cvpr cvpr2013 cvpr2013-350 knowledge-graph by maker-knowledge-mining
Source: pdf
Author: Engin Türetken, Fethallah Benmansour, Bjoern Andres, Hanspeter Pfister, Pascal Fua
Abstract: We propose a novel approach to automated delineation of linear structures that form complex and potentially loopy networks. This is in contrast to earlier approaches that usually assume a tree topology for the networks. At the heart of our method is an Integer Programming formulation that allows us to find the global optimum of an objective function designed to allow cycles but penalize spurious junctions and early terminations. We demonstrate that it outperforms state-of-the-art techniques on a wide range of datasets.
Reference: text
sentIndex sentText sentNum sentScore
1 edu Abstract We propose a novel approach to automated delineation of linear structures that form complex and potentially loopy networks. [sent-7, score-0.48]
2 This is in contrast to earlier approaches that usually assume a tree topology for the networks. [sent-8, score-0.155]
3 At the heart of our method is an Integer Programming formulation that allows us to find the global optimum of an objective function designed to allow cycles but penalize spurious junctions and early terminations. [sent-9, score-0.321]
4 Introduction Networks of curvilinear structures are abundant both in natural and man made systems. [sent-12, score-0.304]
5 They appear at all possible scales, ranging from nanometers in Electron Microscopy images of neurons and meters in aerial images of roads to petameters in dark-matter arbors binding massive galaxy clusters. [sent-13, score-0.303]
6 As a result, their automated reconstruction has been one of the earliest topics addressed by computer vision scientists. [sent-14, score-0.152]
7 Recently, there has been renewed interest in the reconstruction of tree-like structures and significant progress has been achieved by formulating the problem as one of optimizing a global objective function [26, 25]. [sent-16, score-0.136]
8 However, in practice, many interesting networks, such as those formed by the roads and blood vessels depicted by the first row of Fig. [sent-17, score-0.42]
9 1, the imaging resolution is often so low that the branches appear to cross, thus intro∗This work dation. [sent-21, score-0.148]
10 (Confocal) Maximum intensity projection of a confocal image stack of blood vessels, which appear in red. [sent-25, score-0.262]
11 (Brightfield) Minimum intensity projection of a brightfield stack of neurons. [sent-26, score-0.201]
12 (Brainbow) Maximum intensity projection of a brainbow [17] stack. [sent-27, score-0.186]
13 ducing several spurious cycles that can only be recognized as such once the whole structure has been recovered. [sent-29, score-0.23]
14 In fact, this is widely reported as one of the major sources of error [28, 8, 4, 29, 26, 7] and a number of heuristics have been proposed to avoid spurious connections in the presence of such cycles [26, 29, 8]. [sent-30, score-0.23]
15 We will show that, in such cases, it is more effective to relax the tree constraint and to build loopy networks by penalizing the formation of spurious junctions and early branch terminations. [sent-33, score-0.524]
16 Enforcing tree topology results in creation of spurious junctions. [sent-35, score-0.247]
17 The dots depict sample points (seeds) on the centerlines found by maximizing a tubularity measure. [sent-37, score-0.165]
18 In 3D, these branches may be disjoint but the z-resolution is insufficient to see it and only a single sample is found at their intersection, which we color yellow. [sent-38, score-0.148]
19 (b) The sample points are connected by geodesic paths to form a graph. [sent-39, score-0.237]
20 (c,d) The final delineation is obtained by finding a subgraph that minimizes a global cost function. [sent-40, score-0.267]
21 In (d), we allow the yellow point to be used twice, and penalize early terminations and spurious junctions, which yields a better result. [sent-42, score-0.135]
22 voxels that are very likely to belong to the curvilinear structures of interest. [sent-43, score-0.304]
23 We treat them as vertices of a graph and connect those that are within a certain distance of each other by geodesic paths [5] whose quality we assess on the basis of local image evidence. [sent-44, score-0.39]
24 We then look for a subgraph that maximizes a global objective function that combines image-based and geometry-based terms with respect to which edges of the original graph are active. [sent-45, score-0.113]
