cvpr cvpr2013 cvpr2013-286 knowledge-graph by maker-knowledge-mining
Source: pdf
Author: Miaomiao Liu, Richard Hartley, Mathieu Salzmann
Abstract: This paper tackles the problem of reconstructing the shape of a smooth mirror surface from a single image. In particular, we consider the case where the camera is observing the reflection of a static reference target in the unknown mirror. We first study the reconstruction problem given dense correspondences between 3D points on the reference target and image locations. In such conditions, our differential geometry analysis provides a theoretical proof that the shape of the mirror surface can be uniquely recovered if the pose of the reference target is known. We then relax our assumptions by considering the case where only sparse correspondences are available. In this scenario, we formulate reconstruction as an optimization problem, which can be solved using a nonlinear least-squares method. We demonstrate the effectiveness of our method on both synthetic and real images.
Reference: text
sentIndex sentText sentNum sentScore
1 au iu Abstract This paper tackles the problem of reconstructing the shape of a smooth mirror surface from a single image. [sent-8, score-0.933]
2 In particular, we consider the case where the camera is observing the reflection of a static reference target in the unknown mirror. [sent-9, score-0.379]
3 We first study the reconstruction problem given dense correspondences between 3D points on the reference target and image locations. [sent-10, score-0.392]
4 In such conditions, our differential geometry analysis provides a theoretical proof that the shape of the mirror surface can be uniquely recovered if the pose of the reference target is known. [sent-11, score-1.296]
5 Introduction In this paper, we tackle the problem of mirror surface reconstruction from a single image. [sent-16, score-0.873]
6 Traditional 3D recon- struction methods typically perform poorly on mirror objects, since the information captured by the camera comes from the objects’ surroundings rather than from the objects themselves. [sent-17, score-0.616]
7 Methods specifically designed to handle mirror surfaces have been introduced, but usually exploit motion and thus do not apply to the single image scenario [16, 8, 13, 20]. [sent-18, score-0.644]
8 However, existing approaches do not offer theoretical guarantees of the uniqueness of the reconstructed surface. [sent-20, score-0.277]
9 Here, we introduce an approach to reconstructing a mirror surface from a single image with a provably unique solution. [sent-21, score-0.887]
10 Furthermore, we assume that reflection correspondences between 3D points on the reference plane and 2D image locations are given. [sent-25, score-0.572]
11 With dense correspondences, a differential geometry analysis reveals that, for a smooth mirror surface without inter-reflections, reconstruction reduces to solving an initial value problem (IVP) with two partial differential equations (PDEs). [sent-26, score-1.22]
12 We derive a theoretical proof of uniqueness of the solution to this IVP. [sent-27, score-0.392]
13 Furthermore, studying the order of integration of the two PDEs yields a generally unique solution for the starting point of the IVP. [sent-28, score-0.231]
14 This therefore implies uniqueness of the mirror surface reconstruction. [sent-29, score-0.984]
15 To address the more realistic scenario where only sparse reflection correspondences are available, we parametrize the depths of points on the mirror surface as a uniform cubic B-spline. [sent-30, score-1.248]
16 We then formulate reconstruction as an optimization problem that minimizes the 3D error between the points on the reference plane and the image correspondences backprojected to the reference plane via the mirror. [sent-31, score-0.709]
17 In summary, the key contributions of this paper are • • •• A solution to the problem of reconstructing a smooth mAir sroolru tsiournfa tcoe t fhreo mpr a single image given ndegn ase s correspondences between the image and a reference plane with known pose. [sent-32, score-0.505]
18 We demonstrate the effectiveness of our reconstruction method from both dense and sparse correspondences on synthetic and real images, such as those depicted in Fig. [sent-36, score-0.327]
