iccv iccv2013 iccv2013-280 knowledge-graph by maker-knowledge-mining
Source: pdf
Author: Jae-Hak Kim, Yuchao Dai, Hongdong Li, Xin Du, Jonghyuk Kim
Abstract: We present a new multi-view 3D Euclidean reconstruction method for arbitrary uncalibrated radially-symmetric cameras, which needs no calibration or any camera model parameters other than radial symmetry. It is built on the radial 1D camera model [25], a unified mathematical abstraction to different types of radially-symmetric cameras. We formulate the problem of multi-view reconstruction for radial 1D cameras as a matrix rank minimization problem. Efficient implementation based on alternating direction continuation is proposed to handle scalability issue for real-world applications. Our method applies to a wide range of omnidirectional cameras including both dioptric and catadioptric (central and non-central) cameras. Additionally, our method deals with complete and incomplete measurements under a unified framework elegantly. Experiments on both synthetic and real images from various types of cameras validate the superior performance of our new method, in terms of numerical accuracy and robustness.
Reference: text
sentIndex sentText sentNum sentScore
1 It is built on the radial 1D camera model [25], a unified mathematical abstraction to different types of radially-symmetric cameras. [sent-2, score-0.564]
2 We formulate the problem of multi-view reconstruction for radial 1D cameras as a matrix rank minimization problem. [sent-3, score-0.836]
3 Our method applies to a wide range of omnidirectional cameras including both dioptric and catadioptric (central and non-central) cameras. [sent-5, score-0.905]
4 Introduction Having a wide field of view, omnidirectional cameras can be used to reconstruct broad scenes from few views, thus have been widely deployed to applications such as surveillance, robot navigation and 3D modeling of street scene. [sent-9, score-0.436]
5 It is highly desired to have a unified and efficient reconstruction method for omnidirectional cameras. [sent-12, score-0.479]
6 This paper proposes a new multi-view 3D Euclidean reconstruction method for generic types of uncalibrated radially-symmetric cameras. [sent-13, score-0.264]
7 It is built on the radial 1D camera model originally developed by Thirthala and Pollefeys [25]. [sent-14, score-0.471]
8 Our method recovers 3D structure via matrix rank min- imization from general types of uncalibrated radially-symmetric cameras e. [sent-16, score-0.374]
9 fisheye lens cameras, concave shape mirror based catadioptric cameras, noncentral cameras including spherical mirror or any radially-symmetric mirror shape based cameras, and multiple relflection surfaces based Sony RPU camera. [sent-18, score-1.221]
10 First, we formulate the problem of multiview reconstruction for radial 1D cameras as a matrix rank minimization problem, and solve it through convex optimization (and semi-definite programming in particular followed by an efficient alternating direction continuation method). [sent-28, score-0.951]
11 Second, the multi-view reconstruction is upgraded from projective to Euclidean by exploiting the internal constraints. [sent-29, score-0.324]
12 Modeling radially-symmetric cameras Due to the various types of omnidirectional camera de- sign and construction, e. [sent-33, score-0.583]
13 dioptric (lens-based) or catadioptric (mirror-lens system), central or non-central, most 3D reconstruction methods are specially designed for one 11889966 or a few particular types, and thus are not universally applicable to others. [sent-35, score-0.723]
14 Meanwhile, novel types of omnidirectional optical devices are emerging too (e. [sent-36, score-0.25]
15 1), which also calls for a unified 3D reconstruction procedure. [sent-39, score-0.229]
16 There indeed exists a unified mathematical model to represent various types of omnidirectional cameras, so-called generalized camera model (GCM) [19, 22, 11], which models cameras as unconstrained sets of projection rays. [sent-40, score-0.681]
17 However, the GCM does not suggest a unified way to handle 3D reconstruction from uncalibrated cameras. [sent-41, score-0.308]
18 The radial 1D camera model studied in this paper is applicable to both central and non-central cases, and it is in fact a special case (symmetry version) of the GCM (i. [sent-43, score-0.54]
19 In practice, very often omnidirectional cameras will manifest a certain type of symmetry, where radial symmetry being the dominant form. [sent-46, score-0.789]