25 However, unlike in this earlier paper, we let graph vertices be used by several branches, thus allowing cycles as shown in Fig. [sent-47, score-0.251]
26 Unlike earlier graph-based delineation approaches, ours lets vertices be shared among branches while still allowing the recovery of the optimal tree from the resulting loopy subgraph when the result should be acyclical. [sent-49, score-0.748]
27 Related Work There has recently been a resurgence of interest in automated delineation techniques [18, 10, 20, 14] because extracting curvilinear structures automatically and robustly is of fundamental relevance to many scientific disciplines. [sent-57, score-0.572]
28 For example, it has been recognized that “the lack of powerful and effective computational tools to automatically reconstruct neuronal arbors has emerged as a major technical bottleneck in neuroscience research,” as stated on the homepage of the DIADEM challenge [3]. [sent-58, score-0.17]
29 Similar statements could be made about medical research involving the fine modeling of complex blood vessel structures, such as those in the lungs, or automated delineation of linear structures in aerial imagery databases. [sent-59, score-0.718]
30 Most automated approaches involve greedy strategies that start from a set of seed points, incrementally grow branches by evaluating a local tubularity measure—usually based on the Hessian and Oriented Flux matrices [23, 13, 15]—in the vicinity of the initial seeds [7, 27, 4, 2]. [sent-60, score-0.571]
31 High tubularity paths are then iteratively added to the solution and their end points are treated as the new seeds from which the process can be restarted. [sent-61, score-0.35]
32 By contrast, graph-based methods find seed points in the whole image or volume by evaluating the tubularity mea- sure densely and finding its local maxima [12, 22, 27, 26, 25]. [sent-64, score-0.261]
33 The seed points are then connected by paths that follow local maxima of the tubularity measure. [sent-66, score-0.443]
34 This results in a graph that forms an overcomplete representation of the underlying tree structure and the final step is to build a tree by selecting an optimal subset of the edges. [sent-67, score-0.126]
35 As a result, their topology may be wrong and suboptimal post-processing procedures are required to eliminate spurious branches and, possibly, correct topological errors. [sent-70, score-0.332]
36 Furthermore, all these spanning-tree approaches 111888222311 evaluate the quality of an edge between two seed points by integrating a function of the tubularity measure along a connecting path. [sent-73, score-0.37]
37 This quality measure usually fails to distinguish legitimate paths along faint curvilinear structures from those that are shortcuts between high-contrast structures. [sent-74, score-0.517]
38 The MAP formulation [25] addresses these issues by using path classifiers to score the paths and introducing a Mixed Integer Programming approach to guaranteeing optimality of the resulting solution. [sent-75, score-0.215]
39 However, none of these approaches address the delineation problem for loopy structures. [sent-76, score-0.268]
40 Method Our algorithm, like those of [12, 29, 27, 26, 25] starts by building a weighted graph designed to be an overcomplete representation for the underlying network of curvilinear structures, such as the one of Fig. [sent-81, score-0.244]
41 It then finds an optimal subgraph in terms of an appropriately designed objective function. [sent-83, score-0.113]
42 The major difference from these earlier approaches is that instead of constraining the subgraph to be a tree as in Fig. [sent-84, score-0.219]
43 2(d), and penalize spurious junctions and early branch terminations. [sent-86, score-0.302]
44 In the remainder of this section, we introduce our Integer Programming approach to finding an optimal and potentially loopy subgraph. [sent-88, score-0.114]
45 For each pixel or voxel, we first estimate its likelihood ofbeing on the centerline of a curvilinear structure whose radius is within a given range, using a tubularity measure similar to those discussed in Section 2. [sent-93, score-0.371]
46 The paths are obtained by minimizing a geodesic distance in (N+1)-D scale space—N spatial dimensions and the radius—as described in [5]. [sent-95, score-0.192]
47 A loopy graph with a single root vertex v (in red). [sent-97, score-0.238]
48 Allowing vertex c (in green) to be used by the two different branches (denoted by blue and yellow arrows) produces a loopy solution instead of a tree. [sent-98, score-0.339]
