iccv iccv2013 iccv2013-304 knowledge-graph by maker-knowledge-mining
Source: pdf
Author: Philipp Heise, Sebastian Klose, Brian Jensen, Alois Knoll
Abstract: Most stereo correspondence algorithms match support windows at integer-valued disparities and assume a constant disparity value within the support window. The recently proposed PatchMatch stereo algorithm [7] overcomes this limitation of previous algorithms by directly estimating planes. This work presents a method that integrates the PatchMatch stereo algorithm into a variational smoothing formulation using quadratic relaxation. The resulting algorithm allows the explicit regularization of the disparity and normal gradients using the estimated plane parameters. Evaluation of our method in the Middlebury benchmark shows that our method outperforms the traditional integer-valued disparity strategy as well as the original algorithm and its variants in sub-pixel accurate disparity estimation.
Reference: text
sentIndex sentText sentNum sentScore
1 de se l Abstract Most stereo correspondence algorithms match support windows at integer-valued disparities and assume a constant disparity value within the support window. [sent-3, score-0.886]
2 The recently proposed PatchMatch stereo algorithm [7] overcomes this limitation of previous algorithms by directly estimating planes. [sent-4, score-0.306]
3 This work presents a method that integrates the PatchMatch stereo algorithm into a variational smoothing formulation using quadratic relaxation. [sent-5, score-0.427]
4 The resulting algorithm allows the explicit regularization of the disparity and normal gradients using the estimated plane parameters. [sent-6, score-0.707]
5 Evaluation of our method in the Middlebury benchmark shows that our method outperforms the traditional integer-valued disparity strategy as well as the original algorithm and its variants in sub-pixel accurate disparity estimation. [sent-7, score-0.973]
6 Introduction Most stereo matching algorithms are based on the assumption that the pixels within the matching window share the same disparity value. [sent-9, score-0.772]
7 Further very often only discrete disparity values are considered leading to discrete depth layers. [sent-10, score-0.512]
8 One reason for the widespread use of this simplified model is that the number of likelihood evaluations for more precisely sampled disparities and the inclusion of discretized surface orientations quickly becomes intractable. [sent-11, score-0.06]
9 On the other side sub-pixel accurate depth values are necessary to create plausible and precise meshes or point clouds. [sent-12, score-0.078]
10 [7] showed that the PatchMatch algorithm [4, 5] can be applied for stereo matching using slanted support windows so that instead of just estimating a single disparity value for each pixel a complete disparity plane estimation is made. [sent-14, score-1.403]
11 The PatchMatch algorithm does not try to discretize the space of the likelihood function, but rather relies on randomized sampling and propagation of good estimates. [sent-15, score-0.101]
12 This also results in an implicit smoothing model, when good estimates are propagated in the direct neighbourhood. [sent-16, score-0.125]
13 But the implicit smoothing can also lead to problems when wrong or unreliable estimates are propagated. [sent-17, score-0.08]
14 In the Figure 1: Stereo pair taken from [13] and a point cloud created by using the sub-pixel disparity map generated by our algorithm. [sent-18, score-0.578]
15 stereo case this problem can occur in homogeneous untextured regions, regions with repeating structures and extreme sampling choices e. [sent-19, score-0.332]
16 To alleviate these problems an explicit smoothing model based on the combination of PatchMatch and Particle Belief Propagation resulting in the PMBP Algorithm [6] has been recently proposed, leading to improved results compared to the original algorithm. [sent-22, score-0.119]
17 We present an algorithm based on an explicit variational energy formulation combining the PatchMatch stereo algorithm with regularization of the disparity and normal gradients resulting in sub-pixel accurate disparity maps improving the state of the art. [sent-23, score-1.503]
18 Our disparity maps are well suited for the creation of point clouds without discretization or staircasing artifacts as shown in figure 1. [sent-24, score-0.657]
19 Contribution In this paper we show that the projections of scene points belonging to the same planar surface in rectified stereo pairs are fully related by a linear transformation with three degrees of freedom. [sent-27, score-0.421]
20 This has already been shown in [7] for planes in the disparity space and is in the following extended to the real scene space of fully calibrated and rec22336600 tified stereo cameras. [sent-28, score-0.807]
21 Our main contribution is an explicit variational smoothness model for the PatchMatch algorithm using quadratic relaxation [12, 17]. [sent-29, score-0.113]
22 In [17, 14] only the first order derivatives of the optical flow vectors and disparity-values have been considered, but the proposed algorithm allows us to control the smoothness of the first-order and second-order derivatives of the disparities. [sent-30, score-0.139]
23 The second-order derivatives of the disparities are implicitly determined by the gradient of the normals estimated by the PatchMatch algorithm. [sent-31, score-0.223]
24 Evaluation of the proposed method for stereo pairs of the Middlebury benchmark [15] shows its effectiveness in estimating sub-pixel accurate disparity maps. [sent-33, score-0.813]
25 At the time of writing we are currently ranked at position 1out of about 145 algorithms for the sub-pixel error threshold 0. [sent-34, score-0.117]
26 Slanted support windows In [7] the authors showed how planes in the disparity space affect the patch neighbourhood. [sent-39, score-0.528]
27 with the disparity value z0 and a normal n = [nx ny nz ]? [sent-42, score-0.561]
28 we can calculate the d parameter of a plane π = ? [sent-45, score-0.08]