19 Related Work Most existing methods that tackle mirror surface reconstruction exploit temporal information, such as the motion of the camera [13, 20], or that of the environment [8, 16]. [sent-39, score-0.914]
20 Within this class of methods, shape from specular flow has become a popular approach [14, 1, 5, 19]. [sent-40, score-0.189]
21 As an alternative, shape recovery can be performed by exploiting multiple reference planes with known pose relative to the camera. [sent-41, score-0.255]
22 This can be achieved either by utilizing multiple views of the object with a reference plane fixed relative to the camera [3, 12, 2], or with a static camera, but a moving reference plane [4, 11]. [sent-42, score-0.521]
23 In [10], it was shown that the surface shape can be recovered from a single viewpoint when two 3D reference points on the light path are known, which is similar to using a moving reference plane. [sent-43, score-0.56]
24 In this paper, we consider the problem of reconstructing a smooth mirror surface from a single image, and therefore cannot exploit motion. [sent-44, score-0.878]
25 In [9], a method to recover the shape ofthe human cornea from a fixed camera and a static reference plane with known pose was introduced. [sent-46, score-0.372]
26 However, the approach in [9] requires the 3D location of one point on the surface to be known. [sent-48, score-0.253]
27 Here, we provide a theoretical proof of uniqueness of the mirror shape that does not require knowing the position of any surface point. [sent-50, score-1.174]
28 A differential geometry analysis of surface patches was proposed in [17]. [sent-51, score-0.34]
29 While the reconstruction in [17] was limited to local patches, it was extended to modeling a global surface shape in [15]. [sent-52, score-0.353]
30 While we also base our analysis on differential geometry, our formulation naturally extends to the entire mirror surface without requiring stitching patches together. [sent-54, score-0.888]
31 In [18], the shape of the surface was recovered by iteratively estimating the normal at a point from its depth, and the depth of a neighboring point from this normal. [sent-55, score-0.439]
32 Our analytical formulation allows us to establish a theoretical proof of uniqueness of the solution of both the IVP that corresponds to the iterative procedure described above and the starting depth required to solve this IVP. [sent-58, score-0.522]
33 era centred at O is observing a mirror surface P that reflects a reference plane Q in the image I. [sent-67, score-1.064]
34 A point m on Q is reflected to the image point v on I the 3D mirror point p on P. [sent-68, score-0.724]
35 We refer via to m and v as reflection correspondences. [sent-69, score-0.182]
36 R and T denote the pose of the reference plane w. [sent-72, score-0.276]
37 We focus on a rectangular region of interest (ROI) Ix Iy 2 in which the mirror P is visible, non-tangentially. [sent-95, score-0.575]
38 (x m,ye)et ∈s t Ihe ×m iIrror simply (not tangentially) at a point p(x, y) = s(x, y)v, where s(x, y) is referred to as the depth of the mirror at this point. [sent-97, score-0.671]
39 The function s(x, y) therefore determines the shape of the mirror, and finding this function is equivalent to finding the shape of the part of the mirror that lies within the ROI. [sent-98, score-0.685]
40 Differential Geometry Analysis In this section, we present our approach to mirror surface reconstruction given dense reflection correspondences, as well as our proof of uniqueness of a solution. [sent-101, score-1.362]
41 Our analysis of the reconstruction problem relies on the normal n to the mirror at p. [sent-102, score-0.689]
42 Let m be a point on the reference plane 2Ix and Iy are closed sets. [sent-104, score-0.274]
43 Based on via the geometry of reflection, n bisects the angle between the incident ray i = v/| |v| | and the reflected ray l = (m sinvc)id/e| |nmt r−aysv i| |= . [sent-106, score-0.234]
44 This will certainly be true if the mirror is smooth, and with no occluding contours. [sent-117, score-0.575]
45 Under this assumption, the normal to the surface can also be expressed as n = ∂p/∂x ∂p/∂y. [sent-118, score-0.254]
