Iso-charts: Stretch-Driven ParameterizationIso-charts: Stretch-Driven Parameterizationvia Nonlinear Dimension Reductionvia Nonlinear Dimension Reduction

Iso-charts: Stretch-Driven ParameterizationIso-charts: Stretch-Driven Parameterizationvia Nonlinear Dimension Reductionvia Nonlinear Dimension Reduction

Kun Zhou, John Snyder, Baining Guo, Harry ShumKun Zhou, John Snyder, Baining Guo, Harry ShumKun Zhou, John Snyder, Baining Guo, Harry ShumKun Zhou, John Snyder, Baining Guo, Harry Shum

presented at SGP, June 2004presented at SGP, June 2004

Goals of Mesh ParameterizationGoals of Mesh ParameterizationGoals of Mesh ParameterizationGoals of Mesh Parameterization

Large Charts Low Distortion

Stretch-Driven ParameterizationStretch-Driven ParameterizationStretch-Driven ParameterizationStretch-Driven Parameterization AdvantagesAdvantages

measures distortion properly for texturing appsmeasures distortion properly for texturing apps

DisadvantagesDisadvantages requires nonlinear optimization (slow!)requires nonlinear optimization (slow!) provides no help in forming chartsprovides no help in forming charts

– resort to simple heuristics like planarity or compactnessresort to simple heuristics like planarity or compactness

Solution: apply Solution: apply IsomapIsomap (NDR technique) (NDR technique) stretch and Isomap related: both preserve lengthsstretch and Isomap related: both preserve lengths eigenanalysis rather than nonlinear optimizationeigenanalysis rather than nonlinear optimization provides:provides:

– good initial guess for stretch optimizationgood initial guess for stretch optimization

– good chartification heuristic via “spectral clustering”good chartification heuristic via “spectral clustering”

AdvantagesAdvantages measures distortion properly for texturing appsmeasures distortion properly for texturing apps

DisadvantagesDisadvantages requires nonlinear optimization (slow!)requires nonlinear optimization (slow!) provides no help in forming chartsprovides no help in forming charts

– resort to simple heuristics like planarity or compactnessresort to simple heuristics like planarity or compactness

Solution: apply Solution: apply IsomapIsomap (NDR technique) (NDR technique) stretch and Isomap related: both preserve lengthsstretch and Isomap related: both preserve lengths eigenanalysis rather than nonlinear optimizationeigenanalysis rather than nonlinear optimization provides:provides:

– good initial guess for stretch optimizationgood initial guess for stretch optimization

– good chartification heuristic via “spectral clustering”good chartification heuristic via “spectral clustering”

IsoMapIsoMapIsoMapIsoMap

Data points in highdimensional space

[Tenenbaum et al, 2000]

Data points in lowdimensional space

Neighborhoodgraph

Surface Spectral AnalysisSurface Spectral AnalysisSurface Spectral AnalysisSurface Spectral Analysis

12d

Nd2

Nd1

1x

2x

Nx

N

i

N

j

ijji d1 1

2)||(||min yy

Geodesic Distance Distortion (GDD)

2y

Ny

1y

Surface Spectral AnalysisSurface Spectral AnalysisSurface Spectral AnalysisSurface Spectral Analysis

222

21

22

222

221

21

212

211

NNNN

N

N

N

ddd

ddd

ddd

D

1. Construct matrix of squared geodesic distances 1. Construct matrix of squared geodesic distances DDNN

12d

Nd2

Nd1

1x

2x

Nx

Surface Spectral AnalysisSurface Spectral AnalysisSurface Spectral AnalysisSurface Spectral Analysis

2. Perform eigenanalysis on 2. Perform eigenanalysis on DDNN to getto get embedding coords embedding coords yyii

Tnn

T

T

N

v

v

v

yyy

22

11

21

|||

|||

12d

Nd2

Nd1

1x

2x

Nx

Isomap → low stretch (take first two coords)Isomap → low stretch (take first two coords)Isomap → low stretch (take first two coords)Isomap → low stretch (take first two coords)

IsoMap, L2 = 1.04, 2s IsoMap+Optimization, L2 = 1.03, 6s

[stretch, Sander01], L2 = 1.04, 222s [stretch, Sander02], L2 = 1.03, 39s

Isomap → good charts (spectral clustering)Isomap → good charts (spectral clustering)Isomap → good charts (spectral clustering)Isomap → good charts (spectral clustering)

Analysis

Clustering

ResultsResultsResultsResults

19 charts, L2=1.03, running time 98s, 97k faces

ResultsResultsResultsResults

38 charts, L2=1.07, running time 287s, 150k faces

ResultsResultsResultsResults

23 charts, L2=1.06, running time 162s, 112k faces

ResultsResultsResultsResults

11 charts, L2=1.01, running time 4s, 10k faces

Remeshing ComparisonRemeshing ComparisonRemeshing ComparisonRemeshing Comparison

Original model [Sander03], 79.5dB Iso-chart, 82.9dB

Texture Synthesis ResultsTexture Synthesis ResultsTexture Synthesis ResultsTexture Synthesis Results