cvpr cvpr2013 cvpr2013-362 knowledge-graph by maker-knowledge-mining
Source: pdf
Author: Koichiro Yamaguchi, David McAllester, Raquel Urtasun
Abstract: We consider the problem of computing optical flow in monocular video taken from a moving vehicle. In this setting, the vast majority of image flow is due to the vehicle ’s ego-motion. We propose to take advantage of this fact and estimate flow along the epipolar lines of the egomotion. Towards this goal, we derive a slanted-plane MRF model which explicitly reasons about the ordering of planes and their physical validity at junctions. Furthermore, we present a bottom-up grouping algorithm which produces over-segmentations that respect flow boundaries. We demonstrate the effectiveness of our approach in the challenging KITTI flow benchmark [11] achieving half the error of the best competing general flow algorithm and one third of the error of the best epipolar flow algorithm.
Reference: text
sentIndex sentText sentNum sentScore
1 edu e c Abstract We consider the problem of computing optical flow in monocular video taken from a moving vehicle. [sent-4, score-0.563]
2 In this setting, the vast majority of image flow is due to the vehicle ’s ego-motion. [sent-5, score-0.501]
3 We propose to take advantage of this fact and estimate flow along the epipolar lines of the egomotion. [sent-6, score-1.042]
4 Furthermore, we present a bottom-up grouping algorithm which produces over-segmentations that respect flow boundaries. [sent-8, score-0.46]
5 We demonstrate the effectiveness of our approach in the challenging KITTI flow benchmark [11] achieving half the error of the best competing general flow algorithm and one third of the error of the best epipolar flow algorithm. [sent-9, score-2.034]
6 Introduction Optical flow is an important classical problem in computer vision, as it can be used in support of 3D reconstruction, perceptual grouping and object recognition. [sent-11, score-0.46]
7 In this setting, most of the flow can be explained by the vehicle’s ego-motion. [sent-13, score-0.46]
8 As a consequence, once the ego-motion is computed, one can treat flow as a matching problem along epipolar lines. [sent-14, score-1.018]
9 The main difference with stereo vision resides in the fact that the epipolar lines radiate from a single epipole, called the focus of expansion (FOE). [sent-15, score-0.691]
10 A few attempts to utilize these constraints have been proposed [22], mainly in the context of scene flow (i. [sent-16, score-0.46]
11 However, so far, we have not witnessed big performance gains by employing the epipolar constraints. [sent-19, score-0.546]
12 In contrast, we take advantage of recent developments in stereo vision to construct robust solutions to the epipolar flow problem. [sent-20, score-1.122]
13 Our first contribution is to adapt slanted plane stereo models [39, 2] to the problem of monocular epipolar flow estimation. [sent-22, score-1.351]
14 This allow us to exploit global energy minimization methods in order to alleviate problems in texture-less regions and produce dense flow fields. [sent-23, score-0.509]
15 In particular, we represent the problem as one of inference in a hybrid Markov random field (MRF), where a slanted plane represents the epipolar flow for each segment and discrete random variables represent the boundary relations between each pair of neighboring segments (i. [sent-24, score-1.372]
16 The introduction of these boundary variables allows the model to reason about ownerships of the boundary as well as to enforce physical validity of the boundary types at junctions. [sent-27, score-0.194]
17 In order to produce accurate results, slanted plane MRF models require a good over-segmentation of the image, where the planar assumption for each superpixel is approximately satisfied. [sent-28, score-0.303]
18 Towards this goal, our second contribution is an efficient flow-aware segmentation algorithm in the spirit of SLIC [1], but where the segmentation energy involves both image and flow terms. [sent-29, score-0.509]
19 This encourages the segmentation to respect both image and flow discontinuities. [sent-30, score-0.46]
20 Our last contribution is a local flow matching algorithm, inspired by the very successful stereo algorithm semi-global block matching [20], which computes very accurate semi-dense flow fields. [sent-32, score-1.177]
