Geometric analysis of influence of fringe directions on phase sensitivities in fringe projection profilometry

In fringe projection profilometry, phase sensitivity is one of the important factors affecting measurement accuracy. A typical fringe projection system consists of one camera and one projector. To gain insight into its phase sensitivity, we perform in this paper a strict analysis in theory about the dependence of phase sensitivities on fringe directions. We use epipolar geometry as a tool to derive the relationship between fringe distortions and depth variations of the measured surface, and further formularize phase sensitivity as a function of the angle between fringe direction and the epipolar line. The results reveal that using the fringes perpendicular to the epipolar lines enables us to achieve the maximum phase sensitivities, whereas if the fringes have directions along the epipolar lines, the phase sensitivities decline to zero. Based on these results, we suggest the optimal fringes being circular-arc-shaped and centered at the epipole, which enables us to give the best phase sensitivities over the whole fringe pattern, and the quasi-optimal fringes, being straight and perpendicular to the connecting line between the fringe pattern center and the epipole, can achieve satisfyingly high phase sensitivities over whole fringe patterns in the situation that the epipole locates far away from the fringe pattern center. The experimental results demonstrate that our analyses are practical and correct, and that our optimized fringes are effective in improving the phase sensitivities and, further, the measurement accuracies.