iccv iccv2013 iccv2013-234 knowledge-graph by maker-knowledge-mining
Source: pdf
Author: Honghui Zhang, Jingdong Wang, Ping Tan, Jinglu Wang, Long Quan
Abstract: We propose an adaptive subgradient descent method to efficiently learn the parameters of CRF models for image parsing. To balance the learning efficiency and performance of the learned CRF models, the parameter learning is iteratively carried out by solving a convex optimization problem in each iteration, which integrates a proximal term to preserve the previously learned information and the large margin preference to distinguish bad labeling and the ground truth labeling. A solution of subgradient descent updating form is derived for the convex optimization problem, with an adaptively determined updating step-size. Besides, to deal with partially labeled training data, we propose a new objective constraint modeling both the labeled and unlabeled parts in the partially labeled training data for the parameter learning of CRF models. The superior learning efficiency of the proposed method is verified by the experiment results on two public datasets. We also demonstrate the powerfulness of our method for handling partially labeled training data.
Reference: text
sentIndex sentText sentNum sentScore
1 A solution of subgradient descent updating form is derived for the convex optimization problem, with an adaptively determined updating step-size. [sent-3, score-0.847]
2 Besides, to deal with partially labeled training data, we propose a new objective constraint modeling both the labeled and unlabeled parts in the partially labeled training data for the parameter learning of CRF models. [sent-4, score-0.991]
3 We also demonstrate the powerfulness of our method for handling partially labeled training data. [sent-6, score-0.256]
4 Introduction The Conditional Random Field [19] (CRF) offers a powerful probabilistic formulation for image parsing problems. [sent-8, score-0.139]
5 It has been demonstrated in previous works [18, 11, 16] that integration of different types of cues in a CRF model can significantly improve the parsing accuracy, like the smoothness preference and global consistency. [sent-9, score-0.189]
6 However, how to properly combine multiple types of information in a CRF model to achieve excellent parsing performance still remains an open question. [sent-10, score-0.139]
7 For this reason, the parameter learning of CRF models for image parsing tasks has received increasing attention recently. [sent-11, score-0.291]
8 Considerable progress on the parameter learning of CRF models has been made in the past few years. [sent-12, score-0.152]
9 However, the parameter learning of CRF models for the image parsing tasks still remains a challenging problem for several reasons. [sent-13, score-0.291]
10 First, as the CRF models used in many image parsing problems are of large scale and include expressive intervariable interactions, the computational challenges make the parameter learning of CRF models difficult. [sent-14, score-0.331]
11 Given a large number of training images, the learning efficiency would become a critical issue. [sent-15, score-0.152]
12 Second, partially labeled training data could cause the failure of some learning methods, which is common in image parsing. [sent-16, score-0.313]
13 For example, it has been found that the learned parameters involved in the pairwise smoothness potential are forced to tend toward zeros when using partially labeled training data [25]. [sent-17, score-0.411]
14 In this paper, we propose an adaptive subgradient descent method that iteratively learns the parameters of CRF models for image parsing. [sent-18, score-0.655]
15 The parameter learning is iteratively carried out by solving a convex optimization problem in each iteration. [sent-19, score-0.22]
16 The solution for the convex optimization problem gives a subgradient descent updating form with an adaptively determined updating step-size which can well balance the learning efficiency and performance of the learned CRF models. [sent-20, score-1.049]
17 Meanwhile, to deal with partially labeled training images that are common in various image parsing tasks, a new objective constraint for the parameter learning of CRF models is proposed, which models both the labeled and unlabeled parts of partially labeled training images. [sent-21, score-1.21]
18 Related work The parameter learning of CRF models is an active research topic, and investigated in many previous works [7, 27, 23, 20, 12, 2, 21, 15, 9]. [sent-24, score-0.152]
