cvpr cvpr2013 cvpr2013-26 knowledge-graph by maker-knowledge-mining
Source: pdf
Author: Andrew Predoehl, Scott Morris, Kobus Barnard
Abstract: unkown-abstract
Reference: text
sentIndex sentText sentNum sentScore
1 We combine that with a prior model of trail length and smoothness, yielding a posterior distribution for trails, given an image. [sent-7, score-0.889]
2 Our experiments, on trail images and groundtruth collected in the western continental USA, show substantial improvement over those of the previous best trail-finding method. [sent-9, score-0.974]
3 Introduction Recreational trails like the one shown in Figure 1a represent a challenge for computer vision. [sent-11, score-0.164]
4 Automatic identification of trails would be useful not only for bikers, hikers, and land managers, but also as a model problem in other domains that depend on identifying locally-linear structures in images, such as blood vessels or neurons grown in vitro. [sent-15, score-0.244]
5 In this paper we present a statistical model describing trails and trail images, based on a segmentation of images by textures. [sent-16, score-1.032]
6 1c), and, when it does so, the direction in which that trail is oriented (Fig. [sent-19, score-0.884]
7 Example trail (a), trail image (b), and learned characteristics of image textons (c, d). [sent-28, score-1.785]
8 In (c), color indicates the ratio of frequencies, for each superpixel’s texton label, at which that texton generates on- and off-trail pixels. [sent-29, score-0.236]
9 In (d), hue and saturation show the trail direction of each superpixel’s texton, conditioned on it generating a trail pixel. [sent-30, score-1.832]
10 ) In order to infer the route of a latent trail between two known endpoints, we use a sampling approach to search this posterior for a good value. [sent-33, score-0.962]
11 [29], we build a graph from the image evidence, and generate inference proposals as simple paths in this graph. [sent-36, score-0.17]
12 1) we produce a graph in which short paths bgeetw weeeignh hntsod (§e3s . [sent-38, score-0.163]
13 Using our model to judge among the proposals, we retain the most probable route as the best one. [sent-42, score-0.116]
14 However, there are substantial differences between trail extraction and vessel segmentation. [sent-49, score-0.857]
15 In addition, the clutter surrounding trails tends to be more variable, which motivates our approach based on textons and superpixels. [sent-51, score-0.264]
16 In contrast, each invocation of our quasi-Dijkstra procedure produces a random path that is essentially one realization of an implicit decision tree. [sent-59, score-0.144]
17 Third, our image likelihood function uses a statistical model of trail direction learned for the textons. [sent-65, score-0.963]
18 Fourth, our trail proposal scheme differs substantially from theirs. [sent-66, score-0.888]
19 Many road-finding methods are based on solving a dynamic programming or shortest path problem in a pixel or similar lattice. [sent-67, score-0.245]
20 Similarly, our trail proposer finds short, but not necessarily shortest, paths in the superpixel graph, as a heuristic method to explore the space of routes between trail termini. [sent-68, score-1.947]
21 - tion of paths or path lengths bears some similarity to the Figure 2. [sent-72, score-0.242]
22 We use a bank of oriented filters to generate pixel features, which we use to segment the image into superpixels of contiguous texture. [sent-74, score-0.147]
23 We generate independent trail proposals T(i) for i = 1, 2, . [sent-75, score-0.885]
24 by searching for nearly-shortest paths in the superpixel graph, and keep the proposal with the highest posterior measure. [sent-78, score-0.277]
25 stochastic shortest path problem [5, 10, 3 1], but the latter research has more to do with finding an optimal traversal policy. [sent-79, score-0.22]
26 There are a few similarities between our approach to trail-finding and robotic path planning algorithms such as Rapidly-exploring Random Trees (RRTs) [24] and related approaches [4, 22]. [sent-82, score-0.182]
27 In each case, a stochastically-built tree provides a route to a goal point. [sent-83, score-0.111]
28 With ordinary RRTs, as with other standard robotic path planning, candidate paths are cleanly partitioned into feasible and infeasible categories. [sent-85, score-0.31]
29 Also, in robotics, one usually seeks a feasible path that minimizes a cost function; whereas in the present work, we want a path that maximizes a probability distribution. [sent-87, score-0.304]
