iccv iccv2013 iccv2013-436 knowledge-graph by maker-knowledge-mining
Source: pdf
Author: R. Melo, M. Antunes, J.P. Barreto, G. Falcão, N. Gonçalves
Abstract: Estimating the amount and center ofdistortionfrom lines in the scene has been addressed in the literature by the socalled “plumb-line ” approach. In this paper we propose a new geometric method to estimate not only the distortion parameters but the entire camera calibration (up to an “angular” scale factor) using a minimum of 3 lines. We propose a new framework for the unsupervised simultaneous detection of natural image of lines and camera parameters estimation, enabling a robust calibration from a single image. Comparative experiments with existing automatic approaches for the distortion estimation and with ground truth data are presented.
Reference: text
sentIndex sentText sentNum sentScore
1 Unsupervised intrinsic calibration from a single frame using a ”plumb-line” approach R. [sent-1, score-0.468]
2 In this paper we propose a new geometric method to estimate not only the distortion parameters but the entire camera calibration (up to an “angular” scale factor) using a minimum of 3 lines. [sent-8, score-0.885]
3 We propose a new framework for the unsupervised simultaneous detection of natural image of lines and camera parameters estimation, enabling a robust calibration from a single image. [sent-9, score-0.802]
4 Comparative experiments with existing automatic approaches for the distortion estimation and with ground truth data are presented. [sent-10, score-0.328]
5 Introduction We investigate the problem of fully calibrating an image with significant distortion without requiring any type of manual supervision. [sent-12, score-0.336]
6 A solution for automatic, single frame calibration is specially relevant in the case of images mined from the internet, for which knowing the camera parame- ters can be useful for multiple tasks. [sent-13, score-0.619]
7 The article considers the case of cameras with distortion that can be described by the 1-parameter division model (DM) [7, 11], and assumes that the imaged scene has a reasonable number of straight lines. [sent-15, score-0.47]
8 We propose for the first time a calibration algorithm that, given the image of 3 lines, it estimates the distortion, principal point, aspect ratio, and skew. [sent-16, score-0.543]
9 Such result is not surprising if we consider that the division model has obvious resemblances with the stereographic projection used to describe the para-catadioptric sensor [3], and that para-catadioptric cameras can be fully calibrated from a minimum of 3 line images [4]. [sent-17, score-0.346]
10 Nevertheless, and to the best ofour knowledge, the possibility ofcalibrating the intrinsics of dioptric camera with distortion from 3 lines has never been reported. [sent-18, score-0.558]
11 the EXIF tag, the nominal field-of-view) and only knowing the distortion in pixels still enables accurate distortion compensation with the focal length being chosen as a function of the desired resolution for the output perspective [20]. [sent-22, score-0.672]
12 The main contribution of the paper is a processing pipeline that receives as input a set of image contours, selects the ones that are likely to be projections of lines, and outputs both the detected lines and the camera calibration parameters (see Fig. [sent-23, score-0.773]
13 While standard ”plumb-line” calibration requires user intervention for selecting the image edges that are projection of lines, our method carries this operation in a fully automatic manner. [sent-25, score-0.572]
14 For the uncalibrated case we use contour triplets to establish different calibration hypothesis that give raise to different UFL instances. [sent-29, score-0.58]
15 This provides an efficient, robust manner of simultaneously detecting the line contours and finding the camera calibration. [sent-31, score-0.279]
16 The reason for this is that the detected arc contours are usually small and, under strong occlusion, it is difficult to obtain plausible conic estimation [4] and, consequently, plausible calibration hypothesis to be used in the (UFL)-(HFL) framework. [sent-34, score-0.861]
17 Related work The geometric calibration of cameras with distortion is a well-studied topic, with several methods and approaches being described in the literature [22]. [sent-38, score-0.81]
18 However, none of these solutions is well suited for the automatic calibration of images mined from the internet. [sent-39, score-0.525]
19 showed in [1] that it is possible to fully calibrate a camera with distortion using a single image of a chessboard pattern. [sent-42, score-0.421]
20 Since we are addressing the calibration of images of natural scenes, this approach is also no solution for our problem. [sent-43, score-0.468]
