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The actual least-square algorithm to be incorporated in a DLT programs is more complex than [18] of the DLT Method page. Typical least-square 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
and From [10] and [35] of the DLT Method page:
where From [5]:
It was assumed in [6] that L1 - L11 and R are constants and
From [6]:
where 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 L9 - L11. 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 variance-covariance matrix of the parameters can be expressed as
where
MSE in [10] is the mean square error from the least-square 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 variance-covariance matrix of the parameters is symmetric and square (11 x 11):
The diagonal terms are the variances while the off-diagonal 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
where
It was assumed in [16] that u and v are independent from L1 - L11. L1 - L11 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 Close-Range Photogrammetric Systems (pp. 420-476). 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.
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© Young-Hoo Kwon, 1998- |
,
[1]
,
[2]
= the image-plane
coordinates, 
= the object- space coordinates, L1
- L16 = the DLT and additional parameters, n = the number
of control points, and
,
[4]
= the coordinates of the principal point.
,
[5]
= the
random errors involved in the equation, and 
= optical errors.
.
[6]
. [7]
, [8]
= the variances
of the equations, 
= the variances of the image-plane coordinates, and 
= the variances of the object-space coordinates. The variance of the image-plane
coordinates can be obtained by repeating digitization of the control points and
computing the variance of their image-plane coordinates. The variance of the
object-space coordinates can be obtained by repeating measurement of the
real-life coordinates of the control points and computing their variance. It was assumed in [8]
that x, y, z, u, and v are mutually
independent. 
.
[9]
,
[10]
. [11]
. [12]
,
[13]
.
[14]
.
[15]
,
[16]
= the variance/covariance
between the parameters from [12], and
.
[17]
,
[18]