Unknown Radial Distortion Centers in Multiple View Geometry Problems

Abstract

Abstract. The radial undistortion model proposed by Fitzgibbon and the radial fundamental matrix were early steps to extend classical epipolar geometry to distorted cameras. Later minimal solvers have been proposed to find relative pose and radial distortion, given point correspondences between images. However, a big drawback of all these approaches is that they require the distortion center to be exactly known. In this paper we show how the distortion center can be absorbed into a new radial fundamental matrix. This new formulation is much more practical in reality as it allows also digital zoom, cropped images and camera-lens systems where the distortion center does not exactly coincide with the image center. In particular we start from the setting where only one of the two images contains radial distortion, analyze the structure of the particular radial fundamental matrix and show that the technique also generalizes to other linear multi-view relationships like trifocal tensor and homography. For the new radial fundamental matrix we propose different estimation algorithms from 9,10 and 11 points. We show how to extract the epipoles and prove the practical applicability on several epipolar geometry image pairs with strong distortion that -to the best of our knowledge -no other existing algorithm can handle properly.

Radial Distortion Self-Calibration

“…Only few authors have considered to include distortion into the self-calibration problem [10,19,2,23,8,13,3]. Among these, Thirtala and Pollefeys [23] assume the CoD to be known and reason about the shape of the distortion in radial direction.…”

“…Also Fitzgibbon [10] assumed the CoD to be known and then introduced an undistortion model rather than a distortion model allowing to work directly with distorted coordinates. This led to the 4 × 4 radial fundamental matrix with known CoD by Barreto et al [2] that has been generalized to absorb an unknown CoD by Brito et al [3].…”

“…This is an idea that also Devernay and Faugeras had followed [9]. The relations and differences of the models used in [10,8,7,2,3] will be discussed in more detail later.…”

“…This is in particular interesting for cropped images or those with digital zoom, where the CoD may even lie outside the image and no good prior guess is available. We also relate the model of Fitzgibbon [10], Barreto et al [2] and Brito et al [3] to the rational function distortion by Claus et al [8] and point out differences, advantages and drawbacks.…”

“…While these considerations apply to all radial distortion models, we chose to demonstrate the auto-calibration technique using the division distortion model proposed by Fitzgibbon [10], however using the lifted formulation by Brito et al [3] where the CoD need not be known beforehand. The main reason for this is that the set of all epipolar curves is just the row space of the radial fundamental matrix and the search for the straight line becomes a linear problem.…”