21 Given a single image, the algorithm enables to recover the distortion parameters and the principal point whenever the scene has two patterns orthogonal to each other. [sent-46, score-0.362]
22 Contrary to what happens with conventional perspective cameras, in the case of cameras with distortion it is possible to recover calibration information from the projection of 3D lines in random position [22]. [sent-49, score-1.02]
23 Since lines are features that often appear in natural images, with special relevance in the case of man-made environments, line-based calibration is an appealing proposition. [sent-50, score-0.627]
24 The first contributions in camera calibration using the so-called “plumb-line” constraint go back to the 70’s when Brown suggested to model the distortion by a polynomial and estimate its parameters by straightening up the lines in the image [8]. [sent-51, score-1.014]
25 He concluded that, similarly to para-catadioptric cameras, the lines in 3D are projected into a family of conic curves that intersect in two points and satisfy an harmonic conjugate relation with two other points [4]. [sent-55, score-0.594]
26 He also showed that the conic where a line is projected has only two independent degrees-of-freedom (DOF) and that, if the center is known, then it is possible to estimate and correct the image distortion using a single line. [sent-56, score-0.685]
27 proposed an algorithm for computing both the distortion parameter and the principal point from an image of 3 lines [24] using an algebraic interpretation of the division model. [sent-59, score-0.565]
28 [9] suggested an algo- rithm for automatically detecting lines and accomplishing the calibration following the methods of [24]. [sent-62, score-0.627]
29 The structure of the paper is as follows: section 2 introduces some background notions, section 3 addresses the problem of camera calibration from 3 lines and section 4 addresses the problem of line extraction from calibrated images. [sent-64, score-0.884]
30 In section 5 we present the unsupervised calibration algorithm and section 6 shows the experimental results. [sent-65, score-0.534]
31 Points, lines and conics are represented in homogeneous coordinates. [sent-75, score-0.514]
32 Background concepts Throughout this article we will model the camera distortion using the so called division model [7, 11], where ξ is the negative parameter that quantifies the amount of distortion. [sent-78, score-0.454]
33 h() is the radial distortion function that maps undistorted points u in P2 into distorted points d in P2: d ∼ h(u) ∼ ( 2u1 2u2 u3 + ? [sent-79, score-0.436]
34 The distortion function 1transforms a line n into the conic Ω given by [3]: Ω =⎝⎛ξn 2013 ξn20n23 Nn 2 13⎠⎞. [sent-82, score-0.685]
35 (2) It has been shown in [3] that Ω is the distorted image of a world line iff it passes through the circular points I and J, and points r+ and r−are harmonic conjugates [21] with respect to Ω: ⎨⎧⎪rIJT +TΩ IJr−= 0 0 w it h JrI± = = ( ( 1 1i−0i )0±T? [sent-83, score-0.322]
36 Condition for a conic to be the image of a line The conic Ω where a line is imaged is now given by transforming the result of equation 2 by the intrinsic parameters K, as shown in Fig. [sent-113, score-0.905]
37 Since projective transformations preserve incidence and cross-ratio relations, the conic Ω must intersect the line at infinity in points I? [sent-115, score-0.648]
38 Therefore, a conic is the image of a line iff it verifies Φω = 0, with ωT being its representation in P5 and Φ being the 3 6 matrix: Φ =⎡⎣( a cs 2x −+− ia aη2 2) 2 a cs x +−cy ia c1 y2c0 xc0 y0 1⎦⎤(5) with ? [sent-119, score-0.431]
39 If the calibration parameters of the camera are known, then the conic Ω can be estimated from N ≥ 2 image points using constrained least squares [13], with equation 5 giving the set of 3 linear constraints. [sent-125, score-0.92]
40 Minimal solution for the calibration From equation 5, we observe that the images of lines lie in a 2D subspace S of P5 that encodes the calibration. [sent-128, score-0.693]
41 We now show how to recover the calibration parameters from 3 line images ω1, ω2 and ω3 (Fig. [sent-129, score-0.621]
42 If the projection of a 3D line is correctly estimated in the image plane, intersecting it with the line at infinity defines points I? [sent-131, score-0.448]
43 The principal point (cx , cy) and distortion parameter are encoded in the third orthogonal vector to the subspace of 539 (a)? [sent-138, score-0.349]
44 Figure 2: Intersecting projections of 3D lines with the line at infinity. [sent-144, score-0.327]
45 Given tree line images, we can determine this subspace and compute Λ by parametrizing the null space of the lines as follows: N(ω1, ω2, ω3) = K1V1 + K2V2 + K3V3 (7) with Φ ∈ N(ω1 , ω2 , ω3). [sent-146, score-0.353]
