cvpr cvpr2013 cvpr2013-466 knowledge-graph by maker-knowledge-mining
Source: pdf
Author: Brian Potetz, Mohammadreza Hajiarbabi
Abstract: For problems over continuous random variables, MRFs with large cliques pose a challenge in probabilistic inference. Difficulties in performing optimization efficiently have limited the probabilistic models explored in computer vision and other fields. One inference technique that handles large cliques well is Expectation Propagation. EP offers run times independent of clique size, which instead depend only on the rank, or intrinsic dimensionality, of potentials. This property would be highly advantageous in computer vision. Unfortunately, for grid-shaped models common in vision, traditional Gaussian EP requires quadratic space and cubic time in the number of pixels. Here, we propose a variation of EP that exploits regularities in natural scene statistics to achieve run times that are linear in both number of pixels and clique size. We test these methods on shape from shading, and we demonstrate strong performance not only for Lambertian surfaces, but also on arbitrary surface reflectance and lighting arrangements, which requires highly non-Gaussian potentials. Finally, we use large, non-local cliques to exploit cast shadow, which is traditionally ignored in shape from shading.
Reference: text
sentIndex sentText sentNum sentScore
1 com Abstract For problems over continuous random variables, MRFs with large cliques pose a challenge in probabilistic inference. [sent-3, score-0.19]
2 One inference technique that handles large cliques well is Expectation Propagation. [sent-5, score-0.24]
3 EP offers run times independent of clique size, which instead depend only on the rank, or intrinsic dimensionality, of potentials. [sent-6, score-0.311]
4 Here, we propose a variation of EP that exploits regularities in natural scene statistics to achieve run times that are linear in both number of pixels and clique size. [sent-9, score-0.395]
5 We test these methods on shape from shading, and we demonstrate strong performance not only for Lambertian surfaces, but also on arbitrary surface reflectance and lighting arrangements, which requires highly non-Gaussian potentials. [sent-10, score-0.335]
6 Finally, we use large, non-local cliques to exploit cast shadow, which is traditionally ignored in shape from shading. [sent-11, score-0.225]
7 The run time of BP is exponential in the clique size C: each potential requires O(CMC) operations, where Msiz eis C th:e e ncuhm pboteern otifa slt raeteqsu freors eOac(Ch Mvariable. [sent-16, score-0.425]
8 Others have advanced methods of inference which can be applied to probabilistic models over discrete variables with large cliques[10, 24], or large numbers of small cliques [12]. [sent-21, score-0.311]
9 Nevertheless, efficient inference for large cliques remains limited to certain forms of potentials, and remains quadratic or worse in clique size. [sent-23, score-0.51]
10 EP works by approximating a factorized distribution with a simpler, tractable distribution from a family of distributions whose moments can be readily computed. [sent-26, score-0.221]
11 When the approximating family is a product of independent univariate marginals, EP is equivalent to BP [14]. [sent-27, score-0.225]
12 The principal difference between BP and Gaussian EP can thus be summarized by a trade-off in their respective approximating families: BP favors flexible non-Gaussian marginals, while Gaussian EP favors a flexible covariance structure. [sent-30, score-0.246]
13 For example, tree-shaped graphical models can have strong covariance structure, and so the approximating family of BP may be very poor for such models. [sent-32, score-0.351]
14 In a complex graph, however, accurate covariance models can improve performance because updates to one variable immediately affect distant variables known to be correlated. [sent-35, score-0.207]
15 Its running time is independent of clique size, and instead depends polynomially on the rank (or intrinsic dimensionality) of each potential (defined below). [sent-45, score-0.361]
16 In this paper, we propose an efficient inference method that retains the computational advantages of EP, reducing run time and space requirements to linear in the number of pixels, while remaining linear in clique size. [sent-46, score-0.389]
17 This is achieved by limiting EP to efficient families of covariance structures chosen based on the statistics of natural scenes. [sent-47, score-0.223]
