cvpr cvpr2013 cvpr2013-75 knowledge-graph by maker-knowledge-mining
Source: pdf
Author: Zhe Wu, Ping Tan
Abstract: Under unknown directional lighting, the uncalibrated Lambertian photometric stereo algorithm recovers the shape of a smooth surface up to the generalized bas-relief (GBR) ambiguity. We resolve this ambiguity from the halfvector symmetry, which is observed in many isotropic materials. Under this symmetry, a 2D BRDF slice with low-rank structure can be obtained from an image, if the surface normals and light directions are correctly recovered. In general, this structure is destroyed by the GBR ambiguity. As a result, we can resolve the ambiguity by restoring this structure. We develop a simple algorithm of auto-calibration from separable homogeneous specular reflection of real images. Compared with previous methods, this method takes a holistic approach to exploiting reflectance symmetry and produces superior results.
Reference: text
sentIndex sentText sentNum sentScore
1 We resolve this ambiguity from the halfvector symmetry, which is observed in many isotropic materials. [sent-2, score-0.208]
2 Under this symmetry, a 2D BRDF slice with low-rank structure can be obtained from an image, if the surface normals and light directions are correctly recovered. [sent-3, score-0.67]
3 As a result, we can resolve the ambiguity by restoring this structure. [sent-5, score-0.198]
4 We develop a simple algorithm of auto-calibration from separable homogeneous specular reflection of real images. [sent-6, score-0.078]
5 Compared with previous methods, this method takes a holistic approach to exploiting reflectance symmetry and produces superior results. [sent-7, score-0.205]
6 It provides direct access to surface normals, which not only are crucial in photo-realistic rendering, but also help improve 3D reconstruction accuracy [12, 13]. [sent-10, score-0.093]
7 Under unknown directional lighting, it is well known that surface normals of a Lambertian object can only be determined up to a linear ambiguity [10]. [sent-14, score-0.403]
8 Later it is shown [26, 3] that this ambiguity can be reduced to the generalized bas-relief (GBR) ambiguity by enforcing the integrability constraint. [sent-15, score-0.217]
9 Reflectance of an opaque material is described by the bidirectional reflectance distribution function (BRDF), which is a function of the incoming and outgoing light directions in a local coordinate system. [sent-17, score-0.348]
10 isotropy and reciprocity, which provides additional information to resolve the GBR ambiguity. [sent-20, score-0.107]
11 As demonstrated in [20, 22], the GBR ambiguity can be solved using ‘isotropic pairs’ and ‘reciprocal pairs’ identified from one Figure 1. [sent-21, score-0.117]
12 The upper slice, which is obtained from ground truth normals and light direction, is constant along each row, while clearly this structure does not hold for the bottom slice estimated using GBR-distorted normals and light direction. [sent-26, score-0.827]
13 Surface points with specular spike [7, 6] or diffuse maxima [8] can also resolve the GBR ambiguity. [sent-28, score-0.217]
14 However, these methods all require carefully identified special surface points, which are easily affected by image noise. [sent-29, score-0.141]
15 We solve the GBR ambiguity by a holistic analysis of half-vectorsymmetry, which suggests the BRDF value stays unchanged when rotating the incoming and outgoing light directions as a fixed pair around their bisector. [sent-30, score-0.39]
16 This symmetry is closely related to the barycentric parameterization of isotropic BRDFs [19], and can be elegantly expressed in the halfway/difference parameterization [15]. [sent-31, score-0.217]
17 Given the correct surface normals and light directions, we can obtain a 2D BRDF slice from each image of a curved isotropic surface. [sent-32, score-0.668]
18 If the BRDF is half-vector symmetric, this 2D slice should form a special low-rank matrix when it is properly parameterized, as illustrated at the top of Figure 1. [sent-33, score-0.218]
19 However, as we have observed and will present in subsequent sections, such a structure is generally destroyed when normals and light directions are distorted by a GBR transformation, as shown at the bottom of Figure 1. [sent-34, score-0.452]
