cvpr cvpr2013 cvpr2013-208 knowledge-graph by maker-knowledge-mining
Source: pdf
Author: Rui Shi, Wei Zeng, Zhengyu Su, Hanna Damasio, Zhonglin Lu, Yalin Wang, Shing-Tung Yau, Xianfeng Gu
Abstract: Automatic computation of surface correspondence via harmonic map is an active research field in computer vision, computer graphics and computational geometry. It may help document and understand physical and biological phenomena and also has broad applications in biometrics, medical imaging and motion capture. Although numerous studies have been devoted to harmonic map research, limited progress has been made to compute a diffeomorphic harmonic map on general topology surfaces with landmark constraints. This work conquer this problem by changing the Riemannian metric on the target surface to a hyperbolic metric, so that the harmonic mapping is guaranteed to be a diffeomorphism under landmark constraints. The computational algorithms are based on the Ricci flow method and the method is general and robust. We apply our algorithm to study constrained human brain surface registration problem. Experimental results demonstrate that, by changing the Riemannian metric, the registrations are always diffeomorphic, and achieve relative high performance when evaluated with some popular cortical surface registration evaluation standards.
Reference: text
sentIndex sentText sentNum sentScore
1 edu } Abstract Automatic computation of surface correspondence via harmonic map is an active research field in computer vision, computer graphics and computational geometry. [sent-16, score-0.542]
2 Although numerous studies have been devoted to harmonic map research, limited progress has been made to compute a diffeomorphic harmonic map on general topology surfaces with landmark constraints. [sent-18, score-0.923]
3 This work conquer this problem by changing the Riemannian metric on the target surface to a hyperbolic metric, so that the harmonic mapping is guaranteed to be a diffeomorphism under landmark constraints. [sent-19, score-1.374]
4 We apply our algorithm to study constrained human brain surface registration problem. [sent-21, score-0.563]
5 Experimental results demonstrate that, by changing the Riemannian metric, the registrations are always diffeomorphic, and achieve relative high performance when evaluated with some popular cortical surface registration evaluation standards. [sent-22, score-0.597]
6 Among various correspondence research, automatic computation of surface correspondence regulated by certain geometric or functional constraints is an important research field in computer vision and medical imaging. [sent-27, score-0.282]
7 Among various rigid and non-rigid surface registration approaches (e. [sent-29, score-0.378]
8 [3, 5, 16]), harmonic map is one of the most broadly applied methods [27, 32]. [sent-31, score-0.316]
9 In computer vision and medical imaging fields, surface harmonic map has been used to compute spherical conformal mapping [12], image registration [13], high resolution tracking ofnon-rigid motion [27], non-rigid surface registration [17], etc. [sent-33, score-1.296]
10 However, the current state-of-the-art surface harmonic map research has some limitations. [sent-34, score-0.51]
11 A harmonic map combined with landmark matching conditions usually does not guarantee diffeomorphism. [sent-37, score-0.424]
12 All these problems become obstacles to apply harmonic map to solve general non-rigid surface matching problems. [sent-38, score-0.51]
13 In contrast, in current work, we slice along the landmark curves on general surfaces and assign a unique hyperbolic metric on the template surface, such that all the boundaries become geodesics. [sent-39, score-0.944]
14 Then by establishing harmonic mappings, the obtained surface correspondences are guaranteed to be diffeomorphic. [sent-40, score-0.474]
15 In this paper, we apply the proposed method to study human brain cortical surface registration problem. [sent-41, score-0.716]
16 The cortical surface registration may help identify early disease imaging biomarkers, develop new treatments and monitor their effectiveness, as well as lessen the time and cost of clinical trials. [sent-43, score-0.531]
