geometry images
DESCRIPTION
Geometry Images. Xianfeng Gu Harvard University. Steven Gortler Harvard University. Hugues Hoppe Microsoft Research. Irregular meshes. Vertex 1 x 1 y 1 z 1 Vertex 2 x 2 y 2 z 2 …. Face 2 1 3 Face 4 2 3 …. Texture mapping. Vertex 1 x 1 y 1 z 1 Vertex 2 x 2 y 2 z 2 …. - PowerPoint PPT PresentationTRANSCRIPT
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Geometry ImagesGeometry Images
Steven GortlerSteven GortlerHarvard UniversityHarvard University
Xianfeng GuXianfeng GuHarvard UniversityHarvard University
Hugues HoppeHugues HoppeMicrosoft ResearchMicrosoft Research
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Irregular meshesIrregular meshes
Vertex 1 xVertex 1 x11 y y11 z z11
Vertex 2 xVertex 2 x22 y y22 z z22
……
Face 2 Face 2 11 3 3Face 4 2 3Face 4 2 3……
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Texture mappingTexture mapping
Vertex 1 xVertex 1 x11 y y11 z z11
Vertex 2 xVertex 2 x22 y y22 z z22
……
ss11 t t11
ss22 t t22
normal mapnormal mapss
tt
Face 2 Face 2 11 3 3Face 4 2 3Face 4 2 3……
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Complicated rendering processComplicated rendering process
Vertex 1 xVertex 1 x11 y y11 z z11
Vertex 2 xVertex 2 x22 y y22 z z22
……
random access!random access!
random access!random access!
ss11 t t11
ss22 t t22
Face 2 Face 2 11 3 3Face 4 2 3Face 4 2 3……
~40M ~40M ΔΔ/sec/sec
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Semi-regular representationsSemi-regular representations
irregular vertex indicesirregular vertex indices only only semisemi-regular-regular
[Eck et al 1995][Eck et al 1995][Lee et al 1998][Lee et al 1998][Khodakovsky 2000][Khodakovsky 2000][Guskov et al 2000][Guskov et al 2000]……
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Geometry ImageGeometry Image
geometry imagegeometry image257 x 257; 12 bits/channel257 x 257; 12 bits/channel
3D geometry3D geometrycompletely regular samplingcompletely regular sampling
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Basic ideaBasic idea
demodemo
cutcut
parametrizeparametrize
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Basic ideaBasic idea
cutcut
samplesample
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Basic ideaBasic idea
cutcut
[[rr,,gg,,bb] = [] = [xx,,yy,,zz]]
renderrender
storestore
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How to cut ?How to cut ?
sphere in 3Dsphere in 3D2D surface disk2D surface disk
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How to cut ?How to cut ?
Genus-0 surface Genus-0 surface any tree of edges any tree of edges
sphere in 3Dsphere in 3D2D surface disk2D surface disk
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How to cut ?How to cut ?
Genus-Genus-gg surface surface 2g 2g generator loops generator loops minimumminimum
torus (genus 1)torus (genus 1)
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Surface cutting algorithmSurface cutting algorithm
(1) Find topologically-sufficient cut:(1) Find topologically-sufficient cut:22gg loops loops [Dey and Schipper 1995][Dey and Schipper 1995] [Erickson and Har-Peled 2002] [Erickson and Har-Peled 2002]
(2) Allow better parametrization:(2) Allow better parametrization:additional cut pathsadditional cut paths [Sheffer 2002][Sheffer 2002]
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Step 1: Find topologically-sufficient cutStep 1: Find topologically-sufficient cut
(a) retract 2-simplices(a) retract 2-simplices
(b) retract 1-simplices(b) retract 1-simplices
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Results of Step 1Results of Step 1
genus 6genus 6 genus 0genus 0genus 3genus 3
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Step 2: Augment cutStep 2: Augment cut
Make the cut pass through “extrema”Make the cut pass through “extrema” (note: not local phenomena). (note: not local phenomena).
