cvpr cvpr2013 cvpr2013-344 cvpr2013-344-reference knowledge-graph by maker-knowledge-mining
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Author: José Henrique Brito, Roland Angst, Kevin Köser, Marc Pollefeys
Abstract: In cameras with radial distortion, straight lines in space are in general mapped to curves in the image. Although epipolar geometry also gets distorted, there is a set of special epipolar lines that remain straight, namely those that go through the distortion center. By finding these straight epipolar lines in camera pairs we can obtain constraints on the distortion center(s) without any calibration object or plumbline assumptions in the scene. Although this holds for all radial distortion models we conceptually prove this idea using the division distortion model and the radial fundamental matrix which allow for a very simple closed form solution of the distortion center from two views (same distortion) or three views (different distortions). The non-iterative nature of our approach makes it immune to local minima and allows finding the distortion center also for cropped images or those where no good prior exists. Besides this, we give comprehensive relations between different undistortion models and discuss advantages and drawbacks.
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[25] A. Extraction of Lifting Matrix in Rational Undistortion Model Claus and Fitzgibbon [8] also consider the two-sided radial fundamental matrix arising from two images taken by the same camera, i.e. the distortion parameters are the same in both views. Their radial fundamental matrix is a 6-by-6 matrix, since the distorted points are lifted to R6. Specifically, according to Eq. (3), the two-sided radial fundamental looks like = ATFA ∈ R6×6, where F ∈ R3×3 again denotes the standard fFunAda ∈me Rntal ,m wahtreixre. FW ∈hi Rle equation (23) in [8] suggests a non-linear optimization in order to extract the lifting matrix A from a given radial fundamental we argue that this can actually be done in closed-form. Let ∈ R6×6 be the two sided radial fundamental. Let C ∈F FR6 ∈×2 R and R ∈ R6×2 denote the two-dimensional column Rand row space o Rf respectively. Then, since the lifting matrix A has a three-dimensional rowspace which is the underlying space for both the row- and column-space of we have dim (span ([C, R])) = 3. Hence, the three first left singular vectors U = [u1, u2, u3] of [C, R] span the same space as the rows of A, i.e. A = HUT. In the setting used in [8], the radial distortion matrix A is only defined up to an arbitrary projective mapping of the image plane. Hence, the singular vectors in UT represents a perfectly valid solution for the lifting coefficients in A. Fˆ Fˆ, Fˆ Fˆ, Fˆ, 1 1 13 3 37 7 735 3