Pattern-Based System Axis Determination

The pattern-based system axis determination makes use of the pattern-based invariants and cost functions from Sections 5.4.1 and 5.4.2. Since these are independent of the remaining refractive parameters, the orientation of the system axis can be optimized individually. The optimization problem that needs to be solved can be defined as a search for the best system axis by testing hypothetical system axes. This is realized by a self-directed testing strategy. One of the two proposed pattern-based cost functions needs to be used to indicate the accuracy of the tested hypothetical axes. Therefore, a calibration object is needed with a pattern that contains feature points arranged in straight lines. This can be a checker pattern with its squares arranged in i ∈ {1, .., m} rows and j ∈ {1, .., n} columns. The optimization is about finding the hypothetical axis with the minimal function value of the chosen cost function. A vital part is the iterative generation of new hypothetical axes. This generation is self-directed. As will be shown in the following, it is driven by the accuracy of the currently tested candidate axis.

Generation of Hypothetical System Axes. The first steps for the generation of hypothetical system axes are basically the same as for the computation of the PAE and the PFE in Section 5.4.2. Once again, the image points I_ij and I_ij0 form the starting

5.5 Computation of Refractive Parameters

Figure 5.19: Pattern-based system axis determination. Left: Initialization of the iterative

processing and generation of hypothetical system axes hij. Middle: Intermediate

result. Right: Result close to the ideal solution.

point of the computations. The generation of hypothetical system axes is illustrated exemplary on the left side of Figure 5.19and comprises the following steps:

• Compute the VOPs ˇO_ij and ˇO0_ij from pairs of corresponding image points I_ij and

I_ij0 for a hypothetical system axis h₀, as described in Section5.3.1.

• Arrange these VOPs according to the lines in horizontal (l_i) and vertical (l_j) direction on the checker pattern.

• Fit a plane to the set of VOPs per line to get the feature planes in horizontal direction Θ_i : ¯ni· x = di and the feature planes in vertical direction Θj : ¯nj· x = dj,

with ¯n_i and ¯n_j being the normalized normal vectors, x being the position vector to a point on the plane and d_i as well as d_j being the distances of the respective plane to the origin of the coordinate system.

For an erroneous hypothetical system axis h₀, these fitted planes are most likely to deviate from their associated VOPs. This can be utilized for the computation of new hypothetical axes as follows:

• The intersection of the fitted planes Θ_i and Θ_j with the corresponding rays a_ij and a0_ij (the same intersection process as described in Section 5.3.1) results in the position vectors of the points of intersection ˆO_ij and ˆO_ij0 .

• Paired by matching indices, the points ˆO_ij and ˆO0_ij form the hypothetical object axes ˆh_ij with the normalized direction vectors ¯h_ij.

The hypothetical object axes ˆhij with the normalized direction vectors ¯hij represent the

generated hypothetical system axes. The direction vectors computed in this way differ as long as the VOPs ˇO_ij and ˇO_ij0 do not coincide with the points ˆO_ij and ˆO0_ij ideally. Initialization. The initialization phase is illustrated exemplary on the left side of Figure 5.19and comprises the following definitions and initial computations:

• Choose the initial hypothetical system axis h₀ to be the one with a direction vector ¯

5 Calibration of Shared Flat Refractive Systems

• Compute the value E of the PAE or PFE for system axis h₀ as described in Section 5.4.2 and initialize the current global minimal error value E_gmin with: E_gmin= E. • Generate the new hypothetical system axes ˆhij for system axis h0 as described

previously.

• Define the termination conditions:

– Provide a stability counter that terminates the iterative processing if there is no improvement for n iterations.

– Provide a significance counter that terminates the iterative processing if there is no significant amount of improvement ω, compared to a threshold ε, for n iterations.

Iteration. The iteration phase is illustrated exemplary in the middle and on the right side of Figure5.19and comprises the following steps:

1) Compute the error values E_ij for all new hypothetical system axes ˆh_ij as described in Section 5.4.2.

2) Find the local minimum E_lmin = E_ij of this iteration and set h₀= ˆhij accordingly.

3) If a new global minimum is found (E_lmin< E_gmin),

3.1) update the current global minimum (E_gmin= E_lmin),

3.2) reset the stability counter and

3.3) compute the improvement (ω = E_gmin− E_lmin).

3.4) If the improvement is smaller than the significance threshold (ω < ε), increase the significance counter.

3.5) If the improvement is equal or bigger than the significance threshold (ω ≥ ε), reset the significance counter.

4) If no new global minimum is found (E_lmin ≥ E_min), increase the stability counter.

5) If the stability or the significance counter is exceeded, terminate.

6) Generate the new hypothetical system axes for h₀ and repeat from 1).

On the right side of Figure5.19, a result of one iteration is illustrated. It comes close to the ideal result. The hypothetical system axis h₀ almost coincides with the true system axis s₀ and the VOPs ˇOij and ˇO

0

ij almost coincide with the points ˆOij and ˆO

0

ij. The

5.5 Computation of Refractive Parameters