20 Restoring the special structure of 2D BRDF slices can resolve the GBR ambiguity. [sent-35, score-0.14]
21 111444999866 The contribution of this paper mainly lies in: 1) proposing half-vector symmetry as a novel cue to solve the GBR ambiguity; 2) proving that half-vector symmetry resolves the GBR ambiguity; 3) providing a simple auto-calibration algorithm based on a holistic analysis of this symmetry. [sent-37, score-0.209]
22 Related Work Woodham [25] proposed the first photometric stereo method with directional lighting information known a priori. [sent-40, score-0.173]
23 Hayakawa [10] proved that under unknown directional illumination, normals of a Lambertian surface can only be recovered up to a linear transformation. [sent-41, score-0.333]
24 This ambiguity is reduced to the GBR ambiguity by enforcing the integrability constraint [26, 3]. [sent-42, score-0.217]
25 Besides those utilizing interreflection [5] or special lighting configurations [28], the other methods can be divided into two categories. [sent-44, score-0.105]
26 The first category of methods resolves the ambiguity by analyzing reflectance properties. [sent-45, score-0.218]
27 In [7, 6], the authors assumed specular-spike reflectance and showed that the ambiguity is solved by detected specular spots in images. [sent-48, score-0.274]
28 [20, 22, 21] exploited isotropy and reciprocity to recover GBR parameters from carefully identified ‘isotropic pairs’ and ‘reciprocal pairs’ in a single image. [sent-51, score-0.097]
29 In the second category, priors on surface albedos are exploited. [sent-52, score-0.131]
30 [1] recovered the GBR parameters by assuming the true distribution of surface albedo has small entropy. [sent-54, score-0.156]
31 [17] identified surface points with the same albedo but different normals to resolve the ambiguity. [sent-56, score-0.383]
32 In a recent work [8], the authors assumed smoothly varying surface albedos in order to locate the ‘lambertian diffuse maxima’, which are then used in a robust estimation framework to estimate the GBR paramters. [sent-57, score-0.184]
33 Background Assume a Lambertian object is illuminated by distant light sources and imaged by a fixed orthographic camera. [sent-59, score-0.121]
34 If interreflection and shadow are ignored, the intensity ipf of a pixel p under the light source sf is given by ipf = ρpnp? [sent-60, score-0.261]
35 and normalized vector sf indreiscapteec ttivhee light intensity an? [sent-64, score-0.177]
36 For all P pixels in an image under F different light directions, Equation 2. [sent-67, score-0.121]
37 2) Each column of I an image under one of the light sources is and each row is the intensity profile of a pixel under alldifferent illumination conditions. [sent-71, score-0.168]
38 Columns of N are surface normals multiplied by their corresponding albedos, and columns of S are light directions scaled by their intensities. [sent-72, score-0.46]
39 For a smooth surface, normals and light directions can be recovered to an unknown GBR transformation G [3]: I N? [sent-73, score-0.448]
40 3, a norm⎠al n and a light direction s are distorted in the following way nˆ =? [sent-82, score-0.194]
41 Half-Vector Symmetry and GBR In this section, we will introduce half-vector symmetry of BRDFs and present the special low-rank matrix structure enforced by it. [sent-92, score-0.129]
42 After that, we will examine how the GBR ambiguity destroys the structure of this matrix. [sent-93, score-0.117]
43 Half-Vector Symmetry BRDF is a function of incoming and outgoing light directions (ωin, ωout) in a local coordinate system. [sent-96, score-0.249]
44 First of all, the half vector is defined as the bisector of lighting and viewing directions, i. [sent-101, score-0.146]
45 φd indicates the rotation angle of ωin and ωout as a pair around the half vector h. [sent-111, score-0.07]
46 One of the various symmetries widely observed in real-world materials is isotropy, which means BRDF values stay unchanged as the lighting and viewing directions are rotated as a fixed pair around the normal. [sent-113, score-0.271]