17 Introduce a novel algorithm to compute harmonic mappings on hyperbolic metric using nonlinear heat diffusion method and Ricci flow. [sent-45, score-1.082]
18 Develop a novel brain registration method based on hyperbolic harmonic maps. [sent-47, score-1.26]
19 [26] studied brain morphology with Teichm¨ uller space coordinates where the hyperbolic conformal mapping was computed with the Yamabe flow method. [sent-54, score-1.049]
20 Zeng [3 1] proposed a general surface registration method via the Klein model in the hyperbolic geometry where they used the inversive distance curvature flow method to compute the hyperbolic conformal mapping. [sent-55, score-1.968]
21 Various surface registration methods were proposed in computer vision field [7, 20, 16, 18]. [sent-56, score-0.378]
22 To register brain cortical surfaces, a common approach is to compute a range of intermediate mappings to some canonical parameter space [24, 11, 29]. [sent-57, score-0.374]
23 This flow can be constrained using anatomical landmark points or curves [2, 14, 21, 33], by sub-regions of interest [15], by using currents to represent anatomical variation [10], or by metamorphoses [25]. [sent-59, score-0.275]
24 There are also various ways to optimize surface registrations [4, 19, 23]. [sent-60, score-0.26]
25 Overall, finding diffeomorphic mappings between brain surfaces is an important but difficult problem. [sent-61, score-0.404]
26 Theoretic Background This section briefly covers the most relevant concepts and theorems of harmonic maps [22] and surface Ricci flow [31]. [sent-65, score-0.527]
27 g(t) where χ(S) is the Eulder characteristic number of S, A(0) is the total area of the surface at time 0, K(t) is the Gaussian curvature induced by g(t). [sent-70, score-0.402]
28 2 (Hamilton) If χ(S) < 0, then the solution to the normalized Ricci flow equation exists for all t > 0 and converges to a metric with constant curvature 2πAχ((0S)). [sent-72, score-0.279]
29 By running Ricci flow, a hyperbolic metric of the surface can be obtained, which induces −1 Gaussian curvature everywhere. [sent-73, score-1.054]
30 Hyperbolic Plane The Poincar e´’s disk model for the hyperbolic plane H2 is the unit disk on the complex plane {z ∈ C| |z| < 1} with Riemannian metric (1−z ¯z)−2dzd z¯. [sent-74, score-0.97]
31 The hyperbolic rigid motions are M ¨obius transformations φ : z → eiθ (z − z0)/(1 −¯ z 0z) . [sent-76, score-0.611]
32 MTh ¨oeb aiuxiss orafn φ oisr mthea hyperbolic l i→ne through izts f/i(x1ed − points: z1 = limn→∞ φn (z) , z2 = limn→∞ φ−n (z) . [sent-77, score-0.611]
33 Given two non-intersecting hyperbolic lines γ1 and γ2, there exists a unique hyperbolic line τ orthogonal to both of them, and gives the shortest path connecting them. [sent-78, score-1.248]
34 A◦ φnother hyperbolic plane model is the Klein’s disk model, where the hyperbolic lines coincide with Euclidean lines. [sent-80, score-1.375]
35 Furthermore, all the homotopic classes of paths on the surface starting from p form a simply connected surface the projection map p : S˜ → S maps each path to its end point, the projection map i →s a Slo mcala homeomorphism. [sent-88, score-0.542]
36 Suppose S is with a hyperbolic metric, then its universal covering space can be isometrically embedded onto the hyperbolic plane H2. [sent-90, score-1.363]
37 Hyperbolic Pants Decomposition A pair of pants is a genus zero surface with 3 boundaries. [sent-98, score-0.446]
38 Furthermore, each pair of hyperbolic pants can be further decomposed. [sent-102, score-0.819]
39 Assume the pair of pants have three geodesic boundaries {γi , γj , γk}. [sent-103, score-0.291]
40 e T thhee shortest paths divide the surface to two identical hyperbolic hexagons with right inner angles. [sent-105, score-0.988]
41 when mapped to the Klein’s model, the hyperbolic hexagons coincide with convex Euclidean hexagons. [sent-106, score-0.745]
42 Harmonic Map Suppose S is a closed oriented surface with a Riemannian metric g, by running surface Ricci flow, one can obtain a Riemannian metric with constant curvature +1, 0, −1 everywhere. [sent-107, score-0.667]
43 The universal covering space of the s+u1r,fa0c,e− can v beer isometrically uemnivbeerdsadeld c oovnetori nthge s sphere fC th ∪e s{u∞rfa}c, eth cea nE buecl i sdoemane plane C e or tdhdee hyperbolic plane CH 2∪. [sent-108, score-0.775]