29 Therefore the disparity value z of any image point [x y ]? [sent-54, score-0.498]
30 on the plane is given by z =−nxx − nyy + (nnxxz0+ nyy0+ nzz0). [sent-55, score-0.08]
31 (1) We can reformulate this as a linear transformation assuming that the point in the second image is given by p? [sent-56, score-0.074]
32 with normal n and distance d to for a plane π = ? [sent-71, score-0.175]
33 d stereo camera setup the rotation is the identity I the? [sent-76, score-0.306]
34 Figure 2: Illustration of the shearing and scaling transformation induced by disparity and scene planes. [sent-89, score-0.548]
35 In the following n refers to the non overparametrized representation of the normal containing only two components. [sent-94, score-0.095]
36 If needed the normal with three components can directly be calculated from n since we only consider the normals from one half of the unit sphere1 . [sent-95, score-0.205]
37 The derivatives are calculated from grayscale versions of the stereo images. [sent-108, score-0.359]
38 The function w in equation (8) computes a weighting mask based on the color similarity between the center pixel p and the other pixels q inside the patch w(p,q) = e−γ(p,q)||I1(p)−I1(q)||1. [sent-109, score-0.069]
39 ) The reasoning behind the varying γ is that pixels close to the center belong more likely to the same plane and that pixels far away have to be very similar in terms of their colordistance to get the same consideration. [sent-112, score-0.08]
40 N(p) Our regularization term Esmooth imposes spatial smoothness on the disparity-values d and the normals n resulting in Esmooth(d,n) =? [sent-117, score-0.17]
41 (14) As depth and normal discontinuities often occur at strong image gradients we introduce the per-pixel weighting function g(p) with g(p) = e−ζ|∇I1(p)|η. [sent-128, score-0.179]
42 (17) and Σ = diag(σn, σn, σd) being a diagonal matrix weighting the squared distances of the normals and the disparity values. [sent-136, score-0.614]
43 With the previous result the minimization problem of Eaux with respect to dv can be written as arg min Eaux = arg min sup E(dv , pd) dv dv pd =argdv minspudp? [sent-153, score-0.674]
44 (20) 22336622 after the 1st iteration, the final disparity map and two images with different views of a point-cloud generated using the final disparity map. [sent-163, score-0.932]
45 We take the derivative of E(dv , pd) with respect to dv and p and using the divergence theorem we get ∂E(∂ddvv,pd)=g(p)divpd+ θσd(dv− du) (21) ∂E(∂dpv,dpd)=g(p)∇dv−g(? [sent-164, score-0.145]
46 (22) The formulation of the Eaux minimization with respect to nv is analogous and leads to the following derivatives ∂E(∂nnvv,pn)=g(p)divpn+ θσn(nv− nu) ∂E(∂npv,npn)=g(p)∇nv−g(? [sent-166, score-0.151]
47 pn)pn (23) (24) with pn being the dual variable. [sent-167, score-0.05]
48 To solve the energy minimization with respect to Πv we use gradient descent and ascent as in [14] ptd+1βd− ptd=g(p)∇dvt−g(? [sent-168, score-0.056]
49 pd)ptd+1 (25) dvt+1ν−d dvt (26) = − g(p)divptd+1− θσd(dvt+1− du) ptn+1βn− ptn=g(p)∇nvt−g(? [sent-169, score-0.101]
50 (31) nvt+1=ntv+ νn(θσ1n +nu νn−θ gσ(np)divptn+1) (32) where proj projects back onto the unit sphere proj(x) =max(x1,|x|). [sent-176, score-0.093]
51 th Teh step psierze-ssc rβipd,t νd, nβont asn hde νn we use tthioen nv naulumesof ALG3 reported by Chambolle et al. [sent-181, score-0.1]
52 Fixed Πv, solve for Πu Instead of performing an exhaustive search as done in [17, 14] we employ a variant of the PatchMatch stereo algorithm. [sent-185, score-0.306]
53 We do not follow the sequential pixel processing scheme from [7], but use a completely parallel approach. [sent-189, score-0.059]
54 he S set Sview (p) contains the view propagated particles. [sent-196, score-0.074]
55 Ea Tchhe position p has storage for a few view particles and particles from the other view are propagated if storage is still available. [sent-198, score-0.3]
56 22336633 In figure 3 different stages ofour algorithm are shown for a stereo pair and the corresponding final disparity map together with a generated point cloud. [sent-203, score-0.804]
57 The randomized sampling after the initialisation is clearly visible in the image, but already after one iteration the first samples have been successfully propagated in the neighbourhood. [sent-204, score-0.187]
58 Implementation Details We perform the depth and normal map estimation in both images of the stereo pair. [sent-207, score-0.447]
59 This allows us to perform the view propagation of samples and also left-right consistency checking. [sent-208, score-0.061]
60 Especially in the occluded areas arbitrary particles with inconsistent disparity and normal values are very often persistent. [sent-210, score-0.654]
61 Therefore after each PatchMatch iteration - before we apply the Huber-ROF smoothing - we fill the occluded areas with the next non-occluded plane-particle from the same scanline with the more distant depth value at the occluded position as illustrated in figure 4. [sent-211, score-0.254]
62 Our occlusion checking not only uses the depth values but also the plane normals and allows only disparity differences up to 0. [sent-213, score-0.732]
63 For lookup of the plane parameters in the second image we do not use linear interpolation but nearest neighbour sampling. [sent-215, score-0.08]
64 For the initialisation we found it beneficial slsr Figure 4: The occluded gray area in the first view is filled using the plane parameters from position sl. [sent-217, score-0.207]
65 Resulting in disparity values as indicated by the dotted line. [sent-218, score-0.466]
66 sr although also visible in both views is not chosen, because its plane would result in closer disparity values. [sent-219, score-0.546]
67 to draw normal samples more restrictively and the normals of the first PatchMatch iteration are within the 0. [sent-220, score-0.24]
68 To allow propagation and refinement of t|[hen particles ≤in 0 t. [sent-224, score-0.092]
69 We control the values of θ × during the iterations using the smoothstep function. [sent-227, score-0.057]
