Linear sinusoidal phase-shifting method resistant to non-sinusoidal phase error

Abstract

Non-sinusoidal phase error is common in structured light three-dimensional (3D) shape measurement system, thus we perform theoretical and experimental analyses of such error. The number of non-sinusoidal waveform errors in a 2π phase period is the same as the number of steps of the phase-shifting algorithm; no errors occur within the one-phase period. Based on our findings, a new structured light method, the linear sinusoidal phase-shifting method (LSPS), that is resistant to non-sinusoidal phase error is proposed. Experiments show that the non-sinusoidal waveform error is reduced to an almost negligible level (0.001 rad) using the proposed LSPS.OCIS Optical non-contact three-dimensional (3D) shape measurement techniques have been developed to obtain 3D contours. With the recent advancement in digital display technology, 3D shape measurement based on digital projection units has been rapidly expanding [1−2] . However, the challenge remains to be in area of developing a system with an off-the-shelf projector, such as liquid crystal display (LCD) and digital mirror device (DMD), for high-quality 3D shape measurement. One of the major issues is the nonlinear response of the projection engines of projectors [1−10] . Projector gamma calibration is usually needed to perform high-quality 3D shape measurement using a digital fringe projection and phase-shifting method [2−10] . This is because the commonly used commercial video projector is a nonlinear device purposely designed to compensate for human vision [7] . A variety of techniques have been developed for nonlinear phase errors [2−12] . These include the two-step triangular-pattern phase-shifting algorithm and the error compensation method [5−6] , which can reduce periodic measurement errors due to gamma nonlinearity as well as projector and camera defocus. In this letter, the detailed mathematical model for sinusoidal phase shifting was developed to predict the effects of non-unitary gamma on phase-measuring profilometry [3] . Zhang et al. developed a compensation method that needs to calibrate an error look-up table (LUT) and is based on the assumption of periodic phase error [9−10] . In general, the abovementioned methods need an additional compensatory step to reduce phase error. A new method with defocusing binary structured patterns can be used to eliminate nonlinear gamma [7] ; however, controlling the proper defocusing degree to achieve high accuracy is difficult using this method [8] . Guo et al. proposed a gamma correction method using a simple one-parameter gamma function technique by statistically analyzing the fringe images [11] . These techniques significantly reduce the phase error caused by nonlinear gamma. Nevertheless, these compensation methods still have residual error value that cannot be ignored in accurate measurement. In addition, the actual gamma of the projector is very complicated and is not the only factor that causes nonsinusoidal waveforms. Our experiments show that the nonlinear gamma of the projector changes over pixels, and mu...