cvpr cvpr2013 cvpr2013-423 knowledge-graph by maker-knowledge-mining
Source: pdf
Author: Adrien Bartoli, Toby Collins
Abstract: It has been shown that a surface deforming isometrically can be reconstructed from a single image and a template 3D shape. Methods from the literature solve this problem efficiently. However, they all assume that the camera model is calibrated, which drastically limits their applicability. We propose (i) a general variational framework that applies to (calibrated and uncalibrated) general camera models and (ii) self-calibrating 3D reconstruction algorithms for the weak-perspective and full-perspective camera models. In the former case, our algorithm returns the normal field and camera ’s scale factor. In the latter case, our algorithm returns the normal field, depth and camera ’s focal length. Our algorithms are the first to achieve deformable 3D reconstruction including camera self-calibration. They apply to much more general setups than existing methods. Experimental results on simulated and real data show that our algorithms give results with the same level of accuracy as existing methods (which use the true focal length) on perspective images, and correctly find the normal field on affine images for which the existing methods fail.
Reference: text
sentIndex sentText sentNum sentScore
1 We propose (i) a general variational framework that applies to (calibrated and uncalibrated) general camera models and (ii) self-calibrating 3D reconstruction algorithms for the weak-perspective and full-perspective camera models. [sent-4, score-0.666]
2 In the former case, our algorithm returns the normal field and camera ’s scale factor. [sent-5, score-0.367]
3 In the latter case, our algorithm returns the normal field, depth and camera ’s focal length. [sent-6, score-0.938]
4 Our algorithms are the first to achieve deformable 3D reconstruction including camera self-calibration. [sent-7, score-0.369]
5 Experimental results on simulated and real data show that our algorithms give results with the same level of accuracy as existing methods (which use the true focal length) on perspective images, and correctly find the normal field on affine images for which the existing methods fail. [sent-9, score-0.906]
6 Introduction The problem of 3D reconstruction of a deformable surface from monocular video data has been well studied over the past decade. [sent-11, score-0.543]
7 In the template-based setup in particular, where a reference 3D view of the surface is known, 3D reconstruction is carried out from 3D to 2D correspondences established between the template and an input image of the surface being deformed. [sent-12, score-0.93]
8 While this has initially been a reasonable assumption, being able to self-calibrate the camera would grant 3D reconstruction much more flexibility. [sent-16, score-0.311]
9 The most interesting scenario, both in terms of stability and applicability, is where all the intrinsics are known but the focal length which is also allowed to vary in time [13]. [sent-18, score-0.61]
10 This paper proposes a comprehensive framework for 3D reconstruction from a single uncalibrated image under isometric surface deformation. [sent-20, score-0.729]
11 In this context, most existing methods use a fully calibrated perspective camera model [1, 2, 3, 7, 8, 10, 14, 15, 16, 17] and are defeated by affine imaging conditions. [sent-21, score-0.464]
12 The reason is that they do not fully exploit the differential surface constraints, and use the so-called maximum depth heuristic [10], consisting in maximizing the surface’s depth while bounding surface extension [2, 3, 10, 14, 15, 16]. [sent-22, score-0.996]
13 Two exceptions are [1, 5] which use a variational framework with a perspective and an orthographic projection model respectively. [sent-23, score-0.289]
14 In contrast, our general variational framework applies to a general camera model, whether calibrated or uncalibrated. [sent-24, score-0.398]
15 It relates the template to input image warp to the unknown surface embedding. [sent-25, score-0.605]
16 It leads to a general PDE for isometric 3D reconstruction with the camera’s intrinsics as free parameters. [sent-26, score-0.572]
