
The actual leastsquare algorithm to be incorporated in a DLT programs is more complex than [18] of the DLT Method page. Typical leastsquare algorithm involves weights to assure a more stable set of zeros. For this, add the weight matrix to [18] of the DLT Method page:
where
n = number of the equations, and W = diagonal weight matrix. In the camera calibration, matrices X, Y & L shown in [1] are
where = the imageplane coordinates, = the object space coordinates, L_{1}  L_{16} = the DLT and additional parameters, n = the number of control points, and
and = the coordinates of the principal point. From [10] and [35] of the DLT Method page:
where = the random errors involved in the equation, and = optical errors. From [5]:
It was assumed in [6] that L_{1}  L_{11} and R are constants and . [7] From [6]: , [8] where = the variances of the equations, = the variances of the imageplane coordinates, and = the variances of the objectspace coordinates. The variance of the imageplane coordinates can be obtained by repeating digitization of the control points and computing the variance of their imageplane coordinates. The variance of the objectspace coordinates can be obtained by repeating measurement of the reallife coordinates of the control points and computing their variance. It was assumed in [8] that x, y, z, u, and v are mutually independent. It is a common practice to use the reciprocal of the variance as the weight associated with the equation:
Matrix X and Y in [3] both contain R that is a function of L_{9}  L_{11}. For this reason, an iterative approach is typically employed in the camera calibration. Here are a sketch of the iterative approach commonly used in programming:
The variancecovariance matrix of the parameters can be expressed as
where
MSE in [10] is the mean square error from the leastsquare estimation. Since a control point provides two equations, the DOF (degree of freedom) for the MSE computation is 2n  16 as shown in [11]. The variancecovariance matrix of the parameters is symmetric and square (11 x 11):
The diagonal terms are the variances while the offdiagonal terms are the covariances. In the reconstruction, matrices X, Y & L shown in [1] are
where m = the number of cameras, and
Although the weight matrix is again in the form of [2], the actual method to compute the weights is different from [8]. From [5]:
It was assumed in [15] that x, y, and z are constants and . Therefore, the variances of the equations are
where = the variance/covariance between the parameters from [12], and
It was assumed in [16] that u and v are independent from L_{1}  L_{11}. L_{1}  L_{11 }are mutually dependent. The weight matrix for the reconstruction, therefore, becomes
where m = the number of cameras. Again, an iterative approach must be used in solving the system:
References & Related Literature Marzan, G.T. & Karara, H.M. (1975). A computer program for direct lnear transformation solution of the collinearity condition, and some spplications of it. Proceedings of the Symposium on CloseRange Photogrammetric Systems (pp. 420476). Falls Church, VA: American Society of Photogrammetry. Neter, J., Wasserman, W., and Kutner, M.H. (1985). Applied linear statistical models, 2nd Ed. Homewood, IL: Irwin.

© YoungHoo Kwon, 1998 