19 Most current methods for the parameter learning of CRF models can be broadly classified into two categories: maximum likelihood-based methods [19, 17] and max-margin methods [7, 27, 23, 12]. [sent-25, score-0.152]
20 En exhaustive review of the literature is beyond the scope of this paper, and the following review will mainly focus on the max-margin methods in which the parameter learning of CRF models is formulated as a structure learning problem based on the max-margin formulation. [sent-26, score-0.209]
21 Naturally, the max-margin methods for general structure learning can be used for the parameter learning of CRF models, such as the 1-slack and n-slack StructSVM(structural SVM) [27, 12], M3N(max-margin markov network) [7] and Projected Subgradient [23]. [sent-27, score-0.189]
22 The subgradient method [23] is another popular solution for structure learning problems, which is usually efficient and easy to implement. [sent-29, score-0.538]
23 Based on the subgradient method, recent works on the parameter learning of CRF models [15, 24] adopt different decomposition techniques. [sent-30, score-0.633]
24 As the updating step-sizes for the subgradient descent in these methods [23, 15, 24] are predefined and oblivious to the characteristics of the data being observed, to balance the learning efficiency and performance of the learned models, the updating step-sizes need to be carefully chosen. [sent-33, score-0.939]
25 Inappropriate updating step-sizes could lead to bad performance ofthe learned CRF models or slow convergence. [sent-34, score-0.236]
26 This motivates us to improve the subgradient method for the parameter learning of CRF models by adaptively tuning the subgradient descent, termed as adaptive subgradient descent in this paper. [sent-35, score-1.763]
27 However, for the parameter learning problem of CRF models, the optimal value of the objective function is unknown , and how to estimate it is also unclear. [sent-37, score-0.154]
28 Another important but less discussed issue in the previous max-margin methods for the parameter learning of CRF models is related to partially labeled training data, which is common in image parsing tasks. [sent-38, score-0.547]
29 To deal with the partially labeled training data, a maximum likelihood-based method which approximates the partition function with the Bethe free energy function is proposed in [28], with some limitations of the Bethe approximation discussed in [10]. [sent-39, score-0.281]
30 It has been observed that different treatments of the partially labeled data could lead to quite different performance. [sent-40, score-0.212]
31 To deal with the partially labeled training data, we introduce latent variables in the CRF models, inspired by the work [29, 15]. [sent-41, score-0.256]
32 Learning CRF to Parse Images Random field models are widely used to formulate various image parsing problems. [sent-43, score-0.179]
33 Based on this linear model, the parameter learning of CRF models can be cast as a typical structure learning problem, with the corresponding energy functions for the CRF models expressed as: E(x,y) = w · Φ(x,y) = ? [sent-67, score-0.274]
34 c∈C In the following, we review the widely used max-margin formulation for the parameter learning of CRF models: 1-slack and n-slack StructSVM Given a training set {(xn, yn)}nN=1, using the n-slack StructSVM [27], the learning problem can be formulated as [26]: argmwin21? [sent-69, score-0.213]
35 1 The objective constraint in the 1-slack StructSVM is obtained by merging the objective constraints for each sample of the training set {(xn, yn)}nN=1 in the n-slack StructSVM, owfi tthh ex t∗r =in ∪xn, eyt ∗{ (=x ∪yn a)n}d ˆy = ∪ yˆn. [sent-87, score-0.183]
36 The 1-slack and nslack Str=uc ∪tSxVM m=et ∪hoyds iteratively update the parameters to be learned by the cutting plane algorithm. [sent-88, score-0.137]
37 Unconstrained max-margin formulation An unconstrained formulation of (5) is adopted in [23], which uses the projected subgradient method to minimize the following regularized objective function: ρ(w) =12? [sent-90, score-0.616]
38 The parameters to be learned are iteratively updated by: wt+1 = P[wt − αtgw] (10) where gw is the subgradient of the convex function (8), P is the projection operator and αt is the predefined step-size that needs to be chosen carefully. [sent-93, score-0.646]