17 We establish that in the affine case, only the surface normal can be computed but not the absolute depth, while in the perspective case, both the surface normal, absolute depth and focal length can be estimated. [sent-27, score-1.617]
18 It computes the surface normal and the camera’s scale factor (the ratio between the camera’s focal length and the surface’s average depth). [sent-30, score-1.062]
19 It computes the surface normal and depth, and the camera’s focal length. [sent-32, score-0.919]
20 Experimental results support the fact that focal length self-calibration is feasible. [sent-34, score-0.534]
21 n §d§ full-perspective projection respectively, aankdgive solution algorithms for 3D reconstruction including camera self-calibration. [sent-42, score-0.455]
22 State of the Art Reconstructing a deforming surface in the templatebased setting has two main steps: input image to template registration and 3D shape inference. [sent-45, score-0.553]
23 This paper specifically focuses on the 3D shape inference step under isometric surface deformation [1, 2, 3, 5, 10, 14, 15, 16]. [sent-47, score-0.62]
24 The most successful relaxation [2, 14] has been the maximum depth heuristic [10] that consists in maximizing the surface’s depth under inextensibility constraints [2, 14] using Second-Order Cone Programming (SOCP). [sent-49, score-0.396]
25 The fastest results were however obtained by solving a variational for- mulation exploiting the differential structure of local isometry in the perspective [1] and orthographic [5] projection cases. [sent-50, score-0.422]
26 All the previously cited methods make a fundamental assumptions: the camera model is perspective projection and its intrinsics are known (except [5] which uses orthographic projection). [sent-51, score-0.447]
27 These methods are defeated by affine imaging conditions since they do not directly exploit the problem’s full differential structure. [sent-52, score-0.25]
28 In the former case, our algorithm computes the scale factor and the surface normal. [sent-56, score-0.327]
29 In the latter case, our algorithm computes the camera’s focal length, the surface normal and depth. [sent-57, score-0.953]
30 Our method is the first to solve 3D deformable shape reconstruction while performing camera self-calibration. [sent-58, score-0.369]
31 The unknown 3TDhe es uterfmapcela tise parameterized by an is ⊂om Retric embedding of the template, represented by the surface embedding function ϕ : Ω → R3. [sent-62, score-0.457]
32 The camera projection function is written Πtio : Rϕ3 : Ω→ → →R 2R. [sent-63, score-0.255]
33 tion is w→ritte Rn η : Ω → Finally, the unknown surface unit normal function is written ξ : Ω → S3. [sent-65, score-0.526]
34 Existing reconstruction methods compute the surface embedding function ϕ assuming that the camera projection function Π is known. [sent-72, score-0.752]
35 This implies estimating the weak-perspective camera’s scale or self-calibrating the full-perspective camera’s focal length. [sent-74, score-0.391]
36 First, composing the surface embedding and camera projection gives the warp; this is the reprojection constraint: η = Π ◦ ϕ. [sent-76, score-0.721]
37 General Isometric 3D Reconstruction We start from the differential constraint (3), and append the scaled unit surface normal λξ as the rightmost column of this matrix equality: ? [sent-80, score-0.609]
38 the equation is simplified, and gives the general equation of isometric n3 iDs r seicmopnlsitfrieudc,tio ann:d JηJη? [sent-101, score-0.609]
39 )(4) This is a nonlinear PDE in the camera projection Π, the surface embedding ϕ and its normal ξ. [sent-106, score-0.861]
40 Because of its global and parametric nature, camera projection will turn into a set of free unknown parameters when specializing this PDE to a particular camera model. [sent-110, score-0.587]
41 Weak-Perspective Solution We show how the general reconstruction equation (4) is specialized and solved for weak-perspective projection. [sent-112, score-0.318]
42 Specializing the General Equation The general affine camera’s projection function is ΠA(Q) = KASAQ. [sent-115, score-0.223]
43 Defining ξ¯ : Ω → R2, the function giving the first two elements of theξ :u Ωnit → →no Rrmal, as ξ¯ = SAξ, we get the affine equation of isometric 3D reconstruction: JηJη? [sent-125, score-0.571]
44 For a weak-perspective camera, KA = αI where the un- =def df known scale α > 0 is the ratio between the camfocal length f and the surface’s average depth d. [sent-130, score-0.289]
