An improving fringe analysis method based on the accuracy of S-transform profilometry

The S transform, as a simple and popular technique for space\time-frequency analysis, has been introduced in optical three-dimensional surface shape measurement in recent years. Based on the S transform, S transform “ridge” method (STR) and S transform filtering method (STF) have been proposed to extract the phase information from the single deformed fringe pattern. This paper focuses on studying the STR in fringe pattern analysis. In previous researches about the STR, a linear constraint, which assumed that the phase was locally expressed as first-order Taylor expansion with respect to x and y directions, was implicitly added. Actually at least the second-order partial derivatives of the phase in each local area should be taken into account because they are related to the local curvature of the height distribution of the tested object. Therefore, the traditional STR has larger phase measuring errors in those areas with the rapid height variation on the tested object. This paper proposes an improved STR method, in which the phase is approximately expressed as a quadric in each local area. The phase extraction formula based on the quadric is derived, and the phase correction is carried out as well. Both the simulations and the experiments verify that a more accurate phase map can be obtained by the improved method compared with that by the traditional STR, especially in the areas where height variation is steep.