1 efficient simplification of point-sampled geometry from the paper “efficient simplification of...
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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
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outlineintroductionsurface model & local surface analysispoint cloud simplification
hierarchical clustering iterative simplification particle simulation
measuring surface errorcomparisonconclusions
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introduction
3d content creation
acquisition renderingprocessing
many applications require coarser approximationseditingrendering
surface simplification for complexity reduction
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introduction
3d content creation
acquisition renderingprocessing
registration
raw scans
point cloud reconstruction
triangle mesh
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introduction
3d content creation
acquisition renderingprocessing
registration
raw scans
point cloud reconstruction
triangle mesh
simplification
reduced point cloud
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introduction
3d content creation
acquisition renderingprocessing
registration
raw scans
point cloud
simplification
reduced point cloud
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surface model
moving least squares (mls) approximation
Gaussian used for locality
idea: locally approximate surface with polynomial
compute reference planecompute weighted least-squares fit polynomial
implicit surface definition using a projection operator
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surface model
moving least squares (mls) approximation
idea: locally approximate surface with polynomial
compute reference planecompute weighted least-squares fit polynomial
Gaussian used for locality
implicit surface definition using a projection operator
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local surface analysis
local neighborhood (k-nearest neighbors)
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local surface analysislocal neighborhood (e.g. k-nearest)
pp
pp
pp
pp
C
n
T
n
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covariance matrix
eigenvalue problem
lll vvC
pcentroid
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local surface analysislocal neighborhood (e.g. k-nearest)
eigenvectors span covariance ellipsoid
surface variation
smallest eigenvector is normal
in
0)(p
measures deviation from tangent plane curvature
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local surface analysis
example
original mean curvature variation n=20 variation n=50
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surface simplification
incremental clusteringhierarchical clusteringiterative simplificationparticle simulation
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incremental clustering
Clustering by growing regions start with a random seed point successively add nearest points to cluster
until cluster reaches desired maximum size
the growth of clusters can also be limited be surface variation and in that way the curvature adaptive clustering is achieved.
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incremental clustering
Incremental growth leads to some fragmentation. Therefore stray samples need to be added to closest clusters at the end of the run.
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incremental clustering
each cluster is replaced by its centroid
Origina model34,384 points
Simplified model 1,000 pts
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incremental clustering
Results from my incremental clustering implementation.
35,000 pts 1,222 pts
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surface simplification
incremental clusteringhierarchical clusteringiterative simplificationparticle simulation
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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
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hierarchical clustering
2d example
covariance ellipsoid split plane
centroid
root
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hierarchical clustering
2d example
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hierarchical clustering
2d example
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hierarchical clustering
2d example
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hierarchical clustering
4,280 Clusters436 Clusters43 Clusters
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surface simplification
incremental clusteringhierarchical clusteringiterative simplificationparticle simulation
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iterative simplificationiteratively 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
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surface simplification
incremental clusteringhierarchical clusteringiterative simplificationparticle simulation
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particle simulation
Method proposed by Turk G. (for polygonal surfaces)
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
can also be used for up-sampling!
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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
),(max),(max SdSS Q qq
Q
SdQ
SSq
q ),(1
),(avg
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measuring error
d(q,S’) measures the distance of point q to surface S’ using the mls projection operator S
'S
),( Sd q
q
'q
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conclusions
point cloud simplification can be useful to reduce the complexity of geometric models
early in the 3d content creation pipeline create surface hierarchies
References Mark Pauly et al “Efficient Simplification of Point Sampled
Surfaces” Mark Alexa et al “Point Set Surfaces” Levin D. “Mesh-independent surface interpolation”