On the equivalence of moving entrance pupil and radial distortion for camera calibration

Radial distortion for ordinary (non-fisheye) camera lenses has traditionally been modeled as an infinite series function of radial location of an image pixel from the image center. While there has been enough empirical evidence to show that such a model is accurate and sufficient for radial distortion calibration, there has not been much analysis on the geometric/physical understanding of radial distortion from a camera calibration perspective. In this paper, we show using a thick-lens imaging model, that the variation of entrance pupil location as a function of incident image ray angle is directly responsible for radial distortion in captured images. Thus, unlike as proposed in the current state-of-the-art in camera calibration, radial distortion and entrance pupil movement are equivalent and need not be modeled together. By modeling only entrance pupil motion instead of radial distortion, we achieve two main benefits, first, we obtain comparable if not better pixel re-projection error than traditional methods, second, and more importantly, we directly back-project a radially distorted image pixel along the true image ray which formed it. Using a thick-lens setting, we show that such a back-projection is more accurate than the two-step method of undistorting an image pixel and then back-projecting it. We have applied this calibration method to the problem of generative depth-from-focus using focal stack to get accurate depth estimates.