46 (4) Suppose that the reflection correspondence m is known for each point in the ROI (either by dense matching, or modeling) and can therefore be written as a function m(x, y). [sent-138, score-0.254]
47 Assuming that m(x, y) is known for points in Ix Iy, both fx and fy are ultimately functions of x, y and s only. [sent-144, score-0.189]
48 Uniqueness Results ×× The goal of this section is to state certain uniqueness results for the mirror shape, based on the formulation of the problem as the IVP (5). [sent-148, score-0.765]
49 We therefore concentrate on the uniqueness of the solution, which would imply that the true shape of the mirror is the only solution to the IVP. [sent-153, score-0.887]
50 Suppose that for each point (x, y) in a region of interest Ix Iy in an image, the corresponding ray meets a mirror non-tangentially. [sent-156, score-0.689]
51 2, this result will be seen to follow from a standard uniqueness result in Ordinary Differential Equations (ODEs), namely the Picard Lindel¨ of Theorem [6]. [sent-159, score-0.19]
52 However, studying the order of integration of the PDEs in the IVP reveals that, for generic × mirror shapes, there is a single valid in Section 4. [sent-163, score-0.652]
53 Existence and uniqueness of solutions to ODEs are much simpler problems than for PDEs. [sent-172, score-0.222]
54 In particular, the Picard Lindel¨ of Theorem gives the required existence and uniqueness conditions. [sent-173, score-0.19]
55 nis( nonzero, by the assumption that rays meet the mirror non-tangentially, | ? [sent-209, score-0.611]
56 The previous discussion shows that, by holding y0 fixed, one can propagate in the x direction, to find a unique solution s(x, y0) that satisfies Eq. [sent-226, score-0.186]
57 Therefore, the entire visible mirror surface can be reconstructed uniquely given the depth of one starting point. [sent-232, score-0.958]
58 Computing a Starting Depth Although guaranteed to be unique (if it exists), the solution derived in the previous section relies on knowing the depth of one point. [sent-246, score-0.17]
59 In this section, we show that not all depths s0 give a valid solution, and more specifically, that the valid depth of a starting point can be obtained uniquely. [sent-248, score-0.195]
60 , the mirror is a C2 continuous surface), and if s(x, y) satisfies the PDEs in Eqs. [sent-263, score-0.653]
61 Therefore, for generic surfaces, there exists a single valid s0, which, combined with the proof of uniqueness of the solution to the IVP, implies that the mirror surface can be reconstructed uniquely. [sent-316, score-1.217]
62 To conclude, this gives us two ways of computing the shape of the mirror: We can solve the polynomial equation at one image point, and then solve the two PDEs of the IVP (5) sequentially, or we can solve the polynomial equation at each image point. [sent-317, score-0.271]
63 Note that both methods require dense reflection correspondences, both for integration purposes and to compute accurate partial derivatives of the reflection correspondences. [sent-318, score-0.508]
64 Shape Recovery from Sparse Measurements While the previous section describes two possible ways of reconstructing the mirror surface, the solution to the polynomial equation, as well as the integration of the PDEs strongly rely on dense and noise-free correspondences. [sent-320, score-0.855]
65 In a more realistic scenario, reflection correspondences will be sparse and noisy. [sent-321, score-0.325]
66 We first present our surface parametrization, and then describe our reconstruction framework. [sent-323, score-0.298]
67 Surface Representation In the formulation of Section 4, we directly modeled the surface in terms of the depth of image points. [sent-326, score-0.281]
68 Since our proof of uniqueness relies on the surface being C2 continu- ×× ous, we can make use of other parametrizations that encode such a smoothness. [sent-327, score-0.488]
69 This implicitly satisfies the geometric constraint that surface points lie on their respective visual rays. [sent-329, score-0.333]
70 Therefore, a point on the surface p =1 vwecct(oxr, y, u1n)k? [sent-342, score-0.253]
71 Shape Recovery as an Optimization Problem Given our parametric representation of the mirror surface, shape recovery reduces to estimating the depth of the control points c. [sent-355, score-0.808]