44 3 (Harmonic Map) The harmonic energy of the mapping is defined as E(f) =? [sent-117, score-0.338]
45 Sρ(z)(|wz|2+ |w z¯|2)dxdy If f is a critical point of the harmonic energy, then f is called a harmonic map. [sent-118, score-0.56]
46 The necessary condition for f to be a harmonic map is the Euler-Lagrange equation wz ¯z+ρρwwzw z¯≡ 0 The following theorem lays down the theoretic foundation of our proposed method. [sent-119, score-0.367]
47 4 [22] Suppose f : (S1, g1) → (S2 , g2) is a degree one harmonic map, furthermore t)he → Ri (eSmann metric on S2 induces negative Gauss curvature, then for each homotopy class, the harmonic map is unique and diffeomor- phic. [sent-121, score-0.718]
48 Hyperbolic pants decomposition, isometrically embed them to Klein model. [sent-131, score-0.304]
49 Compute harmonic maps using Euclidean metrics between corresponding pairs of pants, with consistent boundary constraints. [sent-133, score-0.325]
50 Use nonlinear heat diffusion to improve the mapping to a global harmonic map on Poincare disk model. [sent-135, score-0.606]
51 Discrete Hyperbolic Ricci Flow Because the Euler characteristic number of the cortical surfaces are negative, they admit hyperbolic metrics. [sent-143, score-0.835]
52 We treat each triangle as a hyperbolic triangle and set the target 222555333311 Figure 1. [sent-144, score-0.637]
53 Decompose M and N into multiple pants, and each pant further decomposed to 2 hyperbolic hexagons. [sent-150, score-0.611]
54 Hyperbolic hexagons on Poincar e´ disk become convex hexagons under Klein model, then a one-to-one map between the correspondent parts of M and N can be obtained. [sent-152, score-0.474]
55 Then we can apply our hyperbolic heat diffusion algorithm to get a global harmonic diffeomorphism. [sent-153, score-0.993]
56 Gauss curvatures for each interior vertex to be zeros, and the target geodesic curvature for each boundary vertex to be zeros as well. [sent-156, score-0.403]
57 Then compute the hyperbolic metrics of the brain meshes using discrete hyperbolic Ricci flow method [31]. [sent-157, score-1.46]
58 The edge length lij of [vi, vj ] is determined by the hyperbolic cosine law: coshlij = coshricoshrj + sinhrisinhrjcosφij 3. [sent-165, score-0.611]
59 The angle related to each corner , is determined by the current edge lengths with the inverse hyperbolic cosine law. [sent-166, score-0.611]
60 Compute the discrete Gaussian curvature Ki of each vertex vi : θjik, Ki=? [sent-168, score-0.304]
61 Hyperbolic Pants Decomposition In our work, the input surface is a genus zero surface with multiple boundary components ∂S = γ0 +γ1 + · · · γn, moreover, the surface is with hyperbolic metric,+ a·n·d· all boundaries are geodesics. [sent-181, score-1.306]
62 The hyperbolic line through the fixed points is the axis of the φγiγj , which is the geodesic in [γiγj] . [sent-184, score-0.707]
63 , γi and γj bound a pants patch, remove th·i γs pants patch from M. [sent-202, score-0.416]
64 The resultant piecewise harmonic mapping is the initial mapping. [sent-210, score-0.338]
65 Assume a pair of hyperbolic pants M with three geodesic boundaries {γi, γj , γk}. [sent-212, score-0.902]
66 On the universal covering space Mo˜u,n γi and γj are lifted to hyperbolic lines, γ˜i and γ˜j respectively. [sent-213, score-0.663]
67 222555333422 In the second stage, each hyperbolic hexagon on the Poincar e´ disk is transformed to a convex hexagon in Klein’s diskusingz → 1+2zz ¯z. [sent-216, score-0.793]
68 It is well known that if the target mapping domain is convex, then planar harmonic maps are diffeomorphic [22]. [sent-220, score-0.476]
69 The boundary conditions need to be consistent, such that the harmonic mappings between hexagons can be glued together to form a homeomorphic initial mapping. [sent-221, score-0.495]
70 Non-linear Heat Diffusion Let (S, g) be a triangle mesh with hyperbolic metric g. [sent-227, score-0.695]
71 The initial mapping is diffused to form the hyperbolic harmonic map. [sent-237, score-0.949]
72 Suppose f : (S1, g1) → (S2 , g2) is the initial map, g1 and g2 are hyperbolic m)e t→rics (. [sent-238, score-0.611]
73 Input: Two surface models M, N with their hyperbolic metric CM and CN on Poincar e´ disk, the one-to-one correspondence (vi, pi) and a threshold ε. [sent-243, score-0.89]
