iccv iccv2013 iccv2013-395 knowledge-graph by maker-knowledge-mining
Source: pdf
Author: Oliver Müller, Michael Ying Yang, Bodo Rosenhahn
Abstract: Inference in continuous label Markov random fields is a challenging task. We use particle belief propagation (PBP) for solving the inference problem in continuous label space. Sampling particles from the belief distribution is typically done by using Metropolis-Hastings (MH) Markov chain Monte Carlo (MCMC) methods which involves sampling from a proposal distribution. This proposal distribution has to be carefully designed depending on the particular model and input data to achieve fast convergence. We propose to avoid dependence on a proposal distribution by introducing a slice sampling based PBP algorithm. The proposed approach shows superior convergence performance on an image denoising toy example. Our findings are validated on a challenging relational 2D feature tracking application.
Reference: text
sentIndex sentText sentNum sentScore
1 We use particle belief propagation (PBP) for solving the inference problem in continuous label space. [sent-4, score-0.609]
2 Sampling particles from the belief distribution is typically done by using Metropolis-Hastings (MH) Markov chain Monte Carlo (MCMC) methods which involves sampling from a proposal distribution. [sent-5, score-0.88]
3 This proposal distribution has to be carefully designed depending on the particular model and input data to achieve fast convergence. [sent-6, score-0.298]
4 We propose to avoid dependence on a proposal distribution by introducing a slice sampling based PBP algorithm. [sent-7, score-0.593]
5 Our findings are validated on a challenging relational 2D feature tracking application. [sent-9, score-0.348]
6 Introduction Markov Random Fields (MRFs) are a powerful tool for modeling relational dependencies among observations. [sent-11, score-0.157]
7 Numerous optimization approaches for discrete labels have been proposed, from binary labeled Graph Cuts [4], to multi-label tree reweighted message passing [17, 7]. [sent-14, score-0.186]
8 In this paper, we deal with continuous labeled MRFs where we use a particle belief propagation (PBP) approach [6]. [sent-15, score-0.573]
9 The efficiency of such particle based approaches highly depends on the sampling scheme used to explore the label space. [sent-16, score-0.42]
10 Previous approaches use Metropolis-Hastings (MH) Markov chain Monte Carlo (MCMC) methods for particle sampling. [sent-17, score-0.316]
11 The performance of these methods depends on a carefully designed proposal distribution. [sent-18, score-0.26]
12 We propose a novel sampling technique for PBP based on slice sampling [12]. [sent-20, score-0.487]
13 This method exploits the structure of the PBP message passing equations for direct sampling from the target distribution and does not de- #1·· #377·· #467·· Figure 1. [sent-21, score-0.341]
14 pend on a proposal distribution which is difficult to tune. [sent-23, score-0.254]
15 Our findings are then verified on a complex 2D relational feature tracking application as shown in Fig. [sent-25, score-0.348]
16 We furthermore provide a publicly available database of image sequences for feature tracking applications including manually labeled groundtruth data [11]. [sent-27, score-0.274]
17 Section 3 introduces notations and definitions used throughout the paper and gives a short introduction to slice sampling. [sent-30, score-0.212]
18 5 we present a thorough evaluation of our method compared to the state-of-the-art and propose a 2D relational feature tracking application. [sent-34, score-0.316]
19 Often such approaches are hard to apply on tasks where a continuous label space would be a more natural choice, such as feature tracking with relational constraints [14, 9]. [sent-39, score-0.393]
20 Loopy belief propagation is a prominent method using a local message passing mechanism for coordinating the optimal labeling of neighboring nodes. [sent-40, score-0.451]
21 Recently, message passing approaches working in continuous rather than discrete label space were proposed 11 112299 Gbr(axtph)icalM Mtos→u dtse(lx (set)xempbl(saxrsy)b(xs)mc samplingxs Figure 2. [sent-43, score-0.263]
