Reconstruction of some segmented and dynamic scenes: trifocal tensors in P^4 : theoretical set up for critical loci, and instability

The context of this work is projective reconstruction of segmented or dynamic scenes from multiple views. More explicitly, the following two cases are considered: 3D points lying on two bodies moving relatively to each other by pure translation and 3D points moving independently along parallel straight lines with constant velocities. These situations are interpreted as static scenes in ℙ4 following Shashua and Wolf, (IJCV 48, 2002). Algorithms to determine suitable tensors used to reconstruct a scene in ℙ4 from three views are implemented in MATLAB®. From the theoretical point of view, a detailed description, with proof, of critical loci in the general context of projections ℙ4 → ℙ2 is given. A number of simulated experiments are performed to prove instability of reconstruction near critical loci in both cases described above. Algorithms for tensor reconstruction allow us to bypass the stratification approach followed in our previous work (ICCV, 2007), thus offering more comprehensive evidence of instability in higher dimension.