39 Inappropriate updating step-size could lead to bad performance of the learned CRF models or slow convergence. [sent-94, score-0.236]
40 Adaptive Subgradient Descent Learning In this section, we propose an adaptive subgradient descent algorithm for the parameter learning of CRF models, as described in the algorithm 1. [sent-96, score-0.706]
41 It is motivated by applying the idea proposed in the proximal bundle method [13] that uses proximal functions to control the learning rate to the subgradient methods in which the learning rate is subtly controlled by the predefined step-sizes. [sent-97, score-0.711]
42 In each iteration, the parameter updating is carried out by solving a convex optimization problem which integrates a proximal term to preserve the previously learned information and the large margin preference to distinguish bad labeling and the ground truth labeling. [sent-98, score-0.444]
43 The solution for the convex optimization problem gives a subgradient descent update form with an adaptively determined updating step-size for the parameter learning, which well balances the learning efficiency and performance of the learned CRF models. [sent-99, score-0.994]
44 A typical training process of using the proposed algorithm to train CRF models for image parsing is shown in Figure 1. [sent-100, score-0.223]
45 Adaptive subgradient descent algorithm In each iteration of the algorithm 1, the adaptive subgradient descent updating is carried out by solving the following convex optimization problem which has a subgradientbased solution with an adaptively determined step-size: wt+1 = argmwin21? [sent-103, score-1.416]
46 Therefore, an objective constraint same as that in the 1-slack StructSVM is used in the optimization problem (11): H(w; x∗, y∗, ˆyt) ≤ ξ (12) yˆt is the merged labeling configuration that most violates the constraint (12) for the current parameter wt. [sent-109, score-0.262]
47 On the other hand, a proximal term that forces the learned parameter wt+1 to stay as close as possible to wt is inserted into the objective function of (11), so that the previous learned information can be preserved. [sent-110, score-0.383]
48 o ξr , 8: Output: wf = arg minwt∈{wt}tM=1 H(wt; x∗ , y∗ , ˆyt) ∈{w}H(w In addition to the objective constraint (12), we add one more constraint that the parameters to be learned are nonnegative, similar to the previous work [26]. [sent-121, score-0.268]
49 0wti− αtdi oift whetir−wis αetdi≥ 0; αt = mαaxL(α),0 ≤ α ≤ C (13) (14) where [d1, d2 , · · · , dK] is the subgradient of the empirical risk (9). [sent-132, score-0.481]
50 Learning with Partially Labeled Image Partially labeled training images are common in image parsing problems, as it is usually very time-consuming to get precise annotations by manual labeling. [sent-167, score-0.293]
51 A typical partially labeled example is shown in Figure 2(a). [sent-168, score-0.212]
52 The unlabeled regions in partially labeled training images are not trivial for the parameter learning of CRF models, as observed in previous works [25, 28]. [sent-169, score-0.587]
53 As evaluating the loss on the unlabeled regions during the learning process is not feasible, discarding the unlabeled regions would be a straightforward choice, which excludes the unlabeled regions from the CRF models built for the partially labeled training images in the learning process. [sent-170, score-1.132]
54 However, without considering the unlabeled regions, the interactions between the labeled 33007836 (a) (b) (c) Figure 2. [sent-171, score-0.29]
55 The unlabeled regions are shown in black; (b) and (c), the pairwise CRF models for the parameter learning with different ways to treat the unlabeled regions in the training image. [sent-173, score-0.707]
56 (b) using the constraint (20), the nodes in the unlabeled regions and links linked to them are shown in green. [sent-174, score-0.274]
57 (c) discarding the unlabeled regions in the parameter learning, with the nodes and links for the unlabeled regions in (b) excluded. [sent-175, score-0.558]
58 regions and the unlabeled regions will not be modeled in the learning process. [sent-176, score-0.335]
59 For example, for the boundaries between the labeled regions and unlabeled regions, as these boundaries are mostly not the real boundaries between different categories, the pairwise smoothness should be preserved on these boundaries. [sent-178, score-0.437]
60 Without the interactions between the labeled regions and the unlabeled regions, the pairwise smoothness constraint on these boundaries will not be encoded in the learning process. [sent-179, score-0.527]