45 This leads to the weak-perspective equation of isometric 3D reconstruction: era’s JηJη? [sent-131, score-0.404]
46 Expanding equation (6), we get the following degree-two polynomial in μ: λ4μ2 =def − λ2tμ +g = 0, (7) =def with t tr ? [sent-154, score-0.237]
47 Criteria such as surface integrability or smoothness [3] can be used (through normal integration) to recover a C1 shape up to scale while disambiguating the normal field. [sent-200, score-0.688]
48 Full-Perspective Solution We here show how the general reconstruction equation (4) is specialized and solved for full-perspective projection. [sent-203, score-0.318]
49 Finally, substituting this expression in the general reconstruction equation (4) we obtain the full-perspective equation of isometric 3D reconstruction: JηJη? [sent-228, score-0.759]
50 We further specialize this equation under twheit assumption that only the focal length f is unknown and the effect of the other intrinsics were undone. [sent-239, score-0.832]
51 The relationship between ξ¯ and γ is quite complex and we thus cannot directly exploit it to solve the variational equation efficiently. [sent-262, score-0.22]
52 Indeed, γ gives the depth and with the reprojection constraint, determines function ϕ, whose first partial derivatives lead to the normal function ξ. [sent-263, score-0.461]
53 We propose the following estimation procedure: (i) sample f over a range of admissible values, (ii) for each candidate f value, solve the equation of isometric 3D reconstruction (11) and (iii) keep the value of f which best satisfies × the global isometric constraint. [sent-264, score-0.817]
54 Note that the template camera’s focpaixl length 5i s× generally unrelated to the runtime camera’s (for instance with printed paper we use the digital texture image as a template). [sent-267, score-0.308]
55 Wofi itthso supte lcoiassl of generality, we here assume λ = 1 (the surface is developable), but the method applies to an arbitrary local scale function λ. [sent-272, score-0.317]
56 More specifically, there are 4 solutions for the normal ξ¯ and 2 for the depth γ. [sent-283, score-0.406]
57 Finding the 1-dimensional affine subspace of solutions, we select the 4 solutions for ξ¯ and the 2 solutions for γ from the quadratic constraint ζ1ζ2 − ζ32 = 0. [sent-286, score-0.275]
58 W ζe do not keep the two ambiguous solutions for the normal field but rather recompute it a posteriori from function γ. [sent-288, score-0.26]
59 For t)hi ∈s point pair we measure the amount of surface extension or shrinking with respect to the template as: |δ(p, p? [sent-294, score-0.451]
60 The surface embedding function ϕ is represented as a linear interpolant of control points positionned on a regular grid. [sent-317, score-0.352]
61 The initial solution is provided by our focal length sampling algorithm. [sent-321, score-0.589]
62 Note that for the weak-perspective solution only the normal error is computable. [sent-327, score-0.296]
63 We first used SIFT [6] to obtain putative keypoint correspondences from which we then estimate a Thin-Plate Spline warp η using a robust method based on spatial consistency [12]. [sent-329, score-0.265]
64 It should be noted that these compared methods assume the focal length to be known. [sent-332, score-0.534]
65 In the case of simulated data this is the groundtruth focal length; it the case of real data it is obtained from static calibration. [sent-333, score-0.67]
66 STAT-PE is an iterative method using the maximum depth heuristic [10]. [sent-334, score-0.216]
67 STAT-SA is a convex solution using the maximum depth heuristic [14]. [sent-335, score-0.271]
68 STAT-BR is a convex SOCP solution using the maximum depth heuristic [2]. [sent-336, score-0.271]
69 We randomly drew m points on the simulated surfaces and projected them with a perspective camera. [sent-352, score-0.237]
70 We varied the simulated focal length (default: 400 pixels), the number of keypoint correspondences (default: 200) and the standard deviation of the gaussiandistributed correspondence noise (default: 1. [sent-354, score-0.768]
71 We observe that SELF-FP degrades with increasing focal length and correspondence noise and improves with increasing number of correspondences. [sent-358, score-0.621]