22 Right: MCMC particle sampling of the belief b(xs) with an exemplary MCMC sampling chain of one particle (blue) and its corresponding histogram (red). [sent-46, score-1.056]
23 To the best of our knowledge, all previously proposed MCMC based belief propagation methods use Metropolis-Hastings (MH) sampling. [sent-49, score-0.296]
24 This sampling strategy consists of two steps: (a) sampling a candidate particle from an easy to sample proposal distribution, and (b) accept or reject the candidate depending on a transition probability [18]. [sent-50, score-0.748]
25 Applying this sampling technique involves a careful design of the proposal distri- bution, which is a compromise between exploring the label space (using a broad proposal distribution) and maximizing the transition acceptance ratio (minimize sample moves) at the same time. [sent-51, score-0.662]
26 Throughout the paper we show that considering alternative sampling techniques can be advantageous. [sent-52, score-0.148]
27 We propose to use slice sampling [12] instead of MH, rendering proposal distribution selection obsolete in the context of PBP. [sent-53, score-0.593]
28 To demonstrate superior performance of our method on a real world problem we propose a relational feature tracking application inspired by [9, 14] in the experiment section. [sent-54, score-0.316]
29 Some related works such as [15, 5] propose to formulate feature tracking as a discrete labeling problem and use global optimization algorithms (i. [sent-55, score-0.19]
30 Closely related methods use belief propagation combined with particle filtering [19, 9, 14], but still use proposal distributions for particle perturbation which introduces sensible optimization parameter tuning. [sent-59, score-1.009]
31 o Fdeors every nod⊂e s th theree s eist a flab neelig xs fbroormin gth neo ldaebsel t space VL. [sent-64, score-0.313]
32 Max-Product Particle Belief Propagation In the following we summarize the max-product particle belief propagation algorithm [8, 3]. [sent-84, score-0.532]
33 The energy term E(x) is approximated by particles such that the label space Ls of iesa caph pnrooxdeim s tine dth bey M paRrtFi cilse represented by a bseelt sopfa particles Ps = ,. [sent-85, score-0.416]
34 Then the estimated belief bsn ) or log disbelief Bsn ) = log(bsn )) of node s at iteration n is calculated as )fo =llo −wlso [g3(]b: {x(s1) (x(si) xs(p) − (x(si) (x(si) Bsn(x(si)) = ψs(x(si)) + ? [sent-89, score-0.595]
35 for xs node s are: ∈ (2) Ps from node t to Mtn→s(xs) =x mt∈inPt[ψs,t(xs,xt)+Btn−1(xt)−Msn→−t1(xt)]. [sent-91, score-0.405]
36 (3) Note that the log disbelief Bsn (xs) and the messages Mtn→s (xs) can be calculated for all continuous values xs ∈ Ls →rasther than only on the particle set Ps. [sent-92, score-0.761]
37 On the othe∈r hLand, the messages from node s to node t are approximated only using the particles xt from the particle set Pt = of node t. [sent-93, score-0.696]
38 (4) The main issue in PBP lies in how to sample new particles xsn ∼ Bsn (xs). [sent-102, score-0.19]
39 This method requires a proposal distribution q where new particles can be easily sampled from. [sent-104, score-0.444]
40 Algorithm 1 summarizes the Metropolis-Hastings based max-product particle belief propagation algorithm (MH-PBP). [sent-107, score-0.532]
41 Typically, q needs to be carefully adjusted to the true belief distribution. [sent-108, score-0.252]
42 In the following we propose to replace the MH sampling step by a slice sampling approach which does not depend on proposal distribution selection. [sent-110, score-0.741]
43 ,p, proposal distribu1: 2: 3: 4: 5: 6: 7: 8: utito:n I pσ Initialize the messages Mt0→s (xs) and log disbelief Bs0 with zero ∀s, t fworit hB zPe riote r∀asti,otn n = 1to N do for each node s and each particle i= 1, . [sent-115, score-0.669]
44 ,p do Initialize sampling chain ← for MCMC iteration m = 1, . [sent-118, score-0.228]
45 , ←M x do Sample ∼ pσ(x | from proposal d pist(rixb |u txion pσ Calc. [sent-121, score-0.216]
46 ) 1, we propose itoo randomly sample one Ldim∈en sRion in each MCMC step and slice sample on this dimension while the other dimensions are held fixed. [sent-133, score-0.191]
47 Assume the unary and/ or binary potential functions ψs and ψst are given as an analytic function. [sent-135, score-0.155]
