cvpr cvpr2013 cvpr2013-218 knowledge-graph by maker-knowledge-mining
Source: pdf
Author: Sagi Katz, Ayellet Tal
Abstract: Point sets are the standard output of many 3D scanning systems and depth cameras. Presenting the set of points as is, might “hide ” the prominent features of the object from which the points are sampled. Our goal is to reduce the number of points in a point set, for improving the visual comprehension from a given viewpoint. This is done by controlling the density of the reduced point set, so as to create bright regions (low density) and dark regions (high density), producing an effect of shading. This data reduction is achieved by leveraging a limitation of a solution to the classical problem of determining visibility from a viewpoint. In addition, we introduce a new dual problem, for determining visibility of a point from infinity, and show how a limitation of its solution can be leveraged in a similar way.
Reference: text
sentIndex sentText sentNum sentScore
1 Presenting the set of points as is, might “hide ” the prominent features of the object from which the points are sampled. [sent-4, score-0.402]
2 Our goal is to reduce the number of points in a point set, for improving the visual comprehension from a given viewpoint. [sent-5, score-0.461]
3 This is done by controlling the density of the reduced point set, so as to create bright regions (low density) and dark regions (high density), producing an effect of shading. [sent-6, score-0.313]
4 This data reduction is achieved by leveraging a limitation of a solution to the classical problem of determining visibility from a viewpoint. [sent-7, score-0.255]
5 In addition, we introduce a new dual problem, for determining visibility of a point from infinity, and show how a limitation of its solution can be leveraged in a similar way. [sent-8, score-0.385]
6 These devices inherently produce 3D point sets from which, in some cases, 3D surfaces are reconstructed. [sent-11, score-0.173]
7 Obviously, one reason is that when drawing all the points, both the fore and the back points are visible, which is confusing. [sent-17, score-0.264]
8 Solving this problem by removing the back points is helpful, but some problems still remain (Figure 1(b)). [sent-18, score-0.232]
9 After removing the hidden points from (a), only the visible points are presented in (b). [sent-28, score-0.63]
10 Drawing every 5th visible point (c) results in loss of the features. [sent-30, score-0.282]
11 In (d), the result of our solution is shown: points on the silhouette and on subtle features, such as on the leaf-shaped relief on the horse’s back, are maintained. [sent-31, score-0.343]
12 Our goal is to perform reduction of points, as viewed from a given viewpoint, such that the object’s fine features become apparent, thus improving the visual comprehension of the entire object, as demonstrated in Figure 1(d). [sent-32, score-0.286]
13 The regions that are desired to appear bright, should have a lower density than regions that are desired to appear dark. [sent-33, score-0.3]
14 The question is how we can characterize these regions and determine the appropriate subset of points. [sent-34, score-0.116]
15 Given a point set P (193,252 points), the visible set of points PV is extracted. [sent-37, score-0.483]
16 Then, silhouette points PSIL and pit points PPIT are extracted as subsets of PV. [sent-38, score-0.688]
17 Some aim at accelerating surface reconstruction, by first reducing the number of input points, and then performing reconstruction on the reduced set [3, 6]. [sent-43, score-0.131]
18 In [8], in addition to the visible points [5], silhouette points, extracted by local reconstruction, are rendered. [sent-46, score-0.514]
19 In both cases, we are given a set of points and one specific point. [sent-52, score-0.201]
20 HPR aims at detecting the visible (or hidden) points, seen from the viewpoint. [sent-53, score-0.201]
21 Conversely, in TPO the goal is to find the subset of points that are visible to outside observers, positioned along the lines that connect these points with a given target point. [sent-54, score-0.773]
22 Second, we show that, interestingly, the false negatives of the operators that approximate HPR and TPO, are the subset of points we wish to preserve for achieving good visual comprehension of the sampled shape. [sent-57, score-0.654]
23 In particular, we show how to extract points that belong to locally-concave regions by determining false negatives of the HPR operator. [sent-58, score-0.56]
24 Similarly, we extract silhouette points by using false negatives of the TPO operator. [sent-59, score-0.534]
25 First and foremost, we present a novel algorithm for meaningful view-dependent data reduction of point sets. [sent-61, score-0.117]
26 Our algorithm is inherently simple and easy to implement, and is flexible in its ability to handle points in various dimensions. [sent-62, score-0.225]
27 Second, we introduce the problem of targetpoint occlusion and present an operator that solves it in the limit and approximates it in the finite case. [sent-64, score-0.307]
28 Finally, we suggest an algorithm for detecting the silhouettes of an object directly from the point cloud. [sent-65, score-0.163]
29 Algorithm Outline Given a point set P and a viewpoint C, our algorithm aims at selecting a subset of P, which preserves the features and the patterns of the object (Figure 2). [sent-67, score-0.198]