21 We demonstrate the effectiveness of our approach in the challenging KITTI flow benchmark [11] achieving half the error of the best competing general flow algorithm and one third of the error of the best competing epipolar flow algorithm. [sent-33, score-2.079]
22 In the remainder of the paper, we first review related work and present our local epipolar flow algorithm. [sent-34, score-0.981]
23 We then discuss our unsupervised segmentation algorithm which preserves epipolar flow discontinuities, and present our slanted plane MRF formulation. [sent-35, score-1.186]
24 Related Work Over the past few decades we have witnessed a great improvement in performance of flow algorithms. [sent-38, score-0.485]
25 While existing many works use a variational approach for continuous flow optimization [21, 5, 6, 41], a number of recent approaches have proposed discrete MRF formulations [26, 35, 14, 25]. [sent-44, score-0.579]
26 The problem is more severe than in stereo, as instead of 1D disparities, a 2D flow field has to be discretized. [sent-46, score-0.46]
27 [14, 25] use a coarse-to-fine approach and sampling, while [26, 35] create a set of candidate flow estimates by standard continuous optical flow algorithms. [sent-47, score-1.049]
28 When dealing with mostly static scenes, optical flow can be expressed as a 3D rigid motion due to the camera motion. [sent-48, score-0.573]
29 The knowledge of this epipolar geometry has been in- troduced as a soft constraint in the energy function [36, 37] or as a hard constraint [33, 22]. [sent-49, score-0.57]
30 In the latter, first the fundamental matrix is calculated and the flow estimation is formulated as a 1D search by restricting a corresponding point to lie on the epipolar line. [sent-50, score-1.046]
31 While a soft constraint can yield less errors in independently moving objects, hard constraints can reduce computational complexity and achieve robust estimation of flow in stationary objects if the fundamental matrix is accurately estimated. [sent-51, score-0.525]
32 In this paper we take the latter approach and adapt the highly successful slantedplane MRF approach to stereo vision for the problem of epipolar flow estimation. [sent-52, score-1.122]
33 Semi-global Block Matching for Flow In this section we extend the popular stereo algorithm, semi-global block matching [20] to tackle the epipolar flow problem. [sent-55, score-1.201]
34 In particular, we first convert the estimation from a 2D matching problem to a 1D search along the epipolar lines, which are defined by the vehicle’s ego-motion. [sent-56, score-0.593]
35 We then define parameterizations and cost functions which are appropriate for epipolar flow. [sent-57, score-0.552]
36 Epipolar Flow as a 1D Search Problem The first step of our algorithm consists on estimating the fundamental matrix that defines the set of epipolar lines. [sent-60, score-0.551]
37 We then estimate the parameters of the flow that is due to camera rotation, and pose the flow problem as a 1D search along the translational flow component. [sent-62, score-1.469]
38 Assuming that the camera rotation between two images is small, uw (p) can be expressed as follows [27], uw(p) = ? [sent-65, score-0.23]
39 xTh =us, x we can yw =rite y uw c(p) as a 5-parameter model. [sent-68, score-0.172]
40 An additional constraint that we can exploit to estimate the rotational component of the flow is given by the fact that uv (p, Zp) is parallel to the epipolar line passing though that point at time t+1. [sent-71, score-1.123]
41 Thus, as uv (p, Zp) being parallel to the epipolar line ? [sent-75, score-0.631]
42 ( p˜ + u˜w(p)) = 0 (2) with u˜w (p) representing uw (p) in homogeneous coordinates. [sent-82, score-0.197]
43 Once this is done, we only need to estimate the flow in the direction of the epipolar lines. [sent-88, score-1.013]
44 Semi-global Block Matching for Flow We now discuss how we can adapt the semi-global block matching stereo algorithm (SGM) [20] to estimate the trans- lational component of flow. [sent-92, score-0.252]
45 We need to define a good parameterization and a good cost function for epipolar flow. [sent-96, score-0.596]
46 In the case of flow, using this parameterization leads to the interaction between the epipolar geometry and the scene depth, as the disparity at each point is a complex non-linear function of depth Zp. [sent-98, score-0.645]
47 (p) a unit vector in the direction of the epipolar line ? [sent-102, score-0.549]
48 (p) and d(p, Zp) the disparity along the epipolar line. [sent-104, score-0.601]
