efficient simplification of point-sampled geometry Mark Pauly Markus Gross Leif Kobbelt ETH Zurich RWTH Aachen

outline introduction surface model & local surface analysis point cloud simplification –hierarchical clustering –iterative simplification –particle simulation measuring surface error comparison conclusions

introduction 3d content creation acquisitionrenderingprocessing many applications require coarser approximations –storage –transmission –editing –rendering  surface simplification for complexity reduction

introduction 3d content creation acquisitionrenderingprocessing registrationraw scanspoint cloudreconstructiontriangle mesh

introduction 3d content creation acquisitionrenderingprocessing registrationraw scanspoint cloudreconstructiontriangle meshsimplification reduced point cloud

introduction 3d content creation acquisitionrenderingprocessing registrationraw scanspoint cloudsimplification reduced point cloud

surface model moving least squares (mls) approximation Gaussian weight function  locality idea: locally approximate surface with polynomial –compute reference plane –compute weighted least- squares fit polynomial implicit surface definition using a projection operator

surface model moving least squares (mls) approximation idea: locally approximate surface with polynomial –compute reference plane –compute weighted least- squares fit polynomial Gaussian weight function  locality implicit surface definition using a projection operator

local surface analysis local neighborhood (e.g. k-nearest)

local surface analysis local neighborhood (e.g. k-nearest) covariance matrix eigenproblem centroid

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

local surface analysis example originalmean curvaturevariation n=20variation n=50

surface simplification hierarchical clustering iterative simplification particle simulation

hierarchical clustering top-down approach using binary space partition recursively split the point cloud if: –size is larger than a user-specified threshold or –surface variation is above maximum threshold split plane defined by centroid and axis of greatest variation replace clusters by centroid

hierarchical clustering 2d example covariance ellipsoid split plane centroid root

hierarchical clustering 2d example

hierarchical clustering 2d example

hierarchical clustering 2d example

hierarchical clustering 4,280 Clusters436 Clusters43 Clusters

surface simplification hierarchical clustering iterative simplification particle simulation

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

compute fundamental quadrics compute initial point-pair contraction candidates iterative simplification 2d example compute edge costs

iterative simplification 2d example priority queue edge cost

iterative simplification 2d example priority queue edge cost

iterative simplification 2d example priority queue edge cost

iterative simplification 2d example priority queue edge cost

iterative simplification 2d example priority queue edge cost

iterative simplification 2d example priority queue edge cost

iterative simplification 2d example priority queue edge cost

iterative simplification 2d example priority queue edge cost

iterative simplification 2d example priority queue edge cost

iterative simplification 2d example priority queue edge cost

iterative simplification 296,850 points2,000 points remaining contraction pairs

surface simplification hierarchical clustering iterative simplification particle simulation

resample surface by distributing particles on the surface particles move on surface according to inter- particle repelling forces particle relaxation terminates when equilibrium is reached (requires damping) can also be used for up-sampling!

mls surface particle simulation 2d example

particle simulation 2d example initialization –randomly spread particles

particle simulation 2d example initialization –randomly spread particles repulsion –linear repulsion force

projection –project particles onto surface particle simulation 2d example initialization –randomly spread particles repulsion –linear repulsion force

particle simulation 2d example initialization –randomly spread particles repulsion –linear repulsion force projection –project particles onto surface

particle simulation original model 296,850 points uniform repulsion 2,000 points adaptive repulsion 3,000 points

measuring error measure distance between two point-sampled surfaces S and S’ using a sampling approach compute set Q of points on S maximum error:  two-sided Hausdorff distance mean error:  area-weighted integral of point-to-surface distances size of Q determines accuracy of error measure

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

comparison: surface error error estimate for Michelangelo’s David simplified from 2,000,000 points to 5,000 points hierarchical clusteringiterative simplificationparticle simulation

comparison: performance execution time as a function of input model size (simplification to 1% of input model size) input size time (sec) hierarchical clustering iterative simplification particle simulation

comparison: performance execution time as a function of target model size (input: dragon, 435,545 points) hierarchical clustering iterative simplification particle simulation target size time (sec)

smoothing effect simplification up-sampling

point cloud vs. mesh simplification simplification  reconstruction 3.5 sec sec reconstruction  simplification sec. 3.5 sec.

conclusions point cloud simplification can be useful to –reduce the complexity of geometric models early in the 3d content creation pipeline –build LOD surface representations –create surface hierarchies the right method depends on the application check out: acknowledgement: European graduate program on combinatorics, geometry, and computation