iccv iccv2013 iccv2013-27 knowledge-graph by maker-knowledge-mining
Source: pdf
Author: Adrien Bartoli, Daniel Pizarro, Toby Collins
Abstract: We study the uncalibrated isometric Shape-fromTemplate problem, that consists in estimating an isometric deformation from a template shape to an input image whose focal length is unknown. Our method is the first that combines the following features: solving for both the 3D deformation and the camera ’s focal length, involving only local analytical solutions (there is no numerical optimization), being robust to mismatches, handling general surfaces and running extremely fast. This was achieved through two key steps. First, an ‘uncalibrated’ 3D deformation is computed thanks to a novel piecewise weak-perspective projection model. Second, the camera’s focal length is estimated and enables upgrading the 3D deformation to metric. We use a variational framework, implemented using a smooth function basis and sampled local deformation models. The only degeneracy which we easily detect– for focal length estimation is a flat and fronto-parallel surface. Experimental results on simulated and real datasets show that our method achieves a 3D shape accuracy – slightly below state of the art methods using a precalibrated or the true focal length, and a focal length accuracy slightly below static calibration methods.
Reference: text
sentIndex sentText sentNum sentScore
1 Our method is the first that combines the following features: solving for both the 3D deformation and the camera ’s focal length, involving only local analytical solutions (there is no numerical optimization), being robust to mismatches, handling general surfaces and running extremely fast. [sent-2, score-0.99]
2 First, an ‘uncalibrated’ 3D deformation is computed thanks to a novel piecewise weak-perspective projection model. [sent-4, score-0.28]
3 Second, the camera’s focal length is estimated and enables upgrading the 3D deformation to metric. [sent-5, score-0.864]
4 We use a variational framework, implemented using a smooth function basis and sampled local deformation models. [sent-6, score-0.203]
5 The only degeneracy which we easily detect– for focal length estimation is a flat and fronto-parallel surface. [sent-7, score-0.725]
6 Experimental results on simulated and real datasets show that our method achieves a 3D shape accuracy – slightly below state of the art methods using a precalibrated or the true focal length, and a focal length accuracy slightly below static calibration methods. [sent-8, score-1.391]
7 Introduction 3D reconstruction from a single image and a template (a known 3D view of the surface) has been researched actively over the past decade. [sent-10, score-0.158]
8 Recovering the 3D deformation is equivalent to recovering the shape as seen in the input image. [sent-12, score-0.226]
9 An important instance of SfT is IsoSfT, where the 3D deformation is distance-preserving, in other words, an isometry. [sent-14, score-0.17]
10 We are here interested in C-IsoSfT, the IsoSfT problem which takes an uncalibrated image as input and includes camera calibration as an unknown. [sent-17, score-0.306]
11 We give a general framework and a detailled solution to the most important practical case where all camera parameters are known (the principal point, aspect ratio and skew) but the focal length. [sent-18, score-0.593]
12 For most applications of SfT, being able to estimate the focal length online is the most important case since it allows one to zoom in and out while filming the deformable surface. [sent-19, score-0.745]
13 More specifically, we contribute with the first robust analytical solution to recover 3D shape and the camera’s focal length. [sent-20, score-0.734]
14 We implemented our theory using putative keypoint correspondences as inputs. [sent-21, score-0.145]
15 Our implementation discards erroneous correspondences and is entirely analytical in that it does not involve numerical optimization. [sent-22, score-0.238]
16 Our analytical solution is based on a variational problem formulation with general template formulation. [sent-25, score-0.38]