18 We then test this approach on a problem with highly non-Gaussian potentials: non-Lambertian shape from shading (SfS). [sent-48, score-0.192]
19 Finally, we use the method to efficiently perform inference over large cliques produced by cast shadows and by global spatial priors. [sent-50, score-0.318]
20 ed and the approximating exponential family is a product of independent univariate discrete distributions, then EP is equivalent to classical belief propagation (BP) [14]. [sent-95, score-0.38]
21 When the elements of the vector are real-valued, the approximating exponential family is nearly always chosen to be Gaussian: G(? [sent-97, score-0.209]
22 Regardless of t)h tei mraen,k r otfh eera tchha potential, the covariance matrix of the posterior S remains full-rank, and must be stored as a D D matrix. [sent-129, score-0.226]
23 If the graphical model underlying equation 1 is sparsely connected, it may alleviate memory requirements to store the inverse covariance matrix S−1 rather than S. [sent-134, score-0.222]
24 Whitened EP For many problems of computer vision, both the number of variables D and the number of potentials N grow linearly with the number of pixels. [sent-152, score-0.213]
25 One desirable property of EP, however, is that the run time is independent of the size of the cliques; only the rank of the potentials affects the run time. [sent-158, score-0.324]
26 Low-rank potentials of large clique size have a wide array of promising applications in computer vision [17, 10]. [sent-159, score-0.38]
27 Also, in a multi-scale setting, potentials at coarse scales require large cliques, but rank remains the same at any scale. [sent-161, score-0.188]
28 Difficulty in performing inference over large-clique potentials has limited the probabilistic models used in computer vision. [sent-163, score-0.243]
29 In this section, we propose an algorithm that achieves both of these goals: run time that is linear in the number of pixels and in clique size. [sent-164, score-0.343]
30 Expectation propagation can be made more efficient by limiting the forms of covariance structure expressible by S. [sent-169, score-0.313]
31 In order for moment matching to correspond to minimizing KL-divergence, the approximating family P˜ must be an exponential family distribution (Eq. [sent-170, score-0.321]
32 expressible covariance structure must include the covariance matrix for natural scenes. [sent-178, score-0.44]
33 Additionally, since scene statistics are typically stationary, we prefer that local covariance structure achievable in one region of an image is also achievable in any region. [sent-179, score-0.222]
34 Let S denote the covariance matrix for natural scenes. [sent-183, score-0.191]
35 However, note that Vi is only non-zero in C columns, where C is the clique size ofthe potential. [sent-201, score-0.243]
36 Thus, each update equation can be 111666777644 performed in time O(K3 + K2C), giving the whitened EP technique a nto ttiaml run (tiKme of O(NK2C) per iteration. [sent-209, score-0.42]
37 Thus, the proposal that Gaussian EP might still work effectively if S−1 was constrained to WD−S1W is equivalent to the proposal that BcoPn might wd toork W effectively sif e messages were approximated by Gaussians as long as the variables were whitened beforehand to reduce correlation. [sent-211, score-0.446]
38 In order to achieve linear time EP with respect to image size, we are not limited solely to diagonal covariance structure in whitened image space. [sent-212, score-0.558]
39 gT Whe covariance structure of the posterior distribution may differ from that of the prior. [sent-222, score-0.194]
40 Shape from Shading Whitened EP permits inference over images in linear time with respect to both pixels and clique size. [sent-226, score-0.353]
41 To achieve this, it constrains the approximating distribution to be Gaussian with a covariance matrix WD−S1W for some diagonal DsiaSn. [sent-227, score-0.288]
42 In particular, we are interested in whether Gaussian message approximation will be effective when the potentials φi are highly non-Gaussian. [sent-229, score-0.177]
43 One highly non-Gaussian problem in computer vision is shape from shading (SfS). [sent-230, score-0.192]
44 The goal of SfS is to estimate 3D shape from a single image, under the assumption that albedo is uniform, lighting originates from a single point from a known direction, and the surface reflectance function is both uniform and known. [sent-231, score-0.356]
45 In recent years, several methods have been developed that solve the classical SfS problem well as long as surface reflectance R is assumed to be Lambertian [19, 17, 6, 3, 7]. [sent-233, score-0.211]