70 Each iteration consists of one PatchMatch iteration followed by several inner iterations for smoothing using the weighted Huber-ROF model. [sent-228, score-0.15]
71 Our PatchMatch sampling strategy is completely parallel in contrast to the original PatchMatch stereo algorithm. [sent-232, score-0.391]
72 The runtime of our algorithm highly varies with the parameter settings and number of iterations. [sent-234, score-0.049]
73 For the highquality settings as used for the Middlebury benchmark evaluation our algorithm has a runtime of about 2 minutes. [sent-235, score-0.09]
74 For the PatchMatch stereo algorithm the authors reported a runtime of about 1 minute for an average Middlebury pair [7]. [sent-236, score-0.355]
75 Different settings for our algorithm allow the estimation of disparity maps in a few seconds. [sent-237, score-0.501]
76 Method Parameters In the following we assume that the values of the stereo image channels are in the range [0, 1] . [sent-241, score-0.306]
77 The value of · σn starts at 0 and goes up to 50 with an additviaolnuael oofff sθet · oσf 5 for the weighted Huber-ROF smoothing of the normals. [sent-263, score-0.08]
78 For the intermediate disparity maps we use a range from 0 to 1, therefore · σd takes values between 0 θ θ manadxidmm5u0axmag alalionw ewditdh i asnpa srpiteyci va l uoef. [sent-264, score-0.501]
79 Evaluation For the evaluation of our algorithm we use the Middlebury stereo benchmark [15, 1]. [sent-268, score-0.347]
80 Our results for the Middlebury stereo benchmark were made using constant parameters as described in the previous section. [sent-269, score-0.347]
81 The maximum allowed disparity was fixed to 60 and used for all four pairs. [sent-270, score-0.466]
82 This shows that our algorithm does not necessarily need to know the disparity range in advance. [sent-271, score-0.466]
83 At the time of writing we are currently ranked at position 1out of about 145 algorithms for the sub-pixel error threshold 0. [sent-274, score-0.117]
84 We achieve results comparable or better than the original PatchMatch stereo implementation [7] and the PMBP method [6] that also has an explicit smoothing model. [sent-276, score-0.425]
85 The final disparity maps and also the error maps for the 0. [sent-277, score-0.536]
86 222 Table 1: First three entries from the Middlebury stereo benchmark [15] and additionally the results from PMBP [6] and the original PatchMatch-Stereo [7] algorithm. [sent-361, score-0.347]
87 Our algorithm is currently ranked at position 1out of about 145 algorithms for the error-threshold 0. [sent-362, score-0.082]
88 From top to bottom: Middlebury stereo pairs [15] Tsukuba, Venus, Teddy and Cones. [sent-365, score-0.306]
89 To empha- size the sub-pixel accuracy of our algorithm we also created point clouds of some Middlebury datasets that contain planar and curved surfaces as depicted in figure 6. [sent-367, score-0.171]
90 Also the curved surface of the platform in the Baby1 scene is very smooth and does not exhibit staircasing or discrete depth layer effects. [sent-369, score-0.122]
91 For the Phong shaded point clouds the normals estimated by our algorithm have been used instead of estimating them using neighbouring vertices. [sent-370, score-0.251]
92 Videos of the point clouds can be found the supplementary material. [sent-371, score-0.109]
93 In order to show that our algorithm also works for more realistic data we tested it using two rectified and downscaled images from Strecha et al. [sent-372, score-0.073]
94 Another point cloud created from our disparity maps is shown in figure 1. [sent-375, score-0.613]
95 Figure 6: Colored and Phong shaded point clouds of the Middlebury datasets Art, Baby1, Cones and Cloth3 [16, 11]. [sent-376, score-0.141]
96 Figure 7: A point cloud created from a rectified stereo pair. [sent-377, score-0.491]
97 Conclusion We presented an new approach to combine the randomized sampling of the PatchMatch algorithm with an explicit variational smoothing method that gives control of the dis- parity and normal gradients. [sent-381, score-0.324]
98 Our evaluation shows that we achieve very good sub-pixel results in the Middlebury benchmark that make our algorithm well suited for the generation of point clouds or meshes. [sent-382, score-0.15]
99 The estimated normals are also maybe useful for depthmap merging and multiview reconstruction. [sent-384, score-0.11]
100 PatchMatch: a randomized correspondence algorithm for structural image editing. [sent-412, score-0.07]
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Abstract: We propose a fundamentally novel approach to real-time visual odometry for a monocular camera. It allows to benefit from the simplicity and accuracy of dense tracking which does not depend on visual features while running in real-time on a CPU. The key idea is to continuously estimate a semi-dense inverse depth map for the current frame, which in turn is used to track the motion of the camera using dense image alignment. More specifically, we estimate the depth of all pixels which have a non-negligible image gradient. Each estimate is represented as a Gaussian probability distribution over the inverse depth. We propagate this information over time, and update it with new measurements as new images arrive. In terms of tracking accuracy and computational speed, the proposed method compares favorably to both state-of-the-art dense and feature-based visual odometry and SLAM algorithms. 