Imposing Differential Constraints on Radial Distortion Correction

Many radial distortion functions have been presented to describe the mappings caused by radial lens distortions in common commercially available cameras. For a given real camera, no matter what function is selected, its innate mapping of radial distortion is smooth, and the signs of its first and second order derivatives are fixed. However, such differential constraints have been never considered explicitly in existing methods of radial distortion correction for a very long time. The differential constraints we claimed in this paper are that for a given real camera, the signs of the first and second order derivatives of the radial distortion function should remain unchanged within the feasible domain of the independent variable, although over the whole domain, or outside of the feasible domain, the signs may change many times. Our method can be somewhat treated as a regularization of the distortion function within the viewing frustum. We relax the differential constraints by using a deliberate strategy, to yield the linear inequality constraints on the unknown coefficients of the radial distortion function. It seems that such additional linear inequalities are not difficult to deal with in recent existing methods of radial distortion correction. The main advantages of our method are not only to ensure the recovered radial distortion function satisfy differential constraints within the viewing frustum, but also to make the recovered radial distortion function working well in case of extrapolation, caused by the features used for distortion correction usually distributed only in the middle part, but rarely near the boundary of the distorted image. The experiments validate our approach.