49 Fig 1 (left) shows the epipolar geometry of two images, where C and C? [sent-105, score-0.521]
50 Adding the rotation flow vector uw (p) to each pixel p means that the image plane at time t is rotated so that its camera direction is the same as the one at time t + 1, as shown in Fig. [sent-109, score-0.803]
51 As a result, the epipole and the epipolar line in the rotated image at time t are exactly the same as those in the image at time t + 1. [sent-111, score-0.601]
52 2 shows the geometric configuration on the epipolar plane, where r and r? [sent-113, score-0.521]
53 We can then compute the disparity as vz vz d(p,Zp) = r? [sent-123, score-0.27]
54 Since the z-component vz of the camera translation is constant for all pixels, the ratio Zvpz, denoted VZ-ratio, depends only on the distance Zp. [sent-127, score-0.155]
55 (3)) represents the scene independent of the epipolar geometry. [sent-129, score-0.521]
56 Next, we need to define a cost function adequate for estimating the epipolar flow. [sent-135, score-0.552]
57 (q, ωq) = q + uw (q) + uv (q, Zq; ωq) is the corresponding pixel in the second image whose VZ-index is ωq, is a constant, W(p) is a window centered at pixel p and G(·) ias ctohen dtairnetc,t Wion(apl) )d iesri av watiinvde oinw th ceen image itn p tihxee ld pire acntdion G (o·)f the epipolar line. [sent-145, score-0.845]
58 Using lower penalties for small changes permits an adaptation to slanted or curved surfaces. [sent-153, score-0.155]
59 The flow can then be estimated by solving for the disparities {ωp} by minimizing the energy in Eq. [sent-154, score-0.577]
60 This provides the sets Ft and Ft+1 of the pixels, whose flow has been estsiemtsat Fed, aanndd FVZ-indices ωˆt (p) and ωˆt+1 (p? [sent-162, score-0.46]
61 111888666422 Algorithm 1 MotionSLIC Init superpixels by sampling pixels in a regular grid for i= 1to #iterations do for all pixel p do sp = argminiE(p, i, θi , μi , ci) end for for all super? [sent-165, score-0.306]
62 Joint Segmentation and Flow Estimation Given an estimate of the flow in a subset of the pixels, we are interested in computing an over-segmentation of the image that respects both flow and image boundaries. [sent-171, score-0.979]
63 This over-segmentation will be used in the next section by our slanted-plane MRF model in order to produce more accurate dense flow estimations. [sent-172, score-0.46]
64 Towards this goal, we represent the VZ-index of each superpixel with a slanted plane, ω(p, θsp) = αspx + βspy + γsp, (4) defined with parameters θsp = (αsp , βsp , γsp ), where sp indexes the superpixel that pixel p belongs to. [sent-173, score-0.553]
65 We frame joint unsupervised segmentation and flow estimation as an energy minimization problem, and define the energy of each pixel as the sum of energies encoding shape, appearance and flow, taking special care into modeling occlusions. [sent-176, score-0.628]
66 The input to our algorithm is the two images as well as our initial (possibly sparse) flow estimate ˆω (see sec- × tion 3). [sent-177, score-0.492]
67 Flow: This potential enforces that the plane parameters should agree with the input flow ωˆt (p) as follows Edtisp(p,θsp) =? [sent-215, score-0.571]
68 2 ioft δhneorw(pis,eθsp) We can define the total energy of a pixel as E = Ectol(p, csp) + Ect+ol1 (p, csp , θsp) + λposEpos(p, μsp) + λdisp λpos ? [sent-222, score-0.212]
69 blem of joint unsupervised segmentation and flow estimation becomes Θm,S,inμ,c? [sent-227, score-0.495]
70 We derive an iterative scheme that works in three steps: first we minimize the energy with respect to the assignments, we update the parameters μsp , csp by simply computing their means, and then compute the plane parameters by using a robust estimator. [sent-231, score-0.255]
71 This algorithm can be extended to stereo vision by simply replacing the two consecutive frames with the left and right images of the stereo pair, and the VZ-ratio with disparity. [sent-233, score-0.282]
72 Our experimental evaluation will demonstrate the effectiveness of our method in both epipolar flow and stereo estimation problems. [sent-234, score-1.157]
73 Recently, [39] proposed a slanted-plane MRF model for stereo vision that reasons about segments as well as occlusion boundaries. [sent-237, score-0.176]
74 Here we follow a similar idea, and represent the epipolar flow estimation problem as inference in a mixed continuous-discrete random field. [sent-238, score-1.016]
