Affected by the height dependent effects, the phase-shifting shadow moire can only be implemented in an approximate way. In the technique, a fixed phase step around pi/2 rad between two adjacent frames is usually introduced by a grating translation in its own plane. So the method is not flexible in some situations. Additionally, because the shadow moire fringes have a complex intensity distribution, computing the introduced phase shift from the existing arccosine function or arcsine function-based phase shift extraction algorithm always exhibits instability. To solve it, we developed a Gram-Schmidt orthonormalization approach based on a three-frame self-calibration phase-shifting algorithm with equal but unknown phase steps. The proposed method using the arctangent function is fast and can be implemented robustly in many applications. We also do optical experiments to demonstrate the correction of the proposed method by referring to the result of the conventional five-step phase-shifting shadow moire. The results show the correctness of the proposed method. (C) 2016 Optical Society of America
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