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235 nips-2011-Recovering Intrinsic Images with a Global Sparsity Prior on Reflectance

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Author: Carsten Rother, Martin Kiefel, Lumin Zhang, Bernhard Schölkopf, Peter V. Gehler

Abstract: We address the challenging task of decoupling material properties from lighting properties given a single image. In the last two decades virtually all works have concentrated on exploiting edge information to address this problem. We take a different route by introducing a new prior on reﬂectance, that models reﬂectance values as being drawn from a sparse set of basis colors. This results in a Random Field model with global, latent variables (basis colors) and pixel-accurate output reﬂectance values. We show that without edge information high-quality results can be achieved, that are on par with methods exploiting this source of information. Finally, we are able to improve on state-of-the-art results by integrating edge information into our model. We believe that our new approach is an excellent starting point for future developments in this ﬁeld. 1

Reference: text

Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 1 Introduction The task of recovering intrinsic images is to separate a given input image into its material-dependent properties, known as reﬂectance or albedo, and its light-dependent properties, such as shading, shadows, specular highlights, and inter-reﬂectance. [sent-12, score-0.44]

2 For example, an image which solely depends on material-dependent properties is helpful for image segmentation and object recognition [11], while a clean image of shading is a valuable input to shape-from-shading algorithms. [sent-14, score-0.855]

3 As in most previous work in this ﬁeld, we cast the intrinsic image recovery problem into the following simpliﬁed form, where each image pixel is the product of two components: I = sR . [sent-15, score-0.504]

4 The fact that shading is only a 1D entity imposes some limitations. [sent-20, score-0.37]

5 For example, shading effects stemming from multiple light sources can only be modeled if all light sources have the same color. [sent-21, score-0.45]

6 The main focus of this paper is on exploring sensible priors for both shading and reﬂectance. [sent-25, score-0.37]

7 (a) Image I “paper1” (b) I (in RGB) (c) Reﬂectance R (d) R (in RGB) (e) Shading s Figure 1: An image (a), its color in RGB space (b), the reﬂectance image (c), its distribution in RGB space (d), and the shading image (e). [sent-30, score-0.949]

8 Omer and Werman [12] have shown that an image of a natural scene often contains only a few different “basis colorlines”. [sent-31, score-0.156]

9 Figure (b) shows a dominant gray-scale color-line and other color lines corresponding to the scribbles on the paper (a). [sent-32, score-0.143]

10 The basis colors are clearly visible in (d), where the cluster for white (top, right) is the dominant one. [sent-34, score-0.14]

11 shading or shadows), and/or properties of the camera, e. [sent-37, score-0.37]

12 The main motivation of our work is to develop a simple, yet powerful probabilistic model for shading and reﬂectance estimation. [sent-44, score-0.387]

13 The idea is to extract those image edges which are (potentially) true reﬂectance edges and then to recover a new reﬂectance image that contains only these edges, using a set of Poisson equations. [sent-47, score-0.412]

14 The next factor is a simple smoothness prior on shading between neighboring image pixels, and has been used by some previous work e. [sent-51, score-0.593]

15 If we use image optimal parameter settings we perform on par with methods that use multiple images as input. [sent-61, score-0.279]

16 2 Related Work There is a vast amount of literature on the problem of recovering intrinsic images. [sent-63, score-0.184]

17 After that the Retinex algorithm was extended to two dimensions by Blake [3] and Horn [8], and later applied to color images [6]. [sent-67, score-0.211]

18 The basic Retinex algorithm is a 2-step procedure: 1) detect all image gradients which are caused by changes in reﬂectance; 2) recover a reﬂectance image which preserves the detected reﬂectance gradients. [sent-68, score-0.395]

19 The basic assumption of this approach is that small image gradients are more likely caused by a shading effect and strong gradients by a change in reﬂectance. [sent-69, score-0.617]

20 For color images this rule can be extended by treating changes in the 1D brightness domain differently to changes in the 2D chromaticity space. [sent-70, score-0.303]