30 Data and Problem Definition We used groundtruth derived from GPS tracks collected by Morris and Barnard [29] from the Great Divide Mountain Bike Route (GDMBR), which traverses the western continental United States. [sent-90, score-0.117]
31 This route was partitioned into trail pieces that each fit into a 2 km square bounding box aligned north-south. [sent-91, score-0.973]
32 Grayscale aerial imagery surrounding each trail piece, originating from the US Geological Survey, was downloaded from Microsoft Research Maps [8]. [sent-92, score-0.887]
33 The result is 1526 trail pieces and images (one trail piece per image), at a resolution of 1m/pixel. [sent-93, score-1.754]
34 Given the image and the endpoints of the corresponding trail piece, how does the trail connect 333333888 those endpoints? [sent-95, score-1.736]
35 We represent a trail piece T as a sequence of eightconnected pixel locations in the image. [sent-97, score-0.947]
36 All other pixel locations in T are determined by using Bresenham’s line algorithm [11] between successive trail vertices. [sent-101, score-0.923]
37 ca Tthioens lik, as long as the path has a well-defined tangent everywhere. [sent-107, score-0.166]
38 Image and trail models We model the textures of the image using a Gaussian mixture (GMM). [sent-109, score-0.84]
39 Each kernel has a sigma of 4 pixels in the narrow direction, which is comparable to the width of typical paths in our image data. [sent-111, score-0.14]
40 Because the oriented Gaussian kernels roughly match the appearance of visible paths, the textons also tend to cluster trail pixels according to the direction of trail. [sent-114, score-0.991]
41 Next we label each image pixel with its most probable texton (GMM mode). [sent-115, score-0.207]
42 During training, we then learn for each texton some additional characteristics: the frequencies with which it generates on-trail and off-trail pixels, and the axial direction (i. [sent-116, score-0.225]
43 The latter we model as a von Mises distribution [27] (an angular distribution) using a maximum-likelihood (ML) fit of axial directions sampled from groundtruth trail pixels. [sent-120, score-1.043]
44 For texton label k, let μk , Σk denote the mean and covariance of the feature vectors generated by the mode. [sent-122, score-0.135]
45 When k does generate an on-trail pixel, its axial direction is modeled by von Mises parameters μk, the expected direction, × and κk, the concentration. [sent-127, score-0.182]
46 We denote the PDF of an axial von Mises distribution evaluated at angle θ ∈ [0, π) by fAM (θ; μ, κ) = 2M (2θ; 2μ, κ) , (1) where M(θ; μ, κ) represents the PDF of an ordinary (radwiahle)r von Mθ;iμse,sκ )d risetprribeusteinotns. [sent-128, score-0.23]
47 A superpixel s is an 8-connected region of pixels sharing a common label, whose size |s| we limit to oatf mpixoestl s16 sh3a8r4in pixels. [sent-136, score-0.129]
48 Mmaoxnim laable clo,n wnheocsteed si regions eo lfi pixels that share a common label but exceed that size are partitioned into multiple superpixels by a 128 128 grid. [sent-137, score-0.172]
49 Segmenting the image by a learned palette of textons lets us relax our independence assumptions, to respect the correlations in the imagery, yet still partition the superpixels into on- and off-trail classes. [sent-145, score-0.249]
50 Trail model Since the trail representation approximates a polygonal path, it has a well-defined tangent direction at every pixel location strictly between two trail vertices: let trail pixel q be between successive vertices vi = [xi , yi]t and vi+1 . [sent-148, score-2.869]
51 We can also define a direction at the trail vertices, as a composite of the directions of the neighboring path edges: at vertex vi with predecessor and successor vertices vi−1 and vi+1, we define the direction at vi as that of vector ? [sent-152, score-1.224]
52 el that prior knowledge by pc(T), a product of von Mises distributions on the differences of successive vertex angles at interior vertices (Fig. [sent-159, score-0.262]
53 , φN−2 discretely approximate the drerivative of path curvature. [sent-169, score-0.144]
54 For a trail T defined by N = 50 vertices, with interior angles φ1 , φ2 , . [sent-171, score-0.883]
55 In addition, we model prior knowledge about a trail’s polygonal path length ? [sent-179, score-0.22]
56 By construction, each trail piece extends across a square bounding box Lmin = 1900 meters on a side. [sent-181, score-0.897]
57 Image likelihood We develop the likelihood p(I|T), where I a grayscale is image surrounding a teralihil hypothesis, wTh. [sent-189, score-0.121]
