Completeness condition for unambiguous profile reconstruction by sub-aperture stitching

We describe the conditions required for a set of displaced sub-aperture measurements to contain sufficient information to reconstruct the stitched mirror profile removing all additive systematic errors of the measuring instrument, independent of the reference surface and of the guidance error of the linear stage used for the translation. We show that even-spaced stitching must be avoided and that the pitch error of the linear stage or the curvature of the reference must be measured, to avoid periodic errors and curvature errors in the reconstructed profile. We show that once these uncertainties are solved, the 1D profile can be reconstructed free of any additive systematic error. The theory is supported by computer simulations and by experimental results using two different instruments.