iccv iccv2013 iccv2013-241 knowledge-graph by maker-knowledge-mining
Source: pdf
Author: Tianfu Wu, Song-Chun Zhu
Abstract: Many object detectors, such as AdaBoost, SVM and deformable part-based models (DPM), compute additive scoring functions at a large number of windows scanned over image pyramid, thus computational efficiency is an important consideration beside accuracy performance. In this paper, we present a framework of learning cost-sensitive decision policy which is a sequence of two-sided thresholds to execute early rejection or early acceptance based on the accumulative scores at each step. A decision policy is said to be optimal if it minimizes an empirical global risk function that sums over the loss of false negatives (FN) and false positives (FP), and the cost of computation. While the risk function is very complex due to high-order connections among the two-sided thresholds, we find its upper bound can be optimized by dynamic programming (DP) efficiently and thus say the learned policy is near-optimal. Given the loss of FN and FP and the cost in three numbers, our method can produce a policy on-the-fly for Adaboost, SVM and DPM. In experiments, we show that our decision policy outperforms state-of-the-art cascade methods significantly in terms of speed with similar accuracy performance.
Reference: text
sentIndex sentText sentNum sentScore
1 edu Abstract Many object detectors, such as AdaBoost, SVM and deformable part-based models (DPM), compute additive scoring functions at a large number of windows scanned over image pyramid, thus computational efficiency is an important consideration beside accuracy performance. [sent-3, score-0.219]
2 In this paper, we present a framework of learning cost-sensitive decision policy which is a sequence of two-sided thresholds to execute early rejection or early acceptance based on the accumulative scores at each step. [sent-4, score-1.199]
3 A decision policy is said to be optimal if it minimizes an empirical global risk function that sums over the loss of false negatives (FN) and false positives (FP), and the cost of computation. [sent-5, score-1.183]
4 While the risk function is very complex due to high-order connections among the two-sided thresholds, we find its upper bound can be optimized by dynamic programming (DP) efficiently and thus say the learned policy is near-optimal. [sent-6, score-0.953]
5 Given the loss of FN and FP and the cost in three numbers, our method can produce a policy on-the-fly for Adaboost, SVM and DPM. [sent-7, score-0.54]
6 In experiments, we show that our decision policy outperforms state-of-the-art cascade methods significantly in terms of speed with similar accuracy performance. [sent-8, score-0.875]
7 During offline training, those scoring functions are learned by minimizing some regularized convex surrogate functions of the 0/1 loss function; and during detection, they are evaluated at a vast number of sliding windows in an image pyramid. [sent-11, score-0.25]
8 In this paper, we present a generic empirical global risk minimization framework and a DP algorithm for learning near-optimal cost-sensitive decision policy online. [sent-13, score-0.976]
9 The policy consists of a sequence oftwo-sided thresholds to execute Figure 1. [sent-14, score-0.592]
10 Illustration of two decision policies learned for a human face AdaBoost classifier. [sent-15, score-0.35]
11 The horizontal axis is the 1, 146 boosted weak classifiers divided in 20 stages, and the vertical axis is the accumulative scores. [sent-16, score-0.235]
12 At each stage the solid histograms show the original positive (red) and negative (green) populations in training dataset, and the dotted histograms show the changed populations after early decisions by the policy. [sent-17, score-0.452]
13 early rejection or early acceptance based on the accumula- tive scores at each stage. [sent-18, score-0.279]
14 By “cost-sensitive”, it means that the decision policy accounts for two factors explicitly: • The loss of misclassification: the losses CFN and CFP fTohre a ofasslse o negative saifnicda a iofanls:e t positive respectively. [sent-19, score-0.789]
15 The number of features and the scoring functions are all learned by the offline training stage as usual. [sent-23, score-0.317]
16 Our policy Θ Θ 775533 does not change these settings and is learned “online” with very flexible design of the configuration of sequential stages (i. [sent-24, score-0.663]
17 The policy is adjustable before the online detection starts. [sent-27, score-0.515]
