1 efficient simplification of point-sampled geometry From the paper “Efficient Simplification of Point-Sampled Surfaces” by Mark Pauly, Markus Gross, Leif Kobbelt Jeffrey Sukharev CMPS260 Final Project

2 outline zintroduction zsurface model & local surface analysis zpoint cloud simplification yhierarchical clustering yiterative simplification yparticle simulation zmeasuring surface error zcomparison zconclusions

3 introduction z3d content creation acquisitionrenderingprocessing many applications require coarser approximations editing rendering surface simplification for complexity reduction

4 introduction z3d content creation acquisitionrenderingprocessing registrationraw scanspoint cloudreconstructiontriangle mesh

5 introduction z3d content creation acquisitionrenderingprocessing registrationraw scanspoint cloudreconstructiontriangle meshsimplification reduced point cloud

6 introduction z3d content creation acquisitionrenderingprocessing registrationraw scanspoint cloudsimplification reduced point cloud

7 surface model zmoving least squares (mls) approximation Gaussian used for locality idea: locally approximate surface with polynomial compute reference plane compute weighted least-squares fit polynomial implicit surface definition using a projection operator

8 surface model zmoving least squares (mls) approximation idea: locally approximate surface with polynomial compute reference plane compute weighted least-squares fit polynomial Gaussian used for locality implicit surface definition using a projection operator

9 local surface analysis zlocal neighborhood (k-nearest neighbors)

10 local surface analysis zlocal neighborhood (e.g. k-nearest) covariance matrix eigenvalue problem centroid

11 local surface analysis zlocal neighborhood (e.g. k-nearest) eigenvectors span covariance ellipsoid surface variation smallest eigenvector is normal measures deviation from tangent plane  curvature

12 local surface analysis zexample originalmean curvaturevariation n=20variation n=50

13 surface simplification zincremental clustering zhierarchical clustering ziterative simplification zparticle simulation

14 incremental clustering zClustering by growing regions ystart with a random seed point ysuccessively add nearest points to cluster until cluster reaches desired maximum size zthe growth of clusters can also be limited be surface variation and in that way the curvature adaptive clustering is achieved.

15 incremental clustering zIncremental growth leads to some fragmentation. Therefore stray samples need to be added to closest clusters at the end of the run.

16 incremental clustering zeach cluster is replaced by its centroid Origina model 34,384 points Simplified model 1,000 pts

17 incremental clustering zResults from my incremental clustering implementation. 35,000 pts 1,222 pts

18 surface simplification zincremental clustering zhierarchical clustering ziterative simplification zparticle simulation

19 hierarchical clustering ztop-down approach using binary space partition zrecursively split the point cloud if: ysize is larger than a user-specified threshold or ysurface variation is above maximum threshold zsplit plane defined by centroid and axis of greatest variation zreplace clusters by centroid

20 hierarchical clustering z2d example covariance ellipsoid split plane centroid root

21 hierarchical clustering z2d example

22 hierarchical clustering z2d example

23 hierarchical clustering z2d example

24 hierarchical clustering 4,280 Clusters436 Clusters43 Clusters

25 surface simplification zincremental clustering zhierarchical clustering ziterative simplification zparticle simulation

26 iterative simplification ziteratively contracts point pairs  each contraction reduces the number of points by one zcontractions are arranged in priority queue according to quadric error metric zquadric measures cost of contraction and determines optimal position for contracted sample zequivalent to QSlim except for definition of approximating planes

27 surface simplification zincremental clustering zhierarchical clustering ziterative simplification zparticle simulation

28 particle simulation zMethod proposed by Turk G. (for polygonal surfaces) zresample surface by distributing particles on the surface zparticles move on surface according to inter- particle repelling forces zparticle relaxation terminates when equilibrium is reached zcan also be used for up-sampling!

29 measuring error zmeasure distance between two point-sampled surfaces S and S’ using a sampling approach zcompute set Q of points on S zmaximum error:  two-sided Hausdorff distance zmean error:  area-weighted integral of point-to-surface distances zsize of Q determines accuracy of error measure

30 measuring error zd(q,S’) measures the distance of point q to surface S’ using the mls projection operator

31 conclusions zpoint cloud simplification can be useful to yreduce the complexity of geometric models early in the 3d content creation pipeline ycreate surface hierarchies zReferences yMark Pauly et al “Efficient Simplification of Point Sampled Surfaces” yMark Alexa et al “Point Set Surfaces” yLevin D. “Mesh-independent surface interpolation”