cvpr cvpr2013 cvpr2013-344 knowledge-graph by maker-knowledge-mining
Source: pdf
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.
Reference: text
sentIndex sentText sentNum sentScore
1 ch Abstract In cameras with radial distortion, straight lines in space are in general mapped to curves in the image. [sent-5, score-0.782]
2 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. [sent-6, score-1.252]
3 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. [sent-7, score-1.289]
4 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. [sent-9, score-0.694]
5 To compensate for such distortion present in many real lens systems, ∗This work was done while this author ment of Computer Science, ETH Z ¨urich was employed by the Depart- Roland Angst∗ Stanford University ETH Z u¨rich, Switzerland rangst @ st anford . [sent-16, score-0.652]
6 ) are (degenerated) circles in general, but those that go through the center of radial distortion are straight lines (l2 and l2? [sent-22, score-1.419]
7 a) When the radial fundamental matrix is obtained with respect to two other images (potentially with different or no distortion) the intersection of the straight epipolar lines reveals the CoD. [sent-24, score-1.052]
8 b) In case the same camera is used to take a pair of images the CoD has to lie on the straight epipolar curves in both images. [sent-25, score-0.583]
9 most importantly radial distortion, several pre-calibration techniques based on calibration objects have been proposed [4, 24, 16, 14, 25, 22] that allow rectification of distorted images prior to further processing. [sent-26, score-0.707]
10 Only few work has addressed autocalibration of radial distortion and virtually all of this focuses on the distortion strength and assumes the distortion center being known. [sent-28, score-2.351]
11 In this contribution we drop this assumption and show how to estimate the center of distortion (CoD) from image data of completely unknown scenes without a calibration object, given a perspective (single center of projection) camera with strictly radial distortion. [sent-29, score-1.422]
12 When projecting the viewing ray of another camera to some point in space into the view of such a camera, one does not obtain an epipolar line, but more generally an epipolar curve. [sent-30, score-0.626]
13 However - under any radial distortion model - the epipolar curve that contains the CoD must be a straight line, since the distortion happens 1 1 13 3 36 6 6 8 6 in direction of the line1 . [sent-31, score-2.136]
14 Given a second straight epipolar curve from a different epipolar geometry (e. [sent-33, score-0.733]
15 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. [sent-37, score-1.759]
16 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. [sent-39, score-1.063]
17 [3] to the rational function distortion by Claus et al. [sent-46, score-0.701]
18 Previous Work Lens distortion has been studied in photogrammetry and computer vision for a long time (cf. [sent-49, score-0.601]
19 Here, radial distortion is one of the dominant distortions [24] and can severely hurt image registration, reconstruction or measurement in images when not being considered. [sent-51, score-1.054]
20 However, given significant distortion most of these methods fail or require pre-rectification with known distortion parameters. [sent-59, score-1.202]
21 Only few authors have considered to include distortion into the self-calibration problem [10, 19, 2, 23, 8, 13, 3]. [sent-60, score-0.601]
22 Among these, Thirtala and Pollefeys [23] assume the CoD to be known and reason about the shape of the distortion in 1Any line through the CoD is a fixed line when radially distorting undistorting an image, although not a line of fixed points. [sent-61, score-0.754]
23 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. [sent-63, score-0.922]
24 After the division undistortion model of [10] also Claus and Fitzgibbon [8] generalize the model, now to a rational function undistortion where up to 17 distortion parameters exist, allowing to represent more general distortion functions, not just radial distortion. [sent-68, score-2.059]
25 Using a similar lifting framework as in [1], our work extends Geyer and Daniilidis’ work [11] from catadioptric to perspective cameras with radial (barrel or pincushion) distortion. [sent-77, score-0.598]
26 Essentially, their idea is that when observing a planar calibration pattern with a camera with radial distortion, then the CoD will act like a focus of expansion or contraction and consequently behave similarly as an epipole. [sent-80, score-0.636]
27 Our solution is based on the observation that lines that go through the CoD are fixed lines under distortion and undistortion. [sent-86, score-0.785]
28 Consequently, we look for straight lines in the set of all possible epipolar curves and argue that the CoD must be on such a line. [sent-87, score-0.607]
29 In order to obtain a parametric representation of the set of all possible epipolar curves, we use the generalization of Fitzgibbon’s division distortion model by Brito et al. [sent-88, score-0.943]
30 On top, we also estimate the distortion strength, which - in sum - is why we call this method radial distortion self-calibration. [sent-91, score-1.636]
31 Relation between Undistortion Models The traditionally used second-order distortion model in computer vision (motivated in [4]) with CoD (dx , dy)T ∈ R2 describes the radial distortion as ? [sent-93, score-1.636]
32 ively, ∈w Rhereas λ ∈ R is the distortion coefficient and r˜2 = ? [sent-104, score-0.601]