74 For each vertex vi of M, embed it’s neighborhood onto Poincar e´ disk, in which vi has coordinate zi; do the same for pi and note it’s coordinate on Poincar e´ disk as wi. [sent-247, score-0.372]
75 Data Preparation We perform the experiments on 24 brain cortical surfaces reconstructed from MRI images. [sent-262, score-0.409]
76 Each cortical surface has about 150k vertices, 300k faces and used in some prior research [21]. [sent-263, score-0.347]
77 On each cortical surfaces, a set of 26 landmark curves were manually drawn and validated by neuroanatomists. [sent-264, score-0.261]
78 Registration Visualization We show the visualized registration result of 2 brain models in Figure 4, with one as 222555333533 target and the other one registered to it. [sent-267, score-0.43]
79 The hyperbolic Ricci flow takes about 120 seconds, the hyperbolic heat diffusion takes about 100 seconds. [sent-270, score-1.377]
80 The complexity of pants decomposition and initial mapping construction depends on the number of land marks. [sent-271, score-0.289]
81 Landmark Variation For general surface registration, it is important to incorporate consistent landmark matching. [sent-277, score-0.302]
82 Our experimental results show that by replacing Euclidean metric by hyperbolic metric on the template cortical surface, the quality of the registrations have been improved prominently, with a slightly increase of time complexity. [sent-291, score-0.983]
83 Registration Flipping One of the most promising advantages of our registration algorithm is that it guarantees the mapping between two surfaces to be diffeomorphic. [sent-294, score-0.313]
84 Curvature Distortion We first evaluated registration accuracy by comparing the alignment of curvature maps between the registered models [21]. [sent-306, score-0.392]
85 We quantified the effects of registration on curvature by computing the difference of curvature maps from the registered models. [sent-308, score-0.565]
86 As Figure 6 shows, we assign each vertex the curvature difference between it’s own curvature and the curvature of it’s correspondent point on the target surface, then build a color map. [sent-309, score-0.635]
87 It is obvious that the current registration method produces less curvature errors. [sent-318, score-0.357]
88 tNedote b yth taht,e i rfa atthieo registration is not diffeomorphic, the local area distortion may go to ∞. [sent-329, score-0.284]
89 Conclusion and Future Work This work introduces a hyperbolic harmonic mapping based algorithm, which automatically establish diffeomorphic surface correspondences between general surfaces. [sent-341, score-1.255]
90 Compared with conventional landmark constrained brain cortical surface registration work, our results are bijective while enforcing the landmark curve matching conditions. [sent-342, score-0.932]
91 To achieve this, the new method changes the Riemannian metric on the target surface and greatly improves the registration quality. [sent-343, score-0.457]
92 Experimental results demonstrate the current method always produces diffeomorphism, and outperforms some existing brain registration methods in terms of curvature difference and local area distortion. [sent-346, score-0.577]
93 Diffeomorphic brain registration [3] [4] [5] [6] [7] [8] [9] [10] under exhaustive sulcal constraints. [sent-369, score-0.369]
94 Inferring brain variability from diffeomorphic deformations of currents: An integrative approach. [sent-456, score-0.297]
95 Genus zero surface conformal mapping and its application to brain surface mapping. [sent-476, score-0.773]
96 Shape-based diffeomorphic registration on hippocampal surfaces using Beltrami holomorphic flow. [sent-546, score-0.367]
97 Comparison of landmark-based and automatic methods for cortical surface registration. [sent-564, score-0.347]
98 Surface fluid registration of conformal representation: Application to detect disease burden and genetic influence on hippocampus. [sent-579, score-0.326]
99 High resolution tracking of non-rigid motion of densely sampled 3d data using harmonic maps. [sent-619, score-0.28]
100 Quantitative evaluation of LDDMM, FreeSurfer, and CARET for cortical surface mapping. [sent-674, score-0.347]
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