20 This is reasonable, because it is convenient to design, to manufacture and to use, an omnidirectional camera with a radially-symmetric field of view. [sent-47, score-0.397]
21 There are two major classes of omnidirectional cam- eras: dioptric and catadioptric. [sent-48, score-0.333]
22 The former one includes a wide-angle lens (fish-eye lens), and the latter one often consists of a perspective camera plus a curved mirror. [sent-49, score-0.29]
23 Most of the commonly used omnidirectional cameras belong to this class, and this is the main focus of this paper. [sent-51, score-0.436]
24 To express radially-symmetric cameras in a unified way, Thirthala and Pollefeys [25] proposed the novel concept of “radial 1D camera” that maps a 3D point to a radial line. [sent-52, score-0.554]
25 The axial camera is an abstraction of stereo systems, non-central catadioptric cameras and pushbroom cameras. [sent-57, score-0.846]
26 Related works To recover structure and motion for cameras with a wide circular field of view, Miˇ cuˇ s ı´k and Pajdla [17] estimated epipolar geometry of radially-symmetric cameras by solving a polynomial eigenvalue problem. [sent-60, score-0.401]
27 Lhuillier [12] presented fully automatic methods for estimating scene structure and camera motion from an image sequence acquired by a catadioptric system, where bundle adjustment is applied to both central and non-central models. [sent-62, score-0.651]
28 When a specific camera model is available, 3D reconstruction can be achieved in a tailor-made style. [sent-63, score-0.332]
29 For catadioptric cameras, with the information of the mirror (model and parameters), calibration and reconstruction can be done through computing the forward and backward projection. [sent-64, score-0.765]
30 Geyer and Daniilidis [8] introduced the circle space representation for an image of points and lines in central catadioptric cameras, from which the epipolar constraint and catadioptric fundamental matrix are derived. [sent-65, score-0.917]
31 Miˇ cuˇ s ı´k and Pajdla [16] developed accurate non-central and suitable approximate central models for specific mirrors, thus allowing to build a 3D metric reconstruction from two uncalibrated non-central catadioptric images. [sent-67, score-0.719]
32 The analytical forward projection leads to 3D reconstruction via bundle adjustment. [sent-70, score-0.329]
33 The plane based calibration for radially-symmetric cameras have also been studied in [20] and [24]. [sent-74, score-0.265]
34 Hartley × and Kang [9] proposed a parameter-free method to simultaneously calibrate the radial distortion function of a camera and other internal calibration parameters by using a planar calibration grid. [sent-76, score-0.733]
35 However, their model is restricted to central cameras and assumes a known calibration grid. [sent-77, score-0.334]
36 When the distortion model is available, radial distortion calibration and multi-view geometry can be solved with algebraic minimization methods such as [7]. [sent-78, score-0.697]
37 Radial 1D camera model The radial 1D camera model [25] (Fig. [sent-80, score-0.618]
38 2) is a much more general mathematical abstraction, which encompasses most of the fisheye cameras, central and non-central catadioptric cameras, perspective and affine cameras. [sent-81, score-0.647]
39 Definition: The radial 1D camera expresses the mapping of a 3D point in P3 onto a radial line in the image plane. [sent-82, score-0.795]
40 e Under the radial 1D camera model, a 3D point Xj = [xj , yj , zj , 1]T is mapped to a distorted image measurement xidj = [uidj, vidj]T by a radial camera Pi ∈ R2×4: PiXj = φijxidj , (1) where φij is a scale factor. [sent-84, score-1.248]
41 11889977 The backward projection of the line is the plane containing the 3D point Xj and the ray passing through center of distortion ci and xidj . [sent-87, score-0.417]
42 , a scaling projection matrix for each radial camera and scaling scene point individually will not change the 2D image measurements. [sent-93, score-0.657]
43 Nevertheless, we can achieve 3D Euclidean reconstruction without ambiguity as shown in the following sections. [sent-94, score-0.233]
44 The radial 1D camera can be thought of as projecting a bundle of planes containing the optical axis onto a bundle of radial lines passing through the radial centre in the image plane (Fig. [sent-96, score-1.253]
