cvpr cvpr2013 cvpr2013-44 knowledge-graph by maker-knowledge-mining
Source: pdf
Author: Zhengyu Su, Wei Zeng, Rui Shi, Yalin Wang, Jian Sun, Xianfeng Gu
Abstract: Brain mapping transforms the brain cortical surface to canonical planar domains, which plays a fundamental role in morphological study. Most existing brain mapping methods are based on angle preserving maps, which may introduce large area distortions. This work proposes an area preserving brain mapping method based on MongeBrenier theory. The brain mapping is intrinsic to the Riemannian metric, unique, and diffeomorphic. The computation is equivalent to convex energy minimization and power Voronoi diagram construction. Comparing to the existing approaches based on Monge-Kantorovich theory, the proposed one greatly reduces the complexity (from n2 unknowns to n ), and improves the simplicity and efficiency. Experimental results on caudate nucleus surface mapping and cortical surface mapping demonstrate the efficacy and efficiency of the proposed method. Conventional methods for caudate nucleus surface mapping may suffer from numerical instability; in contrast, current method produces diffeomorpic mappings stably. In the study of cortical sur- face classification for recognition of Alzheimer’s Disease, the proposed method outperforms some other morphometry features.
Reference: text
sentIndex sentText sentNum sentScore
1 Abstract Brain mapping transforms the brain cortical surface to canonical planar domains, which plays a fundamental role in morphological study. [sent-4, score-0.945]
2 Most existing brain mapping methods are based on angle preserving maps, which may introduce large area distortions. [sent-5, score-0.902]
3 This work proposes an area preserving brain mapping method based on MongeBrenier theory. [sent-6, score-0.902]
4 The brain mapping is intrinsic to the Riemannian metric, unique, and diffeomorphic. [sent-7, score-0.543]
5 The computation is equivalent to convex energy minimization and power Voronoi diagram construction. [sent-8, score-0.38]
6 Experimental results on caudate nucleus surface mapping and cortical surface mapping demonstrate the efficacy and efficiency of the proposed method. [sent-10, score-1.343]
7 Conventional methods for caudate nucleus surface mapping may suffer from numerical instability; in contrast, current method produces diffeomorpic mappings stably. [sent-11, score-0.742]
8 Many prominent approaches, such as conformal mapping [18] and Ricci Flow [20] which have been employed to shape analysis [27, 7] and surface registration [35]. [sent-15, score-0.746]
9 The conformal mapping may bring huge area distortions . [sent-17, score-0.775]
10 In this work, we propose a novel method to compute area preserving mapping between surfaces. [sent-27, score-0.604]
11 We tested our algorithm on cortical and caudate nucleus surfaces extracted from 3D anatomical brain magnetic resonance imaging (MRI) scans. [sent-30, score-0.969]
12 Figure 1demonstrates the unique power that the area preserving mapping provides for brain cortical surface visualization when compared with its counterpart conformal mapping result. [sent-31, score-2.08]
13 On cortical surfaces, the area preserving may also provide good visualization function to visualize those deeply buried sulci areas which otherwise are usually visualized with big area distortions. [sent-32, score-0.976]
14 Comparison The area preserving mapping is based on Optimal Mass Transport (OMT) theory, which has been applied for image registration and warping [19, 26] and visualization [36]. [sent-41, score-0.667]
15 Comparison of geometric mappings for a left brain cortical surface: (a) brain cortical surface lateral view; (b) brain cortical surface medial view; brains are color coded according to functional area definition in [11]; (c) conformal mapping result; (d) area preserving mapping result. [sent-44, score-3.42]
16 The results show that conformal mapping has much more area distortions on the areas close to the boundary while the area preserving mapping provides a map which preserves the area everywhere. [sent-45, score-1.633]
17 Consider a diffeomorphism f : Ω0 → Ω1, which is mass preservation μ0 = |Jf|μ1 ◦ f where Jfis the Jacobian ofthe mapping f. [sent-50, score-0.398]
18 An optimal mass transport map, when it exists, minimizes the mass transport cost. [sent-53, score-0.703]
19 There are two different approaches to prove the existence of the optimal mass transport map, i. [sent-54, score-0.364]
