Construction of Iso-contours, Bisectors and Voronoi Diagrams on ...

IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 33, NO. 8, PAGE 1502-1517, AUGUST 2011 14Fig. 24. Four simulated point patterns on a 2-manifold model. Top: top views <strong>of</strong> color-mapped distance field. Bottom:<strong>Voronoi</strong> diagrams <strong>of</strong> sample points. From left to right: uniform sampling, r<strong>and</strong>om sampling, small cluster sampling(µ = 12, ω = 8) <strong>and</strong> big cluster sampling (µ = 40, ω = 20).box <strong>of</strong> the model. The plots <strong>of</strong> precision versus recall<strong>of</strong> the six approaches are shown in Fig. 23, from whichwe conclude that our voronoi-skeleton matching (V-SKEL), D2 shape distribution [18] <strong>and</strong> bending invariantsignature G2 [14] are robust to noise, while geometricmoment invariants GMT [68], extended Gaussian imagesEGI [25] <strong>and</strong> spin images [26] are more sensitive to noise.This can be interpreted by that noises heavily change thenormals <strong>and</strong> areas <strong>of</strong> models, <strong>and</strong> GMT uses integral <strong>of</strong>area <strong>and</strong> EGIs <strong>and</strong> spin images use normals.6.3 Point Pattern Analysis on MThe <strong>Voronoi</strong> diagrams on triangulated 2-manifolds Mcan also be used to examine whether or not a pattern existsin a set <strong>of</strong> sampling points on M. The sampling mayrepresent the population <strong>of</strong> a state (geography), artefactsin a site (archaeology), subcellular localization in tissues(biology), etc. Using the <strong>Voronoi</strong> diagram constructionalgorithm proposed in this paper, the polygonal basedmethod in [64] can be extended to the domain <strong>of</strong> 2-manifold surfaces M.We use the following methods to generate differentpoint patterns on M:• R<strong>and</strong>om point sampling. An array A is generated withthe number <strong>of</strong> triangles in M, i.e., A[i] correspondsto the triangle t i . Each element in A stores the triangleareas accumulated so far, i.e., A[i] = ∑ ij=1 ∆t j,where ∆t j is the area <strong>of</strong> triangle t j . A r<strong>and</strong>omnumber generator is used to sample between 0 <strong>and</strong>A[n]. Each generated number x corresponds to asample point on M which lies in the triangle t k withA[k − 1] < x ≤ A[k].• Uniform point sampling. The farthest point samplingmethod on M presented in Section 6.1 is used.• Clustering point sampling. First the cluster originso i are r<strong>and</strong>omly distributed. Secondly a numbermeasure uniform r<strong>and</strong>om small cluster big clusterpattern pattern pattern patternARF 0.7614 0.6926 0.6542 0.6311RF H 0.5512 0.5106 0.5404 0.5371AD 0.6835 0.4573 0.1244 −1.2184TABLE 4The mean <strong>of</strong> three measures in ten simulations for thefour different point patterns in Figure 24.measure uniform r<strong>and</strong>om small cluster big clusterpattern pattern pattern patternARF 0.0055 0.007 0.0122 0.0172RF H 0.0341 0.0412 0.0526 0.0547AD 0.0465 0.0451 0.182 0.3757TABLE 5The st<strong>and</strong>ard deviation <strong>of</strong> three measures in tensimulations for the four different point patterns.<strong>of</strong> points are generated for each cluster i from ar<strong>and</strong>om distribution with mean µ. Thirdly the pointsin a cluster i are distributed according to a Gaussianfunction centered at o i <strong>and</strong> with st<strong>and</strong>ard deviationω.Four patterns (one r<strong>and</strong>om, one uniform, two clusterdistributions with different µ, ω) are generated in a 2-manifold model <strong>and</strong> shown in Figure 24. We generatethe <strong>Voronoi</strong> diagrams for different point samples. Foreach <strong>Voronoi</strong> cell V C(p i ), denote its area by A(i) <strong>and</strong>its perimeter by L(i). Three measures are defined below(ARF <strong>and</strong> RF H are adopted from [64]) to test thepattern in the sampling:ARF = 1 nn∑RF (i),i=1RF (i) = 4πA(i)L 2 (i)