58 , s|S| }, and further partition S according tso S Sw =heth {ser a superpixel }in,t aenrdse cftusr tThe. [sent-193, score-0.121]
59 pWaret idtieonno Ste the superpixels containing trail pixels by S1 = {s ∈ S : s ∩ T ∅}. [sent-194, score-0.963]
60 ∈ in which Ds represents the features of superpixel s, l0 (s) represents the likelihood of Ds in a off-trail region, and rl1e (psr;e Tse)n represents tihheo oldik eolfih Dood of Ds when it intersects one or more pixels o thf eT . [sent-204, score-0.199]
61 s∈S l0 (s) is independent of the trail hypothesis T. [sent-206, score-0.84]
62 In both l0 and l1, for each pixel we will account for three characteristics found there: the pixel’s features as conditioned by its texton label, the pixel’s label as conditioned by trail overlap, and the pixel’s label as conditioned by trail directionality. [sent-216, score-2.066]
63 We begin with likelihood l0 (s) of the image data in offtrail superpixel s. [sent-218, score-0.187]
64 Let q be any pixel in s, and let c(q) and c(s) respectively denote the texton label of q and the texton label common to all pixels of s. [sent-219, score-0.346]
65 When s intersects T at pixel q, we assess the likelihood of the directional appearance of Dq by assuming uniform prior ed idstirreicbutitoionnasl aofp pteexatruarnec deior efcDt ion and trail direction, in which case, p(direction of Dq | T) = p(θT (q) | direction of c(s)) . [sent-226, score-1.057]
66 (8) Then the likelihood of image data within superpixel s is a product of the likelihoods of its on-trail and off-trail pixels: l1(s;T) =q1∈? [sent-227, score-0.154]
67 (11) Intuitively the two kinds of ratios in this product can be interpreted as a logical conjunction: not only should the image textures along T “look like” trail (i. [sent-259, score-0.86]
68 We take a random sample of 50 such axes, shown in (b), to which we fit an axial von Mises. [sent-272, score-0.138]
69 Inference Using the prior (3) and likelihood (11), we define posterior distribution p(T|I) =Z1p(T)p(I|T)1/|T| (12) in which exponent 1/|T| makes the likelihood neutral with respect hto e txrpaoiln seinzte, 1 /an|Td| n moarmkeasli tzhaeti loinke flaihcotoord Z n iust rlealft w unknown. [sent-274, score-0.151]
70 (12) for inferring an unknown trail T, given image I the endpoints of T. [sent-276, score-0.896]
71 Instead, we explore the space of trails that bridge between two known endpoints by searching for short paths in a carefully weighted graph of superpixels (an approximately dual problem). [sent-278, score-0.48]
72 Edge weights in superpixel graph Our proposer is inspired by Morris and Barnard [29], who also computed shortest paths for trail inference. [sent-284, score-1.185]
73 We use their same basic idea: an edge that is highly likely to be on a trail should get low weight. [sent-285, score-0.858]
74 However, we face two technical challenges when translating the likelihood ratio of (11) to edge weights: we require an inverse relationship that remains finite, and we lack the directional factor fAM (θT) (i. [sent-286, score-0.121]
75 , while creating a trail proposal, we cannot yet know its tangent). [sent-288, score-0.84]
76 In place of factor fAM (θT), we compute a statistical approximation based on superpixel geometry. [sent-289, score-0.131]
77 corresponds to a trail proposal whose geometry overlaps both superpixels, so we model a random trail proposal connecting s and s? [sent-291, score-1.776]
78 ) of an axial von Mises distribution of a random straight track traversing both s and s? [sent-296, score-0.138]
79 Then we compare this geometric model with the directional model learned for texton c(s? [sent-298, score-0.14]
80 If the two distributions are similar, it is more likely that a path entering s would extend into s? [sent-300, score-0.144]
81 Weight w also needs a dair reeclatt relationship wwit ∼h superpixel size, otherwise the proposer would have a bias in favor of large superpixels. [sent-315, score-0.169]
82 ||1,/ 2ar, ebae coafu sse superpixels tend to be narrow. [sent-321, score-0.115]
83 ) |Isn o| trhdaenr |tso |b alance the sizes of the weights in image regions likely and unlikely to be trail, we include parameters α and γ, which are trained by grid search so as to minimize Hausdorff distance between groundtruth and the shortest path in the superpixel graph. [sent-322, score-0.363]
84 Thus our choice for edge weight between superpixels s and s? [sent-323, score-0.115]
85 Sampling short paths We have developed a variation on Dijkstra’s algorithm to sample paths in this weighted graph that are short but not necessarily the shortest. [sent-334, score-0.299]
