cvpr cvpr2013 cvpr2013-298 knowledge-graph by maker-knowledge-mining
Source: pdf
Author: Michael Kolomenkin, Ilan Shimshoni, Ayellet Tal
Abstract: This paper extends to surfaces the multi-scale approach of edge detection on images. The common practice for detecting curves on surfaces requires the user to first select the scale of the features, apply an appropriate smoothing, and detect the edges on the smoothed surface. This approach suffers from two drawbacks. First, it relies on a hidden assumption that all the features on the surface are of the same scale. Second, manual user intervention is required. In this paper, we propose a general framework for automatically detecting the optimal scale for each point on the surface. We smooth the surface at each point according to this optimal scale and run the curve detection algorithm on the resulting surface. Our multi-scale algorithm solves the two disadvantages of the single-scale approach mentioned above. We demonstrate how to realize our approach on two commonly-used special cases: ridges & valleys and relief edges. In each case, the optimal scale is found in accordance with the mathematical definition of the curve.
Reference: text
sentIndex sentText sentNum sentScore
1 The common practice for detecting curves on surfaces requires the user to first select the scale of the features, apply an appropriate smoothing, and detect the edges on the smoothed surface. [sent-10, score-0.797]
2 In this paper, we propose a general framework for automatically detecting the optimal scale for each point on the surface. [sent-14, score-0.385]
3 We smooth the surface at each point according to this optimal scale and run the curve detection algorithm on the resulting surface. [sent-15, score-0.892]
4 We demonstrate how to realize our approach on two commonly-used special cases: ridges & valleys and relief edges. [sent-17, score-1.035]
5 In each case, the optimal scale is found in accordance with the mathematical definition of the curve. [sent-18, score-0.393]
6 Examples of types of curves include ridges & valleys [19], parabolic curves [8], zero-mean curvature curves [8], demarcating curves [9], and relief edges [10], to name a few. [sent-22, score-2.168]
7 When relief edges [10] are detected using a single scale, some features are missed and others are inaccurate (a)-(c). [sent-33, score-0.442]
8 On the other hand, if the scale is too small, coarse features are localized inaccurately and false features appear. [sent-37, score-0.347]
9 However, most state-of-the-art curve detection algorithms use a single scale [1, 5, 9, 10, 19]. [sent-38, score-0.514]
10 Our goal is to propose a general framework for automatically estimating the optimal scale at each point on the surface. [sent-40, score-0.385]
11 [20] propose a scheme that is designed for a single type of curves, defined as the loci of points whose curvature variation is persistent over all scales. [sent-45, score-0.388]
12 Rather, they essentially apply multi-scale smoothing to the object, hopefully leaving the surface features intact. [sent-49, score-0.414]
13 However, the scale must be proportional not only to the surface features, but also to the curve type. [sent-50, score-0.785]
14 Given a definition of a curve type, we show how to calculate the optimal scale directly from the definition. [sent-55, score-0.59]
15 Briefly, every curve type is associated with a function,whose value indicates the strength of the feature. [sent-56, score-0.428]
16 This function typically depends on the surface curvature and its derivatives. [sent-57, score-0.557]
17 We define the scale at which the maximum is obtained as the optimal scale. [sent-59, score-0.417]
18 Using this local optimal scale, the surface is smoothed in a multiscale manner. [sent-60, score-0.533]
19 On this surface, the curves are detected, utilizing the original curve detection algorithms. [sent-61, score-0.418]
20 We demonstrate the benefit of our approach by applying it to two popular types of curves: ridges & valleys and relief edges. [sent-62, score-1.03]
21 Section 4 describes the algorithm for computing the optimal scale at every point. [sent-68, score-0.357]
22 Background This section describes the background on curve detection on surfaces and on multi-scale processing on surfaces. [sent-72, score-0.343]
23 View-independent curves depend solely on geometric properties of the surface [6, 8, 9, 10, 16, 19]. [sent-76, score-0.506]
24 The first is ridges & valleys [19], which are the extrema of principal curvatures and the sec- ond are relief edges [9, 10], which are the loci of zero crossings of the curvature in the edge direction. [sent-79, score-1.576]
25 Mutli scale processing of surfaces: Multi-scale processing of surfaces can be divided into two separate, yet related, tasks. [sent-80, score-0.382]
26 The first is the creation of scale space, which simultaneously represents the surface at different scales. [sent-81, score-0.546]
27 The second is the optimal scale selection, which automatically selects the scale that best represents the surface locally. [sent-82, score-0.903]
28 While in images, which are defined on a regular grid scale space, smoothing is a matter of consensus [13], for surfaces there are many ways to perform smoothing [3, 11, 17, 20, 21]. [sent-85, score-0.592]