61 Let Rk and Ru denote the labeled regions and the unlabeled regions in the partially labeled training images, yk∗ denote the ground truth label for Rk. [sent-181, score-0.667]
62 Then, the new objective constraint is defined as: H(w; x∗, yt∗, ˆyt) ≤ ξ (20) where the ground truth label yt∗ = yk∗ ∪ yˆtu consists of the ground truth label yk∗ for Rk and the predicted label yˆtu for Ru. [sent-183, score-0.166]
63 Note that when there are no unlabeled regions in the training images, (12) and (20) are the same. [sent-184, score-0.263]
64 A simple pairwise CRF model for a partially labeled training image is shown in Figure 2, with different ways to handle the unlabeled regions in the partially labeled training images illustrated. [sent-185, score-0.779]
65 c∈S = = E(x,y) wu · fu (x, y) + wp · fp (x, y) + wc · fc(x, y) wT · Φ(x, y) (21) ? [sent-191, score-0.144]
66 ,j) ∈E where fu (x, y), fp (x, y) and fc(x, y) are the label dependent feature vectors for the unary potential, pairwise potential and high order potential to enforce label consistency, and Φ(x, y) = [fu (x, y) , fp(x, y) , fc(x, y)] , w = [wu, wp, wc] . [sent-193, score-0.2]
67 The parameters to be learned include wu = [wu], wc = [wc] and wp = [w1p, wp2, · · · , wpL], where L is the number of categories. [sent-194, score-0.207]
68 Similar to ,[2··5·], , wthe unary potential is defined on pixel level, multiplied by the weight parameter wu. [sent-195, score-0.161]
69 The procedure of the evaluation includes three successive steps: 1) unary potential training which is identical in our method, the 1-slack StructSVM method and the Projected Subgradient method; 2) parameter learning of the CRF model; 3) testing of the learned CRF model. [sent-206, score-0.326]
70 As our focus is evaluating the parameter learning algorithms for CRF models, we choose simple patch level features: Texton + SIFT for the unary potential training, − with the Random Forest classifier [6](50 trees, max depth = 32) chosen as the classifier model. [sent-208, score-0.198]
71 The parameter learning of 1As the energy function for the Robust PN model is submodular, no decomposition in [15] is necessary for the model, which makes [15] and the Projected Subgradient method [23] equivalent in this situation. [sent-210, score-0.137]
72 33007847 the Robust PN model starts with the unary classification, with the parameters to be learned initialized as wu = 1, wp = 0, wc = 0 for our method, the 1-slack StructSVM method and Projected Subgradient method. [sent-211, score-0.267]
73 For the performance evaluation, we use two criteria: CAA (category average accuracy, the average proportion of pixels correctly labeled in each category) and GA (global accuracy, proportion of all pixels correctly labeled) same as the previous works on the image parsing [25, 18]. [sent-213, score-0.249]
74 The segmentation accuracy obtained with unary classification, the Robust PN models learned by the 1-slack StructSVM method, Projected Subgradient method and our method on the datasets for the evaluation. [sent-231, score-0.215]
75 Performance of the learned models The segmentation accuracy achieved with the unary classification as well as the Robust PN models learned by our method, the 1-slack StructSVM method and Projected Subgradient method on the two datasets for the evaluation is given in Table 1. [sent-235, score-0.362]
76 These critical parameters include the initial upper bound of the updating step-size in our method, the updating step-sizes and the weight of constraint violation in the Projected Subgradient method, the weight of slack variable in the 1-slack StructSVM method. [sent-237, score-0.236]
77 Several examples of the parsing results obtained by different methods are illustrated in Figure 3. [sent-239, score-0.139]
78 The parsing results obtained by unary classification and the Robust PN models learned by the 1-slack StructSVM [12], Projected Subgradient [23] and our method on the MSRC-21 dataset and the CBCL street scene dataset. [sent-242, score-0.323]
79 The average time cost of the 1-slack StructSVM method, Projected Subgradient method and our method for the parameter learning of the Robust PN models on the MSRC-21 dataset and CBCL StreetScenes dataset. [sent-244, score-0.152]
80 From the comparison in Table 1(a), we find that compared with the unary classification, the segmentation accuracy is improved to varying degrees by the Robust PN models learned by all the three methods. [sent-248, score-0.215]
