Correction of radially asymmetric lens distortion with a closed form solution and inverse function

Abstract

The current paradigm in the lens distortion characterization industry is to use simple radial distortion models with only one or two radial terms. Tangential terms and the optimal distortion centre are also seldom determined. Inherent in the models currently used is the assumption that lens distortion is radially symmetrical. The reason for the use of these models is partly due to the perceived instability of more complex lens distortion models. This dissertation shows, in the first of its three hypotheses, that higher order models are indeed beneficial, when their parameters are determined using modern numerical optimization techniques. They are both stable and provide superior characterization. Although it is true that the first two radial terms dominate the distortion characterization, this work proves superior characterization is possible for those applications that may require it. The third hypothesis challenges the assumption of the radial symmetry of lens distortion. Building on the foundation provided by the first hypothesis, a sample of lens distortion models of similar and greater complexity to those found in literature are modified to have a radial gain, allowing the distortion corrections to vary both with polar angle and distance from the distortion centre. Four angular gains are evaluated, and two provide better characterization. The elliptical gain was the only method to both consistently improve the characterization and not ‘skew’ the corrected images. This gain was shown to improve characterization by as much as 50% for simple (single radial term) models and by 7% for even the most complex models. To create an undistorted image from a distorted image captured through a lens which has had its distortion characterized, one needs to find the corresponding distorted pixel for each undistorted pixel in the corrected image. This is either done iteratively or using a simplified model typically based on the Taylor expansion of a simple (one or two radial coefficients) distortion model. The first method is accurate yet slow and the second, the opposite. The second hypothesis of this research successfully combines the advantages of both methods without any of their disadvantages. It was shown that, using the superior characterization of high order radial models (when fitted with modern numerical optimization methods) together with the ‘side-effect’ undistorted image points created in the lens distortion characterization, it is possible to fit a ‘reverse’ model from the undistorted to distorted domains. This reverse characterization is of similar complexity to the simplified models yet provides characterization equivalent to the iterative techniques. Compared to using simplified models the reverse mapping yields an improvement of more than tenfold - from the many tenths of pixels to a few hundredths.