20 (1) (2) In contrast, Brenier showed there exists a convex function × u : Ω0 → R, such that its gradient map ∇u gives the optimal mass transport map, asn gdr preserves pth ∇e mass: μ0 = det|H(u) |μ1 ◦ ∇u. [sent-63, score-0.449]
21 In our current method, we only discretize the target domain Ω1 to n points, then determine n power weights for them, so that the power Voronoi diagram induced by the points and their power weights gives the optimal mass transport map. [sent-69, score-0.953]
22 Furthermore, the obtained area preserving mapping between two surfaces is solely determined by the surface Riemannian metric, therefore it is intrinsic. [sent-81, score-0.827]
23 Contributions Our major contributions in this work include: a way to compute area preserving mapping between surfaces based on Brenier’s approach in Optimal Mass Transport theory. [sent-84, score-0.679]
24 Thus our method offers a stable way to calculate area preserving mapping in 2D parametric coordinates. [sent-87, score-0.604]
25 To the best of our knowledge, it is the first work to compute area preserving mapping between surfaces based on Brenier’s approach in OMT and apply it to map the profile of differences in surface morphology between healthy control subjects and AD patients. [sent-88, score-1.052]
26 Our experimental results show our work may provide novel ways for shape analysis and improve the statistical detection power for detecting abnormalities in brain surface morphology. [sent-89, score-0.578]
27 Quasi-isometric brain parameterization has been investigated in [13, 6, 12, 3 1]. [sent-93, score-0.345]
28 Conformal brain mapping methods have been well developed in the field, such as circle packing based method in [21], finite element method [3, 23, 32], spherical harmonic map method [17], holomorphic differential method [33] and Ricci flow method [34]. [sent-94, score-0.714]
29 Area preserving mapping has been applied for visualizing branched vessels and intestinal tracts in [36], which combined Kantorovich’s approach with conformal mapping. [sent-95, score-0.759]
30 Optimal mass transport mapping based on Kantorovich’s approach has been applied for image registration in [19]. [sent-96, score-0.613]
31 Comparing to the existing method, our method is based on Monge-Brenier theory to compute the Optimal Transport mapping and achieves the area preserving. [sent-98, score-0.414]
32 The power diagram of {(pi, hi)} is a partition of the Rn iTnhtoe k p ocwelelsr W diia,g sruacmh tohfat { a point x belongs ttioti oWni owfh tehnee vRer Pow(x,pi) = mjinPow(x,pj). [sent-103, score-0.325]
33 We denote the area of Wi as wi, call it the area weight. [sent-104, score-0.338]
34 The dual graph of the power diagram is called the power Delaunay triangulation. [sent-105, score-0.483]
35 The power diagram function is the upper envelope of all supporting hyper planes u(x) := miax{? [sent-107, score-0.325]
36 +21hi} (3) Hence the power diagram on P corresponds, by vertical projection, to the graph of u(x). [sent-111, score-0.325]
37 Fix a finite point set P = {p1, p2, · · · ,pk}, the powers are h = (h1, h2, · · · , hk), {thpe power diagram hofe { p(opwi,e ehrsi)} a ien hRn = partitions ,Ω· t·o , chells t{hWe1 p , Ww2e , ·d ·i ·a , rWamk} o, ft {he( areas are w = (w1, w2 , · · · , wk). [sent-141, score-0.445]
38 Furthermore, we treat the areas w as the function of the powers h, then the mapping h → w is a diffeomorphism. [sent-147, score-0.335]
39 th Fei area pvoeicntotr se ats Pso,c giiavteend ato p tohwee convex rc hell ∈ ∈de Hc,olm e-t position of Ω induced by the power diagram for {(pi, hi)}, tphoesnit itohne mapping w = by tφh(eh) p : eHr →ia Wram mis f a diffeom)or}-, phism. [sent-160, score-0.826]
40 Let the power diagram for h be Dh, the dual Power Delaunay triangulation be Th. [sent-163, score-0.391]
41 From the minimizer h, we construct the power Voronoi diagram Dh, which partitions to convex polygonal cells {W1, W2 , · · · , Wk}, and the power Voronoi diagram feullnsct {iWon u(x) using Eqn. [sent-189, score-0.705]
42 The conformal factor defines a measure on the unit disk μ = e2λdxdy. [sent-195, score-0.394]
43 eTxhies composition mapping pτp−in1 g◦ τ τφ : : S, →xd D) →is an area preserving mapping. [sent-197, score-0.604]
44 In practice, the surface is approximated by a triangle mesh M, normalized by a scaling such that the total area is π. [sent-198, score-0.344]
45 The conformal mapping φ : M → D can be achieved using Tdhiescr ceotnef oRrimccail fl moawp pminetghφo d : [ M34] . [sent-199, score-0.569]
46 Then compute the dual power Delaunay triangulation, compute the lengths of edges in the diagram and triangulation, form the Hessian matrix H using Eqn. [sent-208, score-0.351]