49 Describing the crossing in terms of edge pairs {i, c, j} and {k, c, l} being active and all other edge pairs containing c, nsduc {hk as {k, c, j}, being ainnadc atilvle o tmheakre esd gite possible ntoeventually recover t {hek ,trc,eej topology inneavcetirvtheel mesaks. [sent-99, score-0.439]
50 This produces a graph G = (V, E), whose vertices V = {vi} represent the seed points and pairs of oppositely Vdir e=cte {dv edges Ese =t t {eij =d (ovini,t vj) , eji r=s (vj , vpio)s}i tehley paths linking tsh Eem. [sent-100, score-0.397]
51 Disallowing cycles prevents vertices from being shared by separate branches, as is required for successful reconstruction cases such as the one of Fig. [sent-103, score-0.246]
52 However, in some cases, such as when delineating the neural structures of the Brightfield and Brainbow images of Fig. [sent-107, score-0.136]
53 One approach would be to first recover the subgraph defined by the active edges and then attempt to assess its topology. [sent-111, score-0.221]
54 However, to consistently enforce geometric constraints on branches even at junctions, we do both simultaneously by reasoning in terms of whether consecutive pairs of edges belong to the final delineation or not. [sent-112, score-0.353]
55 3, edge pairs (eic, ecj) and (ekc, ecl) should belong but neither (eic, ecl) nor (ekc, ecj). [sent-114, score-0.16]
56 2(d), edge pairs (eij , ejk) and (emj , ejl) are both active and vertex j belongs to both branches. [sent-118, score-0.307]
57 We w{xill say thhea tc eijk piso nacdtiivneg vine tchtoer rso olfu itniodnic aift xijk =ria b1. [sent-120, score-0.185]
58 k (2) ∈F where wijk is a cost term that accounts for the quality of the geodesic paths associated with the edge pair eijk. [sent-128, score-0.342]
59 As discussed in Section 2, we have found it more effective at distinguishing legitimate paths from spurious ones than more standard methods, such as those that integrate a tubularity measure along the path. [sent-130, score-0.513]
60 anedijn o∈uFtgXoi njgn, ed wgheic phair dsen inot oe out of edge eij . [sent-135, score-0.392]
61 However, not all choices of binary values for the indicator variables give rise to a connected subgraph that represents a plausible delineation. [sent-157, score-0.158]
62 To avoid having to process them sequentially in a greedy manner, which may result in some branches being “stolen” by the first structure we reconstruct and therefore a suboptimal solution, we connect them all. [sent-162, score-0.195]
63 Assuming we are given a set R of seed vertices, one for each structure of interest, we create a virtual root vertex vv and connect it to each vr ∈ R by zero cost edge pairs containing all other vertices to wh∈ic Rh vr is connected. [sent-163, score-0.87]
64 4 are such that seed vertices are not isolated, branches are edge-disjoint, potential crossovers are consistently handled, and all active edge pairs are connected. [sent-165, score-0.587]
65 Non-isolated Seeds: We require the seed vertices vr ∈ R to be connected to at least one vertex other than the vir∈tu Ral root vv and to have no incoming edge other than evr. [sent-166, score-0.769]
66 (5) Disjoint Edges: For each edge eij ∈ E, we let at most one edge pair be active among all tho∈se tEh,at w weieth leert caot nmtaoisnt eij or sufficiently overlap with it. [sent-177, score-0.854]
67 We do the first by preventing the number of active incoming edge pairs into an edge to be more than one. [sent-178, score-0.403]
68 Let tij denote 111888222533 the geodesic path corresponding to edge ? [sent-180, score-0.242]
69 In the example depicted by the figure above, among all the edge pairs incoming to the edges eij, ek1l1 and ek2l2 , only one can be active in the final solution. [sent-193, score-0.343]
70 For those curvilinear structures that are inherently trees, these constraints make their recovery from the resulting subgraph possible by starting from the terminal vertices and following the active edge pairs along the paths that lead to the root vertices. [sent-194, score-0.944]
71 Crossover Consistency: A potential crossover in G is a vertex, which is adjacent to at least four other vertices and whose in- and out-degrees are greater than one. [sent-195, score-0.151]
72 =∈vEi: ∀eij ∈ E ∀emq ∈ C(vj) : vm = , vi C(vj) = {enk ∈ E | vk = vj ∨ (vj ∈ tnk, vn (7) = vj , vk = vj)} These constraints are only active when dealing with structures that inherently are trees, such as the neural structures of Fig. [sent-208, score-0.468]