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We use the term visual odometry as supposed to SLAM, as for simplicity we deliberately maintain only information about the currently visible scene, instead of building a global world-model. – – 1.1. Related Work Feature-based monocular SLAM. In all feature-based methods (such as [4, 8]), tracking and mapping consists of two separate steps: First, discrete feature observations (i.e., their locations in the image) are extracted and matched to each other. Second, the camera and the full feature poses are calculated from a set of such observations disregarding the images themselves. While this preliminary abstrac– tion step greatly reduces the complexity of the overall problem and allows it to be tackled in real time, it inherently comes with two significant drawbacks: First, only image information conforming to the respective feature type and parametrization typically image corners and blobs [6] or line segments [9] is utilized. Second, features have to be matched to each other, which often requires the costly computation of scale- and rotation-invariant descriptors and robust outlier estimation methods like RANSAC. – – Dense monocular SLAM. To overcome these limitations and to better exploit the available image information, dense monocular SLAM methods [11, 17] have recently been proposed. The fundamental difference to keypoint-based approaches is that these methods directly work on the images 11444499 instead of a set of extracted features, for both mapping and tracking: The world is modeled as dense surface while in turn new frames are tracked using whole-image alignment. This concept removes the need for discrete features, and allows to exploit all information present in the image, increasing tracking accuracy and robustness. To date however, doing this in real-time is only possible using modern, powerful GPU processors. Similar methods are broadly used in combination with RGB-D cameras [7], which directly measure the depth of each pixel, or stereo camera rigs [3] greatly reducing the – complexity of the problem. Dense multi-view stereo. Significant prior work exists on multi-view dense reconstruction, both in a real-time setting [13, 11, 15], as well as off-line [5, 14]. In particular for offline reconstruction, there is a long history of using different baselines to steer the stereo-inherent trade-off between accuracy and precision [12]. Most similar to our approach is the early work of Matthies et al., who proposed probabilistic depth map fusion and propagation for image sequences [10], however only for structure from motion, i.e., not coupled with subsequent dense tracking. 1.2. Contributions In this paper, we propose a novel semi-dense approach to monocular visual odometry, which does not require feature points. The key concepts are • a probabilistic depth map representation, • tracking based on whole-image alignment, • the reduction on image-regions which carry informattihoen (esdeumctii-odenn osen), i manadg • the full incorporation of stereo measurement uncertainty. To the best of our knowledge, this is the first featureless, real-time monocular visual odometry approach, which runs in real-time on a CPU. 1.3. Method Outline Our approach is partially motivated by the basic principle that for most real-time applications, video information is abundant and cheap to come by. Therefore, the computational budget should be spent such that the expected information gain is maximized. Instead of reducing the images to a sparse set of feature observations however, our method continuously estimates a semi-dense inverse depth map for the current frame, i.e., a dense depth map covering all image regions with non-negligible gradient (see Fig. 2). It is comprised of one inverse depth hypothesis per pixel modeled by a Gaussian probability distribution. This representation still allows to use whole-image alignment [7] to track new orignalimagesemi-densedepthmap(ours)clfoasre keypointdepthmap[8]densedepthmap[1 ]RGB-Dcamera[16] Figure 2. Semi-Dense Approach: Our approach reconstructs and tracks on a semi-dense inverse depth map, which is dense in all image regions carrying information (top-right). For comparison, the bottom row shows the respective result from a keypoint-based approach, a fully dense approach and the ground truth from an RGB-D camera. frames, while at the same time greatly reducing computational complexity compared to volumetric methods. The estimated depth map is propagated from frame to frame, and updated with variable-baseline stereo comparisons. We explicitly use prior knowledge about a pixel’s depth to select a suitable reference frame on a per-pixel basis, and to limit the disparity search range. The remainder of this paper is organized as follows: Section 2 describes the semi-dense mapping part of the proposed method, including the derivation of the observation accuracy as well as the probabilistic data fusion, propagation and regularization steps. Section 3 describes how new frames are tracked using whole-image alignment, and Sec. 4 summarizes the complete visual odometry method. A qualitative as well as a quantitative evaluation is presented in Sec. 5. We then give a brief conclusion in Sec. 6. 2. Semi-Dense Depth Map Estimation One of the key ideas proposed in this paper is to estimate a semi-dense inverse depth map for the current camera image, which in turn can be used for estimating the camera pose of the next frame. This depth map is continuously propagated from frame to frame, and refined with new stereo depth measurements, which are obtained by performing per-pixel, adaptive-baseline stereo comparisons. This allows us to accurately estimate the depth both of close-by and far-away image regions. In contrast to previous work that accumulates the photometric cost over a sequence of several frames [11, 15], we keep exactly one inverse depth hypothesis per pixel that we represent as Gaussian probability distribution. This section is comprised of three main parts: Sec11445500 reference small baseline medium baseline large baseline tcso0120 .050.10.150.20.2sl5m areagdleiulm0.3 inverse depth d Figure 3. Variable Baseline Stereo: Reference image (left), three stereo images at different baselines (right), and the respective matching cost functions. While a small baseline (black) gives a unique, but imprecise minimum, a large baseline (red) allows for a very precise estimate, but has many false minima. tion 2. 1 describes the stereo method used to extract new depth measurements from previous frames, and how they are incorporated into the prior depth map. In Sec. 2.2, we describe how the depth map is propagated from frame to frame. In Sec. 2.3, we detail how we partially regularize the obtained depth map in each iteration, and how outliers are handled. Throughout this section, d denotes the inverse depth of a pixel. 