45 The radial 1D camera model encompasses most ofthe central and non-central omnidirectional cameras. [sent-98, score-0.79]
46 This is because the only essential requirement in this model is that all points lie in one plane, of the bundle around the optical axis, project onto the same radial line (passing through the radial centre). [sent-99, score-0.727]
47 Multi-view reconstruction upto projectivity In this paper, we target at multi-view Euclidean reconstruction from arbitrary radially-symmetric cameras. [sent-101, score-0.441]
48 First, we achieve multi-view reconstruction upto projectivity through factorization, which offers great simplicity and elegancy. [sent-103, score-0.256]
49 Second, by exploiting intrinsic constraints, the projective reconstruction is upgraded to Euclidean reconstruction. [sent-104, score-0.324]
50 In this way, we do not need any specific camera and distortion model other than the radially-symmetric condition. [sent-105, score-0.251]
51 Thus, we have reached a factorization formulation similar to the factorization model for perspective cameras. [sent-120, score-0.351]
52 Actually, a perspective camera model with or without distortion falls exactly into the radial 1D camera model. [sent-121, score-0.771]
53 -(3) can handle non-central cameras as well since it is a projection model by radial 1D cameras in [25]. [sent-123, score-0.75]
54 Hadamard factorization based solution Recall that the multi-view factorization model for radially symmetric cameras is expressed as: PX = Φ ? [sent-126, score-0.61]
55 For general wide view angle omnidirectional cameras, the coefficient φij is positive. [sent-129, score-0.336]
56 Taking all the constraints into consideration, mathematically multi-view factorization for radially symmetric cameras is formulated as: Problem 3. [sent-130, score-0.459]
57 × Once the scaling matrix Φ is recovered, the multi-view factorization problem can be solved via the singular value decomposition (SVD) as W = PX. [sent-134, score-0.24]
58 Minimization objective: Under the affine camera model, multi-view factorization achieves the maximum likelihood estimation (MLE) [10]. [sent-137, score-0.298]
59 For the perspective camera model, multi-view factorization is actually minimizing an algebraic error, which can be viewed as an approximation to the geometric reprojection error. [sent-138, score-0.401]
60 Under the radial 1D camera model, we cannot measure the geometric reprojection error but only an angular error. [sent-139, score-0.586]
61 We define the angular error corresponding to a distorted image measurement xidj as eij, which measures the angle between the measured ray and the reconstructed ray, i. [sent-140, score-0.502]
62 Column-wise and row-wise normalization of φij deal with scale ambiguity with P and X respectively, which has been used in projective factorization problems e. [sent-190, score-0.338]
63 In solving the multi-view factorization problem, we are actually relaxing the objective function to an algebraic error to approximate the geometric angular error. [sent-193, score-0.32]
64 To achieve a good approximation, we propose to normalize all the image measurements xidj to unit norm xidj ← xidj/ ? [sent-194, score-0.553]
65 , th aes factorization formulation expresses as: PiXj = φijxidj , where xidj is a unit norm vector (direction). [sent-198, score-0.414]
66 Multi-view Euclidean Reconstruction In this section, we upgrade the multi-view reconstruction from radially-symmetric camera to Euclidean reconstruction by exploiting constraints on the intrinsic camera matrix. [sent-210, score-0.664]
67 Once we have recovered the scaling matrix Φ and the weighted measurement matrix W, the projection matrix P and scene structure X can be recovered through SVD as W = PX. [sent-214, score-0.388]
68 If a reasonable upgrading matrix H is achieved, the Euclidean structure and motion can be recovered from structure matrix Xˆ and projection matrix Pˆ. [sent-218, score-0.293]
69 Efficient Implementations In this section, we propose to solve multi-view projective reconstruction Problem 3. [sent-232, score-0.288]
70 Semi-definite Programming Under noiseless and complete measurement case, our multi-view factorization Problem 3. [sent-236, score-0.234]
71 Recently, nuclear norm minimization has been widely used in low-rank modeling such as projective factorization [5, 3] and robust principal component analysis [4]. [sent-249, score-0.379]