18 1 shows two examples of decision policies learned for the human face AdaBoost classifier based on different parameter settings: the top one is more cautious, while the bottom one is more aggressive in early rejection/acceptance. [sent-30, score-0.465]
19 Learning decision policy is formulated under the risk minimization framework with two phases. [sent-32, score-0.9]
20 (i) Off-line learning: We assume that an additive scoring Θ × function (AdBoost, SVM or DPM) consisting of T basic additive terms is trained on a dedicated training data set, and remains fixed. [sent-33, score-0.291]
21 We divide the T basic terms into N sequential stages, and compute N histograms of accumulative scores (with K bins) on the positive (see the red histograms in Fig. [sent-35, score-0.308]
22 4) projected from the trajectories so that we can count the number of FPs and FNs, and calculate the computational cost for any decision policy recursively. [sent-39, score-0.809]
23 (ii) On-line learning: We derive an upper bound of the risk to address the optimization issue caused by the highorder connections between the two-sided thresholds. [sent-40, score-0.491]
24 Then, we propose a DP algorithm to minimize the upper bound efficiently with Θ = (CFN, CFP, λ) given to produce the near-optimal decision policy. [sent-41, score-0.375]
25 So, a fast online algorithm is entailed for automatically adapting the policy to different contexts and goals. [sent-47, score-0.491]
26 This, together with the proposed global risk function, its tight upper bound and the DP solution, distinguishes our work from many others in the literature [3–5, 10, 12, 16, 18, 20, 23]. [sent-48, score-0.404]
27 In boosting cascade [26], the rejection thresholds of each stage are learned independently, and the training process needs time-consuming trial-and-error tuning. [sent-51, score-0.557]
28 Soft cascade [3] needs a sufficiently large validation set to calibrate the rejection thresholds. [sent-56, score-0.277]
29 In [7], the cascade is learned for a off-line trained DPM. [sent-57, score-0.238]
30 The rejection threshold of each stage is selected so that the probability of a misclassification occurring is bounded from above by a predefined small positive value. [sent-58, score-0.299]
31 (ii) The coarse-to-fine methods [1, 2, 9, 21] require top- down coarse-to-fine partition of the pose space of an object class, which is often done manually and then used to organize all the tests into a hierarchy (such as a decision tree). [sent-59, score-0.205]
32 This paper makes three main contributions to rapid object detection: (i) It presents a cost-sensitive formulation to learn the decision policy accounting for the loss of misclassification and the computational cost explicitly, applicable to object detectors with additive scoring functions. [sent-65, score-1.01]
33 (ii) It derives an upper bound of the empirical global risk function and presents a DP algorithm to minimize it efficiently. [sent-66, score-0.48]
34 (iii) In experiments, it shows the learned policies of AdaBoost, SVM and DPM outperform the state-of-the-art cascade methods significantly in speed with similar accuracy. [sent-68, score-0.319]
35 Definition of the decision policy Denote by f(x) = gt(x) the off-line trained additive scoring function w? [sent-72, score-0.861]
36 accumulative score a ≤t th ie ≤ ≤i-t Nh stage is defined by fi(x) = ? [sent-79, score-0.245]
37 ecision policy): A N-stage decision policy of f(x) is define? [sent-82, score-0.666]
38 In testing, ΠN makes three possible actions at stage iin terms of accumulative score fi(x) : • Reject x, if fi(x) < τi− , • Accept x, if fi(x) τi+, and • Continue to compute Gi+1 (x), otherwise. [sent-87, score-0.245]
39 The risk function of a decision policy Denote by R(ΠN; Θ) the global risk function of ΠN, R(ΠN; Θ) = L(ΠN; CFP, CFN) λ · C(ΠN), (2) ≥ + where L(·) is the expected loss and C(·) the expected computational )cio sst th. [sent-93, score-1.192]
40 Denote by c(GTi)h eth eex computational fc coosmt opf an itniodnivi Cd(uΠal stage i(which is fixed after f(x) is trained offline). [sent-101, score-0.204]
41 For a given sample x, let n(x) be the number of stages tested before a decision is made by ΠN (i. [sent-106, score-0.316]
42 =N1 pˆ(i;ΠN) · Ci (4) where pˆ(i; ΠN) is the empirical probability of making an early decision at stage i(to be calculated in Sec. [sent-111, score-0.447]
43 4), ˆp(i; ΠN) = pˆ(ΠN classifies x at stage i|x ∈ D) = #Si/S, where #Si is the number of examples in D which are classified at stage iby ΠN. [sent-112, score-0.34]
44 By substituting the three empirical probabilities into the global risk function in Eqn. [sent-113, score-0.333]
45 Given parameters Θ = (CFP, CFN, λ), the optimal decision policy Π∗N (Θ) is sought by minimizing the empirical global risk in Eqn. [sent-120, score-1.005]
