Iso-charts: Stretch-driven Mesh Parameterization using Spectral Analysis Kun Zhou, John Snyder*, Baining Guo, Heung-Yeung Shum Microsoft Research Asia Microsoft Research* Kun Zhou, John Snyder*, Baining Guo, Heung-Yeung Shum Microsoft Research Asia Microsoft Research*

Parameterizing Arbitrary 3D Meshes ChartificationTexture Atlas

Goals of Mesh Parameterization Large Charts Low Distortion

Iso-chart Algorithm Overview l Surface spectral analysis l Stretch optimization l Recursively split charts n until stretch criterion is met l Recursively split charts n until stretch criterion is met l Surface spectral clustering l Optimize chart boundaries Input: 3D mesh, user-specified stretch threshold Output: atlas having large charts with bounded stretch

IsoMapIsoMap Data points in high dimensional space [Tenenbaum et al, 2000] Data points in low dimensional space Neighborhood graph Analyze geodesic distance to uncover nonlinear manifold structure

Surface Spectral Analysis Geodesic Distance Distortion (GDD)

Surface Spectral Analysis Construct matrix of squared geodesic distances D N

Surface Spectral Analysis Perform centering and normalization to D N

Surface Spectral Analysis Perform eigenanalysis on B N to get embedding coords y i

GDD-minimizing Parameterization Parametric coordinates [Zigelman et al, 2002] Texture mapping l Produces triangle flips l Only handles single-chart (disk-topology) models

Stretch-minimizing Parameterization   2D texture domain surface in 3D linear map singular values: γ, Γ [Sander et al, 2001]

Stretch Optimization IsoMap, L 2 = 1.04, 2sIsoMap+Optimization, L 2 = 1.03, 6s [Sander01], L 2 = 1.04, 222s[Sander02], L 2 = 1.03, 39s

Surface Spectral Clustering Analysis Clustering

Surface Spectral Clustering l Get top n (≥ 3) eigenvalues/eigenvectors n where n maximizes l For each vertex n compute n-dimensional embedding coordinates l For each of the n dimensions n find two extreme vertices n set them as representatives l Remove representatives that are too close l Grow charts from representatives l Get top n (≥ 3) eigenvalues/eigenvectors n where n maximizes l For each vertex n compute n-dimensional embedding coordinates l For each of the n dimensions n find two extreme vertices n set them as representatives l Remove representatives that are too close l Grow charts from representatives

Surface Spectral Clustering n=3 n=4

Surface Spectral Clustering n=1: 2 charts n=2: 4 charts n=4: 8 charts n=3: 6 charts

Optimizing Partition Boundaries l create nonjaggy cut, through “crease” edges [Katz2003] l minimize embedding distortion

Optimizing Partition Boundaries Angular capacity alone [Katz et al, 2003] Distortion capacity aloneCombined capacity

Special Spectral Clustering l Avoid excessive partition for simple shapes n n n > 2 n = 2 1 st dimension n = 2 2 nd dimension n = 2 3 rd dimension l Special clustering for tabular shapes

Signal-Specialized Atlas Creation l Signal-specialized parameterization [Sander02] l Combine geodesic and signal distances geometry stretchsignal stretch

Implementation Details l Acceleration n Landmark IsoMap [Silva et al, 2003] n Only compute the top 10 eigenvalues l Acceleration n Landmark IsoMap [Silva et al, 2003] n Only compute the top 10 eigenvalues l Merge small charts as a post-process

Partition Process

ResultsResults 19 charts, L 2 =1.03, running time 98s, 97k faces

ResultsResults 38 charts, L 2 =1.07, running time 287s, 150k faces

ResultsResults 23 charts, L 2 =1.06, running time 162s, 112k faces

ResultsResults 11 charts, L 2 =1.01, running time 4s, 10k faces

ResultsResults 11 charts, L 2 =1.02, running time 90s, 90k faces

ResultsResults 6 charts, L 2 =1.03, running time 17s, 40k faces

Geometry Remeshing

Remeshing Comparison Original model [Sander03], 79.5dBIso-chart, 82.9dB

LOD Generation for Texture Synthesis 32x3264x64128x128

Texture Synthesis Results

Signal-Specialized Atlas Creation Original Geometry stretch SAE = 20.8 Signal param SAE = 17.9 Signal chart&param SAE = 16.5

Signal-Specialized Atlas Creation Original Geometry stretch SAE = 18.7 Signal param SAE = 11.5 Signal chart&param SAE = 9.7

ConclusionConclusion l Surface spectral analysis n for parameterization –provides good starting point for stretch minimization n for chartification –separates global features well –optimizes chart boundaries –yields special partition for tubular shapes l Surface spectral analysis n for parameterization –provides good starting point for stretch minimization n for chartification –separates global features well –optimizes chart boundaries –yields special partition for tubular shapes l Signal-specialized atlas creation l Iso-chart: a fast and effective atlas generator