An exact formula for inverse radial lens distortions
Julien Lefèvre, Pierre Drap

We present here a new approach to calculating the inverse of radial distortions. The method presented here provides a model of reverse radial distortion, currently modeled by a polynomial expression, that proposes another polynomial expression where the new coefficients are a function of the original ones. The proposed method is based on a formal calculus involving a power series used to deduce a recursive formula for the new coefficients. We present several implementations of this method and describe the experiments conducted to assess the validity of the new approach. Such an approach, non-iterative, using another polynomial expression, able to be deduced from the first one, can actually be interesting in terms of performance, reuse of existing software, or bridging between different existing software tools that do not consider distortion from the same point of view.
One of the applications for using such a formula to compute the inverse distortion coefficient in function of k1..k4 is to convert distortion models between two software programs that use the inverse distortion model, as for example PhotoModeler and PhotoScan from Agisoft. Indeed PhotoModeler uses the Brown distortion model to compense observations made on images and so to obtain a theoretical observation without distortion effect. In contrary, PhotoScan from Agisoft uses a similar model but it adds the distortion to a point projected onto the image. To convert a distortion model from PhotoModeler to PhotoScan, or vice versa, we need to compute the inverse distortion model. We need to take in consideration the unit used to express the 2D point coordinate: in photomodeler the points are measured in millimeters and their range is limited to the camera frame; whereas in PhotoScan, the points are normalized by the focal length.

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