Robust and Accurate Principal Point Estimation Using the Five-Point Perspective Model of Fish-Eye Radial Distortion

Radial lens distortion is the displacement of image points along a radial axis from a single point on the image plane. This point is known as the principal point, and does not always necessarily align with the image sensor center. Therefore, in order to be able to fully model and correct radial lens distortion, it is necessary to accurately determine the principal point. Physically, the principal point is the point at which the optical axis of the camera lens system intersects the image plane. Inaccurate estimation of the principal point will introduce additional radial distortion and tangential distortion into the image when undistortion is applied. The estimation of the principal point is only relevant in wide-angle camera systems that display reasonable degrees of radial lens distortion. In fact, according to Ruiz et al. [1], the location of the principal point in cameras with small to moderate fields-of-view can be considered negligible. Several methods exist for finding the principal point of a wide-angle or fish-eye camera. However, these use complex systems using external lasers [2], an iterative average over several calculations [3], [4] or by adding coefficients (and thus complexity) to the distortion model [5], [6]. To determine the principal point, we use a different model of fish-eye distortion from the standard mathematical fish-eye models. We use a five-point perspective model to describe the fish-eye distortion. In the standard pin-hole perspective model, parallel sets of lines converge at a single point, known as the vanishing point (VP). In fish-eye perspective, parallel sets of lines converge at two VPs. The principal point lies on the line between the two VPs (the horizon line). Therefore, in order to find the principal point, we will need at least two sets of parallel lines in the test diagram. We use a checkerboard pattern and extract the parallel lines from it. Using the fact that lines in 3-d coordinates are transformed to circles in the image plane in a fish-eye camera [7], we fit circles to each line using a standard least-squares method. It is then simply a matter of finding the intersection points of the two horizon lines. This is demonstrated in figure 1. Additionally, we created a set of synthetic test images with varying levels of radial distortion, noise, light falloff, perspective, rotation and edge modular transfer function as well as a variable principal point. We used these to measure the accuracy and robustness of our estimation algorithm. We found that our average error was 3.98 pixels. We compared this with the commercially available ImaTest software [8], and found that ImaTest had an average error of 8.71 pixels in its estimation of the principal point.