Variational Shape Approximation of Point Set...
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M. Skrodzki, E. Zimmermann, and K. Polthier Freie Universit¨ at Berlin Corresponding author contact: [email protected] Variational Shape Approximation of Point Set Surfaces Variational Shape Approximation (VSA) The VSA procedure [1] partitions a surface S ⊆ R 3 into k ∈ N disjoint regions R i ⊆ S , R i = S , where each region is asso- ciated a linear proxy P i =(C i ,N i ) ∈ R 3 × S 2 , where C i de- notes the center and N i denotes an associated unit-length normal, i.e. every proxy appears as a plane. The proxies are fitted to the input by minimizing E ({(R i ,P i ) | i =1,...,k })= k X i=1 L 2,1 (R i ,P i ), (1) where L 2,1 (R i ,P i )= X t j n(t j ) - N i 2 |t j | . (2) For point sets, the area of a triangle |t j | is not available and the normals are located at the points [2]. Minimization is performed via a k -means clustering approach separated into three steps: 1. Flood: From chosen centers C i , propagate the normals N i . 2. Proxy Update: Calculate proxy normals as arithmetic mean of normals in their regions. 3. Seeds: Find new center C i in the respective region R i from which to start flooding again. Example: Growth in Error Functional n points n points 1 point angle 135 ◦ ( -1 1 ) ( 0 1 ) N N = 1 n+2 n · ( 0 1 ) + ( -1 0 ) + N ( -1 0 ) (a)Setup for growing error functional. (b) Segmentation after first flood. (c) Segmentation after second flood. Example for a growth in the error measure after a flood and proxy update. Evaluating (1) for the situation in Figures 1(b) and 1(c) gives an error of E 1 ≈ 1.9802 and E 2 ≈ 39.395 respectively, when choosing n = 100 points. Goals Obtain a version of VSA for both meshes and point sets with guaranteed convergence. Also, reduce the dependency on the a priori chosen number of seeds and their positions. To achieve this, we introduce a new user-given parameter κ ∈ R ≥0 to measure a proxy’s flatness. New Operation: Split For any proxy P i with L 2,1 (R i ,P i ) >κ, use weighted principal component analysis [3] to compute the most spread direction of R i and split it into two new regions R 1 i , R 2 i thus increasing the number of regions k . New Operation: Merge Consider a pair P i , P j of neighboring proxies with their respective normals N i , N j . If the region R = R i R j with normal N = N i +N j 2 achieves an error measure (2) strictly less than κ, replace them by their union, decreasing k . New Operation: Switch For all elements, consider their nearest neighbors. Assume a neighbor is assigned to a different proxy, compute the change of (1) resulting from switching this assign- ment. Reassign the element such that the error measure is reduced maximally. Experimental Results: Bunny From left to right: The Bunny model segmented by our method, by automatic VSA, and by VSA with manual seed placement. Experimental Results: Fandisk From left to right: The Fandisk model segmented by our method, by automatic VSA, and by VSA with manual seed placement. Future Work I Optimize runtime of the switch operation I Find a general framework for patch reconstruction I Compare to other segmentation approaches References [1] D. Cohen-Steiner, P. Alliez, and M. Desbrun. Variational Shape Approximation. In ACM Transactions on Graphics (TOG), 2004. [2] K. Lee and P. Bo. Feature curve extraction from point clouds via developable strip intersection. Journal of Computational Design and Engineering, 2016. [3] P. Harris, C. Brunsdon, and M. Charlton. Geographically weighted principal components analysis. International Journal of Geographical Information Science, 2011.
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M. Skrodzki, E. Zimmermann, and K. PolthierFreie Universitat Berlin
Corresponding author contact: [email protected]
Variational Shape Approximation of Point Set SurfacesVariational Shape Approximation (VSA)The VSA procedure [1] partitions a surface S ⊆ R3 into k ∈ Ndisjoint regions Ri ⊆ S, tRi = S, where each region is asso-ciated a linear proxy Pi = (Ci, Ni) ∈ R3× S2, where Ci de-notes the center andNi denotes an associated unit-length normal,i.e. every proxy appears as a plane. The proxies are fitted to theinput by minimizing
E({(Ri, Pi) | i = 1, . . . , k}) =k∑i=1
L2,1(Ri, Pi), (1)
whereL2,1(Ri, Pi) =
∑tj
‖n(tj)−Ni‖2 |tj| . (2)
For point sets, the area of a triangle |tj| is not available and thenormals are located at the points [2]. Minimization is performedvia a k-means clustering approach separated into three steps:
1. Flood: From chosen centers Ci, propagate the normals Ni.2. Proxy Update: Calculate proxy normals as arithmetic mean
of normals in their regions.3. Seeds: Find new center Ci in the respective region Ri from
which to start flooding again.
Example: Growth in Error Functional
npoin
ts
n points1 point
angle 135◦
(−11
)(0
1
)N
N = 1n+2
(n ·(0
1
)+(−1
0
)+N
)
(−10
)
(a) Setup for growing error functional.
(b) Segmentation after first flood. (c) Segmentation after second flood.Example for a growth in the error measure after a flood and proxy update.
Evaluating (1) for the situation in Figures 1(b) and 1(c) gives anerror of E1 ≈ 1.9802 and E2 ≈ 39.395 respectively, whenchoosing n = 100 points.
GoalsObtain a version of VSA for both meshes and point sets with guaranteed convergence. Also, reduce the dependency on the a priori chosennumber of seeds and their positions. To achieve this, we introduce a new user-given parameter κ ∈ R≥0 to measure a proxy’s flatness.
New Operation: SplitFor any proxy Pi withL2,1(Ri, Pi) > κ, use weightedprincipal component analysis [3] tocompute the most spread direction of Ri
and split it into two new regions R1i , R2
i
thus increasing the number of regions k.
New Operation: MergeConsider a pair Pi, Pj of neighboringproxies with their respective normals Ni,Nj. If the region R′ = Ri t Rj withnormal N ′ = Ni+Nj
2 achieves an errormeasure (2) strictly less than κ, replacethem by their union, decreasing k.
New Operation: SwitchFor all elements, consider their nearestneighbors. Assume a neighbor is assignedto a different proxy, compute the changeof (1) resulting from switching this assign-ment. Reassign the element such that theerror measure is reduced maximally.
Experimental Results: Bunny
From left to right: The Bunny model segmented by our method, byautomatic VSA, and by VSA with manual seed placement.
Experimental Results: Fandisk
From left to right: The Fandisk model segmented by our method, byautomatic VSA, and by VSA with manual seed placement.
Future WorkI Optimize runtime of the switch operationI Find a general framework for patch reconstructionI Compare to other segmentation approaches
References[1] D. Cohen-Steiner, P. Alliez, and M. Desbrun. Variational Shape Approximation. In ACM Transactions
on Graphics (TOG), 2004.[2] K. Lee and P. Bo. Feature curve extraction from point clouds via developable strip intersection. Journal
of Computational Design and Engineering, 2016.[3] P. Harris, C. Brunsdon, and M. Charlton. Geographically weighted principal components analysis.
International Journal of Geographical Information Science, 2011.