47 Then the power diagram for {(φ(vi) , hi} partitions D to convex polygonal rc deilalgs r{aWmi} fo, rt {he( φB(vrenier map itist given by ·μ + : vWexi p→o yφg(ovni)a. [sent-211, score-0.41]
48 The area preserving mapping is given by τ−1 ◦ φ(vi) = ci. [sent-213, score-0.604]
49 τ 222222333866 Algorithm 1 Area Preserving Mapping Input: Input triangle mesh M, total area π and area dif- ference threshold δw. [sent-216, score-0.365]
50 Output: A unique diffeomorphic area preserving mapping f : M → D, where D is a unit disk. [sent-217, score-0.718]
51 The area wi of epainchg cfel :l MWi → →∈ DD, i sw hcleorsee D Dto i sth ae target area wi. [sent-218, score-0.429]
52 Assign emaecthh osidte [ 3φ4(],v φi) :∈ M MD → →wi Dth, power hDi s= a 0u aint dd target area wi = μ(vi) de)f ∈ined D a wbiotvhe p. [sent-221, score-0.392]
53 Compute the power diagram and calculate the area wi of each cell Wi. [sent-224, score-0.585]
54 Compute the dual power Delaunay triangulation, and compute the lengths of edges in the diagram and triangulation to form the Hessian matrix H using Eqn. [sent-226, score-0.391]
55 Then the area preserving mapping is given by τ−1 ◦φ(vi) = ci, where τ is the Brenier map τ : Wi → φ(vi). [sent-239, score-0.634]
56 Experimental Results We applied our area preserving mapping method to various anatomical surfaces extracted from 3D MRI scans of the brain. [sent-241, score-0.718]
57 The caudate nucleus is a nucleus located within the basal ganglia of human brain. [sent-250, score-0.385]
58 Figure 2 (a) shows the triangular mesh of a reconstructed left caudate surface segmented by FIRST. [sent-252, score-0.423]
59 For example, a conformal mapping on slim surface usually introduces area distortions at the exponential level and may cause big numerical problems. [sent-254, score-0.962]
60 In contrast, our method evenly embeds the caudate surface to the parametric domain and keeps the area element unchanged. [sent-255, score-0.538]
61 Figure 2 (b) shows that most parts of conformal mapping result shrink towards the center, while the area preserving method shown in Figure 2 (c) gives a good mapping, keeping the same area element, without much numerical error. [sent-257, score-1.097]
62 Figure 3 are the histograms of area distortion of result surface triangles to original surface triangles for conformal mapping and area preserving mapping, respectively. [sent-258, score-1.432]
63 It shows that conformal mapping cause up to 220 times shrinkage, while area preserving mapping almost keep the same area. [sent-259, score-1.173]
64 In Figure 4, we put circle textures on both conformal mapping result and area preserving result, it gives a direct visualization of our method’s correctness. [sent-260, score-1.021]
65 Although multi-subject studies are clearly necessary, this demonstrates our area preserving method may potentially be useful to study some morphometry change to classify and compare different subcortical structure surfaces. [sent-261, score-0.426]
66 Comparison of geometric mappings for caudate surface: (a) original caudate surface represented by a triangular mesh; (b) conformal mapping result; (c) area preserving mapping result. [sent-263, score-1.836]
67 The area preserving mapping method evenly maps the surface to the unit disk and eliminates the big distortions close to the upper tip area in (a). [sent-264, score-1.028]
68 Histogram of area distortion: (a) area distortion of conformal mapping; (b) area distortion of area preserving mapping. [sent-266, score-1.268]
69 The area preserving mapping result shows a much smaller area distortion. [sent-267, score-0.773]
70 Application of Alzheimer’s Disease Diagnosis For Alzheimer’s disease, structural MRI measurements of brain shrinkage are one of the best established biomarkers of AD progression and pathology. [sent-270, score-0.333]
71 Circle packing of different geometric mappings: (a) circle packing of conformal mapping. [sent-272, score-0.487]
72 The parameterizations are illustrated by the texture map of a uniformly distributed circle patterns on the caudate surface, the circle texture is shown in the upper left corner. [sent-274, score-0.369]
73 brain mapping may offer advantages over volume-based brain mapping [5] to study structural features of brain, such as cortical gray matter thickness, complexity, and patterns of brain change over time due to disease or developmental process. [sent-277, score-1.744]
74 According to prior AD researches [15, 14], the brain atrophy is an important biomarker of AD. [sent-278, score-0.44]
