A calibration method immune to the projector errors in fringe projection profilometry

Abstract

In fringe projection technique, system calibration is a tedious task to establish the mapping relationship between the object depths and the fringe phases. Especially, it is not easy to accurately determine the parameters of the projector in this system, which may induce errors in the measurement results. To solve this problem, this paper proposes a new calibration by using the cross-ratio invariance in the system geometry for determining the phase-to-depth relations. In it, we analyze the epipolar eometry of the fringe projection system. On each epipolar plane, the depth variation along an incident ray induces the pixel movement along the epipolar line on the image plane of the camera. These depth variations and pixel movements can be connected by use of the projective transformations, under which condition the cross-ratio for each of them keeps invariant. Based on this fact, we suggest measuring the depth map by use of this cross-ratio invariance. Firstly, we shift the reference board in its perpendicular direction to three positions with known depths, and measure their phase maps as the reference phase maps; and secondly, when measuring an object, we calculate the object depth at each pixel by equating the cross-ratio of the depths to that of the corresponding pixels having the same phase on the image plane of the camera. This method is immune to the errors sourced from the projector, including the distortions both in the geometric shapes and in the intensity profiles of the projected fringe patterns.The experimental results demonstrate the proposed method to be feasible and valid.