30 The naive approach of reducing the number of points by taking every kth point, blurs the silhouettes and the fine features, as it treats all the points similarly (Figure 1(c)). [sent-68, score-0.573]
31 Dense regions appear dark, whereas sparse regions appear bright. [sent-70, score-0.198]
32 Formally, we seek a visually-comprehensible subset PC of the visible points PV , PC ⊆ PV ⊆ P. [sent-71, score-0.452]
33 hInav oer tdheer f tooll porwo-ing two properties: First, in order to highlight the outline of the object, the likelihood of a point pi ∈ PV to belong to PC should be proportional to the angle b∈etw Peen the surface normal and the line-of-sight. [sent-73, score-0.321]
34 For instance, points on the silhouette should belong to PC. [sent-74, score-0.351]
35 Second, to enhance the shading effect on the viewed point set, the likelihood of a point pi ∈ PV to belong to PC should be large if it belongs to a deep concavity and small if it belongs to a locally-convex region. [sent-75, score-0.379]
36 This is based on the observation that deep concavi111222222 ties tend to be less illuminated, compared to convex regions, due to a reduced amount of light that reaches them. [sent-76, score-0.107]
37 The first operator is the HPR operator of [5] (Section 3), used for approximating the set of visible points, PV, from a viewpoint. [sent-78, score-0.675]
38 The second operator is used for approximating the set of occluding points from infinity (Section 4). [sent-79, score-0.689]
39 Both operators have a curvature-related limitation, which is leveraged for extracting the following two subsets (Section 5): PSIL ⊂ PV is the set of points that are located near silhouette regions, as viewed from the given viewpoint. [sent-80, score-0.513]
40 PPIT ⊂ PV is the set of points that are located near concave regions, as viewed from the viewpoint. [sent-81, score-0.353]
41 Hence, PC contains points that are either near silhouette regions or near concave regions, as desired: PC = PPIT ∪ PSIL. [sent-83, score-0.492]
42 The Hidden-Point Removal (HPR) Operator Given a set of points P and a viewpoint C, the goal is to find the set of points PV, which would be visible if the surface from which the points were sampled, were known. [sent-85, score-0.976]
43 Determining the visibility of a point set is an intriguing problem, as points cannot occlude each other. [sent-86, score-0.488]
44 The most common way to compute visibility is therefore to reconstruct the surface [2] and determine visibility on it. [sent-87, score-0.379]
45 Since both rays and points are singular primitives, the algorithms assume either “thick” rays [13] or “finitearea” points [4, 12]. [sent-90, score-0.534]
46 Unfortunately, assumptions need to be made regarding the thickness of the rays or the area of the points and normals must be estimated. [sent-91, score-0.295]
47 Inversion: An inversion function maps every point pi ∈ P to an inverted domain. [sent-96, score-0.301]
48 Many inversion functions are possible ann idn vine e[5d] dFo(mpia)i =. [sent-97, score-0.12]
49 Assuming, without loss of generality, that C is at the origin, the inversion function is defined as −rs Fγ(pi) =? [sent-100, score-0.12]
50 Convex hull construction: The convex hull of the transformed set of points and the viewpoint is calculated. [sent-106, score-0.48]
51 The main result of [5] is that the points that reside on the convex hull of Step 2 are the images of the visible points. [sent-107, score-0.532]
52 Intuitively, “how much” a point is visible, depends on the size of the empty region that lies between the point and the viewpoint. [sent-108, score-0.19]
53 It is proved that points that fall on the convex hull in the inverted domain are those that are associated with large empty regions. [sent-110, score-0.392]
54 While the HPR operator is powerful, deep concavities in the object result in false negatives (i. [sent-111, score-0.504]
55 , points that should be visible but are not detected as such by the operator). [sent-113, score-0.402]
56 This can be explained by the following lemma, which relates between the correct detection of the visible points to the curvature κ, the distance between the surface and the viewpoint d, and the angle β between the surface normal and the line of sight. [sent-114, score-0.7]
57 1 Point p ∈ {HPRγ (P)} if p is visible and the Lcuermvamtuare 3 κ 1a tP p nsatt pisf ∈ies {: κ <γ(1 −d γ()γs2isni(nβ2()β(c)o +s2 c(oβs)2 −(β γ)s)i23n2(β) . [sent-116, score-0.201]
58 This lemma, whose proof for the inversion function that we use, is given in the appendix, implies that the HPR operator cannot resolve visibility in regions of high curvature. [sent-117, score-0.559]
59 It is guaranteed to detect the points correctly only if the curvature κ is below a threshold, i. [sent-118, score-0.27]
60 , the local surface is convex or the local surface is concave and the curvature is small enough. [sent-120, score-0.322]
61 Given a set of points P and a target point C, the goal is to find the subset of P, which would occlude C from outside observers positioned at ∞, if the surface from which the points were sampled, dex aitst ∞ed,. [sent-124, score-0.868]
62 In the HPR problem, the rays coming out ofviewpoint C, intersect the visible points. [sent-126, score-0.326]
63 In the TPO problem, on the other hand, the rays come towards C from infinity and intersect the occluding points before reaching the target point. [sent-127, score-0.579]