29 Formally, given a surface S(u, v) : R2 → R3, its scale space representation is S(u, v,t) : R2 → R3, where t is the scale parameter, which is proportional to the amount of smoothing applied to the object. [sent-86, score-0.956]
30 Optimal scale selection: In images, the optimal scale selection was developed for edge detection and point-based features by Lindeberg [13]. [sent-89, score-0.737]
31 He proved that a function of the image spatial derivatives, which is normalized in a certain way, obtains a single maximum in scale space. [sent-90, score-0.511]
32 The scale at which the maximum is obtained is termed the optimal scale. [sent-91, score-0.417]
33 Then, the γ-normalized function of the derivatives of the image, defined as tγ/2f(δxkL(x; t)) (1) obtains a single maximum at the optimal scale to. [sent-93, score-0.609]
34 Optimal scale for surfaces was mostly used for interest point detection. [sent-95, score-0.41]
35 At these points, a function of some surface properties, normalized by the scale parameter, obtains a maximum both in the spatial and in the scale domains. [sent-96, score-1.03]
36 For instance, [20] compute the ratio between the eigenvalues; [14] compute the area with the minimal descriptor length; [12] compute the optimal size of the support region for computing surface normals. [sent-100, score-0.377]
37 Definition of the Optimal Scale Intuitively, the optimal scale at point p is the scale at which the likelihood that a curve passes through p is max- 222222666 imal. [sent-102, score-0.845]
38 Hence, the optimal scale is reformulated as the scale at which the normalized curve strength obtains a maximum in scale space. [sent-106, score-1.492]
39 Definition: Let curve c have an associated strength function f(K), which depends on the curvature and the curvature derivatives K. [sent-107, score-0.985]
40 Let t be the scale parameter, which is related to the amount of smoothing applied to the surface. [sent-108, score-0.368]
41 Then, the optimal scale sc of curve c is defined as the scale at which the function obtains a maximum: sc = arg matx tγf(K(t)) , (2) where tγ is the normalization coefficient, similar to what is used in Equation (1). [sent-110, score-0.971]
42 The strength function f(K(t)) monotonically decreases with scale, since as the surface becomes smoother, the features appear less prominent. [sent-112, score-0.524]
43 This single-maximum property is proved in Section 5 for two commonly-used spatial curve types: ridges & valleys and relief edges. [sent-115, score-1.203]
44 To get a feeling why this is true, recall that a surface can be presented locally, at every point on the surface, as a third degree polynomial (the Monge form) defined on the point’s tangent plane. [sent-116, score-0.335]
45 The surface curvature is proportional to the polynomial second derivative. [sent-117, score-0.599]
46 According to Equation (2), the normalized second derivative obtains a single maximum in the scale space. [sent-118, score-0.687]
47 Therefore, the surface curvature should also obtain a maximum in scale space. [sent-119, score-0.88]
48 Thus, the optimal scale (the maximum) is undefined there. [sent-121, score-0.357]
49 However, since there exist no curves in a featureless region and our goal is to detect curves, any scale chosen is valid and our definition holds. [sent-122, score-0.559]
50 First, we compute the optimal scale at every point on the surface. [sent-126, score-0.385]
51 This computation depends on the type of curve we want to detect, and specifically on its corresponding strength function f. [sent-127, score-0.428]
52 On this smoothed surface, the curves are finally computed us- ing their original detection algorithms. [sent-130, score-0.347]
53 Computing the optimal scale at every point: For points with features, we apply Equation (2) as is and obtain the optimal scale. [sent-132, score-0.451]
54 For featureless points, any value of scale is valid, and yet we need to choose a single scale. [sent-133, score-0.385]
55 The scale at a point with high strength is equal to the solution of Equation (2). [sent-136, score-0.482]
56 Let p be a point on the surface and ti(p) be the scale that solves Requirement 1. [sent-142, score-0.574]
57 The final scale t(p) is found by solving the following system: w(pt)(Δpt)(p =) t =i(p 0), (3) where w(p) is a weight function inversely proportional to the strength at p and Δ is the Laplacian. [sent-143, score-0.496]
58 Creating an optimally-smoothed surface: After the optimal scale at each point has been computed, we create a surface in which each point is smoothed to its optimal scale. [sent-149, score-0.942]
59 First, the surface is smoothed uniformly for each scale, using [23]. [sent-151, score-0.409]
60 Next, we create a surface for which the coordinates of every vertex are taken from the surface of its corresponding scale. [sent-153, score-0.592]
61 Finally, the resulting surface is smoothed in order to remove artifacts. [sent-154, score-0.409]
62 Specific Cases In this section we demonstrate how to apply our method to commonly-used curves: ridges & valleys and relief edges. [sent-159, score-1.006]
63 Recall that the normalization coefficient γ should be chosen such that to ensure a single maximum of the normalized strength function. [sent-161, score-0.361]
