iccv iccv2013 iccv2013-389 knowledge-graph by maker-knowledge-mining

389 iccv-2013-Shortest Paths with Curvature and Torsion


Source: pdf

Author: Petter Strandmark, Johannes Ulén, Fredrik Kahl, Leo Grady

Abstract: This paper describes a method of finding thin, elongated structures in images and volumes. We use shortest paths to minimize very general functionals of higher-order curve properties, such as curvature and torsion. Our globally optimal method uses line graphs and its runtime is polynomial in the size of the discretization, often in the order of seconds on a single computer. To our knowledge, we are the first to perform experiments in three dimensions with curvature and torsion regularization. The largest graphs we process have almost one hundred billion arcs. Experiments on medical images and in multi-view reconstruction show the significance and practical usefulness of regularization based on curvature while torsion is still only tractable for small-scale problems.

Reference: text


Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 We use shortest paths to minimize very general functionals of higher-order curve properties, such as curvature and torsion. [sent-6, score-0.866]

2 To our knowledge, we are the first to perform experiments in three dimensions with curvature and torsion regularization. [sent-8, score-1.181]

3 Experiments on medical images and in multi-view reconstruction show the significance and practical usefulness of regularization based on curvature while torsion is still only tractable for small-scale problems. [sent-10, score-1.428]

4 Introduction In differential geometry, the fundamental theorem of curves states that any regular curve in three dimensions with non-zero curvature has its shape completely determined by its curvature and torsion [12]. [sent-12, score-1.895]

5 Therefore, for curve recon- struction problems, it makes sense to regularize the solution with curvature and torsion priors. [sent-13, score-1.251]

6 In this paper, we demonstrate that with today’s modern CPUs and clever implementation choices, it is actually possible to solve inverse problems involving both curvature and torsion priors at reasonable running times. [sent-16, score-1.225]

7 In [10], it is shown that Euler’s elastica – the line integral of the squared curvature conforms better to intuitive completion of boundary curves. [sent-18, score-0.703]

8 Other applications where curvature plays an important role include saliency [17], inpainting [9], stereo [21], region-based image segmentation [15] and surface reconstruction [19]. [sent-19, score-0.682]

9 In many applications, the reconstructed – long with high curvature, but have low torsion as they lie approximately in a plane. [sent-21, score-0.587]

10 The coronary arteries are neither short nor have low curvature, but they do have low torsion. [sent-26, score-0.176]

11 We follow a classic approach for computing the global minimum of an active contour model based on shortest paths [2, 3]. [sent-29, score-0.194]

12 Most of × these methods use only weighted length regularization, even though the idea of applying shortest paths to higher-order functionals involving curvature is not new. [sent-31, score-0.921]

13 An extension of the live-wire framework with curvature priors is presented in [20]. [sent-36, score-0.594]

14 Many nice examples of the benefits with curvature are given in both [11] and [20], but no running times are reported. [sent-39, score-0.656]

15 Our contribution is a global method to find a curve of minimal curvature between two points using line graphs. [sent-44, score-0.72]

16 We are the first to demonstrate practical curvature experiments in 3D and our two-dimensional running times are in the order of seconds. [sent-45, score-0.637]

17 Shortest Paths with Curvature and Torsion We are interested in finding a curve γ in two or three dimensions which minimizes an image-dependent functional of its length, curvature and torsion, or more generally: iγn,Lf? [sent-55, score-0.664]

18 (2) Here κ(s) and τ(s) denote the curvature and the torsion, respectively; see Section 2. [sent-68, score-0.594]

19 The scalars ρ, σ and ν are weighting factors of length, curvature and torsion, respectively and can be made image-dependent as well. [sent-70, score-0.594]

20 Figure 2: The same path through a mesh (shown in black) repre- sented in three different ways. [sent-74, score-0.213]

21 The shortest path computation is carried out in a graph, consisting of nodes and arcs. [sent-78, score-0.238]

22 The mesh is stored explicitly in memory and is the same no matter which regularization is used. [sent-81, score-0.267]

23 The simplest case is length regularization (σ = ν = 0), for which the graph is identical to the mesh. [sent-84, score-0.284]

24 It follows that we need to consider at least three points to determine the curvature cost. [sent-89, score-0.631]

25 To achieve this, a line graph is constructed in which each node corresponds to an edge in the mesh. [sent-90, score-0.171]

26 In this manner we can compute the curvature cost of any path through the original mesh. [sent-92, score-0.699]

27 Because three points always lie in the same plane, at least four points are needed to determine the torsion cost. [sent-95, score-0.661]

28 Figure 2c shows the same path as before, with each node in the graph corresponding to an edge pair in the mesh. [sent-97, score-0.253]

