant velocities. More explicitly, algorithms to determine suitable tensors used to reconstruct a scene in P

4:

from three views are implemented with MATLAB. A number of simulated experiments are finally performed in order to prove instability of reconstruction near critical loci in both cases described above.