2.1. Stereo-Based Depth Map Update It is well known [12] that for stereo, there is a trade-off between precision and accuracy (see Fig. 3). While many multiple-baseline stereo approaches resolve this by accumulating the respective cost functions over many frames [5, 13], we propose a probabilistic approach which explicitly takes advantage of the fact that in a video, smallbaseline frames are available before large-baseline frames. The full depth map update (performed once for each new frame) consists of the following steps: First, a subset of pixels is selected for which the accuracy of a disparity search is sufficiently large. For this we use three intuitive and very efficiently computable criteria, which will be derived in Sec. 2. 1.3. For each selected pixel, we then individually select a suitable reference frame, and perform a onedimensional disparity search. Propagated prior knowledge is used to reduce the disparity search range when possible, decreasing computational cost and eliminating false minima. The obtained inverse depth estimate is then fused into the depth map. 2.1.1 Reference Frame Selection Ideally, the reference frame is chosen such that it maximizes the stereo accuracy, while keeping the disparity search range as well as the observation angle sufficiently cur ent framepixel’s “age” -4.8 s -3.9 s -3.1 s -2.2 s -1.2 s -0.8 s -0.5 s -0.4 s Figure 4. Adaptive Baseline Selection: For each pixel in the new frame (top left), a different stereo-reference frame is selected, based on how long the pixel was visible (top right: the more yellow, the older the pixel.). Some of the reference frames are displayed below, the red regions were used for stereo comparisons. small. As the stereo accuracy depends on many factors and because this selection is done for each pixel independently, we employ the following heuristic: We use the oldest frame the pixel was observed in, where the disparity search range and the observation angle do not exceed a certain threshold (see Fig. 4). If a disparity search is unsuccessful (i.e., no good match is found), the pixel’s “age” is increased, such that subsequent disparity searches use newer frames where the pixel is likely to be still visible. 2.1.2 Stereo Matching Method We perform an exhaustive search for the pixel’s intensity along the epipolar line in the selected reference frame, and then perform a sub-pixel accurate localization of the matching disparity. If a prior inverse depth hypothesis is available, the search interval is limited by d 2σd, where d and σd de,e nthoete s etharec mean avnadl ssta lnimdaiterdd d beyv dia ±tion 2σ σof the prior hypothesis. Otherwise, the full disparity range is searched. In our implementation, we use the SSD error over five equidistant points on the epipolar line: While this significantly increases robustness in high-frequent image regions, it does not change the purely one-dimensional nature of this search. Furthermore, it is computationally efficient, as 4 out ± of 5 interpolated image values can be re-used for each SSD evaluation. 2.1.3 Uncertainty Estimation In this section, we use uncertainty propagation to derive an expression for the error variance σd2 on the inverse depth d. 11445511 In general this can be done by expressing the optimal inverse depth d∗ as a function of the noisy inputs here we consider the images I0, I1 themselves, their relative orientation ξ and the camera calibration in terms of a projection function π1 – d∗ = d(I0, I1, ξ, π) . The error-variance of d∗ is then given by σd2 = JdΣJdT, (1) (2) where Jd is the Jacobian of d, and Σ the covariance of the input-error. For more details on covariance propagation, including the derivation of this formula, we refer to [2]. For simplicity, the following analysis is performed for patchfree stereo, i.e., we consider only a point-wise search for a single intensity value along the epipolar line. For this analysis, we split the computation into three steps: First, the epipolar line in the reference frame is computed. Second, the best matching position λ∗ ∈ R along it (i.e., the disparity) is determined. Third, the i∈nv eRrse al depth d∗ is computed from the disparity λ∗ . The first two steps involve two independent error sources: the geometric error, which originates from noise on ξ and π and affects the first step, and the photometric error, which originates from noise in the images I0, I1 and affects the second step. The third step scales these errors by a factor, which depends on the baseline. Geometric disparity error. The geometric error is the error ?λ on the disparity λ∗ caused by noise on ξ and π. While it would be possible to model, propagate, and estimate the complete covariance on ξ and π, we found that the gain in accuracy does not justify the increase in computational complexity. We therefore use an intuitive approximation: Let the considered epipolar line segment L ⊂ R2 be deLfineted th by L := ?l0 + λ?llyx? |λ ∈ S? , (3) where λ is the disparity with search interval S, (lx , ly)T the normalized epipolar line direction and l0 the point corresponding to infinite depth. We now assume that only the absolute position of this line segment, i.e., l0 is subject to isotropic Gaussian noise ?l . As in practice we keep the searched epipolar line segments short, the influence of rotational error is small, making this a good approximation. Intuitively, a positioning error ?l on the epipolar line causes a small disparity error ?λ if the epipolar line is parallel to the image gradient, and a large one otherwise (see Fig. 5). This can be mathematically derived as follows: The image constrains the optimal disparity λ∗ to lie on a certain isocurve, i.e. a curve of equal intensity. We approximate 1In the linear case, this is the camera matrix K – in practice however, nonlinear distortion and other (unmodeled) effects also play a role. FiguLre5.Geo?l mλetricDigs,palrityEroL?rl:Influe?nλceofgasmla posi- tioning error ?l of the epipolar line on the disparity error ?λ . The dashed line represents the isocurve on which the matching point has to lie. ?