46 Our hope is that whitened EP, by permitting efficient inference over large cliques, will enable new MRF models capable of tackling generalized depth inference problems. [sent-237, score-0.56]
47 In this section, we demonstrate how whitened EP handles several of these issues. [sent-238, score-0.352]
48 MRF Data Likelihood In the past, MRF models for SfS have inferred surface normals rather than depth [17]. [sent-239, score-0.243]
49 m Haolw toev beer, c nonotall surface normal maps correspond to a valid surface z. [sent-242, score-0.2]
50 Methods that infer surface normals must include additional MRF potentials that encourage p and q to obey this relationship. [sent-244, score-0.356]
51 Enforcing integrability is often the largest computational bottleneck of probabilistic inference because it requires a clique size of at least four variables [17]. [sent-245, score-0.439]
52 This has been difficult to do using belief propagation because it requires a clique size for φR of at least three. [sent-250, score-0.351]
53 Belief propagation is exponential in clique size, and φR is not eligible for computational shortcuts such as the linear constraint node simplification. [sent-251, score-0.394]
54 In contrast, whitened EP can either infer surface normals or infer depth directly, and the two objectives require similar run times. [sent-253, score-0.706]
55 To infer depth, whitened EP operates over a MRF whose variable nodes correspond to the whitened surface depth. [sent-254, score-0.851]
56 Let zw (x, y) = Wz refer to the whitened surface depth, where W(x ,isy t)he = li Wnezar whitening etr wanhsitfeornmed. [sent-255, score-0.554]
57 Then, for each pixel (x, y), we can enforce that the surface normal at that point is consistent with the known pixel intensity i(x, y) with the potential φR(vp · z, vq · z | i), where vp and vq are the derivatives of inverse whitening fii)l,te wr centered at point (x, y). [sent-257, score-0.435]
58 The clique size of this potential is the size of the support of vp and vq, and the rank of the potential is two. [sent-258, score-0.471]
59 Because whitened EP is linear in both clique size and rank, inference over this potential is efficient. [sent-259, score-0.74]
60 Alternatively, if whitened EP is used to infer surface normals p and q, the clique size would be twice the support of the inverse whitened filter. [sent-260, score-1.167]
61 In our experiments, we will use whitened EP to infer depth directly. [sent-266, score-0.451]
62 We use a Laplace distribution for φR to penalize depth maps z that are not consistent with the known pixel intensity: φR(vp ·z, vq ·z|i) = φR(p, q|i) = e−|R(p,q)−i)|/b (15) where R(p, q) = iis the reflectance map given by the known surface BRDF and lighting. [sent-267, score-0.304]
63 MRF Spatial Prior The SfS problem is highly ambiguous: even when lighting direction and albedo are known, one image is consistent with large families of possible 3D surfaces which all render identically [6]. [sent-269, score-0.208]
64 × Methods that allow spatial priors with larger cliques have produced substantial performance gains [24, 17]. [sent-272, score-0.208]
65 However, these methods are limited in the size and form of cliques achievable by the method. [sent-273, score-0.191]
66 First, inference is linear in clique size, which could allow the use of large clique spatial priors such as Fields of Experts [20], which consists of 5 5 potentials of rank one. [sent-275, score-0.798]
67 sian potential requires no computational cost, regardless of rank or clique size. [sent-277, score-0.361]
68 In all following SfS experiments, we use a spatial prior that is implemented as a Gaussian with zero mean and covariance matrix equal to the covariance structure of natural range images S. [sent-284, score-0.355]
69 This prior has full rank and clique size rDan, making eits impractical tioor implement using dB cP. [sent-285, score-0.294]
70 rIenq audirdeisti oonnl tyo O Oit(s1 e)ff i nci eeancth run taitmioen, unifying many pairwise potentials into one large potential increases the fidelity of the Bethe approximation implicit in message passing algorithms [25]. [sent-287, score-0.272]
71 Finally, this approach allows us to match the full covariance structure of natural scenes, including distant non-local covariances. [sent-288, score-0.191]
72 As mentioned earlier, a pairwise-connected MRF produces a restricted covariance structure whose inverse matrix S−1 only contains elements along three unique diagonals [15]. [sent-289, score-0.197]
