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This work presents a stable noise-robust numerical integration technique derived from a gradient representation of the Q-Forbes polynomials for surfaces with axial symmetry. This modal-integration technique uses an orthogonalization process through the Householder reflections to obtain a numerically orthogonal set for the surface slopes that is used to reconstruct the surface shape. It is shown that for typical Deflectometry measurements, the resulting random component of the uncertainty after numerical integration has a root mean square error well below 1nm.
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Ana H. Ramirez-Andrade, Rosario Porras-Aguilar, Konstantinos Falaggis, "Numerical integration of slope data with application to deflectometry," Proc. SPIE 11490, Interferometry XX, 1149009 (21 August 2020); https://doi.org/10.1117/12.2570600