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# Noise propagation during stitching

Strictly speaking, the noise propagation law changes with different stitching algorithms. However, most algorithms are based on the same mathematics, which is finally a linear LS problem. Solving the linear equations Am = b gives the LS estimation of misalignment coefficients included in the vector m. b is the measured height difference in the overlapping area of two neighboring subapertures. Suppose the RMS error of each measured height distribution is a; the variance of height difference distribution is 2a2 for uncorrelated noises. From the LS solution,

We can get the variance of coefficients analytically, but it is rather sophisticated, especially when more kinds of misalignments are included, as in Eq. (30).

For simple stitching with only piston and tip-tilt corrected, Otsubo et al.13 presented an analytical but general model to estimate the noise propagation during stitching. Their conclusion is drawn on two square subapertures of N x N pixels with an overlapping area of w x N pixels. The standard deviations of tip, tilt, and piston are then estimated as follows:

Smith and Burge66 extended the above equation to the subaperture geometry of an annular ring with circular subapertures equally spaced and equally sized. They again made a big step to tolerancing by relating the variances of coefficients to the final surface height change and radial slope change, which are of greater concern to optical engineers. In their paper, the constrained random walk is utilized to describe the height and slope changes from one overlap to the next. The analytical model explains that “the presence of noise in subaperture measurements is more likely to produce errors with tilt or astigmatism-like shapes than higher-order fluctuations.”66 More subaperture overlaps are beneficial but also result in worse repeatability of measurements because a longer time is consumed.

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