75 The continuous variables represent 3D planes encoding the VZ-ratio, while the discrete variables encode the type of boundaries between pairs of superpixels. [sent-239, score-0.217]
76 Our approach takes as input epipolar flow as well as an oversegmentation of the image. [sent-240, score-0.981]
77 In particular, we employ the epipolar flow fields and segmentations estimated by MotionSLIC (see section 4). [sent-241, score-1.043]
78 Let oi,j ∈ {co, hi, lo, ro} be a discrete random variable representing {wchoe,thhei,r tow,or neighboring planes are coplanar, f roerpmre a hinge or an occlusion boundary. [sent-246, score-0.202]
79 Here, lo implies that plane ioccludes plane j, and ro the opposite. [sent-247, score-0.156]
80 We define our hybrid conditional random field in terms of all slanted-planes and boundary variables and encode potentials over sets of continuous, discrete or mixture of both types of variables. [sent-248, score-0.175]
81 VZ-ratio: We define truncated quadratic potentials for each segment encoding that the plane should agree with the epipolar flow estimated using the algorithm from section 3. [sent-250, score-1.19]
82 Boundary: We employ 3-way potentials linking our discrete and continuous variables expressing the fact that when 111888666644 O u rs MSPGCotBiMPo-nFSlLoIwCN76o. [sent-251, score-0.191]
83 two neighboring planes are hinge or coplanar they should agree on the boundary, and when a segment occludes another, the boundary should be explained by the occluder. [sent-276, score-0.314]
84 Color similarity: This potential encodes the fact that we expect segments which are coplanar to have similar color statistics, while the entropy is higher when the planes form an occlusion boundary or a hinge. [sent-284, score-0.222]
85 As shown in Table 1, our approach significantly outperforms all approaches, yielding approximately half the error of the best general flow algorithm, and a third of the error of the best epipolar flow algorithm, i. [sent-312, score-1.529]
86 The error of the best oracle match along the epipolar line when employing our estimated FOE is also very small. [sent-355, score-0.733]
87 In “Oracle GT”, ground truth flow vectors are converted into VZ-index values using the epipolar lines estimated from ground truth, and VZ-index planes are fitted to the superpixel segments, which are generated by motionSLIC. [sent-356, score-1.181]
88 In “Oracle estimated”, flow vectors of ground truth are converted to VZ-index values using our estimated epipolar lines. [sent-357, score-1.008]
89 Our first assumption is that most of the flow is due to the ego-motion. [sent-362, score-0.485]
90 When utilizing our estimated FOE (via SIFT matching and 8-point algorithm with RANSAC), the error of the best oracle match along the epipolar line is also very small. [sent-364, score-0.77]
91 Note that given the ground truth epipolar lines (“Oracle GT”), the piece-wise planar assumption is fairly accurate. [sent-367, score-0.575]
92 When the epipolar lines are estimated by our ego-motion estimation (“Oracle estimated”), the piecewise planar assumption becomes worse, but is still a good fit. [sent-368, score-0.637]
93 Stereo: Our MotionSLIC algorithm can be utilized for stereo vision in order to compute disparities and segmentations that respect depth boundaries. [sent-373, score-0.182]
94 Conclusion and Future Work We have presented a slanted-plane MRF model for the problem of epipolar flow estimation which utilizes a robust data term as well as an over-segmentation of the image that respects flow boundaries. [sent-380, score-1.503]
95 We have demonstrated the effectiveness of our approach in the challenging KITTI flow benchmark, achieving half the error of the best competing general flow algorithm and one third of the error of the best competing epipolar flow algorithm. [sent-381, score-2.079]
96 High accuracy optical flow estimation based on a theory for warping. [sent-417, score-0.574]
97 Large displacement optical flow: Descriptor matching in variational motion estimation. [sent-422, score-0.155]
98 Hierarchical scan line dynamic programming for optical flow using semi-global matching. [sent-488, score-0.567]
99 Optical flow estimation on coarse-to-fine region-trees using discrete optimization. [sent-524, score-0.525]
100 Optic flow goes stereo: a variational method for estimating discontinuitypreserving dense disparity maps. [sent-579, score-0.579]
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