17 This allows us to derive an operator which locally maps an image warp to an uncalibrated solution to 3D shape. [sent-27, score-0.272]
18 In a second step, we robustly solve for the focal length and upgrade 3D shape globally. [sent-28, score-0.75]
19 Both steps involve analytical solutions and are extremely fast to compute. [sent-29, score-0.245]
20 State of the Art Existing SfT methods can be broadly classified into three categories: (C1) analytical solutions, (C2) convex optimization and (C3) nonconvex optimization. [sent-54, score-0.173]
21 None of [2, 4] copes with mismatches, and none lends itself to camera calibration (perspective projection does not allow one to factor out the focal length, and thus to compute an uncalibrated solution). [sent-58, score-0.864]
22 The most successful relaxation is the so-called inextensibility, which upper bounds the Euclidean distance between a pair of points by its true geodesic distance computed from the template [8]. [sent-62, score-0.155]
23 Methods in (C3) estimate a quasi-isometric deformation, which is a nonconvex constraint, while minimizing the reprojection error [3]. [sent-68, score-0.231]
24 Finally, a recent paper has also shown that the focal length could be calibrated in SfT [1]. [sent-70, score-0.737]
25 The key idea is to sample a set of admissible focal lengths, solve SfT for each of them and keep the one minimizing some consistency measure. [sent-71, score-0.501]
26 This is achieved by first solving the problem locally to get an initial uncalibrated shape and then estimating the focal length. [sent-73, score-0.749]
27 Each formulation has a data constraint called the reprojection constraint and a prior called the deformation constraint. [sent-78, score-0.448]
28 Modeling and 3D Formulation In the SfT problem, one has a 3D shape template R ⊂ R3 wnh tihceh S may rboeb represented as a parametric msuprlfaatcee Rwi t⊂h an embedding ζ ∈ C2 (Ω, R3) from a parameterization space bΩe d⊂i Rg2 ζ. [sent-85, score-0.267]
29 ∈Giv Cen one image of the deformed 3D shape SΩ, ⊂the Runknowns are (i) the 3D deformation Ψ ∈ Csh2a p(Re, SR,3 )th etha utn brings Rs a rtoe S (i,) a tnhde ( 3iiD) th deef camera projection( Rfu,nRctio)n th hΠat ∈ b rPin. [sent-86, score-0.351]
30 Oeruar practical solution estimates solely the focal length which is the most important intrinsic of the pinhole camera. [sent-89, score-0.759]
31 We model them as an image warp η ∈ C2 (Ω, R2), though they may eall tsohe e bme represented by keypoint matches [9]. [sent-92, score-0.157]
32 The reprojection constraint (1) is obvious by construction (see figure 1). [sent-96, score-0.187]
33 The deformation constraint (2) means that the Jacobian matrix of Ψ in the tangent plane at any point of R has to be a columntohrteh toannogremntal mlaantreix a. [sent-97, score-0.252]
34 t Tanhyis mouinstt ohfo lRd f hoars sΨ t oto b represent an isometric deformation of R. [sent-98, score-0.306]
35 The template shape may have an arbitrary topology and parameterization. [sent-109, score-0.184]
36 Tη = ϕ = Π ◦T ζ ϕ ( rde pfro rjmeact io n ) ( 45) We observe that the result does not depend on the actual template’s embedding but on its metric tensor only. [sent-112, score-0.21]
37 The reprojection constraint (4) is obtained by substituting the definition (3) of ϕ in the reprojection constraint (1). [sent-114, score-0.374]
38 The deformation constraint (5) is obtained by differentiating the definition (3) of ϕ, giving ∇ϕ = (∇Ψ ◦ ζ)∇ζ, and multiplying it by its transpose to ∇η adj (T ζ)∇η? [sent-115, score-0.404]
39 , where adj is the adjugate matrix (the transpose of the cofactor matrix, adj (A) = det(A)A−1). [sent-125, score-0.282]
40 The reprojection constraints (4) and (6) are just the same. [sent-127, score-0.138]
41 As for the deformation constraint (7), we invoke Cholesky decomposition of the metric =def tensor T ζ. [sent-128, score-0.259]
42 996633 We differentiate the reprojection constraint (6) and multiply it to the right by Γ−1 : ∇η Γ−1 = (∇Π ◦ ϕ)∇ϕΓ−1. [sent-140, score-0.225]
43 We then multiply each side of the equation by its transpose to the right, yielding: (∇Π +(∇Π ◦ ◦ ϕ)(∇Π ◦ ϕ)? [sent-144, score-0.171]