46 We are only able to determine the ratio η between ξ and f2, that can be understood as the distortion parameter expressed in pixels rather than in mm. [sent-151, score-0.308]
47 Nevertheless this coupled parameter enables to rectify the image distortion (as shown in the experiments). [sent-152, score-0.301]
48 We can verify that, considering a fourth line projection ω4, puts no further constraints to the calibration problem. [sent-154, score-0.658]
49 The conic curve must satisfy two linear con- straints since it must pass by points I? [sent-155, score-0.363]
50 This means that only 2 of the 5 DOF of the conic curve are really independent and they refer to the orientation of the plane containing the original line in 3D (see [3]). [sent-158, score-0.407]
51 Thus, we conclude that line images ωi, with i > 3, bring no additional information about the camera calibration and it is impossible to decouple the focal length f from the distortion parameter ξ using exclusively line features. [sent-159, score-1.162]
52 The calibration solution demonstrated above enables to determine the back-projection directions up to an angular multiplicative factor. [sent-160, score-0.521]
53 The joint effect of noise and strong partial occlusion makes the estimation of the conic very uncertain [2]. [sent-168, score-0.303]
54 2c we can see that the arrangement of the initial conics does not comply with the constrains derived in section 3. [sent-170, score-0.355]
55 In this case the conics do not intersect Ωi(0) the line at infinity in two unique points and the harmonic relations with respect to points r? [sent-172, score-0.707]
56 We start by estimating the likely location of the conics intersection with the line at infinity (steps 1 to 3) and then the conics a re-estimated from the corresponding image points enforcing the incidence with points I? [sent-176, score-1.024]
57 The calibration estimation of steps 1 to 6 is sub-optimal and is used as initialization for a final iterative optimization step. [sent-180, score-0.52]
58 (8) × with G being a 6 3 matrix that encodes the calibration parameters and m being the 3 1 vector encoding the orientation of the plane that contains the line [3]. [sent-188, score-0.621]
59 Given the conics ωi and the matrix G, computed with the calibration initialization, the corresponding vector mi is determined linearly. [sent-189, score-0.823]
60 Let be contour point j = 1···Ni belonging to the ith conic wi. [sent-190, score-0.303]
61 The bundle adjustment of the calibration parameters is carried by minimizing the function of equation 9: qj(i) f =a,s,cxm,ciyn,η,mi ? [sent-191, score-0.524]
62 the camera is skewless and has square pixels), then the 3 first steps can be skipped and the calibration carried trough 4 to 7. [sent-200, score-0.58]
63 Line extraction from a calibrated image Let us assume a calibrated image with distortion for which we want to detect projections of world straight lines. [sent-202, score-0.461]
64 We start by applying a standard edge detector [10], followed by a connected components algorithm in order to obtain several contours ei that are line projection candidates. [sent-203, score-0.38]
65 We aim at identifying the contours ei that support conics ωj lying on the 2D subspace S ∈ P5 defined by the calibration parameters. [sent-204, score-1.025]
66 This can be seen as a multi-model fitting problem where the models are the conics ωj consistent with the calibration and we want to assign to each contour ei a model (or discard the contour in case it does not fit any model). [sent-205, score-0.986]
67 Consider a set V0 comprising M possible facility locations, the cost ci0j for assigning the facility ωj0 to the customer ei and the cost vj0 for opening the particular facility ωj0. [sent-210, score-1.207]
68 Intersect each conic with the line at infinity and obtain Ii? [sent-217, score-0.469]
69 (Re)-estimate the conics using constrained least squares [13], forcing them to intersect I? [sent-227, score-0.41]
70 Given the conics Ωi=1···3 compute a basis for the null space N and determine the Λ vector by solving equation 7. [sent-231, score-0.42]
71 Refine the calibration result by minimizing equation 8 using iterative optimization (equation 9). [sent-235, score-0.502]
72 goal of the UFL problem is to select a subset of V0 such that each customer is served by one facility and the sum of the customer-facility costs plus the sum of facility opening costs is minimized. [sent-236, score-0.87]
73 The objective is to assign to each segment ei an image conic ωj0 ∈ V0 using as few unique models as possible. [sent-258, score-0.385]
74 Consider that the segments ei are the customers and the putative conics ωj0 are the facilities. [sent-259, score-0.565]