46 The two-sided thresholds at a stage igenerate the three numbers (#FNi, #FPi, #Si). [sent-136, score-0.228]
47 All the thresholds are fully dependent in the sense that the positive and negative sub-populations observed at a stage iare affected by the thresholds at all previous stages. [sent-137, score-0.406]
48 High-order connections between {τi}’s in ΠN Before presenting the method of minimizing the risk in Eqn. [sent-153, score-0.325]
49 2), which are affected by all the previous stages j (j < i) as early thresholds τj ’s change the populations through their decisions. [sent-160, score-0.329]
50 structure makes the problem of minimizing the risk in Eqn. [sent-164, score-0.263]
51 By using the derived recursions, the empirical risk in Eqn. [sent-169, score-0.31]
52 (5) is divided into two parts: one is the sum of risks caused by individual stages (i. [sent-170, score-0.206]
53 , measured independently), and the other accounts for all the double-counting occurred by removing the oFifg positive (red da ×sh Ned m lines) ac rneda negative (blue ones) training examples (created for the human face AdaBoost classifier in Fig. [sent-172, score-0.227]
54 2 and only keeping the sequential chain connections, we derive an upper bound for the empirical risk which is then minimized by a DP algorithm efficiently. [sent-181, score-0.567]
55 It is an upper bound because certain populations could be counted more than once in the empirical risk. [sent-182, score-0.314]
56 We also show the tightness of the upper bound empirically (see Fig. [sent-183, score-0.252]
57 Offline learning: computing the risk We first compute two histograms of the accumulative ×× scores fi(x) at each stage ion D+ and D−respectively. [sent-189, score-0.536]
58 At stage i, the set of accumulative scores is denoted by A(i) = {fi(x) ;∀x ∈ D}. [sent-190, score-0.276]
59 The trajectories of examples over the sequence of accumulative scores remain relatively steady, as illustrated in Fig. [sent-200, score-0.213]
60 In other words, if τi−1 classified x based on fi−1 (x), its scores fi(x) in the next steps unlikely jump inside [τi− , τi+), thus the policy will not regret its decisions. [sent-203, score-0.522]
61 4) which lead to the recursions of counting (#FNi, #FPi, #Si) in the empirical global risk in Eqn. [sent-205, score-0.368]
62 Furthermore, we use the notation of a threshold itself 775566 ×× (such as τi+ or τi−) in the decision policy to denote its bin index in the K N matrix without confusion. [sent-210, score-0.714]
63 We denote the sub-population of positive examples falling into bin k at stage iby, D+(ki) = {x : bin(x, i) = k, x ∈ D+}, (9) and its cardinality by S+ (ki) = |D+ (ki) | (e. [sent-217, score-0.232]
64 Denote the entries below the rejection line of Π,i· (·f·r o,τnt part in ΠN) by, Πi= (τ1−,··· ,τi−), and the entries above the acceptance line by, (11) Πi = (τ1+, ··· ,τi+). [sent-231, score-0.206]
65 i (16) i This set of examples will be counted more than once if we compute the empirical risk as the sum of risks caused by individual stages. [sent-253, score-0.43]
66 j=1 (18) The number of examples in D which are classified at stage iby Πi is defined by, |D(τi−) ∪ D(τi+)| − S(τi−; Πi−1)] + [S(τi+) − S(τi+; S(τi) ? [sent-274, score-0.218]
67 i) All the statistics are computed once at the offline stage, and can be used to compute the decision policy online for any given Θ = (CFP, CFN, λ). [sent-276, score-0.743]
68 This figure shows the tightness of the upper bound (Eqn. [sent-278, score-0.252]
69 (5), it is divided into two parts: (i) the sum of risks caused by the assignment of an individual τi, R(τi;Θ) =S1·[λ · Ci· (S(τi−) + S(τi+)) + S+(τi−) · CFN+ S−(τi+) · CFP], (20) and (ii) the sum of risks caused by examples which are double-counted at stage iw. [sent-285, score-0.337]
70 (21) make the problem of minimizing the risk very hard in general. [sent-295, score-0.263]
71 (22), we know that we can obtain an upper bound of the risk by deriving a lower bound for R(τi , Πi−1 ; Θ) (due to the minus operator). [sent-300, score-0.502]
72 2), and only consider the sequential chain connections (black arrows) to obtain the lower bound of R(τi, Πi−1 ; Θ). [sent-302, score-0.247]
73 Proposition 2 (The upper bound of risk): By substituting R(τi, Πi−1 ; Θ) with its lower bound R(τi, τi−1 ; Θ), we obtain an upper bound of R(ΠN; Θ), R? [sent-303, score-0.438]
74 The intuitive idea is that R(τi , τi−1 ; Θ) consider the risk caused by examples double-counted at τi w. [sent-315, score-0.284]