λ is small if the epipolar line is parallel to the image gradient (left), and a large otherwise (right). this isocurve to be locally linear, i.e. the gradient direction to be locally constant. This gives l0 + λ∗ ?llxy? =! + γ?−gxgy?, g0 γ ∈ R (4) where g := (gx , gy) ?is the image gradient and g0 a point on the isoline. The influence of noise on the image values will be derived in the next paragraph, hence at this point g and g0 are assumed noise-free. Solving for λ gives the optimal disparity λ∗ in terms of the noisy input l0: λ∗(l0) =?g,g?g0,−l? l0? (5) Analogously to (2), the variance of the geometric disparity error can then be expressed as σλ2(ξ,π)= Jλ∗(l0)?σ0l2 σ0l2?JλT∗(l0)=?gσ,l 2?2, (6) where g is the normalized image gradient, lthe normalized epipolar line direction and σl2 the variance of ?l. Note that this error term solely originates from noise on the relative camera orientation and the camera calibration π, i.e., it is independent of image intensity noise. ξ Photometric disparity error. Intuitively, this error encodes that small image intensity errors have a large effect on the estimated disparity if the image gradient is small, and a small effect otherwise (see Fig. 6). Mathematically, this relation can be derived as follows. We seek the disparity λ∗ that minimizes the difference in intensities, i.e., λ∗ = mλin (iref − Ip(λ))2, (7) where iref is the reference intensity, and Ip(λ) the image intensity on the epipolar line at disparity λ. We assume a good initialization λ0 to be available from the exhaustive search. Using a first-order Taylor approximation for Ip gives λ∗(I) = λ0 + (iref − Ip(λ0)) g−p1, (8) where gp is the gradient of Ip, that is image gradient along the epipolar line. For clarity we only consider noise on iref and Ip(λ0) ; equivalent results are obtained in the general case when taking into account noise on the image values involved in the computation of gp. The variance of the pho11445522 ?i Ip?λ ?iiIp?λλ Figure 6. Photometric Disparity Error: Noise ?i on the image intensity values causes a small disparity error ?λ if the image gradient along the epipolar line is large (left). If the gradient is small, the disparity error is magnified (right). tometric disparity error is given by σλ2(I) = Jλ∗(I)?σ0i2 σ0i2?Jλ∗(I) =2gσ2pi2, (9) where σi2 is the variance of the image intensity noise. The respective error originates solely from noisy image intensity values, and hence is independent of the geometric disparity error. Pixel to inverse depth conversion. Using that, for small camera rotation, the inverse depth d is approximately proportional to the disparity λ, the observation variance of the inverse depth σd2,obs can be calculated using σd2,obs = α2 ?σ2λ(ξ,π) + σλ2(I)? , (10) where the proportionality ?constant α in th?e general, nonrectified case – is different for each pixel, and can be calculated from – α :=δδdλ, (11) where δd is the length of the searched inverse depth interval, and δλ the length of the searched epipolar line segment. While α is inversely linear in the length of the camera translation, it also depends on the translation direction and the pixel’s location in the image. When using an SSD error over multiple points along the epipolar line – as our implementation does – a good upper bound for the matching uncertainty is then given by ?min{σ2λ(ξ,π)} + min{σλ2(I)}? σd2,obs-SSD ≤ α2 , (12) where the min goes over all points included in the? SSD error. 2.1.4 Depth Observation Fusion After a depth observation for a pixel in the current image has been obtained, we integrate it into the depth map as follows: If no prior hypothesis for a pixel exists, we initialize it directly with the observation. Otherwise, the new observation is incorporated into the prior, i.e., the two distribu- tions are multiplied (corresponding to the update step in a Knoailsmya onb fsieltrvera)t:io Gniv Nen(do a, pσrio2o)r, d thiest priobsutetiroionr N is( gdipv,eσnp2 b)y and a N?σ2pdσo2p++ σ σo2o2dp,σ2σpp2+σo2 σo2?. 2.1.5 (13) Summary of Uncertainty-Aware Stereo New stereo observations are obtained on a per-pixel basis, adaptively selecting for each pixel a suitable reference frame and performing a one-dimensional search along the epipolar line. We identified the three major factors which determine the accuracy of such a stereo observation, i.e., • the photometric disparity error σλ2(ξ,π), depending on tphheo magnitude sofp trhiet image gradient along the epipolar line, • the geometric disparity error σλ2(I) ,depending on the athnegl gee bometewtereinc dthisep image gradient and the epipolar line (independent of the gradient magnitude), and • the pixel to inverse depth ratio α, depending on the camera etlra tons ilantvioenrs, eth dee pfothcal r length ,a dndep tehned pixel’s position. These three simple-to-compute and purely local criteria are used to determine for which pixel a stereo update is worth the computational cost. Further, the computed observation variance is then used to integrate the new measurements into the existing depth map. 2.2. Depth Map Propagation We continuously propagate the estimated inverse depth map from frame to frame, once the camera position of the next frame has been estimated. Based on the inverse depth estimate d0 for a pixel, the corresponding 3D point is calculated and projected into the new frame, providing an inverse depth estimate d1 in the new frame. The hypothesis is then assigned to the closest integer pixel position to eliminate discretization errors, the sub-pixel accurate image location of the projected point is kept, and re-used for the next propagation step. For propagating the inverse depth variance, we assume the camera rotation to be small. The new inverse depth d1 can then be approximated by – d1(d0) = (d0−1 − tz)−1, (14) where tz is the camera translation along the optical axis. The variance of d1 is hence given by σd21= Jd1σd20JTd1+ σp2=?dd01?4σd20+ σp2, (15) where σp2 is the prediction uncertainty, which directly corresponds to the prediction step in an extended Kalman filter. It can also be interpreted as keeping the variance on 11445533 in