21 Note, 2 Note, a gradient in chromaticity can only be caused by differently colored light sources, or inter-reﬂectance. [sent-72, score-0.178]

22 By doing so we directly extend previous intrinsic image models which makes evident the gains that can be attributed to a global sparse reﬂectance term alone. [sent-86, score-0.386]

23 The key idea in their work is to perform a pre-processing step where the (normalized) reﬂectance image is partitioned into a few clusters. [sent-89, score-0.176]

24 However, the differences are that they do not formulate this idea as a joint probabilistic model over latent reﬂectance “basis colors” and shading variables. [sent-93, score-0.407]

25 Also, they need a Retinex type of edge term to avoid the trivial solution of s = 1. [sent-95, score-0.103]

26 on learning low-level vision [5] formulates a probabilistic model for intrinsic images. [sent-98, score-0.202]

27 In essence, they build a patch-based prior jointly over shading and reﬂectance. [sent-99, score-0.398]

28 In a new test image the best explanation for reﬂectance and shading is determined. [sent-100, score-0.526]

29 Since no large-scale ground database was available at that time, they only train and test on computer generated images of blob-like textures. [sent-102, score-0.163]

30 This idea is derived from the Laplacian matrix used for image matting [10]. [sent-107, score-0.176]

31 3 3 A Probabilistic Model for Intrinsic Images The model outlined here falls into the class of Conditional Random Fields, specifying a conditional probability distribution over reﬂectance R and shading S components for a given image I p(s, R | I) ∝ exp (−E(s, R | I)) . [sent-110, score-0.549]

32 Thus Ii is an image pixel (vector of dimension 3), Ri a reﬂectance vector (a 3-vector), si the shading (a scalar). [sent-113, score-0.592]

33 There are two ways to use the relationship (1) to formulate a model for shading and reﬂectance, corresponding to two different image likelihoods p(I | s, R). [sent-121, score-0.526]

34 One possible way is to relax the relation (1) and for example assume a Gaussian likelihood p(I | s, R) ∝ exp(− I − sR 2 ) to account for some noise in the image formation process. [sent-122, score-0.177]

35 Since Ii = si Ri has to hold of all color channels c = {R, G, B}, the unknown variables are speciﬁed up to scalar multipliers, in other words the direction of Ri is already known. [sent-125, score-0.166]

36 We rewrite Ri = ri Ri , with Ri = Ii / Ii , leaving r = (r1 , . [sent-126, score-0.149]

37 The shading components can be computed using si = Ii /ri . [sent-132, score-0.411]

38 The latter reduction is commonly exploited by intrinsic image algorithms in order to simplify the model [7, 14, 4] and in the remainder we will also make use of it. [sent-134, score-0.3]

39 We will describe the three components and their inﬂuence in greater detail below, ﬁrst we write the optimization problem that corresponds to a MAP solution in its most general form min ws Es (r) + wr Eret (r) + wcl Ecl (r, α). [sent-143, score-0.333]

40 ,n Note, the global scale of the energy is not important, hence we can always ﬁx one non-zero weight ws , wr , wcl to 1. [sent-147, score-0.392]

41 Shading Prior (Es ) We expect the shading of an image to vary smoothly over the image and we encode this in the following pairwise factors −1 −1 ri Ii − rj Ij Es (r) = 2 , (4) i∼j where we use a 4-connected pixel graph to encode the neighborhood relation which we denote with i ∼ j. [sent-148, score-0.933]

42 Gradient Consistency (Eret ) As discussed in the introduction, the main idea of the Retinex algorithm is to disambiguate between edges that are due to shading variations from those that are caused by material reﬂectance changes. [sent-154, score-0.472]

43 Assume that we already know, or have classiﬁed, that an edge at location i, j in the input image is caused by a change in reﬂectance. [sent-156, score-0.233]