73 Gaussian potentials permit EP to capture the second order statistics of all higher-order derivatives and any other linear feature. [sent-292, score-0.193]
74 Results of whitened EP under several reflectances and lighting conditions. [sent-314, score-0.467]
75 In all experiments, whitened EP was run for 10 iterations, which is typically near to convergence. [sent-330, score-0.42]
76 was found numerically at each potential by sampling over a 26 26 discrete reflectance map (resembling fig. [sent-333, score-0.178]
77 We also implemented classical EP with a sparse inverse covariance matrix; results are shown in 1d. [sent-340, score-0.197]
78 Among these methods, whitened EP is fastest and admits a wider class of MRF models. [sent-346, score-0.352]
79 For the 128 128 penny image, whitened EP required 8d emlsi. [sent-347, score-0.415]
80 8×G1H28z pXeneonyn, amndag er u,n w-thiimteen grows linearly with the number of pixels, linearly with the clique size of the potentials, and linearly in their rank. [sent-349, score-0.342]
81 The BP result required 24 hours, grows linearly with the number of pixels, quadratically in the clique size, and exponentially in rank. [sent-350, score-0.301]
82 Our results suggests that the constrained covariance family used by whitened EP provided a sufficient approximation of the full covariance structure inferred by standard EP. [sent-367, score-0.811]
83 While there has been some success in applying methods such as Lax-Friedrichs and fastmarching to non-Lambertian reflectance [1, 23], these generalizations must proceed on a case-by-case basis for each class of reflectance functions. [sent-369, score-0.254]
84 Example potentials φR are shown along the left column of figure 2, and are highly non-Gaussian. [sent-373, score-0.177]
85 When cast shadows are present and lighting originates from a single point source, we must enforce two rules. [sent-384, score-0.194]
86 First, pixels lying in shadow must be occluded from the lighting direction. [sent-385, score-0.202]
87 Because we are inferring depth directly (as opposed to surface normals), this can be enforced simply by a pairwise potential of rank one. [sent-386, score-0.27]
88 Suppose that lighting comes from the left, and suppose zunlit is the depth of a shadowed pixel, and zlit is the depth at the nearest unshadowed pixel to its left. [sent-387, score-0.35]
89 Secondly, we must also enforce that pixels that are lit within the image are unshadowed in the inferred shape. [sent-391, score-0.264]
90 Note that this approach to enforcing shadow cues would be expensive using BP because the potential φL is realvalued with a clique size of three, and is not eligible for LCN computational shortcuts. [sent-403, score-0.423]
91 Traditional Gaussian EP becomes inefficient whenever shadow cues are enforced because non-local connectivity produces an inverse covariance matrix that is no longer sparse. [sent-404, score-0.275]
92 Importantly, the shadow constraint is satisfied completely by the inferred surface: all pixels that are lit in the input image are unshadowed in the inferred surface, and all black pixels in the input are shadowed in the output. [sent-406, score-0.42]
93 Conclusions The methods in this paper reduce the run time ofEP from cubic to linear in the number of pixels for visual inference, while retaining a run time that is linear in clique size. [sent-408, score-0.411]
94 This is a substantial improvement over BP, which is exponential in clique size. [sent-409, score-0.29]
95 The computational expense of inference for large cliques has prohibited the investigation of complex probabilistic models for vision. [sent-410, score-0.268]
96 Our hope is that whitened EP will facilitate further research in these directions. [sent-411, score-0.352]
97 Results for whitened EP on SfS shows that the sacrifice in performance for this approach is small, even in problems with highly non-Gaussian potentials. [sent-412, score-0.417]
98 We expect that efficient inference with large cliques will be especially beneficial for depth inference, where multi-scale representations, complex spatial priors, shadows, occlusions, and the simultaneous inference of unknown global scene attributes all necessitate potentials with large cliques. [sent-414, score-0.507]
99 High-frequency shape and albedo from shading using natural image statistics. [sent-431, score-0.216]
100 Shape from shading with a linear triangular element surface model. [sent-492, score-0.228]
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