72 Given a set of sparse reflection correspondences between image points {v1, v2 , . [sent-356, score-0.361]
73 As mentioned earlier, a 3D point on the mirror corresponding to image point (xi, yi) can be expressed as pi = wic(xi, yi, 1)? [sent-364, score-0.643]
74 The pose of the reference plane relative to the camera is determined by the ro- tation matrix R and the translation vector T. [sent-376, score-0.347]
75 er parameter c of the surface by solving a non-linear least-squares problem that minimizes the error between our backprojections to the plane and the real points on the plane. [sent-399, score-0.405]
76 (17) Note that Section 4 shows that only the correct mirror surface corresponds to reflected rays that intersect the reference plane at the observed points. [sent-407, score-1.117]
77 3: We initialize the unknown mirror P as a plane, and seek for its pose such that the camera can best see the reflection of the reference plane Q. [sent-414, score-1.105]
78 convergence, of the reflection rays with a convex, resp. [sent-428, score-0.218]
79 concave, mirror, Tv defines an upper bound for a convex mirror and a lower bound for a concave one. [sent-429, score-0.575]
80 within the bounds [0 Tv], or [Tv 3Tv], and take the mirror shape P that gives the smallest energy value in Eq. [sent-431, score-0.63]
81 Synthetic Data For our synthetic experiments, we used a UCBS, an ellipsoid and a sphere as mirror surfaces. [sent-437, score-0.733]
82 The reflection correspondences were obtained by backprojecting all image pixels to the reference plane. [sent-438, score-0.421]
83 More than 4M reflection correspondences were used in our synthetic experiments. [sent-439, score-0.334]
84 Recall that we have 3 possible ways of reconstructing the surface: With dense correspondences, we can either solve a degree 2 polynomial equation for each pixel, or solve this equation for a single pixel and solve the IVP (5). [sent-440, score-0.231]
85 Red dots denote the surface reconstructed by solving polynomial equations, Cyan dots the surface obtained by solving the PDEs in Order A, and Magenta dots in Order B. [sent-443, score-0.761]
86 4 compares with ground truth the shapes obtained by solving either a polynomial equation at each pixel independently, or the IVP. [sent-457, score-0.172]
87 5 depicts the reconstruction and self-consistency errors as a function ofthe percentage of correspondences used for reconstruction for the approach by solving IVP. [sent-468, score-0.307]
88 The reconstruction error is computed as the mean 3D point-to-point distance between the reconstructed shape and ground-truth. [sent-469, score-0.165]
89 In the case of the ellipsoid, representing the surface as a UCBS introduces approximation errors, depending on the number of control points used. [sent-479, score-0.296]
90 Experiments on Real Data To evaluate our approach on real surfaces, we used the stainless steel spoon and gravy boat depicted in Fig. [sent-493, score-0.223]
91 This yielded 1919 and 2029 correspondences for the spoon and gravy boat, respectively. [sent-496, score-0.267]
92 The pose of the reference plane relative to the ×× camera was calibrated with the Matlab Calibration Toolbox. [sent-497, score-0.317]
93 The spoon and gravy boat were approximated by a UCBS with 20 20 and 10 10 control points, respectively. [sent-499, score-0.235]
94 ri35n0o4f the spoon and gravy boat depicted in Fig. [sent-534, score-0.223]
95 theoretical proof of uniqueness of the solution in the presence of dense reflection correspondences. [sent-538, score-0.612]
96 Furthermore, we have introduced an optimization framework to reconstruct the mirror surface when only sparse correspondences are available. [sent-539, score-0.937]
97 Currently, our approach requires the pose of the reference plane to be known. [sent-540, score-0.276]
98 Reconstructing curved surfaces from specular reflection patterns using spline surface fitting of normals. [sent-611, score-0.576]
99 A theory of refractive and specular 3d shape by light-path triangulation. [sent-617, score-0.189]
100 Dense mirroring surface recovery from 1d homographies and sparse correspondences. [sent-650, score-0.314]
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