A Method of Obtaining the Relative Positions of 4 Points from 3 Perspective Projections

According to Ullman's Structure-from-Motion Theorem [U79], three orthogonal projections of four points in a rigid non-planar configuration uniquely determine their structure, and the relative orientations of the three views, up to a reflection in the image plane. It is here shown that a corresponding result holds for the more general "para-perspective" case, and leads to a rapidly convergent algorithm for the fully perspective case. Unless the four points are nearly coplanar, or the images closely similar, the output of this algorithm is not unduly sensitive to errors in the image coordinates. 1 The orthogonal case We adopt one of the points PQ as origin and denote the 3D coordinates of the others by (Xn , Yn , Zn), (Xn ' ,Yn ' ,Zn ') and (Xn",Yn",Zn") in the three projection frames, (Xn, Yn), (Xn' Yn') and (Xn",Yn") being the (relative) image coordinates and Z ^ Z n ' and Zn" the (relative) depth coordinates. Then there will exist rigid rotation matrices U = [uy] and V = [vy] such that X n ' = u l l x n + 12n + 13n' x n " = v l l x n + v 1 2 Y n + 13n' (1\1") Yn ' = «21 n + "22Yn + u 2 3 Z n , Yn" = v 2 1 X n + v 2 2 Y n + v^Z , , , (2\2") Z n = "31n + 32n + U33n. z n " = 31n + 32n + The problem is to find U and V and the depth coordinates from the three sets of image coordinates. As each rotation involves 3 unknown parameters, and there are 9 depth coordinates, we have 18 equations for only 15 unknowns, and may expect to encounter 3 consistency conditions, useful for checking purposes. Elimination of Zj, between (I) and (2') gives the three equations x n " 3 2 Y n u 3 l + x n ' u 2 3 Y n ' u 1 3 = 0, (n = 1, 2, 3), (4) BMVC 1991 doi:10.5244/C.5.12 87 from which one can obtain the ratios of the "border elements" U32 : U31 : U23 : U13. (The computation fails if either (i) the four points are coplanar, in which case equations (4) are no longer independent, or (ii) the Z and Z axes coincide, in which case U33 = ± 1 and all four border elements vanish.) For U to be a rigid rotation of the first frame into the second, its border elements must satisfy U 1 3 2 + U 23 2 ( l ~ 33) = U 3 1 2 + 32This is one of the three consistency conditions mentioned above, and can be checked as soon as the ratios U32 : U31 : U23 : n^ have been obtained from (4). Introducing the normalized quaternion Q = ip + jq + kr + s, p+q+r+s = 1, (5) related to U by the equation u l l 12 13 p-<i-r+s, 2(pq-rs) , 2(pr + qs) U(Q) = u 2 1 u 2 2 «23 = (Pq + s). -p 2 +q 2 r 2 +s 2 , 2(qr-ps) (6) 31 32 33 2(pr-qs), 2(qr + ps), -p 2 -q 2 +r 2 +s 2 we see that the ratios U32 :113^ : U23 : UJ3 determine the ratio of p to q and the ratio of r to s, but not the ratio of p to r. It follows that if Q is written in the parametric form Q = (i sin A + j cos A) sin C + (k sin B + cos B) cos C, (7) then the two images (Xn, Yn) and (Xn',Yn') yield the values of the two parameters A and B, but not the "vergence" parameter C (equal to half of the angle between the Z and Z' axes). To compute C we need all three images, and the A and B parameters of the rotations connecting them, which we now denote by Uj (= U), U2 (= V~*) and U 3 (=VU~ 1 ) , satisfying (see figure at top of next page):