86 Our idea is to alter the priority queue to support an EXTRACT-NEAR-MIN operation that, at each iteration, extracts a vertex (superpixel) selected randomly, with a preference for vertices of smaller distance. [sent-338, score-0.223]
87 For a vertex s with distance d(s) in the queue, its probability of being the next vertex removed from the queue U is given by a power law, Pr(s will be drawn next) =? [sent-339, score-0.133]
88 The stochastic priority queue is implemented with a redblack tree that stores non-normalized probability mass d(s)−β with entry s, and maintains at each tree node a sum of all subtree nodes’ probability masses. [sent-345, score-0.173]
89 Thus we can perform an EXTRACT-NEAR-MIN operation in time O(log |U|), where |U| is the number of vertices (superpixeOls()l oing t|Uhe| priority queue. [sent-346, score-0.121]
90 tSheinc nue tmhebe superpixel graph eisr pla333444111 Figure 5. [sent-347, score-0.13]
91 Examples of intermediate and final results of inference process: (a) the groundtruth pixel footprint of superpixels of a typical trail piece, i. [sent-348, score-1.073]
92 , all pixels of the superpixels touching groundtruth; (b) 200 short paths generated by the quasi-Dijkstra method; (c) shortest path found by Dijkstra’s algorithm; (d) short path approximately maximizing eq. [sent-350, score-0.636]
93 ) nar, sampling a short path with this implementation uses time O(|S| log |S| ). [sent-354, score-0.182]
94 ail proposal, we use either this quasiDijkstra algorithm, or the unmodified Dijkstra’s algorithm, to find a short path P in the superpixel graph bridging between endpoints. [sent-356, score-0.312]
95 P is a simple path of superpixels, P = (si1, si2, . [sent-357, score-0.144]
96 Because our prior model requires a polygonal path with a fixed number of vertices, we take some additional steps to reduce the superpixel path into a polygonal path of pixels. [sent-364, score-0.666]
97 rtest path in the pixel graph, using 8-way adjacency and Euclidean distance. [sent-371, score-0.194]
98 Results and Conclusions We use the Hausdorff distance metric between groundtruth and inferred path for evaluation; a perfectly inferred path will have a distance of zero. [sent-377, score-0.353]
99 6 shows the success rate of three methods of generating and assessing trail proposals. [sent-379, score-0.84]
100 The top curve, labeled “QD-MAP,” denotes the success rate when we generate 200 trail proposals by the quasi-Dijkstra procedure described above, and keep the proposal with Table 1. [sent-381, score-0.933]
wordName wordTfidf (topN-words)
[('trail', 0.84), ('trails', 0.164), ('fam', 0.15), ('path', 0.144), ('texton', 0.108), ('superpixel', 0.103), ('paths', 0.098), ('superpixels', 0.097), ('route', 0.094), ('dijkstra', 0.091), ('textons', 0.081), ('vvii', 0.077), ('axial', 0.073), ('vertices', 0.069), ('mises', 0.068), ('proposer', 0.066), ('groundtruth', 0.065), ('von', 0.065), ('piece', 0.057), ('endpoints', 0.056), ('dq', 0.056), ('polygonal', 0.055), ('queue', 0.055), ('priority', 0.052), ('shortest', 0.051), ('likelihood', 0.051), ('pixel', 0.05), ('conditioned', 0.05), ('hausdorff', 0.049), ('proposal', 0.048), ('barnard', 0.045), ('morris', 0.045), ('proposals', 0.045), ('direction', 0.044), ('independence', 0.042), ('short', 0.038), ('vessels', 0.035), ('gdmbr', 0.033), ('lmin', 0.033), ('offtrail', 0.033), ('predoehl', 0.033), ('roadfinding', 0.033), ('successive', 0.033), ('directional', 0.032), ('ds', 0.031), ('vertex', 0.031), ('continental', 0.029), ('kobus', 0.029), ('radians', 0.029), ('recreational', 0.029), ('rrts', 0.029), ('hue', 0.029), ('lets', 0.029), ('pdf', 0.029), ('saturation', 0.029), ('aerial', 0.028), ('posterior', 0.028), ('statistical', 0.028), ('label', 0.027), ('ona', 0.027), ('graph', 0.027), ('ordinary', 0.027), ('vi', 0.026), ('pixels', 0.026), ('gmm', 0.025), ('pc', 0.025), ('traversal', 0.025), ('neurons', 0.024), ('characteristics', 0.024), ('arizona', 0.023), ('concentration', 0.023), ('numerator', 0.023), ('western', 0.023), ('probable', 0.022), ('tangent', 0.022), ('roads', 0.022), ('interior', 0.022), ('partitioned', 0.022), ('fp', 0.022), ('blood', 0.021), ('kb', 0.021), ('angles', 0.021), ('footprint', 0.021), ('prior', 0.021), ('ratio', 0.02), ('ratios', 0.02), ('ml', 0.02), ('intersects', 0.019), ('planning', 0.019), ('robotic', 0.019), ('surrounding', 0.019), ('sse', 0.018), ('tso', 0.018), ('edge', 0.018), ('tree', 0.017), ('pieces', 0.017), ('substantial', 0.017), ('probability', 0.016), ('narrow', 0.016), ('preference', 0.016)]
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