29 The graph grows very rapidly in size, but we never construct it explicitly; only the mesh is explicitly created and stored. [sent-98, score-0.176]

30 squared curvature as well as the length element ds = ? [sent-114, score-0.749]

31 Figure 3 graphs an example curve (a helix), for which the exact curvature and torsion are well known: they are both = det ? [sent-131, score-1.278]

32 Figure 4 shows the error when approximating the curvature and torsion using three and four points, respectively. [sent-142, score-1.181]

33 Figure 4: Error when computing the curvature and torsion using the middle point of a spline fit to two or three edges on the curve in Figure 3. [sent-144, score-1.319]

34 Our Implementation Dijkstra’s algorithm for computing the shortest path is well known. [sent-147, score-0.197]

35 (ii) An oracle that given a node returns its neighbors and arc weights. [sent-150, score-0.167]

36 Each edge and edge pair in the mesh then require 8 and 12 bytes, respectively. [sent-172, score-0.302]

37 The array of edges is sorted after the mesh is created, which makes it possible to find the index of a given edge in logarithmic time. [sent-173, score-0.299]

38 Then it looks up the edge index for each pair of points in the sorted edge array. [sent-176, score-0.207]

39 Instead, the neighbors of an edge pair are computed via the edge neighbors of its last edge as above. [sent-180, score-0.335]

40 For example, the curvature cost between two edges is the same even if both edges are translated by the same amount. [sent-182, score-0.706]

41 Thus, the integrated curvature needs only • – be computed once for each configuration. [sent-183, score-0.612]

42 The cache has to be fast, since when working with torsion there are millions of different configurations of two edge pairs. [sent-184, score-0.722]

43 Computing the shortest path between two nodes efficiently in parallel is not a trivial task. [sent-189, score-0.238]

44 The curvature or torsion cache is filled before the computation starts. [sent-193, score-1.241]

45 First, we would like to demonstrate the differences between length, curvature, and torsion with a couple of synthetic experiments. [sent-197, score-0.587]

46 54 s) Table 1: Number of nodes visited using Dijkstra’s (first row) and A* (second row) for the curvature experiments in Figure 6. [sent-214, score-0.658]

47 Naturally, the heuristic works best for low curvature regularization. [sent-215, score-0.614]

48 All of our 2D (3D) experiments are performed with a mesh whose points are arranged in a square (cubic) lattice. [sent-223, score-0.169]

49 We manually sampled the river image in Figure 6 at different locations and calculated the data cost for every pixel as the shortest (Euclidean) distance in L*a*b* space to any of the color samples of the river delta. [sent-230, score-0.226]

50 Curvature regularization yields a long path which does not turn much. [sent-233, score-0.216]

51 No amount of length regularization is able to find that path as indicated by Figure 6a. [sent-234, score-0.341]

52 Figure 7 shows an experiment where some edges have been removed around the start and end sets in such a way that the optimal analytical solution is a circle of maximum radius. [sent-236, score-0.172]

53 This experiment shows that low regularization radii introduce significant bias when minimizing the squared curvature. [sent-237, score-0.19]

54 The underlying mesh has 220,443 points (373 591) and 3,509,756 edges (16-connected). [sent-246, score-0.236]

55 Figures (c) and (d) show the order in which the nodes were visited for medium curvature with and without A*, from blue (early) to red (late). [sent-249, score-0.677]

56 ference between a curve with curvature regularization and torsion regularization is given. [sent-251, score-1.521]

57 High torsion regularization forces the curve to stay within a plane, but within the plane, the curve may have high length and curvature. [sent-252, score-0.967]

58 The graph used for the torsion regularization had about 75 billion arcs. [sent-253, score-0.791]

59 2D Retinal Images Automated segmentation of vessels in retinal images is an important problem in medical image analysis, which, for example, can be used when screening for diabetic retinopathy. [sent-257, score-0.256]

60 We have investigated whether curvature regularization is useful for this task on a publicly available dataset [18]. [sent-258, score-0.729]

61 We iteratively computed shortest paths from a user-provided start point to any point on the image boundary. [sent-260, score-0.221]

62 A curve could start anywhere along the previously found curves, but not end close to a previous end point. [sent-261, score-0.2]

63 We performed experiments with different amounts of length regularization (Figures 8b–8e). [sent-262, score-0.24]

64 No or low length regularization resulted in noisy, wiggling paths. [sent-263, score-0.24]

65 This problem expectedly disappeared for medium regularization, but sharp turns were still present in the solution (sharp turns usually change direction in the vessel tree). [sent-264, score-0.302]

66 When using too high regularization (Figure 8e), the solution almost ignores the data term and prefers paths along the image boundary. [sent-265, score-0.194]