64 Derive the expression for the scale at which the normalized strength obtains a maximum. [sent-180, score-0.646]
65 Ridges & valleys The most prevalent curves on surfaces are ridges & valleys [19]. [sent-186, score-1.39]
66 A ridge (valley) point is a point on a manifold, where the positive (negative) principal curvature obtains a maximum (minimum) along its principal direction. [sent-188, score-0.58]
67 We now discuss the four steps mentioned above, for the case of ridges, whereas the case of valleys is similar. [sent-189, score-0.401]
68 Surface representation: We approximate a ridge s(x√; t0) of scale t0 with a Gaussian of standard deviation σ = √t0: s(x;t0) = √21πt0e−x2/2t0. [sent-192, score-0.337]
69 We assume that the scale space is represented by convolutions with Gaussians with smoothing parameter t: s(x; t + t0) = s(x; t0) ∗ g(x; t) . [sent-194, score-0.368]
70 (5) Thus, the ridge s(x; t0) at scale t is s(x;t + t0) =? [sent-195, score-0.337]
71 to show that the normalized curvature of s(x; t0) obtains a single maximum in scale space at x = 0. [sent-199, score-0.785]
72 (6) Then, we take its derivative with respect to t and prove that this derivative is equal to zero only at a single t. [sent-201, score-0.407]
73 The curvature κ(x) of a curve s(x) is known to be: κ(x) = −(1 +s s? [sent-203, score-0.471]
74 We need to compute the derivatives ofthe surface (curve) s? [sent-207, score-0.332]
75 Results: Figure 3 shows ridges and valleys on objects consisting offeatures ofdifferent scales. [sent-246, score-0.676]
76 When we increase the scale (c), the curves become smoother, but some features disappear. [sent-248, score-0.483]
77 See the 222222999 (a) Input Accuracy: (b) fine scale 85% (c) coarse scale 77% (d) coarsest scale 42% Figure 4. [sent-251, score-0.906]
78 (e) multi scale 91% The noisy cylinder has both fine and coarse valleys (a). [sent-253, score-0.778]
79 It is a cylinder with ridges and valleys of a couple of different scales, to which we added noise, as shown in Fig- ure 4. [sent-258, score-0.768]
80 In terms of error, we computed the percentage of the groundtruth valleys for which there exist real valleys within a predefined distance. [sent-263, score-0.802]
81 In practice, we counted the number of faces with ground truth valleys for which there exists a face with a detected valley within 0. [sent-264, score-0.44]
82 While the accuracy of the multi-scale results is 91%, for the single scale algorithm the results are between 79% for the finest scale to 42% for the coarsest, The results are: Scale 0 (finest): 79%, Scale 1: 85%, Scale 2: 77%, Scale 3 (coarsest): 42%, and Multiscale: 91%. [sent-266, score-0.583]
83 Relief edges Relief edges are defined as zero crossings of the curvature in the direction of the step edge model that best approximates the surface locally [10]. [sent-269, score-0.829]
84 They run on the slopes between ridges and valleys and are parallel to them. [sent-270, score-0.7]
85 Strength function f: As a strength function f, we employ the curvature derivative with respect to the arclength λ in the edge direction, as proposed in [10]. [sent-271, score-0.729]
86 Surface representation: A relief edge is represented, by definition, by a smoothed step edge. [sent-273, score-0.52]
87 Let s(x; t0) be a relief edge of an initial scale t0 (an ideal step edge smoothed with a Gaussian of standard deviation σ0 = √t0): s(x;t0) = √21πt0? [sent-274, score-0.847]
88 We want to show that the normalized curvature derivative of s(x; t0) obtains a single maximum in scale space at x = 0. [sent-278, score-0.961]
89 To compute ∂k/∂x, we take the derivative of the curvature defined in Equation (7). [sent-283, score-0.45]
90 When the scale is small (b), the resulting single scale curves are noisy. [sent-296, score-0.747]
91 When the scale is big (c), the single scale curves are smooth but not accurate enough. [sent-297, score-0.747]
92 We now proceed to compute the curvature κ(x) κ(x) =(−t0x+e− tx)23//22(√t0+2tπ)·[1 + e−x2/(t0+t)/1(2π(t0+ t))]3/2 and the curvature derivative ∂ kx? [sent-300, score-0.724]
93 Taking the derivative with respect to t and setting it to zero, we obtain a quadratic equation whose roots are: tmax = (A where ± B)/C, (15) A B C = = = 2γ − 6πt0 + 8πγt0 + 1, (4γ2 + 40πγt0 + 4γ + 36π2t02 − 12πt0 + 4(3π − 2πγ) . [sent-326, score-0.348]
94 Results: Figure 5 depicts relief edges on surfaces having features of various scales. [sent-332, score-0.534]
95 It compares two single-scale relief edges with our multi-scale edges. [sent-333, score-0.389]
96 This is so, since the depth of the relief varies, thus a single scale is insufficient. [sent-336, score-0.62]
97 Conclusion This paper presented a framework for automatic estimation of the optimal scale for curve detection on surfaces. [sent-343, score-0.581]
98 It can be applied to any curve type, as long as the curve has a strength function based on the curvature and its derivatives. [sent-344, score-0.859]
99 Edge detection and ridge detection with automatic scale selection. [sent-422, score-0.391]
100 Maxima of normalized curvature derivative of a step edge Here, we find the values of t at which Equation (14) obtains maximum. [sent-484, score-0.675]
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