44 This idse tt(heT general equation of C-IsoSfT w atith e general template parameterization. [sent-151, score-0.198]
45 Local Uncalibrated Solution We now derive a practical solution to uncalibrated reconstruction which can be computed locally. [sent-153, score-0.26]
46 We use a pinhole camera with unknown focal length f ∈ R+. [sent-154, score-0.775]
47 Our method first solves for α as an uncalibrated solution to C-IsoSfT, then calibrates f, and finally returns ϕ. [sent-168, score-0.23]
48 It allows us to solve for ϕX = ηαx and ϕY = ηαy while the deformation constraint (7) allows us to =def solve for α. [sent-174, score-0.219]
49 (8) Theorem 1 Equation (8) has a unique solution for α and at most two solutions for ν, each of them corresponding to a solution for the normal ξ, given by: α = ν± = ξ± = ? [sent-178, score-0.161]
50 We will use the normal as a clue to avoid local degeneracies when estimating the focal length. [sent-198, score-0.588]
51 (u v) and the deformation (coRnust)ra×in(tR R(2v)) )t o= fi dnealti(zRe )thRe de(riuva×tiovn). [sent-220, score-0.17]
52 Focal Length Calibration Our main result in this section is to compute the focal length analytically from the uncalibrated solution α. [sent-262, score-0.949]
53 Basic Equations Starting from the point-tangent formulation (proposition 2), we use the reprojection constraint (4) to establish: ϕ =α1? [sent-265, score-0.229]
54 ation in the deformation constraint (5) then leads to: f2T α−=α ? [sent-283, score-0.219]
55 4 l2e we nodb t∇aαin the following analytical ∇soαluTtio αn∇ foαr f:= 5. [sent-298, score-0.139]
56 (15) Criterion (14) is derived from the isometric deformation constraint. [sent-311, score-0.306]
57 It expresses the fact that at every point, the length of an infinitesimal step in any direction is preserved. [sent-312, score-0.222]
58 To be more general, we can prove that it preserves the length of every 2D curve lying on the template shape. [sent-313, score-0.321]
59 This is easily shown using the definition (3) of deformation constraint (2): ? [sent-322, score-0.219]
60 Degenerate cases arise when the focal length cannot be estimated uniquely from the data. [sent-332, score-0.694]
61 αT =his 0 was a kisn otow sna degenerate case in plane-based camera calibration [11]. [sent-337, score-0.192]
62 be greater than some minimal angle r ∈ R+ for a point to stably contribute to focal length negstliem ra t∈ioRn . [sent-340, score-0.694]
63 r oAmt l 5o%cal t osc 5a0le% sh, the support region Ω¯k,h ⊂ Ω is circular with diameter s and centred on the template keypoint qk. [sent-391, score-0.195]
64 aTrh we iltohca dil ascmaelete trrsa d aensdoff stability and deformation complexity: larger scale improves stability but increase sensitivity to high-frequency surface deformation. [sent-392, score-0.261]
65 We estimate the PWP scale factor ak,h and a focal length estimate fk,h for every warp in the pool. [sent-399, score-0.788]
66 This is obtained by fitting a TPS to= an ∇esαtimate of) αk,h at each keypoint in Ω¯k,h computed using the equation directly above. [sent-406, score-0.137]
67 We have a large number of candidate focal length estimates. [sent-425, score-0.694]
68 ρg We typically use 1% of the image size for g, and solve the problem by sampling focal length estimates. [sent-431, score-0.694]
69 For every putative feature match k, we select the largest local scale sh whose local focal length estimate fk,h agrees with the robust estimate by testing ρg fk,h). [sent-440, score-0.877]
70 Compared Methods and Measured Errors We compared 7 methods built on 3 base methods from the 3 categories outlined in §2: (C1) PWP (our proposed analytical tfergaomrieewso orukt)l,i n(Ced2) in nS §L2Z: ((Ca convex method [10]) and (C3) REF (iterative nonlinear refinement [3]). [sent-445, score-0.165]
71 For a method, the leading letter may be U or C: U means that the focal length is estimated by our method or refined and C means that the true focal length (for simulated data) or the focal obtained by static calibration (for real data) is used. [sent-446, score-1.999]
72 × We measured the average depth error in mm and the relative focal length error in %. [sent-448, score-0.793]