75 Let the cost ci0j be the root mean square geometric distance between points of ei and conic ωj0. [sent-260, score-0.468]
76 The goal is to select a subset of conics in V0 such that sum of the consistency measures ci0j and the costs vj0 is minimized, which corresponds to the minimization of Eq. [sent-262, score-0.407]
77 It can be seen that the line extraction al- gorithm successfully identifies the correct lines and clusters disconnected segments in the same line. [sent-266, score-0.329]
78 Unsupervised Plumb-line calibration using RANSAC-UFL In the previous section we presented an algorithm that, given the calibration, detects and estimates distorted world lines projections. [sent-268, score-0.69]
79 This section considers the unsupervised calibration of the camera, which consists in simultaneously determining a suitable set of calibration parameters along with the corresponding world line projections in the image. [sent-269, score-1.226]
80 Consider a set of M facility locations V0 and L storage facilities V1. [sent-274, score-0.425]
81 In addition to the costs vj0 and ci0j described in the previous section, we now add the cost vj1 for opening the storage facility Γk1, and the cost cj1k associated with the facility Γk1 supplying the facility ωj0. [sent-275, score-1.136]
82 11 are that if a facility ωj0 is closed in layer 0, then ωj0 will not need to be stocked by a storage facility Γj1, whereas if a facility ωj0 is open, then it must be stocked by a facility in layer 1. [sent-289, score-1.524]
83 Γk1 are calibration hypothesis and ωj0 are conics estimated from segments ei constrained by the associated calibration. [sent-292, score-1.0]
84 Since only one calibration Γk1 is desirable, the penalization vk1 should be very high. [sent-301, score-0.499]
85 For each generated Γk1 we compute the M conics ωj0 that are consistent with the calibration Γk1 and minimize the geometric distance of ei to ωj0 (the method described in section 3. [sent-303, score-0.962]
86 Our HFL formulation retrieves a single calibration by setting the connection costs cj1k between ωj0 and Γk1 as cj1k=? [sent-325, score-0.52]
87 Being formulated as a HFL problem, the unsupervised calibration algorithm can be computationally intensive if the number of segments ei and/or the number of calibration hypothesis Γk is high. [sent-329, score-1.179]
88 We show that our HFL problem can be efficiently solved as a minimization over a calibration dependent function fΓ1k (x0), which in turn is the result of solving the UFL problem (please refer to the supplementary material for details): N fΓ1k(x0) = mx0in? [sent-330, score-0.468]
89 The RANSAC-UFL randomly samples triplets of connected components, generating calibration hypothesis Γk1. [sent-340, score-0.55]
90 The calibration with the lowest UFL energy fΓ1k (x0) is the output of the unsupervised calibration. [sent-343, score-0.534]
91 Experimental results To evaluate our unsupervised calibration accuracy we compared the camera parameters estimation in 8 image of a cluttered environment against ground truth calibration obtained with [1]. [sent-345, score-1.138]
92 7, we compare our approach with [9] by correcting the radial distortion in some images dominated by straight world lines, showing that out approach outperforms [9] for the estimation of the center and amount of distortion in both robustness and accuracy. [sent-351, score-0.714]
93 For each scene we show the segment ei on the top and the resulting distortion correction in the bottom. [sent-365, score-0.428]
94 Conclusion In this work we have proposed a new method for the calibration of a camera using a minimum of 3 natural lines in a single image. [sent-367, score-0.714]
95 Our work is based on a solid geometric interpretation of the line projection under the division model in perspective cameras and is able to estimate the principal point, aspect ratio, skew angle and a coupled parameter of the distortion and focal distance. [sent-368, score-0.809]
96 For unsupervised camera calibration, we devised a framework for the joint line detection and calibration parameters estimation from a single image, that has been tested in challenging situations. [sent-369, score-0.801]
97 General central projection systems, modeling, calibration and visual servoing. [sent-386, score-0.548]
98 Solving the uncapacitated facility location problem using message passing algorithms. [sent-474, score-0.366]
99 A new solution for camera calibration and real-time image distortion correction in medical endoscopy - initial technical evaluation. [sent-496, score-0.874]
100 A simple method of radial distortion correction with centre of distortion estimation. [sent-525, score-0.66]
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