72 However, the f-error is kept below about 15% and is of a few percents for most simulated configurations. [sent-359, score-0.268]
73 This is comparable with an excellent static camera calibration. [sent-361, score-0.259]
74 As with the ferror, we observe that SELF-FP degrades with increasing focal length and correspondence noise and improves with increasing number of correspondences. [sent-364, score-0.621]
75 It gives normal estimates which are almost always more accurate than SELF-FP’s despite the significant amount of perspective in short focal 111555 111 686 MMeet h o d s u us i n g s se l f - c a l i b r a t i o n MMeetthhooddss uussiinngg ssttaatti cc ccaalli bbrraatti oonn Figure 2. [sent-371, score-0.691]
76 The first column shows the result when changing the simulated focal length. [sent-375, score-0.51]
77 Therefore, the accuracy of some methods based on the maximum depth heuristic (STAT-SA and STAT-PE) degrades significantly. [sent-377, score-0.262]
78 The focal length and depth also become ill-constrained as only their ratio can be measured, explaining why we observe that their estimates by SELF-FP degrades. [sent-378, score-0.721]
79 The surface normal how- ever is still well-constrained, as can be observed from SELFFP’s normal estimates. [sent-379, score-0.688]
80 This can be seen from equation (12): when f grows large the depth γ becomes ill-constrained but not the normal ξ. [sent-380, score-0.522]
81 We observe that the f-error and the depth error for SELF-RE increase with the focal length but much less than for SELF-FP, while the normal error is kept to its lower bound provided by STAT-RE. [sent-381, score-1.08]
82 These two examples respectively use a short and long focal length. [sent-392, score-0.391]
83 We now describe the short focal length example in details. [sent-393, score-0.534]
84 The bar-plot in figure 3 shows the true and estimated focal length as the level of zoom varies. [sent-400, score-0.618]
85 The focal length is fixed and its groundtruth value is 528 pixels. [sent-409, score-0.601]
86 We observe on the left graph that SELF-FP produces a depth error slightly larger that the other methods, but of the same order of magnitude. [sent-411, score-0.227]
87 On the other hand, SELF-RE achieves a depth error comparable to methods using static calibration. [sent-412, score-0.279]
88 The middle graph shows that both SELF-FP and SELF-RE overestimate the focal length by a few dozens of pixels. [sent-413, score-0.534]
89 The template here is in 3D since it is non-developable (the cap cannot be isometrically flattened to a plane). [sent-420, score-0.419]
90 The groundtruth focal length from static calibration is 2040 pixels. [sent-422, score-0.743]
91 We first reconstructed the visible part of the cap using template-based deformable 3D reconstruction. [sent-426, score-0.272]
92 We then transferred the hidden part of the cap from the template by extrapolating the transformation obtained for the reconstructed visible part. [sent-428, score-0.379]
93 The estimated focal length was 1890 pixels for SELF-FP and 2118 pixels for SELF-RE, which means an f-error of 7. [sent-435, score-0.534]
94 Conclusion The main conclusion of our paper is that focal length self-calibration in template-based isometric deformable 3D reconstruction is feasible. [sent-440, score-1.005]
95 Our initialization algorithm facilitates accurate 3D reconstruction for small to medium focal length values while our nonlinear refinement algorithm handles small to large focal length values extremely well, being as accurate as methods using static calibration. [sent-442, score-1.426]
96 When the focal length grows too large it cannot be computed. [sent-443, score-0.573]
97 We showed how the surface normal can however still be accurately estimated with the weak-perspective projection model. [sent-444, score-0.576]
98 On template-based reconstruction from a single view: Analytical solutions and proofs of well-posedness for developable, isometric and conformal surfaces. [sent-461, score-0.535]
99 Self-calibration and metric reconstruction in spite of varying and unknown internal camera parameters. [sent-537, score-0.35]
100 Linear local models for monocular reconstruction of deformable surfaces. [sent-547, score-0.257]
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