73 The default parameters were a focal length of 800 pixels, an image noise of 1. [sent-452, score-0.694]
74 We observe in the first column of graphs that the focal length error is always below 10% for the proposed analytical solution U-PWP. [sent-456, score-0.904]
75 It has a minimum for a focal length of around 600 pixels. [sent-458, score-0.694]
76 explained by the fact that for short focal lengths the PWP approximation tends to be less accurate, while large focal lengths tend to be ill-constrained since they cancel perspective. [sent-551, score-1.058]
77 Shape and focal length refinement by U-REF-SHAPE-F always improve on the results of U-PWP. [sent-552, score-0.72]
78 Moreover, it minimizes the reprojection error, which is physically meaningful. [sent-555, score-0.138]
79 We observe that the depth error of uncalibrated methods increases with the focal length while the error of calibrated methods is approximately steady or decreases. [sent-556, score-1.028]
80 All methods are sensitive to the focal length accuracy. [sent-558, score-0.694]
81 The first one is U-PWP, U-SLZ and U-REF-SHAPE, which use the focal length estimated by U-PWP. [sent-560, score-0.694]
82 They are outperformed by UREF-SHAPE-F which refines this focal length estimate, and forms the second group. [sent-561, score-0.694]
83 We detected 2923 SIFT keypoints in the template and 9472 in the input image, from which we obtained 2923 putative matches, filtered down to 617 after spatial consistency was enforced. [sent-571, score-0.237]
84 For the input image, the groundtruth focal length was 2727. [sent-572, score-0.765]
85 The histogram of local focal length estimates is shown in figure 3. [sent-574, score-0.694]
86 The focal length we estimated with U-PWP is 2668. [sent-575, score-0.694]
87 Once the focal length was robustly estimated we took the number of matches down to 612 by checking that their focal length es- timate was correct at one scale at least. [sent-581, score-1.482]
88 We observe that the isolated matches which were kept have a large local scale, while matches in dense keypoint areas may have a smaller scale, especially if the deformation is important. [sent-583, score-0.333]
89 FfTro uemcalfUleo-ncPalWgtlPehn gth5%(in%oftemplatesize)50% Histogram of local focal length estimates Selected local feature scales Figure 3. [sent-592, score-0.694]
90 The groundtruth shape and constant focal length were provided. [sent-597, score-0.821]
91 We observe that for U-PWP the focal length error is generally below 10%, while for U-REF-SHAPE-F it is generally below 5%. [sent-598, score-0.727]
92 U-REF-SHAPE-F is the best performing of the uncalibrated methods, reaching almost the same accuracy as the calibrated methods, except at those frames where the pose is degenerate. [sent-606, score-0.235]
93 996677 Groundtruth Proposed analytical solution (U-PWP) Color-coded depth er or in Ω0 12 0 m mmmmm Figure 4. [sent-607, score-0.21]
94 Ω 60 groundtruth U-PWP 90 groundtruth U-PWP Figure 5. [sent-610, score-0.142]
95 Conclusion We have proposed the first method which solves the Isometric Shape-from-Template problem analytically while recovering the camera’s focal length. [sent-613, score-0.526]
96 Our experimental results show that the method gives sensible estimates: the focal length error was less than 10% in most cases. [sent-615, score-0.727]
97 We showed that using the true focal length with our PWP model leads to a depth error comparable to state of the art algorithms, including nonlinear refinement ofthe reprojection error (we recall that the proposed method does not use numerical optimization). [sent-616, score-1.032]
98 The focal length estimate is sensitive to noise in near degenerate configurations. [sent-617, score-0.798]
99 Template-based isometric deformable 3D reconstruction with sampling-based focal length self-calibration. [sent-627, score-0.86]
100 On template-based reconstruction from a single view: Analytical solutions and proofs of well-posedness for developable, isometric and conformal surfaces. [sent-634, score-0.21]
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