33 ctually describes the distorted point in explicit form: given the undistorted point (xu, yu)T and the distortion parameters λ and (dx , dy)T, the distorted point can be computed easily by evaluating the right-hand side of Eq. [sent-115, score-1.194]
34 For auto-calibration, or for direct estimation based on distorted measurements, an undistortion model rather than a distortion model is required. [sent-117, score-0.922]
35 2 The division undistortion model In [10], Fitzgibbon argues that an undistortion model can be equally powerful as a distortion model and compares several (un-)distortion functions. [sent-121, score-0.924]
36 This is the model also used in the radial fundamental matrix with lifted coordinates [2] and later by minimal solvers with radial distortion like [17]. [sent-123, score-1.669]
37 The undistorted distance to the CoD is plotted as a function of the distorted distance d for different λ according to the division distortion model (Eq. [sent-125, score-1.043]
38 This leads to interesting consequences: Although every point can be undistorted according to this model, only those close to the distortion center can be distorted. [sent-128, score-0.859]
39 However, since non-monotonic distortion curves do not make much sense, the one closer to the distortion center is the interesting one, since this is on the (useful) monotonic part of the curve and the other one can be considered an artifact by the model. [sent-132, score-1.378]
40 represents the squared distance to the radial distortion center, this center has to be known when working with these models. [sent-133, score-1.128]
41 All of these approaches can cover the same family of radial distortion functions and thus the same family of lenses. [sent-134, score-1.083]
42 Having different coefficients for the x2 and y2 terms also means that is no longer a radially symmetric distortion model. [sent-141, score-0.646]
43 On top, the parameters can be recovered only up to a homography and in the end, although epipolar lines are straightened, there is significant distortion left in their example images. [sent-143, score-0.977]
44 Division undistortion with unknown center In [3] it has been shown that the lifting and the radial fundamental matrix can be reformulated such that one can work with (r? [sent-144, score-0.849]
45 Rather, it is absorbed in the lifting matrix, or ultimately, in the radial fundamental matrix. [sent-146, score-0.583]
46 In the Claus and Fitzgibbon framework this would mean that the second column of A is zero (no mixed terms), the first and third column are equal and that all entries depend only on the distortion center (dx , dy) and λ. [sent-147, score-0.726]
47 CoD From Straight Epipolar Lines We now consider a pair of images A and B, at least one of them (say image A) having radial distortion (but not necessarily being described parametrically by any of the above Figure 3. [sent-155, score-1.035]
48 Visualization of the radial distortion due to the division distortion model: The CoD is visualized in red. [sent-156, score-1.751]
49 The distorted epipolar line becomes an epipolar circle (orange). [sent-158, score-0.764]
50 Recall that always all epipolar curves go through the epipole and now we look at the one that also includes the distortion center. [sent-163, score-1.068]
51 Distortion happens in radial direction from the center to the epipole and thus within the line, but does not change the line as a whole. [sent-164, score-0.719]
52 Radial Fundamental Matrix We now choose the distortion model of Brito et al. [sent-168, score-0.601]
53 and compute the radial fundamental matrix [3] between our image A and some other image (depending on the other image this could be the single-sided or two-sided radial fundamental matrix). [sent-169, score-1.071]
54 As argued before the distortion center must be on that line. [sent-183, score-0.716]
55 1 Degenerate Cases In case image A does not have distortion, then λ is zero and consequently the last column of the radial fundamental will be zero. [sent-186, score-0.571]
56 In that case all epipolar curves are straight and the distortion center is not defined. [sent-187, score-1.175]
57 All epipolar curves are also straight if the epipole coincides with the distortion center. [sent-188, score-1.208]
58 CoD from three (different) images As argued before, from a pair of images A and B, we can constrain the position of the distortion center to a line in the image plane. [sent-195, score-0.73]
59 To obtain the full coordinates of the distortion center we can intersect the line with another line. [sent-196, score-0.795]
60 Consequently, the straight epipolar curve will be different and by intersecting the two straight epipolar curves we obtain the distortion center. [sent-198, score-1.552]
61 For practical reasons, the lines should intersect ideally at a right angle, so the epipoles should be at different directions when viewed from the distortion center. [sent-200, score-0.793]
62 CoD from two images of the same camera In case we have a video or multiple images taken by the same camera we can even extract the distortion center from a pair of two images. [sent-209, score-0.854]
63 In each of the images we obtain the constraint that the distortion center must be on the straight epipolar line. [sent-210, score-1.139]
64 However, in case we know that it is the same distortion center in both images, we can just intersect these two lines. [sent-211, score-0.735]
65 Note that this is a similar setting as studied by Fitzgibbon [10], however we also estimate the distortion center. [sent-215, score-0.601]
66 As before, straight lines in the image without distortion are mapped to (circular) curves in the distorted view. [sent-223, score-1.098]