64 Hence, each of these points occludes C from an observer, which is located in the direction from which the ray is coming, but farther than the occluding point. [sent-128, score-0.508]
65 Obviously, when a ray reaches C, the sample point on the ray is not occluded by any other point in this direction. [sent-129, score-0.317]
66 1 Occluding point: Given a point set P and a target point C, a point pi ∈ P is an occluding point to C, 111222333 (a) From-point visibility (b) Target-point occlusion Figure 3. [sent-131, score-0.805]
67 We are given a point set P (sampled from the gray surface) and a point C (blue). [sent-134, score-0.162]
68 The black points are the visible points from C in (a) and the points occluding C from observers at infinity in (b). [sent-135, score-1.116]
69 if pi is visible to some observer located in O where t > suppj∈P ? [sent-136, score-0.363]
70 To approximate the solution to the target-point occlusion problem, we present an operator which, similarly to the HPR operator, is composed of two stages: (1) point transformation and (2) convex hull computation. [sent-139, score-0.517]
71 In particular, in Equation (1), we use γ < 0 for from-point visibility and 0 < γ < 1for target-point occlusion; see Figure 4. [sent-141, score-0.149]
72 The correctness of this operator is expressed in the following lemmas, whose proofs are provided in the supplementary material. [sent-142, score-0.224]
73 Specifically, we show that in the limit, when the point set is an infinite sampling of the surface, the operator extracts the occluding subset accurately. [sent-143, score-0.547]
74 Let P be an infinite sampling of a surface S, O ⊆ S be tLheet s Pet bofe points itnhiatet o scacmlupdlein tghe o target point SC, (O i. [sent-144, score-0.432]
75 ⊆, th Se ground truth), and TPOγ (P) ⊆ S be the set of points extracted by our operator. [sent-146, score-0.201]
76 , every point p marked occluding by the operator ⊆is Oind,e ie. [sent-150, score-0.471]
77 2 When γ → +0, the set of points marked by the TLPeOm operator, hise equal to the set of occluding points. [sent-154, score-0.367]
78 1 suggests that in the limit, the number of false positives of the operator is 0. [sent-156, score-0.311]
79 2 adds that the number of false negatives is also 0 when γ → +0. [sent-158, score-0.221]
80 ) We are particularly interested in understanding the locations where false negatives appear. [sent-160, score-0.221]
81 0or1 from-point visibility & target-point occlusion for a 2D point set. [sent-167, score-0.286]
82 Blue points were detected visible (occluding), while green points were detected as invisible (nonoccluding). [sent-168, score-0.603]
83 3 p ∈ {TPOγ (P)} if p is visible to an observer positioned on t h∈e ray pO-C(P, farther away bthlean to any point einr P, and the curvature κ at p satisfies κ >γ(1 −d γ()γs2isni(nβ2()β(c)o +s2 c(oβs)2 −(β γ)s)i23n2(β) . [sent-171, score-0.558]
84 Therefore, while the HPR operator is limited in its ability to correctly identify visible points around locally-concave regions, in respect to the viewpoint, the TPO is limited in its ability to correctly identify occluding points around locallyconvex regions. [sent-172, score-0.993]
85 These dual limitations are the basis for our point reduction method. [sent-173, score-0.168]
86 Controlled Reduction of Points Our goal is to extract a subset of the visible points, which satisfies the two requirements discussed in Section 2. [sent-175, score-0.275]
87 Recall that we look for (1) points at locally-concave regions, which create a natural shading effect, and (2) points around the silhouettes regions, which by definition, have normals perpendicular to the line of sight. [sent-176, score-0.557]
88 1, which states that when applying the HPR operator with γ < 0, the operator will produce false negatives around locally-concave regions. [sent-179, score-0.669]
89 But these are precisely the points we seek after. [sent-180, score-0.201]
90 We term these false negatives as pit points and define them as: Ppits = PV \ HPRγp (P). [sent-181, score-0.566]
91 3, we learn that the TPO op- erator will produce false negatives around locally-convex regions. [sent-183, score-0.221]
92 These false negatives are the tip points, defined as: Ptips = PV \ TPOγt (PV). [sent-184, score-0.286]
93 (3) Since, as illustrated in Figure 4, the tip points need not necessarily be a subset of the visible points (e. [sent-185, score-0.718]
94 of the bunny), we apply the TPO operator on the set of visible points, rather than on the entire point set. [sent-190, score-0.506]
95 Since we wish to extract points with a small β (i. [sent-193, score-0.201]
96 Due to the fact that this data consists of samples that are seen from a given viewpoint (scanner), it provides the visible point set from the scanner location. [sent-201, score-0.39]
97 Therefore, in this case PV = P and the pit points are simply Ppits = P \ HPRγp (P) . [sent-202, score-0.345]
98 Similarly, the set of silhouette points is defined as Psils = P \ TPOγt (P). [sent-203, score-0.313]
99 Figure 5 shows the pit points and the tip points for a range data set, for γt = −γp, which are points on locallyconcave regions and on locally-convex regions, respectively. [sent-204, score-0.878]
100 While the details of the facade of the church cannot be identified in the input, they are clearly perceived when rendering only the tip or only the pit points. [sent-205, score-0.209]
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