Robustness of reduced temporal phase unwrapping in the measurement of shape

The predictions of success rate and depth uncertainty for the negative exponential sequence used for temporal phase unwrapping of shape data are generalized to include the effect of a reduced sequence and speckle noise in single-channel and multichannel systems, respectively. To cope with the reduction of the sequence, a scaling factor is introduced. A thorough investigation is made of the performance of this algorithm, called the reduced temporal phase-unwrapping algorithm. Two different approaches are considered: a single-channel approach in which all the necessary images are acquired sequentially in time and a multichannel approach in which the three channels of a color CCD camera are used to carry the phase-stepped images for each fringe density in parallel. The performance of these two approaches are investigated by numerical simulations. The simulations are based on a physical model in which the speckle contrast, the fringe modulation, and random noise are considered the sources of phase errors. Expressions are found that relate the physical quantities to phase errors for the single-channel and the multichannel approaches. In these simulations the single-channel approach was found to be the most robust. Expressions that relate the measurement accuracy and the unwrapping reliability, respectively, with the reduction of the fringe sequence were also found. As expected, the measurement accuracy is not affected by a shorter fringe sequence, whereas a significant reduction in the unwrapping reliability is found as compared with the complete negative exponential sequence.