The Rank 4 Constraint in Multiple (>=3) View Geometry

the projective representation of the 3D scene. In other words, the projective representation of the scene can undergo a projective transformation (which eeectively translates the scene along the optical axes of the rst view) which is interchangeable with the camera motion from view to view. The in-terchangeability point is eeectively contained in the arguments of 3] about the geometric content of each trilinearity. The camera transformation between images and 0 is represented by p 0 = 1 0 ?x 0 0 1 ?y 0 It can be veriied by inspection that p 0 = A; v 0 ]P can be represented by the following two equations: s l k v 0 k + p i s l k a k i = 0; (4) with the standard summation convention that an index that appears as a subscript and superscript is summed over (known as a contraction). Note that we have two equations because l = 1; 2 is a free index. Similarly,the camera transformation between views and 00 is p 00 = B; v 00 ]P: Likewise, let r m j be the matrix, r = 1 0 ?x 00 0 1 ?y 00 And likewise, r m j v 00 j + p i r m j b j i = 0; (5) Note that k and j are dummy indices (are summed over) in equations 4 and 5, respectively. We used diierent dummy indices because now we are about to eliminate and combine the two equations together. Likewise, l; m are free indices, therefore in the combination they must be separate indices. We eliminate and after some rearrangement and grouping we obtain: s l k r m j p i (v 0 k b j i ? v 00 j a k i) = 0; and the term in parenthesis is the trilinear tensor.