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The measurement of an object's shape using projected fringe patterns needs a relation between the measured phase and the object's height. Among various methods, the Fourier transform profilometry proposed by Takeda and Mutoh Appl. Opt.22, 3977-3982 (1983) is widely used in the literature. Rajoub et al. have shown that the reference relation given by Takeda is erroneous J. Opt. A. Pure Appl. Opt.9, 66-75 (2007). This paper follows from Rajoub's study. Our results for the phase agree with Rajoub's results for both parallel- and crossed-optical-axes geometries and for either collimated or noncollimated projection. Our two main results are: (i) we show experimental evidence of the error in Takeda's formula and (ii) we explain the error in Takeda's derivation and we show that Rajoub's argument concerning Takeda's error is not correct.
Maurel et al. (Wed,) studied this question.