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

acquisition renderingprocessing

many applications require coarser approximations– storage– transmission– editing– rendering

surface simplification for complexity reduction

introduction

3d content creation

acquisition renderingprocessing

registration

raw scans

point cloud reconstruction

triangle mesh

introduction

3d content creation

acquisition renderingprocessing

registration

raw scans

point cloud reconstruction

triangle mesh

simplificationreduced point

cloud

introduction

3d content creation

acquisition renderingprocessing

registration

raw scans

point cloud

simplificationreduced 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)

pp

pp

pp

pp

C

n

T

n

11

covariance matrix

eigenproblem

lll vvC

pcentroid

local surface analysis

local neighborhood (e.g. k-nearest)

eigenvectors span covariance ellipsoid

surface variation

smallest eigenvector is least-squares normal

in

0)(p

measures deviation from tangent plane curvature

local surface analysis

example

original mean curvature variation n=20 variation 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 pairseach 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

6 0.02

2 0.03

14 0.04

5 0.04

9 0.09

1 0.11

13 0.13

3 0.22

11 0.27

10 0.36

7 0.44

4 0.56

priority queue

edge cost

iterative simplification

2d example

6 0.02

2 0.03

14 0.04

5 0.04

9 0.09

1 0.11

13 0.13

3 0.22

11 0.27

10 0.36

7 0.44

4 0.56

priority queue

edge cost

iterative simplification

2d example

2 0.03

14 0.04

5 0.06

9 0.09

1 0.11

13 0.13

3 0.25

11 0.27

10 0.36

7 0.49

4 0.56

priority queue

edge cost

iterative simplification

2d example

14 0.04

5 0.06

9 0.09

1 0.11

13 0.13

3 0.25

11 0.27

10 0.36

7 0.49

4 0.56

priority queue

edge cost

iterative simplification

2d example

14 0.04

5 0.06

9 0.09

1 0.11

13 0.13

3 0.25

11 0.27

10 0.36

7 0.49

4 0.56

priority queue

edge cost

iterative simplification

2d example

5 0.06

9 0.09

1 0.11

13 0.13

3 0.25

11 0.27

10 0.36

7 0.49

4 0.56

priority queue

edge cost

iterative simplification

2d example

5 0.06

9 0.09

1 0.11

13 0.13

3 0.25

11 0.27

10 0.36

7 0.49

4 0.56

priority queue

edge cost

iterative simplification

2d example

9 0.09

1 0.11

13 0.13

3 0.25

11 0.27

10 0.36

7 0.49

4 0.56

priority queue

edge cost

iterative simplification

2d example

9 0.09

1 0.11

13 0.13

3 0.25

11 0.27

10 0.36

7 0.49

4 0.56

priority queue

edge cost

iterative simplification

2d example

11 0.27

10 0.36

7 0.49

4 0.56

priority queue

edge cost

iterative simplification

296,850 points 2,000 points remaining

contraction pairs

surface simplification

hierarchical clustering iterative simplification particle simulation

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

)()()( iii rkF ppppp

projection– project particles onto

surface

particle simulation

2d example initialization

– randomly spread particles

repulsion– linear repulsion force

)()()( iii rkF ppppp

particle simulation

2d example initialization

– randomly spread particles

repulsion– linear repulsion force

)()()( iii rkF ppppp

projection– project particles onto

surface

particle simulation

original model296,850 points

uniform repulsion2,000 points

adaptive repulsion3,000 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

),(max),(max SdSS Q qq

Q

SdQ

SSq

q ),(1

),(avg

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

comparison: surface error

error estimate for Michelangelo’s David simplified from 2,000,000 points to 5,000 points

0046.0max 4

avg 1014.6

hierarchical clustering iterative simplification particle simulation

0052.0max 4

avg 1043.5 0061.0max

4avg 1069.5

comparison: performance

execution time as a function of input model size (simplification to 1% of input model size)

0

50

100

150

200

250

300

350

400

450

500

0 500 1000 1500 2000 2500 3000 3500

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)

0

10

20

30

40

50

60

70

020406080100120140160180

hierarchical clustering

iterative simplification

particle simulation

target size

time (sec)

smoothing effect

simplification up-sampling

point cloud vs. mesh simplification

simplification reconstruction3.5 sec. 2.45 sec

reconstruction simplification112.8 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: www.pointshop3d.com

acknowledgement: European graduate program on combinatorics, geometry, and computation