81 The segmentation accuracy achieved with the Robust PN model learned by our method outperforms that achieved with the model learned by the 1-slack StructSVM method and is comparable to that achieved with the Robust PN model learned by the Projected Subgradient method. [sent-249, score-0.352]
82 The segmentation accuracy achieved with the Robust PN models learned by our method, the 1-slack StructSVM method and Projected Subgradient method is given in Table 1(b). [sent-252, score-0.178]
83 Similar to the result on the MSRC-21 dataset, the Robust PN models learned by all the three methods improve the segmentation accuracy to varying degrees. [sent-253, score-0.155]
84 The number of iterations used to train the Robust PN models in the Projected Subgradient method [23] and our method to the corresponding parsing error achieved with the learned Robust PN models on the MSRC-21 dataset and the CBCL StreetScenes dataset. [sent-260, score-0.351]
85 The parsing error is measured by the loss of GA (global accuracy) and CAA (category average accuracy). [sent-261, score-0.139]
86 also find that the segmentation accuracy achieved with the Robust PN model learned by our method is slightly better than that achieved with the models learned by the 1-slack StructSVM method and Projected Subgradient method. [sent-262, score-0.285]
87 The learning efficiency of our method and the Projected Subgradient method depends on the predefined numbers of iterations used to train the CRF models, such as M in the algorithm 1. [sent-266, score-0.138]
88 In Figure 4, we find that on the test set of both datasets for the evaluation, the parsing errors of the Robust PN model learned by our method become stable rapidly, after only five iterations. [sent-268, score-0.223]
89 By contrast, on the test sets of both datasets for the evaluation, the parsing errors of the Robust PN model learned by the Projected Subgradient method decreased gradually, approximately stable after 200 iterations. [sent-269, score-0.223]
90 The segmentation accuracy achieved on the MSRC-21 and CBCL dataset with the Robust PN models learned by our method, the 1-slack StructSVM [12] and Projected Subgradient method [23], using different ways to treat the unlabeled regions. [sent-289, score-0.39]
91 The parsing results obtained with the Robust PN models learned by our method, with different ways to treat the unlabeled regions in partially labeled training images. [sent-291, score-0.79]
92 This indicates that modeling the interactions between the labeled regions and unlabeled regions of the partially labeled training images in the learning process is important for our method and the Projected Subgradient method. [sent-294, score-0.721]
93 Please note that the results of our method and the Projected Subgradient method reported in Table 1 are also obtained by using the modified objective constraint (20), as many images in the two datasets for the evaluation are partially labeled. [sent-295, score-0.224]
94 For the 1-slack StructSVM method, the learned models using the modified objective constraint (20) achieve better performance on the CBCL StreetScenes dataset and worse performance on the MSRC21 dataset. [sent-296, score-0.246]
95 Several examples of the parsing results obtained with the Robust PN models learned by our method are illustrated in Figure 6, with different ways to treat the unlabeled regions in partially labeled training images. [sent-297, score-0.79]
96 Conclusion We present an adaptive subgradient descent method to learn parameters of CRF models for image parsing. [sent-299, score-0.634]
97 In each iteration of the algorithm, the adaptive subgradient descent updating is carried out by solving a simple convex optimization problem which has a subgradient-based solution with an adaptively determined step-size. [sent-300, score-0.865]
98 The adaptively determined updating step-size can well balance the learning efficiency and performance of the learned CRF models. [sent-301, score-0.36]
99 Meanwhile, the proposed method is capable of handling partially labeled training data robustly, with a new objective constraint modeling both the labeled and unlabeled parts in the partially labeled training images for the parameter learning. [sent-302, score-0.934]
100 Scene segmentation with crfs learned from partially labeled images. [sent-469, score-0.366]
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