73 For inherently loopy ones, such as the roads and blood vessels, we deactivate them to allow creation of junctions that are parts of legitimate cycles. [sent-210, score-0.553]
74 Connectedness: We require all the active edge pairs to be connected to the virtual root vv. [sent-211, score-0.361]
75 An edge pair eijk is said to be connected if there exists a path in G, starting at vv and containing eijk, along which all the edge pairs are active. [sent-212, score-0.64]
76 =es t)he b enu am nobner- noefg gdaistitvinec cto odnitreincuteodu sp faltohws i nv athriesolution, from the virtual root vv to vertex vl, that traverse = the edge eij , . [sent-214, score-0.7]
77 yjlk, ∀vj,vl ∈ V \ {vv} : vj = vl yill ≥ ∀eilk ∈ F , ei? [sent-226, score-0.322]
78 xkil, ∀eil ∈ E : vi = vv , , (9) (10) (11) (12) ek? [sent-232, score-0.145]
79 The first two constraints guarantee that the amount of flow outgoing from virtual vertex vv to true vertex vl is equal to the incoming flow to vl, which must be smaller than the in degree of vl . [sent-238, score-0.72]
80 Finally, the last three constraints bind the flow variables to the binary ones, ensuring that a connected network formed by the non-zero flow variables is also connected in the active edge pair variables. [sent-240, score-0.307]
81 Note that, since we are looking for possibly cyclic subgraphs, there can be multiple paths incoming to a vertex. [sent-241, score-0.276]
82 We then present our results on the first two, which contain cyclic networks, and finally on the next two, which contain true trees but whose optical resolution is so poor that they look like cyclic graphs. [sent-252, score-0.202]
83 1 and described in more detail below: • • Aerial: Aerial images of loopy road networks. [sent-257, score-0.114]
84 We only considered the red channel of these stacks since it is the only one used to label the blood vessels. [sent-261, score-0.229]
85 Brightfield: Six image stacks were acquired by brightfBierildg microscopy mfraogme biocytin-stained riaretd d br bayin bsr. [sent-262, score-0.14]
86 Brainbow: Neurites were visualized by targeting Bmriacein primary visual cortex using the brainbow technique [17] so that each neuronal structure has a distinct color. [sent-264, score-0.31]
87 Many roads of the Aerial dataset are partially occluded by trees while the images from the other datasets are very noisy, making the delineation task challenging in all cases. [sent-266, score-0.344]
88 Roads and Blood Vessels The roads and blood vessels of the Aerial and Confocal datasets form graphs in which there are many real cycles. [sent-278, score-0.371]
89 In the case of the blood vessels, this is because there are capillaries that connect the arteries to the veins and irrigate the cells along the way. [sent-279, score-0.265]
90 DIADEM [3] scores for our results (L-QMIP) and those of [25] (HGD-QMIP) for the delineations of 3 Brainbow stacks and the 3 Brightfield ones. [sent-295, score-0.165]
91 short roads and a few roads dead-ending because the connecting path to the closest junction is severely occluded. [sent-314, score-0.354]
92 The quality of the blood vessel delineations depicted by the second row is much harder to assess on the printed page but becomes clear when looking at the rotating volumes that we supply as supplementary material. [sent-317, score-0.39]
93 Neural Structures The neurites of the Brightfield and Brainbow datasets form tree structures without cycles. [sent-320, score-0.254]
94 However, because of the low z-resolution, branches that really are disjoint appear to cross. [sent-321, score-0.148]
95 Because the ground truth tracings are trees, we were able to compute the DIADEM scores [3] for the four delineations depicted by the bottom two rows of Fig. [sent-322, score-0.16]
96 Bottom Rows: For each dataset, two minimal or maximal projections and overlaid delineation results. [sent-332, score-0.154]
97 Each connected curvilinear structure network is shown in a distinct color. [sent-333, score-0.289]
98 Conclusion We have presented a graph-based approach to delineating complex linear structures in 2D images and 3D image stacks. [sent-338, score-0.136]
99 Unlike most earlier ones, it explicitly handles the fact that they may be cyclic and builds graphs in which vertices may belong to more than one branch. [sent-339, score-0.231]
100 3-d image pre-processing algorithms for improved automated tracing of neuronal arbors. [sent-470, score-0.342]
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