Triangulation

point of the common perpendicular to the two rays (the midpoint method). Perhaps a better choice would be to In this paper, we consider the problem of finding the position of a point in space given its position in two images taken with divide the common perpendicular in proportion to the cameras with known calibration and pose. This process requires distance from the two camera centers, since this would the intersection of two known rays in space and is commonly more closely equalize the angular error. Nevertheless, this known as triangulation. In the absence of noise, this problem method will not give optimal results, because of various is trivial. When noise is present, the two rays will not generally approximations (for instance, the angles will not be premeet, in which case it is necessary to find the best point of cisely equal in the two cases). In the case of projective intersection. This problem is especially critical in affine and reconstruction, or affine reconstruction, however, the camprojective reconstruction in which there is no meaningful metric era matrices will be known in a projective frame of referinformation about the object space. It is desirable to find a ence, in which concepts such as common perpendicular or triangulation method that is invariant to projective transformamidpoint (in the projective case) have no sense. In this tions of space. This paper solves that problem by assuming a case, the simple midpoint method here will not work. Gaussian noise model for perturbation of the image coordinates. The importance of a good method for triangulation is The triangulation problem may then be formulated as a leastsquares minimization problem. In this paper a noniterative clearly shown by Beardsley et al. who demonstrate that solution is given that finds the global minimum. It is shown the midpoint method gives bad results. In [2, 3] they suggest that in certain configurations, local minima occur, which are an alternative method based on ‘‘quasi-Euclidean’’ reconavoided by the new method. Extensive comparisons of the new struction. In this method, an approximation to the correct method with several other methods show that it consistently Euclidean frame is selected and the midpoint method is gives superior results.  1997 Academic Press carried out in this frame. The disadvantage of this method is that an approximate calibration of the camera is needed. It is also clearly suboptimal. 1. THE TRIANGULATION PROBLEM This paper is an extended version of [9] which describes a new algorithm that gives an optimal global solution to We suppose that a point x in R3 is visible in two images. the triangulation problem, equally valid in both the affine The two camera matrices P and P9 corresponding to the and the projective reconstruction cases. The solution relies two images are supposed known. Let u and u9 be projecon the concepts of epipolar correspondence and the fundations of the point x in the two images. From these data, mental matrix [4]. The algorithm is noniterative and simple the two rays in space corresponding to the two image points in concept, relying on techniques of elementary calculus may easily be computed. The triangulation problem is to to minimimze the chosen cost function. It is also moderate find the intersection of the two lines in space. At first sight in computation requirements. In a series of experiments, this is a trivial problem, since intersecting two lines in space the algorithm is extensively tested against many other does not present significant difficulties. Unfortunately, in methods of triangulation and found to give consistent supethe presence of noise these rays cannot be guaranteed to rior performance. No knowledge of camera calibration is cross, and we need to find the best solution under some needed. assumed noise model. A commonly suggested method [2] is to choose the midThe triangulation problem is a small cog in the machin-