Nonlinear Correction for Fringe Projection Profilometry With Shifted-Phase Histogram Equalization

Abstract

Fringe projection profilometry has been widely used for 3-D measurement, but the gamma nonlinearity severely affects the accuracy. Ideally, the histogram of the undistorted phase is uniform, but that of the distorted phase becomes nonuniform. In practice, adequate sampling pixels are required to calculate the phase histogram from modulated fringe patterns. Therefore, this article proposes an effective shifted-phase histogram equalization (SHE) method for nonlinear correction. Based on the finding that the distorted histogram contains three symmetric valleys in theory, if we shift the wrapped phase by <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\pm 2\pi $ </tex-math></inline-formula> /3, the shifted phases will share the same histogram as the original phase. Without additional patterns, we can easily obtain two shifted phases. Combining the original and two shifted phases, the number of sampling pixels will increase by three times, and we can obtain more accurate histogram than the original phase. To make the histogram of the distorted phase be uniform as that of the undistorted phase, we directly apply histogram equalization on the distorted phase to estimate the undistorted phase. Moreover, this article proposes a spline interpolation optimization (SIO) method to eliminate the discretization error caused by histogram equalization. Simulations and experiments have been carried out, and their results indicate that the proposed SHE method performs better than the conventional phase histogram equalization (PHE) method, and the proposed SIO method can further reduce the discretization error.