47 Many isotropic materials also satisfy the half-vector symmetry, which suggests that BRDF values are invariant with rotation of lighting and viewing directions around the half vector. [sent-115, score-0.299]
48 Halfway/difference parameterization of BRDF [15] depend on φd and is further reduced to a bivariate function f(θh, θd). [sent-117, score-0.152]
49 This kind of bivariate BRDF model is reported in previous works. [sent-118, score-0.126]
50 [19] studied several traditional parametric reflectance models and showed that they are bivariate. [sent-120, score-0.099]
51 [18] further used a biquadratic function to represent bivariate BRDFs. [sent-122, score-0.126]
52 Besides, the same bivariate BRDF representation has already been adopted in calibrated photometric stereo [2] and reflectometry [14]. [sent-125, score-0.217]
53 Structured 2D BRDF Slice The pixel intensity of a general isotropic surface is calculated as I f(θh, θd, φh) (n · s). [sent-128, score-0.177]
54 c camera aen ads shuommpotigoenneous surface reflectance, pixel intensities in an image are determined as I fθd (θh , φd) (n · s) , = (3. [sent-135, score-0.115]
55 2) where fθd (θh, φd) = f(θh, θd, φd) is a 2D slice of the original BRDF. [sent-136, score-0.19]
56 a sphere, when both the normals n and light direction s are known, we can estimate a 2D slice of the BRDF, namely fθd (θh, φd), based on Equation 3. [sent-139, score-0.526]
57 This BRDF slice can be arranged into a matrix form in the range θh ∈ [0, π2], φd ∈ [0, 2π] . [sent-141, score-0.19]
58 GBR-Distorted 2D BRDF Slice When surface normals and light direction are distorted by a GBR transformation as in Equation 2. [sent-148, score-0.521]
59 This observation motivates us to resolve the GBR ambiguity by restoring the low-rank structure of BRDF slices. [sent-152, score-0.218]
60 In the special case that the lighting and viewing directions coincide, the low-rank property is preserved by the classic bas-relief ambiguity, i. [sent-153, score-0.2]
61 A black ellipse corresponds to a row in the BRDF slice of Figure 1. [sent-159, score-0.285]
62 Top row shows the effect of a GBR transformation (μ, ν 0); bottom row sshhoowwss tthhee case ot fo a cal GasBsiRc tb raasn-srfeoliremf attraionnsf (oμrm,νat? [sent-160, score-0.094]
63 Before GBR transformation, the values on an ellipse are the same, while this is not true after GBR transformation. [sent-162, score-0.091]
64 In fact, for a general bivariate BRDF, we are able to prove the following proposition. [sent-164, score-0.126]
65 Any GBR transformation cannot simultaneously perserve the special low-rank structure of bivariate BRDF slices estimated from two images whose light directions are not coplanar with the viewing direction. [sent-166, score-0.532]
66 , light direction s, half vector h and every surface normal n can find their corresponding points on this plane. [sent-171, score-0.314]
67 Given the lighting and surface normal directions, a BRDF value can be estimated at each pixel of an image. [sent-175, score-0.181]
68 By encoding these BRDF values into colors and mapping them to the projective plane according to the normal at each pixel, we obtain a ‘BRDF map’ shown in Figure 3, where red indicates larger values. [sent-176, score-0.065]
69 First of all, as marked by the black ellipse in Figure 3, points with the same θh form an ellipse around h, with one of its symmetry axis being the line vs connecting v and s. [sent-177, score-0.263]
70 For a bivariate BRDF, the BRDF value should be constant along each ellipse, since θh is fixed for those points on the same ellipse and θd is fixed for all pixels in the same image. [sent-178, score-0.201]
71 These ellipses correspond to rows in the matrix representation of the 2D BRDF slice in Figure 1. [sent-179, score-0.218]
72 The GBR transformation moves normals and the light direction in different ways, as shown in Equation 2. [sent-181, score-0.41]
73 In fact, the transformed light direction will still lie on the line vs, according to the equation s 111445990088 s =? [sent-183, score-0.254]
74 eOtrna tnhseother hand, a GBR transformation will translate all normals (along with the BRDF map associated them) by a displacement (μ, ν) and scale them by λ. [sent-193, score-0.234]