67 When looking for those curves that are straight lines, we obtain a 1D family of lines that all intersect at the distortion center. [sent-225, score-0.965]
68 In projective space this 1D family of lines spans a 2D subspace, and since the distortion center must lie on all of the lines, it is the orthogonal complement of the 2D subspace. [sent-226, score-0.845]
69 Since distortion happens in radial direction, we can compute the distances d1, d2 of both distorted epipoles to the distortion center and study the distortion as a 1D problem, where u is the undistorted distance: u =1 +d1 λd12=1 +d λ2d22. [sent-238, score-2.775]
70 (10) Here u is the distance of the undistorted epipole to the distortion center. [sent-239, score-0.876]
71 Higher order distortion parameters As long as we have a truly radially symmetric distortion model, we can always play the same trick with the intersection of the straight epipolar curve to determine the distortion center. [sent-243, score-2.296]
72 Therefore the above mentioned algorithms for extracting the distortion center remain valid even for higher order radial distortion models and one can also think of a higher order radial fundamental. [sent-244, score-2.163]
73 For all of the exper- iments we estimate the (two-sided) radial fundamental matrix using direct linear transformation on normalized feature coordinates (similar to [12]) using all inliers. [sent-247, score-0.57]
74 Synthetic Data First, we verify whether it is actually possible to estimate the distortion center with synthetic data (see Fig. [sent-248, score-0.694]
75 As can be seen, in an ideal setting with no noise we can always extract the distortion center, which proves our general idea of finding straight epipolar curves. [sent-251, score-1.024]
76 It is clear that in case the effects from noise come close to one of the other phenomena then the radial fundamental estimate (and thus the distortion center estimation) will be highly fragile. [sent-254, score-1.219]
77 Synthetic experiments for stereo pairs of the same camera with moderate distortion (|λ| = 0. [sent-258, score-0.681]
78 Then the two-sided radial fundamental is estimated from 100 correspondences and the distortion center is extracted. [sent-264, score-1.24]
79 First we calibrate the distortion center using a chessboard by the method from Hartley and Kang [13] . [sent-269, score-0.694]
80 From each pair we extract the distortion center and distortion coefficient as plotted in Fig. [sent-273, score-1.313]
81 Although we can see that the distortion centers cluster in a believable position, there is still a substantial offset to the position reported by the calibration-target-based method of Hartley and Kang. [sent-275, score-0.623]
82 We compute the radial distortion center from pairs of views, where one is from the start of the sequence and the other from the end of the sequence. [sent-280, score-1.128]
83 The centers cluster around (1108;360) close to the center of the image (also visualized) while we obtained a distortion center using the chessboard-based method of [13] at (983;530). [sent-281, score-0.809]
84 We do not show the distortion centers when estimated without normalization here, because they are not useful and often very far away from the image, however we do plot the estimated λ showing that normalization is important. [sent-282, score-0.645]
85 that the camera’s actual distortion curve does not fit to the division distortion model. [sent-284, score-1.304]
86 For the center image the estimated radial distortion center is visualized by showing the two straight epipolar lines. [sent-287, score-1.682]
87 As compared to a chessboard-based calibration by the method of [13] the distortion center is 2. [sent-288, score-0.769]
88 We then compute two radial fundamental matrices to the center camera and obtain a distortion center that is 2. [sent-295, score-1.392]
89 Conclusion Based on the observation that straight lines through the distortion center are fixed lines under any radial distortion model, we have derived constraints on the radial distortion center from epipolar geometry. [sent-301, score-3.446]
90 Essentially, by intersecting two lines that must include the distortion center, its coordinates are revealed. [sent-302, score-0.752]
91 For high distortion, it is important to choose the appropriate distortion model, maybe with more than one parameter or directly aim at a parameter-free representation as in [13] (which then comes at the cost of requiring more views and many more correspondences). [sent-304, score-0.62]
92 Unknown radial distortion centers in multiple view geometry problems. [sent-328, score-1.077]
93 A plumbline constraint for the rational function lens distortion model. [sent-354, score-0.779]
94 Parameter-free radial distortion correction with center of distortion estimation. [sent-387, score-1.729]
95 A non-iterative method for correcting lens distortion from nine-point correspondences. [sent-418, score-0.652]
96 Multi-view geometry of 1d radial cameras and its application to omnidirectional camera calibration. [sent-450, score-0.611]
97 the distortion parameters are the same in both views. [sent-463, score-0.601]
98 Their radial fundamental matrix is a 6-by-6 matrix, since the distorted points are lifted to R6. [sent-464, score-0.79]
99 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. [sent-467, score-0.625]
100 In the setting used in [8], the radial distortion matrix A is only defined up to an arbitrary projective mapping of the image plane. [sent-474, score-1.056]
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