Generalized Pupil-centric Imaging and Analytical Calibration for a Non-frontal Camera

Abstract

We consider the problem of calibrating a small field of view central perspective non-frontal camera whose lens and sensor planes may not be parallel to each other. This can be due to manufacturing defects or intentional tilting. Thus, as such all cameras can be modeled as being non-frontal with varying degrees. There are two approaches to model non- frontal cameras. The first one based on rotation parameterization of sensor non-frontalness/tilt increases the number of calibration parameters, thus requiring heuristics to initialize a few calibration parameters for the final non-linear optimization step. Additionally, for this parameterization, while it has been shown that pupil-centric imaging model leads to more accurate rotation estimates than a thin-lens imaging model, it has only been developed for a single axis lens-sensor tilt. But, in real cameras we can have arbitrary tilt. The second approach based on decentering distortion modeling is approximate as it can only handle small tilts and cannot explicitly estimate the sensor tilt. In this paper, we focus on rotation based non-frontal camera calibration and address the aforementioned problems of over-parameterization and inadequacy of existing pupil-centric imaging model. We first derive a generalized pupil-centric imaging model for arbitrary axis lens-sensor tilt. We then derive an analytical solution, in this setting, for a subset of calibration parameters including sensor rotation angles as a function of center of radial distortion (CoD). A radial alignment based constraint is then proposed to computationally estimate CoD leveraging on the proposed analytical solution. Our analytical technique also estimates pupil-centric parameters of entrance pupil location and optical focal length, which have typically been done optically. Given these analytical and computational calibration parameter estimates, we initialize the non-linear calibration optimization for a set of synthetic and real data captured from a non-frontal camera and show reduced pixel re-projection and undistortion errors compared to state of the art techniques in rotation and decentering based approaches to non-frontal camera calibration.