48 Image Denoising For analyzing the random walk behaviour of our method we have chosen the application of image denoising due to its relatively simple model structure. [sent-142, score-0.163]
49 (18) For minimizing particle noise in the final estimation re- sult an annealing scheme is used where the target belief distribution is modified to where Tn = T0 · (TN/T0)n/N is the temperature at PBP iteration n, T0 is the· start temperature, and TN the end temperature. [sent-144, score-0.619]
50 We further compared the efficiency of the slice sampling method to the Metropolis-Hastings sampling applied on the image denoising problem. [sent-161, score-0.544]
51 An MCMC chain of M = 500 samples is generated for each particle and xθ(i) 11 113322 Figure 4. [sent-163, score-0.316]
52 Comparison of the empirical risk for with different proposal distributions. [sent-175, score-0.288]
53 For the MH-PBP proposal distribution the family of Gaussian distributions pσ(x | x? [sent-178, score-0.254]
54 In order to provide a fpai[r− comparison the proposal distribution is adapted to the current temperature by using pσ(x | x? [sent-185, score-0.342]
55 Figure x5 s xhows a comparison of the empirical risk for different MH-PBP proposal distributions. [sent-188, score-0.288]
56 This effect can be significantly reduced by averaging over particles instead of only selecting the best one as stated in Eq. [sent-194, score-0.212]
57 For comparing the random walk behavior of the MCMC sampling chains from S-PBP and MH-PBP, the normalized autocorrelation function ρk=? [sent-196, score-0.261]
58 It can be observed that the MH-PBP method produces a much higher autocorrelation than the S-PBP method, thus the MCMC chain mixing behaviour of S-PBP outperforms MH-PBP. [sent-213, score-0.185]
59 Relational Feature Tracking We propose to apply our S-PBP algorithm on a 2D relational feature tracking system inspired by [9, 14] as a more complex application. [sent-220, score-0.316]
60 The model is separated into two parts: (a) the unary potentials are derived from a feature patch matching model, and (b) the binary potentials encode the relative positioning of the features to each other. [sent-224, score-0.177]
61 Teh iem oagrieentation vector os encodes two aspects: the feature patch rotation (rotation of os, i. [sent-233, score-0.216]
62 The modifications include an additional particle resampling step, where for each frame the initial set of particles are sampled with replacement from the set of particles }i=1,. [sent-258, score-0.639]
63 For the slice sampling approach we need to define the boundary functions Aψs (u) and Axψts,t (u). [sent-269, score-0.339]
64 An analytic description of the unary potential is not available thus we have to define the boundary manually. [sent-272, score-0.155]
65 ps ∈ [1, W] [1, H], where W and H are the image wid∈th [a1n,dW height respectively, Wan adn tdo H res atrreic tt os mtoos ∈ [−10, 10] [−10, 10] . [sent-275, score-0.228]
66 This way it is ensured that the sampling space i×s large enough. [sent-276, score-0.148]
67 iOsn w tahye oitt hiser e n hsaunrde,d particles sampled outside the true (sub-)bounds are automatically rejected by the algorithm. [sent-277, score-0.19]
68 In order to provide a fair comparison of our slice sampling approach to the stateof-the-art MH-PBP approach, the design of the proposal distribution has to be done very carefully. [sent-279, score-0.593]
69 The label space can be divided into two parts, the feature position ps ∈ R2 and orthogonal feature transformation os ∈ R2. [sent-281, score-0.368]
70 The∈ proposal distributionforps | = I2×2· σxy), where N(μ, Σ) is a Ga|uspsian pdf with mean μ and covaria),n wceh Σere. [sent-282, score-0.216]
71 The sequences have a spatial resolution of 960 px 540 px and csoeqnusiesnt oefs 5h6av3e a an dsp 7a2ti6a lfr raemsoelust respectively. [sent-298, score-0.345]
72 The similar appearing features were chosen to stress the relational structure of our tracker model. [sent-300, score-0.256]
73 The PAPER1 sequence consists of five feature patches with a carefully chosen position pattern which allows unique identification of the features by only having knowledge about the relative distances of the features to each other. [sent-302, score-0.162]