75 5, we show tightness of the upper bound empirically on the human face AdaBoost classifier which is consistent across different settings of the loss of misclassifica- tion (we set CFP ∈ [0. [sent-320, score-0.361]
76 L(eΠt Bi [τi] be the risk of the best assignment to the ifirst ? [sent-331, score-0.234]
77 (ii) The backward step for finding the near-optimal decision policy Π∗N = (τ1∗ , · · · , τN∗), where we first take, τN∗ = arg minτN BN [τN] an,d· t·h·e n,τ trace back τi∗ = Ti+1 [τi∗+1] in the order of decreasing i= N − 1, · · · , 1. [sent-336, score-0.666]
78 Experiments In the experiments, we learn decision policies for AdaBoost, SVM and DPM, and compare with their corresponding popular cascade methods [3, 7, 26]. [sent-339, score-0.472]
79 × Settings: In all the comparison experiments, we learned the decision policies by specifying (α, β), and the corresponding (CFP, CFN) are searched. [sent-340, score-0.315]
80 4t12s6yo075P9neurtWpinfoms the cascade methods in speed sigificantly on both the classification testing dataset and the CMU+MIT accuracy performance. [sent-354, score-0.209]
81 For example, in detection, the two cascade methods and our decision policy obtain similar APs, 0. [sent-357, score-0.849]
82 809 respectively, but our decision policy saves about 10 haar feature evaluations per sliding window on average. [sent-360, score-0.712]
83 Decision Policy for AdaBoost Classifier In this experiment, we learn the decision policy for the AdaBoost classifier trained for human face detection. [sent-363, score-0.776]
84 We compare with the original “hard” cascade method [26] and the soft cascade method [3] which are the two of the most popular cascade methods for face detection. [sent-364, score-0.619]
85 We use the same 4 types ofHaar features used in [26], and adopt the decision stump as the weak classifier. [sent-371, score-0.256]
86 So, the computational cost for each weak classifier is the same and the computational cost for each stage is proportional to the number of weak classifiers included in that stage. [sent-373, score-0.456]
87 We train the cascade consisting of 20 stages for both the “hard” and soft cascade using the publicly available OpenCV package. [sent-375, score-0.512]
88 The trained “hard” cascade consists of 2497 boosted weak classifiers in total and the soft cascade consists of 2135 boosted weak classifiers (note that the soft cascade needs an additional validation dataset to tune the thresholds, and we use 5000 positives and 40,000 negative examples). [sent-376, score-0.95]
89 To learn the decision policy, we first divide the 1146 boosted weak classifier into 20 subsets (as is shown in the bottom of Fig. [sent-379, score-0.339]
90 Note that the configuration of the decision policy is not particularly chosen, but can be very flexible for different situations (which can not be easily specified in training the cascade). [sent-381, score-0.692]
91 Overall, the decision policy outperforms the two cascade methods in speed ×× × with similar accuracy performance. [sent-384, score-0.875]
92 Decision policies for SVM classifiers and DPMs We learn decision policies for linear SVM classifier and DPM trained on INRIA person dataset [6]. [sent-387, score-0.472]
93 We compare with the probably approximate admissible (PAA) threshold [7], which is used in the cascade of DPM and selects the rejection thresholds based on τi− = min{x;f(x)≥t1 ,x∈D+} fi(x), where t1is predefined to focus on high-scoring positive examples. [sent-388, score-0.42]
94 So, we h daavteas e15t,0 r ecseull sti ning t ointa 1l5 5w ×hic 5h × are wreeoigrdhetr epdar abamseetde on tShoe, statistical power/computational cost ratio [2], and then organized into 20 subsets with the size of the first 19 stages being 7 cells and the last stage being 17 cells. [sent-392, score-0.337]
95 So, the computational cost of a stage is proportional to the number of cells. [sent-395, score-0.203]
96 Our decision policy outperforms the PAA method in speed in both clas775599 Table2. [sent-407, score-0.692]
97 Conclusion This paper presents a framework of learning cost- sensitive decision policy for popular object detectors with additive scoring functions (such as AdaBoost, SVM and DPM). [sent-420, score-0.855]
98 The decision policy explicitly accounts for the loss of misclassification and the cost of computation. [sent-421, score-0.814]
99 The learning is formulated under the risk minimization framework and the upper bound of the risk function is solved by an efficient DP algorithm. [sent-422, score-0.638]
100 By comparing with the state-of-art cascade methods, our decision policy outperforms them in speed significantly with similar accuracy in experiments. [sent-423, score-0.875]
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