75 In this work, we proposed to use Beltrami coefficients [16] computed from area preserving mapping result to conformal mapping result, as a shape signature to analyze the human brain cortical surfaces among AD patients and CTL subjects. [sent-281, score-1.932]
76 f Wore a uusteodmatic skull stripping, tissue classification, cortical surface extraction, vertex correspondences across brain surfaces and cortical parcellations. [sent-297, score-1.029]
77 According to work [11], we labeled the functional areas of a left brain cortical surface shown in Figure 5 (a) and (b). [sent-298, score-0.809]
78 1 ± Cortical Surface Parameterization Results Figure 5 (c)-(f) are the conformal mapping results and area preserving mapping results of the left brain cortical surfaces of a healthy control subject and an AD patient. [sent-301, score-1.957]
79 After the cutting, the remaining cortical surface becomes a genus zero surface with one open boundary. [sent-303, score-0.55]
80 Both algorithms compute a diffeomorphism map between the cortical surface and a unit disk. [sent-304, score-0.497]
81 The results show that the conformal mapping results have much more area distortion on the areas close to the boundary while the area preserving mapping provides a map which preserves the area of each individual functional area. [sent-305, score-1.689]
82 The area preserving mapping has a potential to better visualize certain sulci areas which are deeply buried under gyri, and hence to provide a tool for a more accurate manual landmark delineation operation. [sent-306, score-0.764]
83 (a) and (b) illustrate the functional areas on the left brain cortex [11]. [sent-308, score-0.437]
84 (c) and (e) are conformal mapping results of a CTL subject and an AD patient, respectively; (d) and (f) are area preserving mapping results of a CTL subject and an AD patient, respectively. [sent-311, score-1.173]
85 The area preserving mapping may provide a better visualization tool for tracking sulci landmark curves on cortical surfaces. [sent-312, score-0.939]
86 We tested the discrimination ability of our shape signature 222222334088 on a set of left and right brain surfaces of 50 CTL subjects and 50 AD patients. [sent-317, score-0.449]
87 The histograms show the norm of Beltrami coefficients of cortical surfaces of AD patients are obviously larger than those of healthy control subjects. [sent-321, score-0.58]
88 It means that AD patients may have larger conformality distortion in both area and shrinkage directions because AD patients may suffer a more serious atrophy of brain structures which result from a combination of neuronal atrophy, cell loss and impairments in myelin turnover and maintenance [14]. [sent-322, score-0.903]
89 The AD result demonstrated a stronger and more anisotropic deformation due to a more serious atrophy of brain structures. [sent-326, score-0.44]
90 For the classification experiment, 80% of each category of both left and right brain cortical surfaces are set to be training samples and the remaining 20% as testing samples. [sent-331, score-0.627]
91 F coorm area d ba asreeda method, we computed the surface areas for the base domain and 3 regions mentioned above on each hemisphere as a sig- ± MethodRate % VAorleuame6720. [sent-341, score-0.372]
92 Conclusions and Future Work In this paper, we presented a method to compute area preserving mapping between surfaces based on Brenier’s approach in Optimal Mass Transport theory. [sent-361, score-0.679]
93 Therefore, our method offers a stable and effective way to compute area preserving mapping in 2D parametric coordinates. [sent-364, score-0.604]
94 Our experimental results show our work may provide novel ways for shape analysis and improve the statistical power for detecting abnormalities in brain surface morphology. [sent-365, score-0.578]
95 Correlation between rates of brain atrophy and cognitive decline in AD. [sent-537, score-0.44]
96 Genus zero surface conformal mapping and its application to brain surface mapping. [sent-566, score-1.163]
97 Cortical cartography using the discrete conformal approach of circle packings. [sent-593, score-0.383]
98 3D maps localize caudate nucleus atrophy in 400 Alzheimer’s dis- [26] [27] [28] [29] [30] [3 1] [32] [33] ease, mild cognitive impairment, and healthy elderly subjects. [sent-652, score-0.567]
99 Fast optimal mass transport for 2D image registration and morphing. [sent-660, score-0.393]
100 Conformal slit mapping and its applications to brain surface parameterization. [sent-737, score-0.691]
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