the top right shows the new frame I2 (x) without depth information. Middle: Intermediate steps while minimizing E(ξ) on different pyramid levels. The top row shows the back-warped new frame I2 (w(x, d, ξ)), the bottom row shows the respective residual image I2 (w(x, di,ξ)) − I1 (x) . The bottom right image shows the final pixel-weights (black = small weight). Small weights mainly correspond to newly oc,cξl)ud)e −d or disoccluded pixel. tWhe z fo-cuonodrtd hina t uesi onfg a sm poailnlt v failxue ds, fo i.re. σ,p2 sedteticnrgea σsez2s0 d=rift σ,z2 a1s. it causes the estimated geometry to gradually ”lock” into place. Collision handling. At all times, we allow at most one inverse depth hypothesis per pixel: If two inverse depth hypothesis are propagated to the same pixel in the new frame, we distinguish between two cases: 1. if they are statistically similar, i.e., lie within 2σ bounds, they are treated as two independent observations of the pixel’s depth and fused according to (13). 2. otherwise, the point that is further away from the camera is assumed to be occluded, and is removed. 2.3. Depth Map Regularization For each frame – after all observations have been incorporated – we perform one regularization iteration by assign- ing each inverse depth value the average of the surrounding inverse depths, weighted by their respective inverse variance. To preserve sharp edges, if two adjacent inverse depth values are statistically different, i.e., are further away than 2σ, they do not contribute to one another. Note that the respective variances are not changed during regularization to account for the high correlation between neighboring hypotheses. Instead we use the minimal variance of all neighboring pixel when defining the stereo search range, and as a weighting factor for tracking (see Sec. 3). Outlier removal. To handle outliers, we continuously keep track of the validity of each inverse depth hypothesis in terms of the probability that it is an outlier, or has become invalid (e.g., due to occlusion or a moving object). For each successful stereo observation, this probability is decreased. It is increased for each failed stereo search, if the respective intensity changes significantly on propagation, or when the absolute image gradient falls below a given threshold. If, during regularization, the probability that all contributing neighbors are outliers i.e., the product of their individual outlier-probabilities rises above a given threshold, the hypothesis is removed. Equally, if for an “empty” pixel this product drops below a given threshold, a new hypothesis is created from the neighbors. This fills holes arising from the forward-warping nature of the propagation step, and dilates the semi-dense depth map to a small neighborhood around sharp image intensity edges, which signifi– – × cantly increases tracking and mapping robustness. 3. Dense Tracking Based on the inverse depth map of the previous frame, we estimate the camera pose of the current frame using dense image alignment. Such methods have previously been applied successfully (in real-time on a CPU) for tracking RGB-D cameras [7], which directly provide dense depth measurements along with the color image. It is based on the direct minimization of the photometric error ri (ξ) := (I2 (w(xi, di , ξ)) − I1 , (16) where the warp function w : Ω1 R R6 → Ω2 maps each point xi ∈ Ω1 in the reference× image RI1 →to Ωthe respective point w(x∈i, Ωdi, ξ) ∈ Ω2 in the new image I2. As input it requires the 3D,ξ pose Ωof the camera ξ ∈ R6 and uses the reestqiumiraetesd t hienv 3erDse p depth fd it ∈e cRa mfore rthae ξ pixel in I1. Note that no depth information with respect t toh Ie2 p i sx required. To increase robustness to self-occlusion and moving objects, we apply a weighting scheme as proposed in [7]. Further, we add the variance of the inverse depth σd2i as an additional weighting term, making the tracking resistant to recently initialized and still inaccurate depth estimates from 11445544 (xi))2 Figure 8. Examples: Top: Camera images overlaid with the respective stimated semi-dense inverse depth map. Bot om: 3D view of tracked scene. Note the versatility of our approach: It accurately reconstructs and tracks through (outside) scenes with a large depth- variance, including far-away objects like clouds , as well as (indoor) scenes with little structure and close to no image corners / keypoints. More examples are shown in the attached video. the mapping process. The final energy that is minimized is hence given by E(ξ) :=?iα(rσid2(iξ))ri(ξ), (17) where α : R → R defines the weight for a given residual. Minimizing t h→is error can b thee interpreted as computing uthale. maximum likelihood estimator for ξ, assuming independent noise on the image intensity values. The resulting weighted least-squares problem is solved efficiently using an iteratively reweighted Gauss-Newton algorithm coupled with a coarse-to-fine approach, using four pyramid levels. Figure 7 shows an example of the tracking process. For further details on the minimization we refer to [1]. 4. System Overview Tracking and depth estimation is split into two separate threads: One continuously propagates the inverse depth map to the most recent tracked frame, updates it with stereocomparisons and partially regularizes it. The other simultaneously tracks each incoming frame on the most recent available depth map. While tracking is performed in real- time at 30Hz, one complete mapping iteration takes longer and is hence done at roughly 15Hz if the map is heavily populated, we adaptively reduce the number of stereo comparisons to maintain a constant frame-rate. For stereo observations, a buffer of up to 100 past frames is kept, automatically removing those that are used least. We use a standard, keypoint-based method to obtain the relative camera pose between two initial frames, which are then used to initialize the inverse depth map needed for tracking successive frames. From this point onward, our method is entirely self-contained. In preliminary experiments, we found that in most cases our approach is even able to recover from random or extremely inaccurate initial depth maps, indicating that the keypoint-based initialization might become superfluous in the future. Table 1. Results on RGB-D Benchmark position drift (cm/s) rotation drift (deg/s) ours [7] [8] ours [7] [8] – fr2/xyz fr2/desk 0.6 2.1 0.6 2.0 8.2 - 0.33 0.65 0.34 0.70 3.27 - 5. Results We have tested our approach on both publicly available benchmark sequences, as well as live, using a hand-held camera. Some examples are shown in Fig. 8. Note that our method does not attempt to build a global map, i.e., once a point leaves the field of view of the camera or becomes occluded, the respective depth value is deleted. All experiments are performed on a standard consumer laptop with Intel i7 quad-core CPU. In a preprocessing step, we rectify all images such that a pinhole camera-model can be applied. 