Approach: parametrize and look for “bad” areas.Approach: parametrize and look for “bad” areas.
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Step 2: Augment cutStep 2: Augment cut
……iterate while parametrization improvesiterate while parametrization improves
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Results of Steps 1 & 2Results of Steps 1 & 2
genus 1genus 1 genus 0genus 0
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Parametrize boundaryParametrize boundary
Constraints:Constraints: cut-pathcut-path mates identical length mates identical length endpointsendpoints at grid points at grid points
aaa’a’
aa
a’a’
no cracksno cracks
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Parametrize interiorParametrize interior
– optimizes point-sampled approx. optimizes point-sampled approx. [Sander et al 2002][Sander et al 2002]
Geometric-stretch metricGeometric-stretch metric– minimizes undersampling minimizes undersampling [Sander et al 2001][Sander et al 2001]
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Previous metricsPrevious metrics (Floater, harmonic, uniform, …)(Floater, harmonic, uniform, …)
Stretch parametrizationStretch parametrization
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SampleSample
geometry image
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RenderingRendering
(65x65 geometry image)(65x65 geometry image)
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renderingrendering
geometry image geometry image 2572572 2 x 12b/chx 12b/ch
normal-map image normal-map image 5125122 2 x 8b/chx 8b/ch
Rendering with attributesRendering with attributes
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Advantages for hardware renderingAdvantages for hardware rendering
Regular sampling Regular sampling no vertex indices. no vertex indices. Unified parametrization Unified parametrization no texture coordinates. no texture coordinates.
Raster-scanRaster-scan traversal of traversal of sourcesource data: data: geometry & attribute samples in lockstep. geometry & attribute samples in lockstep.
SummarySummary: compact, regular, no indirection: compact, regular, no indirection
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normal mapnormal map512x512; 8b/ch512x512; 8b/ch
Normal-Mapped DemoNormal-Mapped Demo
geometry imagegeometry image129x129; 12b/ch129x129; 12b/ch
demodemo
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demodemocolor mapcolor map
512x512; 8b/ch512x512; 8b/ch
Pre-shaded DemoPre-shaded Demo
geometry imagegeometry image129x129; 12b/ch129x129; 12b/ch
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ResultsResults
257x257257x257
normal-map 512x512normal-map 512x512
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ResultsResults
257x257257x257
color image 512x512color image 512x512
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Mip-mappingMip-mapping
257x257257x257 129x129129x129 65x6565x65
boundary constraintsboundary constraintsset for size 65x65set for size 65x65
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Hierarchical cullingHierarchical culling
view-frustum cullingview-frustum culling
backface cullingbackface culling
geometry imagegeometry image
normal-map imagenormal-map image
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CompressionCompression
1.5 KB1.5 KB
+ topological sideband (12 B)+ topological sideband (12 B)fused cutfused cut295 KB295 KB
Image wavelet-coderImage wavelet-coder
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Compression resultsCompression results
1.5 KB1.5 KB 3 KB3 KB 12 KB12 KB 49 KB49 KB
295 KB 295 KB
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4045505560657075808590
100 1000 10000 100000File Size (bytes)
PS
NR
(dB
)
Khodakovsky
geometry image
Rate distortionRate distortion
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Some artifactsSome artifacts
aliasingaliasing anisotropic samplinganisotropic sampling
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SummarySummary
Simple rendering:Simple rendering: compact, no indirection, raster-scan stream. compact, no indirection, raster-scan stream.
Mipmapped geometryMipmapped geometry Hierarchical cullingHierarchical culling CompressibleCompressible
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Future workFuture work
Better cutting algorithmsBetter cutting algorithms
Feature-sensitive remeshingFeature-sensitive remeshing
Tangent-frame compressionTangent-frame compression
Bilinear and bicubic renderingBilinear and bicubic rendering
Build hardwareBuild hardware
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