74 The PAPER2 sequence is more challenging since the number of features is increased to 70 and the features are arranged in a grid structure allowing local relational ambiguities. [sent-303, score-0.182]
75 The sequences have a spatial resolution of 352 px s2i8o8n px T(FheAC seEqOuCeCnc1e) sa hndav v3e2 0a px t×i 2l4 r0es px (tiFoAnC oEfOC 35C22p) xan ×d both consist of 8188) afnrdam 3e2s0 pexac×h. [sent-305, score-0.568]
76 The sequences and tracking results are shown in Fig. [sent-307, score-0.175]
77 For the HOG features we set the smallest scale pyramid resolution to 50 px 50 px. [sent-313, score-0.177]
78 We use N = 20 PBP iterations and p = 10 particles for each node. [sent-317, score-0.243]
79 Since we compare the overall sampling behaviour of the proposed method rather than the belief propagation convergence behaviour selecting these parameters should be uncritical. [sent-319, score-0.57]
80 We consider the distance εtrack between the estimated feature position and the groundtruth (manually labeled) position as a quality measure. [sent-321, score-0.167]
81 For MH-PBP, the MH sampling parameters {σxy, σr, σφ} are chosen (from thes asmetp {lin0g. [sent-330, score-0.172]
82 We have evaluated the tracking performance for different MCMC iterations M = 2 to 5. [sent-364, score-0.177]
83 This is mainly due to a much higher overall sampling noise of the MH-PBP method compared to S-PBP. [sent-367, score-0.148]
84 We observed that the sampling noise of S-PBP is much less than with MH-PBP at feature positions with high confidence (i. [sent-368, score-0.183]
85 On the other hand the sampling noise of S-PBP increases for uncertain feature positions. [sent-371, score-0.209]
86 The RMSD in sequence PAPER2 and FACEOCC 1is higher for S-PBP than for MH-PBP due to temporal tracking failures. [sent-372, score-0.149]
87 These tracking failures are caused by strong local deformations or by occlusions of many feature points. [sent-373, score-0.194]
88 Typical tracking failures are depicted in the bottom row of Fig. [sent-374, score-0.159]
89 It can be observed in such cases that S-PBP leads to much higher tracking error than MH-PBP due to broader particle sampling in uncertain feature positions. [sent-376, score-0.569]
90 Figure 9 shows an evaluation of MH-PBP under differing (non-optimal) sampling parameters. [sent-377, score-0.188]
91 In order to visualize both the performance differences for nearoptimal parameters and tracking failures, the error values below and above the 15 px mark are shown with a differing vertical axis scaling. [sent-380, score-0.331]
92 It can be observed that the tracking performance of MH-PBP strongly depends on careful parameter selection. [sent-384, score-0.172]
93 The parameter σxy has the highest impact on the tracking performance and the optimal parameter value varies strongly between sequences (σxy = 5 for PAPER1 and σxy = 0. [sent-385, score-0.225]
94 The computational complexity for MH-PBP is O(NSpM (1+ V p)) and for S-PBP is O(NSpM(3 + 2Vi ps) O) given tMhe number of PBP iterationsi sN O, nNoSdpesM MS(,3 particles p, vMenCM thCe iterations M and the average number of neighbors per node V . [sent-391, score-0.289]
95 Relational feature tracker evaluation results showing the overal RMSD (for MCMC iterations from 2 to 5) and box plots over the error distance to groundtruth for selected MCMC iterations. [sent-401, score-0.227]
96 Note that the vertical axis is stretched for error values lower than 15 px in order to better visualize performance differences. [sent-405, score-0.167]
97 Conclusion We presented a novel particle belief propagation algorithm using slice sampling (S-PBP) instead of MetropolisHastings. [sent-408, score-0.871]
98 We exploit the message passing equations to compute the slice sampling bounds, provided the unary and binary potentials are defined by analytic functions or can be bounded by one. [sent-409, score-0.642]
99 Furthermore we showed that our approach performs equally well or better than MH-PBP on challenging relational feature tracking sequences. [sent-411, score-0.316]
100 Pmbp: Patchmatch [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] belief propagation for correspondence field estimation. [sent-434, score-0.296]
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