5.1. RGB-D Benchmark Sequences As basis for a quantitative evaluation and to facilitate reproducibility and easy comparison with other methods, we use the TUM RGB-D benchmark [16]. For tracking and mapping we only use the gray-scale images; for the very first frame however the provided depth image is used as initialization. Our method (like any monocular visual odometry method) fails in case of pure camera rotation, as the depth of new regions cannot be determined. The achieved tracking accuracy for two feasible sequences that is, sequences which do not contain strong camera rotation without simultaneous translation is given in Table 1. For comparison we also list the accuracy from (1) a state-of-the-art, dense RGB-D odometry [7], and (2) a state-of-the-art, keypointbased monocular SLAM system (PTAM, [8]). We initialize PTAM using the built-in stereo initializer, and perform a 7DoF (rigid body plus scale) alignment to the ground truth trajectory. Figure 9 shows the tracked camera trajectory for fr2/desk. We found that our method achieves similar accu– – 11445555 era the the the trajectory (black), the depth map of the first frame (blue), and estimated depth map (gray-scale) after a complete loop around table. Note how well certain details such as the keyboard and monitor align. racy as [7] which uses the same dense tracking algorithm but relies on the Kinect depth images. The keypoint-based approach [8] proves to be significantly less accurate and robust; it consistently failed after a few seconds for the second sequence. 5.2. Additional Test Sequences To analyze our approach in more detail, we recorded additional challenging sequences with the corresponding ground truth trajectory in a motion capture studio. Figure 10 shows an extract from the video, as well as the tracked and the ground-truth camera position over time. As can be seen from the figure, our approach is able to maintain a reasonably dense depth map at all times and the estimated camera trajectory matches closely the ground truth. 6. Conclusion In this paper we proposed a novel visual odometry method for a monocular camera, which does not require discrete features. In contrast to previous work on dense tracking and mapping, our approach is based on probabilistic depth map estimation and fusion over time. Depth measurements are obtained from patch-free stereo matching in different reference frames at a suitable baseline, which are selected on a per-pixel basis. To our knowledge, this is the first featureless monocular visual odometry method which runs in real-time on a CPU. In our experiments, we showed that the tracking performance of our approach is comparable to that of fully dense methods without requiring a depth sensor. References [1] S. Baker and I. Matthews. Lucas-Kanade 20 years on: A unifying framework. Technical report, Carnegie Mellon Univ., 2002. 7 [2] A. Clifford. Multivariate Error Analysis. John Wiley & Sons, 1973. 4 sionpito[m ]− 024 2 0 s1xzy0s20s30s40s50s60s Figure 10. Additional Sequence: Estimated camera trajectory and ground truth (dashed) for a long and challenging sequence. The complete sequence is shown in the attached video. [3] A. Comport, E. Malis, and P. Rives. Accurate quadri-focal tracking for robust 3d visual odometry. In ICRA, 2007. 2 [4] A. Davison, I. Reid, N. Molton, and O. Stasse. MonoSLAM: Real-time single camera SLAM. Trans. on Pattern Analysis and Machine Intelligence (TPAMI), 29, 2007. 1 [5] D. Gallup, J. Frahm, P. Mordohai, and M. Pollefeys. Variable baseline/resolution stereo. In CVPR, 2008. 2, 3 [6] C. Harris and M. Stephens. A combined corner and edge detector. In Alvey Vision Conference, 1988. 1 [7] C. Kerl, J. Sturm, and D. Cremers. Robust odometry estimation for RGB-D cameras. In ICRA, 2013. 1, 2, 6, 7, 8 [8] G. Klein and D. Murray. Parallel tracking and mapping for small AR workspaces. In Mixed and Augmented Reality (ISMAR), 2007. 1, 2, 7, 8 [9] G. Klein and D. Murray. Improving the agility of keyframebased SLAM. In ECCV, 2008. 1 [10] M. Pollefes et al. Detailed real-time urban 3d reconstruction from video. IJCV, 78(2-3): 143–167, 2008. 2, 3 [11] L. Matthies, R. Szeliski, and T. Kanade. Incremental estimation of dense depth maps from image image sequences. In CVPR, 1988. 2 [12] R. Newcombe, S. Lovegrove, and A. Davison. DTAM: Dense tracking and mapping in real-time. In ICCV, 2011. 1, 2 [13] M. Okutomi and T. Kanade. A multiple-baseline stereo. Trans. on Pattern Analysis and Machine Intelligence (TPAMI), 15(4):353–363, 1993. 2, 3 [14] T. Sato, M. Kanbara, N. Yokoya, and H. Takemura. Dense 3-d reconstruction of an outdoor scene by hundreds-baseline stereo using a hand-held camera. IJCV, 47: 1–3, 2002. 2 [15] J. Stuehmer, S. Gumhold, and D. Cremers. Real-time dense geometry from a handheld camera. In Pattern Recognition (DAGM), 2010. 1, 2 [16] J. Sturm, N. Engelhard, F. Endres, W. Burgard, and D. Cremers. A benchmark for the evaluation of RGB-D SLAM systems. In Intelligent Robot Systems (IROS), 2012. 2, 7 [17] A. Wendel, M. Maurer, G. Graber, T. Pock, and H. Bischof. Dense reconstruction on-the-fly. In ECCV, 2012. 1 11445566
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simIndex simValue paperId paperTitle
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