72 Therefore, we obtain an alternating direction continuation based algorithm for Hadamard factorization formulation for radially symmetric camera reconstruction, which alternates between updating U, V and Φ. [sent-336, score-0.506]
73 Dealing with incomplete measurements With a wide field of view camera, we can reconstruct 3D scene from a few images, which will create an incomplete measurement matrix in general and the SVD based method cannot be applied. [sent-340, score-0.365]
74 × In this subsection, we extend our Hadamard factorization formulation from complete measurements case to incomplete measurements case. [sent-341, score-0.349]
75 With missing data setting, we are given an incomplete measurement matrix M = [xidj] , where the missing elements are completed with 0. [sent-342, score-0.332]
76 e, With these notations, the imaging process for radiallysymmetric cameras with missing data can be compactly expressed as: ? [sent-345, score-0.278]
77 Additionally, our alternating adtirreixct wiointh hc nonot einn-uation based efficient implementation can also be extended to incomplete measurement case directly, achieving missing points handling ALM method (MALM). [sent-363, score-0.263]
78 To generate the synthetic data we created 100 3D points placed randomly on 3 walls intersecting each other in 90 degrees, and a catadioptric system consisting of a perspective camera and a spherical mirror axially aligned is placed to capture the image of the 3D points on the walls. [sent-370, score-0.908]
79 The catadioptric system is moved to capture 20 images at different views. [sent-371, score-0.386]
80 Note that this reflection by the spherical mirror creates axially aligned radial distortion in the image. [sent-373, score-0.702]
81 The 2D angle error is the angle between the estimated radial line and the input radial line in the image. [sent-383, score-0.782]
82 DTh peo sikntews e error tise obtained from the estimated camera calibration matrix. [sent-386, score-0.27]
83 (Left) A catadioptric system consists of a spherical mirror and a camera (green frustrum) in 6 textured walls. [sent-400, score-0.715]
84 (Third and fourth) The same views of 3D reconstruction by MALM and missing points are recovered. [sent-418, score-0.267]
85 We generated a dataset ‘MirrorBall’ which is an image sequence rendered by a computer graphics application to simulate a catadioptric system in the configuration shown in Fig. [sent-425, score-0.421]
86 A spherical mirror is placed in front of a perspective camera inside 6 textured walls, and the reflection on the spherical mirror is rendered. [sent-427, score-0.612]
87 Sony RPU camera module consisting of multiple reflection surfaces and a camera to create a panoramic image was used to capture a ‘rooftop’ sequence of 17 images. [sent-480, score-0.346]
88 An omnidirectional image sequence captured by a catadioptric camera (Kumotek VS-C14U-80-ST) was used in our experiment. [sent-487, score-0.783]
89 4(Lef6t) An image of a cylinder shape of a paper captured by the omnidirectional camera (catadioptric). [sent-490, score-0.397]
90 A top view of the reconstruction by SIESTA (171 frames, 40 points) and the area of the reconstruction in the building map. [sent-499, score-0.411]
91 der shape, then the catadioptric camera was inserted inside the rolled paper. [sent-503, score-0.569]
92 Image sequences were captured by Canon fisheye lens at indoor scenes such as corridors and a room as shown in Fig. [sent-508, score-0.237]
93 The reports show that our proposed ALM achieves less than 5% error of 3D reconstruction compared with the ground truth. [sent-530, score-0.229]
94 With multi-view input, our method overcomes the previous theoretical boundary of 3D reconstruction from radial 1D cameras. [sent-536, score-0.509]
95 one view from a fish eye lens and another view from catadioptric camera), theoretically this is identical and solvable. [sent-539, score-0.562]
96 A current limitation of our method is that the centre of distortion needs to be known or estimated from lens outer edges. [sent-540, score-0.234]
97 Analytical forward projection for axial non-central dioptric and catadioptric cameras. [sent-549, score-0.601]
98 Beyond Alhazen’s problem: Analytical projection model for non-central catadioptric cameras with quadric mirrors. [sent-556, score-0.626]
99 Parameter-free radial distortion correction with center of distortion estimation. [sent-597, score-0.532]
100 Automatic scene structure and camera motion using a catadioptric system. [sent-613, score-0.533]
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