Depth recovering method immune to projector errors in fringe projection profilometry by use of cross-ratio invariance

Abstract

In fringe projection profilometry, the projector parameters are difficult to calibrate accurately thus inducing errors in measurement results. To solve this problem, this paper analyzes the epipolar geometry of the fringe projection system, revealing that, on an epipolar plane, the depth variation along an incident ray induces the pixel movement along the epipolar line on the image plane of the camera. The depth variation and the pixel movement can be related to each other by using projective transformations. Under this condition, their cross-ratio keeps invariant. By use of this cross-ratio invariance, we suggest a depth recovering method immune to projector errors. To calibrate the measurement system, we shift a reference board perpendicularly to three positions with known depths and measure its phase maps as the reference phase maps. 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. Experimental results demonstrate that, with this technique, the errors associated with the projector, including its errors in geometry parameters, its lens distortions, and its luminance nonlinearity, do not affect the measurement results.