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s P 2 n s 2 . use none none encoder todisplay none in none Microsoft Office Excel Website 22.4 Three-view critical configurations Note that the condition that P = Q is not any restriction of generality, since the projective frames for the two configurations (P , P1, P2) and (Q , Q1, Q2) are independent. One may easily choose a projective frame for the second configuration in which this condition is true. This assumption is made simply so that one may consider the point P in relation to the projective frame of the second set of cameras.

The extra condition that the point P does not lie on the plane of camera centres CQ is necessary, however the reader is referred to [Hartley-OOb] for justification of this claim. Note that this case will usually not arise, however, since the intersection point of the three quadrics with the trifocal plane will be empty, or in special cases consist of a finite number of points. Where it does arise is through the possibility that the three camera centres CQ, CQ and C2, are collinear, in which case any other point is coplanar with these three camera centres.

Proof. The first statement follows directly from lemma 22.10(p542).

For the second part, the fact that the points P and Q lie on the intersections of the three quadrics follows (as pointed out before the statement of the theorem) from lemma 22.10(p542) applied to each pair of cameras in turn. To prove the final assertion, suppose that P lies on the intersection of the three quadrics.

Then from lemma 22.10(p542), applied to each of the three quadrics SpJ, there exist points Qlj such that the following conditions hold: P P = Q Q01 P P = Q Q02 PXP = C^Q12 PXP = Q V 1 P 2 P = Q2Q02 P 2 P = Q2Q12..

It is easy to none none be confused by the superscripts here, but the main point is that each line is precisely the result of lemma 22.10(p542) applied to one of the three pairs of camera matrices at a time. These equations may be rearranged as p p == Q Q 01 =Q Q 02.

pXp P2P == == Q1Q01 = Q1Q12 Q2Q02 = Q2Q12 Now, the cond none none ition that Q1Q01 = QXQ12 means that the points Q01 and Q12 are collinear with the camera centre C^ of Q1. Thus, assuming that the points Q u are distinct, they must lie in a configuration as shown in figure 22.7.

One sees from the diagram that if two of the points are the same, then the third one is the same as the other two. If the three points are distinct, then the three points Q y and the three camera centres Cq are coplanar, since they all lie in the plane defined by Q01 and the line joining Q02 to Q12. Thus the three points all lie in the plane of the camera centres CQ.

However, since P P = Q Q01 = Q Q02 it follows that P must lie along the same line as Q01 and Q02, and hence must lie in the same plane as the camera centres CQ. Thus, this result shows that the points in a 3-view critical configuration lie on the intersection of three quadrics, whereas the camera centres lie on the intersections of. 22 Degenerate Configurations Fig. 22.7.

Co none none nfiguration of the three camera centres and the three ambiguous points, if the three points Q y are distinct, then they all lie in the plane of the camera centres CQ.. pairs of the quadrics. In general, the intersection of three quadrics will consist of eight points. In this case, the critical set with respect to the two triplets of camera matrices will consist of these eight points, less any such points that lie on the plane of the three cameras Q\ In fact, it has been shown (in a longer unpublished version of [Maybank-98]) that of the eight points of intersection of three quadrics, only seven are critical, since the eighth point lies on the plane of the three cameras.

In some cases, however, the camera matrices may be chosen such that the three quadric surfaces meet in a curve. This will occur if the three quadrics SpJ are linearly dependent. For instance if Sp2 = aSp11 + /JSp12, then any point P that satisfies P T S 1 P = 0 and P T S 2 P = 0 will also satisfy P T S P 2 P = 0.

Thus the intersection of the three quadrics is the same as the intersection of two of them. In this case, the three cameras must also lie on the same intersection curve. We define a non-degenerate elliptic quartic to be the intersection curve of two non-degenerate ruled quadrics, consisting of a single curve.

This is a fourth-degree space curve. Other ways that two quadrics can intersect include a twisted-cubic plus a line, or two conies. Examples of elliptic quartics are shown in figure 22.

8. Example 22.29.

Three-view critical configuration - the elliptic quartic. We consider the quadrics represented by matrices A and B = B + B , where 0 1 0 0 0 0 0 0 0 0 - 1 0-1 0 1 0 V 0 0 0 Q r 0 0.
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