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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”