75 Besides, BRDF values are also changed since they are estimated from pixel intensities and the GBR transformed shading ˆn ? [sent-194, score-0.156]
76 In general, the different motions of normals and the light direction will make the BRDF value change along the transformed ellipse(consists of points forming the same half angle with ˆh). [sent-197, score-0.469]
77 As a result, the low-rank structure of the 2D BRDF slice in Figure 1 will be destroyed. [sent-198, score-0.21]
78 The transformed BRDF map has varying values along the ellipse around hˆ. [sent-200, score-0.154]
79 n t hthei csl case, bthase- GrelBieRf transformation moves the light direction s = [sx, sy , sz]? [sent-208, score-0.23]
80 So the transformed BRDF value is no longer constant along the transformed ellipse. [sent-214, score-0.126]
81 Given multiple images taken under varying lightings and a fixed viewpoint, it is relatively easy to separate the diffuse and specular components using existing techniques [16, 23]. [sent-218, score-0.131]
82 From the diffuse images, we are able to recover surface normals and light directions up to a GBR transformation by the uncalibrated photometric stereo method [26]. [sent-219, score-0.675]
83 By assuming the specular BRDF is bivariate, it is guaranteed by our earlier discussion that surface normals are correctly recovered iff the low-rank structure in estimated specular BRDF slices is restored. [sent-220, score-0.529]
84 One natural idea of restoring the low-rank structure is to use the established TILT technique [27], which recovers a low-rank pattern via domain transformation. [sent-221, score-0.083]
85 In our case, however, both the position of each point (corresponding to a normal) on the 2D BRDF slice and its associated BRDF value are changed by a GBR transformation. [sent-222, score-0.205]
86 We formulate a simple optimization algorithm to estimate the GBR parameters given normals and light directions up to a GBR ambiguity, together with a set of specular images. [sent-224, score-0.445]
87 We first define an objective function to measure how well the estimated 2D BRDF slice satisfies the special ‘low-rank’ constraint, which in our case means each row is constant. [sent-225, score-0.269]
88 Firstly, with a real image of limited resolution, some entries of the 2D BRDF slice are missing because the corresponding normals are not observed in the image. [sent-229, score-0.37]
89 Thus we give higher weights to rows of a larger number of valid observations, since the variances calculated from those rows are more reliable. [sent-231, score-0.072]
90 the number of input specular images and we can estimate a 2D BRDF slice fi from each image. [sent-248, score-0.268]
91 The integer ki,θh is the number of valid observations in row θh of slice i. [sent-249, score-0.21]
92 One of the recovered BRDF slice(only the range θh ∈ Five materials from the database are [0◦ ,40◦] is shown) is also shown above the rendered sphere. [sent-264, score-0.071]
93 Its intensity is smaller than that of m1 and their transformed shading values are equal: m1? [sent-275, score-0.131]
94 On the other hand, these two points have the same transformed half angle and thus they belong to the same row of the GBR distorted 2D BRDF slice. [sent-279, score-0.191]
95 Even if (μ, ν) is parallel to vs, the GBR transformation will still break the low-rank structure of the BRDF slice from another image whose lighting direction is not coplanar with v and s. [sent-281, score-0.371]
96 Since the normals nv = v and ns = s have the same half angle: θh = θd, they should have the same BRDF value coIs((n2vθd))=cIo(sn(0s)). [sent-287, score-0.283]
97 1) A similar relation holds for the GBR transformed case coIˆs((n2vθˆ)d)=cIˆo(sn(0 sˆ)) Iˆ (A. [sent-289, score-0.063]
98 2) s, θˆd where is the transformed intensity map, n sˆ = and is the transformed half angle for nv and n sˆ. [sent-290, score-0.262]
99 On the other hand, n sˆ is transformed from nx, which lies even further from v: = I(nx). [sent-295, score-0.063]
100 Iˆ(n sˆ) Iˆ(n sˆ) away from the fixed viewing direction v and lighting direction s: I(nx) < I(ns). [sent-297, score-0.16]
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