44 Using the fact log( Ii ) = log(Ii )−log(Ri ) (for all channels c) and assuming a squared deviation around the log gradient magnitude, this translates into the following Gaussian MRF term on the reﬂectances 2 (log(ri ) − log(rj ) − gij (I)(log( Ii ) − log( Ij ))) . [sent-158, score-0.143]

45 Eret (r) = (5) i∼j It remains to specify the classiﬁcation function g(I) for the image edges. [sent-159, score-0.156]

46 For each pixel i and a neighbor j we compute the gradient of the intensity image and the gradient of the chromaticity change. [sent-161, score-0.359]

47 The two parameters which are the thresholds θg , θc for the intensity and the chromaticity change are then estimated using leave-one-out-cross validation. [sent-164, score-0.117]

48 It is worth noting that this term is qualitatively different from the smoothness prior on shading (4) even for pixels where gij (I) = 0. [sent-165, score-0.544]

49 Here, the log-difference is penalized whereas the shading smoothness does also depend on the intensity values Ii , Ij . [sent-166, score-0.441]

50 Global Sparse Reﬂectance Prior (Ecl ) Motivated by the ﬁndings of [12] we include a term that acts as a global potential on the reﬂectances and favors the decomposition into some few reﬂectance clusters. [sent-169, score-0.105]

51 Every reﬂectance component ri belongs to one of the clusters and we denote its cluster membership with the variable αi ∈ {1, . [sent-174, score-0.237]

52 This is summarized in the following energy term 4 Figure 2: A crop from the image “panther”. [sent-178, score-0.228]

53 Left: input image I and true decomposition (R, s). [sent-179, score-0.175]

54 Note, the colors in reﬂectance image (True R) have been modiﬁed on purpose such that there are exactly 4 different colors. [sent-180, score-0.21]

55 The second column shows a clustering (here from the solution with ws = 0), where each cluster has an arbitrary color. [sent-181, score-0.27]

56 The remaining columns show results with various settings for C and ws (left reﬂectance image, right shading image). [sent-182, score-0.537]

57 Top row is the result for C = 4 and bottom row for C = 50 clusters, columns are results for ws = 0, 10−5 , and 0. [sent-183, score-0.211]

58 Below the images is the corresponding LMSE score (described in Section 4. [sent-185, score-0.119]

59 This represents a global potential, since the cluster means depend on the assignment of all pixels in the image. [sent-190, score-0.155]

60 The cluster means Rc 1 ˜ are optimally determined given r and α: Rc = |{i:αi =c}| i:αi =c ri Ri . [sent-192, score-0.211]

61 We use a simpliﬁed model (2), namely Ecl + ws Es , and vary ws as well as the number of clusters. [sent-194, score-0.352]

62 Let us ﬁrst consider the case where ws = 0 (third column). [sent-195, score-0.167]

63 Hence the shading within one cluster looks reasonable, but is not aligned across clusters. [sent-198, score-0.432]

64 If we were to give each pixel its own cluster this would no longer be true and we would get the trivial solution of s = 1. [sent-202, score-0.133]

65 Finally, results deteriorate when the smoothing term is too strong (last column ws = 0. [sent-203, score-0.225]

66 Note, that for this simple toy example the smoothness prior was not important, however for real images the best results are achieved by using a non-zero ws . [sent-205, score-0.334]

67 de/mkiefel/projects/intrinsic 5 comment Color Retinex no edge information Col-Ret+ global term full model Es - Ecl - Eret - - LOO-CV 29. [sent-218, score-0.145]

68 The column “best-single” is the parameter set that works best on all 16 images jointly, “image opt. [sent-232, score-0.12]

69 ” is the result when choosing the parameters optimal for each image individually, based on ground truth information. [sent-233, score-0.224]

70 For instance, we virtually always achieve a lower energy compared to using the ground truth r as initial start point. [sent-235, score-0.138]