67 Art laln pdo ienntds points have been chosen in such a way that the analytical solution when minimizing the integrated squared curvature is a circle with cost π/10. [sent-270, score-0.753]

68 In contrast, curvature regularization is able to more cor- rectly capture the vessel tree. [sent-271, score-0.938]

69 3D Coronary Artery Centerline Extraction Finding the centerlines of coronary arteries is of high clinical importance [14], but is very time consuming when performed manually. [sent-275, score-0.176]

70 Each segmentation is initialized by manually specifying the start and end of each vessel information which is included in the dataset. [sent-278, score-0.329]

71 To model the vessel we adopt the speed image S from [5], which combines the probability of a voxel being a vessel, measured by “vesselness” [13] and a soft threshold of the image. [sent-280, score-0.231]

72 Figure 8: Segmenting a vessel tree with 8 leaves in a 2D retinal image from the DRIVE [18] database. [sent-300, score-0.271]

73 The underlying mesh has 329,960 points (584 565) and 10,496,766 edges (32-connected). [sent-302, score-0.236]

74 oAnlnl length regularizations (b-d) have various issues and curvature regularization (e) finds a reasonable tree. [sent-305, score-0.864]

75 ble 2 presents quantitative result on the training data of [14] and compares length to curvature regularization. [sent-306, score-0.699]

76 A example where curvature regularization outperforms length is given in Figure 9. [sent-307, score-0.834]

77 We reconstructed a tree iteratively in the same way as in Figure 8 and, as expected, length regularization introduces similar artifacts. [sent-313, score-0.259]

78 Integrating the image along the projected 3D edge gives the edge cost for a single view. [sent-314, score-0.174]

79 2 · (3 · 10−4) 10 Table 2: Length and curvature regularization for coronary vessel segmentation. [sent-319, score-1.075]

80 We segmented each vessel in the training set of [14] with length and curvature regularization. [sent-320, score-0.927]

81 For both length and curvature we choose the strength giving the best result and applied it to all vessels. [sent-322, score-0.699]

82 The second column reports the number of vessels for which length and curvature performed best, respectively, not counting any vessel with less than 50% overlap (6 out of 28 vessels) or ties (5 vessels). [sent-323, score-1.014]

83 Figure 9: Example of coronary artery segmentation in 3D using length (left) and curvature (right). [sent-331, score-0.938]

84 The mesh has 841,662 points (114 107 69) and s60ol,u2t7io1,n57 (848 edges. [sent-339, score-0.169]

85 We run the segmentation with either very high curvature (yellow) or very high torsion regularization (red). [sent-349, score-1.359]

86 The underlying mesh has 32,560 points (22 74 20), 4,118,200 edges (146connected) a2n,5d6 053 p8o,i7n1t7s,7 (2922 edge pairs. [sent-350, score-0.311]

87 6 s for torsion (harder problems will take longer time to solve). [sent-353, score-0.587]

88 Using length regularization gives similar artifacts (sharp turns) as in Figure 8. [sent-355, score-0.24]

89 The underlying mesh has 125,000 points and 24,899,888 edges (218-connectivity). [sent-356, score-0.236]

90 Conclusions This paper has demonstrated the possibility of incorporating curvature and torsion to shortest path problems. [sent-362, score-1.378]

91 Although all of our methods find the global optimum in polynomial time, using torsion can not be considered feasible, as we have only used it for quite small problems. [sent-365, score-0.587]

92 On the other hand, using curvature is not that expensive for many problems and we believe we have demonstrated its usefulness for medical imaging problems, both in 2D (Figure 8) and in 3D (Figure 9). [sent-366, score-0.684]

93 The “vesselness” measures [4, 13] commonly used for blood vessel segmentation use the eigenvalues of the Hessian. [sent-372, score-0.284]

94 An interesting avenue for future research is to include the eigenvectors with anisotropic curvature, resulting in a direction-aware curvature regularization. [sent-373, score-0.594]

95 A review of 3d vessel lumen segmentation techniques: models, features and extraction schemes. [sent-430, score-0.275]

96 Three-dimensional multi-scale line filter for segmentation and visualization of curvilinear structures in medical images. [sent-456, score-0.172]

97 Standardized evaluation methodology and reference database for evaluating coronary artery centerline extraction algorithms. [sent-460, score-0.267]

98 A linear framework for region-based image segmentation and inpainting involving curvature penalization. [sent-467, score-0.68]

99 The elastic ratio: Introducing curvature into ratio-based globally optimal image segmentation. [sent-474, score-0.636]

100 Ridge based vessel segmentation in color images of the retina. [sent-490, score-0.252]


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tfidf for this paper:

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