Multiple-View Geometry and Photometry

1 ; 2 ; > (running over all views). Each additional view contributes linearly 12 parameters and its tensor with 1 ; 2 can be determined linearly using 6 matching points. More details on the material presented in this section can be found in 14]. 6 summary This paper has presented results on the goal of capturing the interrelationship , geometrically and pho-tometrically, across multiple perspective views. The main analysis vehicle is the \trilinear tensor" which captures in a very simple and straightforward manner the basic structures associated with this problem of research. We have not described in detail the particular applications to reconstruction, recognition and animation , but these can be found in some of the references provided in the text. Stratiication of three-dimensional vision: projective, aane and metric representations. appear in the constraint equation we have a para-metric constraint, which means that every pixel with non-vanishing spatial gradient contributes an equation with a xed number of unknowns. Moreover, the constraint equation is linear in the elements of jk i. Therefore, in principle, we can obtain a linear least-squares solution for the elements of jk i just by measuring spatial and temporal derivatives. More details can be found in 15]. 5 Rank 4: Properties of the Tensor Manifold The ultimate goal is to nd the means for combining together the contribution of any number of views, not only three. To this end we suggest to start with investigating the space of all trilinear tensors and look for rank deeciencies in that space. Any nding of that sort is extremely useful because it readily allows a statistical way of putting together many views, simply by means of factorization. The main result is that trilin-ear tensors across m > 3 views are embedded in a low dimensional linear subspace. Consider the following arrangement: we are given views 1 ; 2 ; :::: m+2 , m 1. For each (ordered) triplet of views there exists a unique trilinear ten-sor. Rather than considering all triplets of views, we consider the m triplets that contain 1 each of the tensors as a vector of 27 components (arrange the components arbitrarily, but stick with this arrangement for all tensors) and concatenate all these vectors as columns of a 27m matrix. The question is what is the rank of this matrix when m 27 ? Clearly, if the rank is smaller than 27 we obtain a …