71 From these constraints we can derive that Ii ≥ ri ≥ 3. [sent-240, score-0.149]

72 Experiments For the empirical evaluation we use the intrinsic image database that has been introduced in [7]. [sent-256, score-0.325]

73 This dataset consists of 16 different images for all of which the ground truth shading and reﬂectance components are available. [sent-257, score-0.561]

74 In all experiments we compare against Color Retinex which was found to be the best performing method among those that take a single image as input. [sent-260, score-0.156]

75 The method from [19] yields better results but requires multiple input images from different light variations. [sent-261, score-0.121]

76 The ﬁrst metric is the average of the localized mean squared error (LMSE) between the predicted and true shading and predicted and true reﬂectance image. [sent-264, score-0.37]

77 the weights wcl , ws , wr and the gradient thresholds θc , θg have been chosen using a leave-one-out estimate (LOO-CV). [sent-269, score-0.362]

78 Due to the high variance of the scores for the images we used the median error to score the parameters. [sent-270, score-0.119]

79 Thus for image i the parameter was chosen that leads to the lowest median error on all images except i. [sent-271, score-0.281]

80 Additionally we record the best single parameter set that works well on all images, and the score that is obtained when using the optimal parameters on each image individually. [sent-272, score-0.175]

81 For models using both the cluster and shading smoothness terms, we select from ws ∈ {0. [sent-276, score-0.638]

82 1}, for models that use the cluster and Color Retinex term wr ∈ {0. [sent-279, score-0.164]

83 When all three terms are non-zero, we vary ws as above paired with wr ∈ ×{0. [sent-283, score-0.249]

84 The ﬁrst observation is that the Color Retinex algorithm (1st row) performs about similar to the system using a shading smoothness prior together with the global factor Ecl (2nd row). [sent-293, score-0.485]

85 The lower value for the image optimal setting of 18. [sent-296, score-0.156]

86 With knowledge about the optimal image parameter it yields a lower LMSE score (16. [sent-304, score-0.175]

87 0 we believe that they are informative, given the parameter estimation Table 2: Method comparison with other intrinsic image algoproblems due to the diverse and rithms also compared in [7]. [sent-338, score-0.318]

88 For entries ’-’ we had no individproves over all the compared methual results (and no code), the two numbers marked ∗ are estiods that use only a single image as mated from Fig4. [sent-343, score-0.156]

89 The full model is even better on 6/16 images than the Weiss algorithm [19] that uses multiple images. [sent-350, score-0.117]

90 For example note that the method BAS that either attributes all variations to shading (r = 1) or to reﬂectance alone (s = 1) already yields a LMSE of 36. [sent-360, score-0.387]

91 6, if for every image the optimal choice between the two is made. [sent-361, score-0.156]

92 We have also tested our method on various other real-world images and results are visually similar to [15, 4]. [sent-364, score-0.155]

93 Without the global term (Color Retinex with LOO-CV and image optimal) the result is imperfect. [sent-368, score-0.242]

94 With a global term (remaining three results) the images look visually much better. [sent-372, score-0.241]

95 7 Figure 3: Various results obtained with different methods and settings (more in supplementary material); For each result: left reﬂectance image, right shading image Note that the third row shows an extreme variation for the full model when switching from image optimal setting to LOO-CV setting. [sent-373, score-0.721]

96 Note, in this case we chose for both methods the image optimal settings to illustrate the potential of each model. [sent-382, score-0.156]

97 5 Discussion and Conclusion We have introduced a new probabilistic model for intrinsic images that explicitly models the reﬂectance formation process. [sent-383, score-0.282]

98 Another reﬁnement would be to replace the Gaussian cluster term with a color line term [12]. [sent-387, score-0.249]

99 Ground-truth dataset and baseline evaluations for intrinsic image algorithms. [sent-439, score-0.3]

100 Intrinsic images decomposition using a local and global sparse representation of reﬂectance. [sent-483, score-0.167]

similar papers computed by tfidf model

tfidf for this paper:

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