Least Square Method
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Weighted Least Square Method
Camera Calibration
Reconstruction
References & Related Literature

Weighted Least Square Method

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:

,    [1]

where

,    [2]

n = number of the equations, and W = diagonal weight matrix.

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Camera Calibration

In the camera calibration, matrices X, Y & L shown in [1] are

,    [3]

where = the image-plane coordinates, = the object- space coordinates, L1 - L16 = the DLT and additional parameters, n = the number of control points, and

,    [4]

and = the coordinates of the principal point.

From [10] and [35] of the DLT Method page:

,    [5]

where = the random errors involved in the equation, and = optical errors.

From [5]:

.    [6]

It was assumed in [6] that L1 - L11 and R are constants and

.    [7]

From [6]:

,    [8]

where = 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. 

It is a common practice to use the reciprocal of the variance as the weight associated with the equation:

.    [9]

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:

  1. In the first iteration, it is impossible to compute R since L9, L10, and L11 are not available. The coordinates of the principal point are not available as well. Therefore, the standard DLT with 11 parameters must be used. Set the weight matrix to the identity matrix 1.
  1. From the second iteration on, compute R using the L9, L10, and L11 obtained from the previous iteration. Compute [uo, vo] based on the L's obtained from the previous iteration. See [27] of the DLT Method page for details. Form the normal equation shown in [1] and solve the system for L1 to L16.
  1. Repeat this procedure until all L's converge sufficiently. In either case, one must set a reasonable tolerance level for the convergence check.
  1. Compute the variance-covariance matrix for L1 - L11. Since the system has only 10 independent factors, the parameters are mutually dependant. (See the Modified DLT page for details.) In other words, covariances among the parameters exist. The variance-covariance matrix for the L's will be used later in the reconstruction to compute the weights for the equations.

The variance-covariance matrix of the parameters can be expressed as

,    [10]

where

.    [11]

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):

.    [12]

The diagonal terms are the variances while the off-diagonal terms are the covariances.

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Reconstruction

In the reconstruction, matrices X, Y & L shown in [1] are

,    [13]

where m = the number of cameras, and

.    [14]

Although the weight matrix is again in the form of [2], the actual method to compute the weights is different from [8]. From [5]:

.    [15]

It was assumed in [15] that x, y, and z are constants and . Therefore, the variances of the equations are

,    [16]

where = the variance/covariance between the parameters from [12], and

.    [17]

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

,    [18]

where m = the number of cameras.

Again, an iterative approach must be used in solving the system:

  1. In the first iteration, use the identity matrix (1) as the weight matrix. Solve the system for x, y & z.

  2. From the second iteration on, compute the weight vectors (17). Use the x, y & z from the previous iteration in computing the weights and R (14).

  3. Stop iteration when x, y & z sufficiently converge. Set a proper tolerance level for this.

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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-