  We present a description of our method to process a set of
autocollimator-based deflectometer 1-dimensional line-scans taken over a large
optical surface and reconstruct them to a best-fit conic-section surface. The
challenge with our task is that each line-scan is in a different (unknown)
coordinate reference frame with respect to the other line-scans in the set.
This problem arises due to the limited angular measurement range of the
autocollimator used in the deflectometer and the need to measure over a greater
range; this results in the optic under measurement being rotated (in pitch and
roll) between each scan to bring the autocollimator back into measurement range
and therefore each scan is taken in a different coordinate frame. We describe
an approach using a 6N+2 dimension optimisation (where N is the number of scan
lines taken across the mirror) that uses a gradient-based non-linear least
squares fitting combined with a multi-start global search strategy to find the
best-fit surface. Careful formulation of the problem is required to reduce
numerical noise and allow the routine to converge on a solution of the required
accuracy.
