Multi-chart Geometry ImagesMulti-chart Geometry ImagesMulti-chart Geometry ImagesMulti-chart Geometry Images

Pedro SanderPedro SanderHarvardHarvard

Pedro SanderPedro SanderHarvardHarvard

Hugues HoppeHugues HoppeMicrosoft ResearchMicrosoft Research

Hugues HoppeHugues HoppeMicrosoft ResearchMicrosoft Research

Steven GortlerSteven GortlerHarvardHarvard

Steven GortlerSteven GortlerHarvardHarvard

John SnyderJohn SnyderMicrosoft ResearchMicrosoft Research

John SnyderJohn SnyderMicrosoft ResearchMicrosoft Research

ZoZoë Woodë WoodCaltechCaltech

ZoZoë Woodë WoodCaltechCaltech

Geometry representationGeometry representationGeometry representationGeometry representation

semi-regularsemi-regularirregularirregular completely regularcompletely regular

Basic ideaBasic ideaBasic ideaBasic idea

cutcut

parametrizeparametrize

Basic ideaBasic ideaBasic ideaBasic idea

cutcut

samplesample

Basic ideaBasic ideaBasic ideaBasic idea

cutcut

[[rr,,gg,,bb] = [] = [xx,,yy,,zz]]

simple traversalsimple traversalto renderto render

storestore

Benefits of regularityBenefits of regularityBenefits of regularityBenefits of regularity

Simplicity in renderingSimplicity in rendering No vertex indirectionNo vertex indirection No texture coordinate indirectionNo texture coordinate indirection

Hardware potentialHardware potential Leverage image processing tools for geometric Leverage image processing tools for geometric

manipulationmanipulation

Simplicity in renderingSimplicity in rendering No vertex indirectionNo vertex indirection No texture coordinate indirectionNo texture coordinate indirection

Hardware potentialHardware potential Leverage image processing tools for geometric Leverage image processing tools for geometric

manipulationmanipulation

Limitations of single-chartLimitations of single-chartLimitations of single-chartLimitations of single-chart

Unavoidable distortion and undersamplingUnavoidable distortion and undersampling

long extremitieslong extremities high genushigh genus

Limitations of semi-regularLimitations of semi-regularLimitations of semi-regularLimitations of semi-regular

Base “charts” effectively constrained to be Base “charts” effectively constrained to be equal size equilateral trianglesequal size equilateral triangles

piecewisepiecewise regular regular400x160irregularirregular

Multi-chart Geometry ImagesMulti-chart Geometry ImagesMulti-chart Geometry ImagesMulti-chart Geometry Images

defineddefineddefineddefined

undefinedundefinedundefinedundefined

Multi-chart Geometry ImagesMulti-chart Geometry ImagesMulti-chart Geometry ImagesMulti-chart Geometry Images

Simple reconstruction rules;Simple reconstruction rules;for each 2-by-2 quad of MCGIM samples:for each 2-by-2 quad of MCGIM samples: 3 defined samples 3 defined samples render 1 triangle render 1 triangle 4 defined samples 4 defined samples render 2 triangles render 2 triangles

(using shortest diagonal) (using shortest diagonal)

Simple reconstruction rules;Simple reconstruction rules;for each 2-by-2 quad of MCGIM samples:for each 2-by-2 quad of MCGIM samples: 3 defined samples 3 defined samples render 1 triangle render 1 triangle 4 defined samples 4 defined samples render 2 triangles render 2 triangles

(using shortest diagonal) (using shortest diagonal)

Multi-chart Geometry ImagesMulti-chart Geometry ImagesMulti-chart Geometry ImagesMulti-chart Geometry Images

Simple reconstruction rules;Simple reconstruction rules;for each 2-by-2 quad of MCGIM samples:for each 2-by-2 quad of MCGIM samples: 3 defined samples 3 defined samples render 1 triangle render 1 triangle 4 defined samples 4 defined samples render 2 triangles render 2 triangles

(using shortest diagonal) (using shortest diagonal)

Simple reconstruction rules;Simple reconstruction rules;for each 2-by-2 quad of MCGIM samples:for each 2-by-2 quad of MCGIM samples: 3 defined samples 3 defined samples render 1 triangle render 1 triangle 4 defined samples 4 defined samples render 2 triangles render 2 triangles

(using shortest diagonal) (using shortest diagonal)

Cracks in reconstructionCracks in reconstructionCracks in reconstructionCracks in reconstruction

Challenge: the discrete sampling will cause Challenge: the discrete sampling will cause cracks in the reconstruction between chartscracks in the reconstruction between charts

Challenge: the discrete sampling will cause Challenge: the discrete sampling will cause cracks in the reconstruction between chartscracks in the reconstruction between charts

““zippered”zippered”

MCGIM Basic pipelineMCGIM Basic pipelineMCGIM Basic pipelineMCGIM Basic pipeline

Break mesh into charts Break mesh into charts Parameterize chartsParameterize charts Pack the chartsPack the charts Sample the charts Sample the charts Zipper chart seams Zipper chart seams Optimize the MCGIMOptimize the MCGIM

Break mesh into charts Break mesh into charts Parameterize chartsParameterize charts Pack the chartsPack the charts Sample the charts Sample the charts Zipper chart seams Zipper chart seams Optimize the MCGIMOptimize the MCGIM

Mesh chartificationMesh chartificationMesh chartificationMesh chartificationGoal: planar charts with compact boundariesGoal: planar charts with compact boundaries

Clustering optimization - Lloyd-Max Clustering optimization - Lloyd-Max (Shlafman 2002)(Shlafman 2002):: Iteratively grow chart from given seed face.Iteratively grow chart from given seed face.

(metric is a product of distance and normal)(metric is a product of distance and normal) Compute new seed face for each chart.Compute new seed face for each chart.

(face that is farthest from chart boundary)(face that is farthest from chart boundary) Repeat above steps until convergence.Repeat above steps until convergence.

Goal: planar charts with compact boundariesGoal: planar charts with compact boundaries

Clustering optimization - Lloyd-Max Clustering optimization - Lloyd-Max (Shlafman 2002)(Shlafman 2002):: Iteratively grow chart from given seed face.Iteratively grow chart from given seed face.

(metric is a product of distance and normal)(metric is a product of distance and normal) Compute new seed face for each chart.Compute new seed face for each chart.

(face that is farthest from chart boundary)(face that is farthest from chart boundary) Repeat above steps until convergence.Repeat above steps until convergence.

Mesh chartificationMesh chartificationMesh chartificationMesh chartificationBootstrappingBootstrapping

Start with single seedStart with single seed Run chartification using increasing number of Run chartification using increasing number of

seeds each phaseseeds each phase Until desired number reachedUntil desired number reached

BootstrappingBootstrapping Start with single seedStart with single seed Run chartification using increasing number of Run chartification using increasing number of

seeds each phaseseeds each phase Until desired number reachedUntil desired number reached

demodemodemodemo

Chartification ResultsChartification ResultsChartification ResultsChartification Results

Produces planar charts with compact boundariesProduces planar charts with compact boundaries Produces planar charts with compact boundariesProduces planar charts with compact boundaries

Sander et. al. 2001Sander et. al. 200180% stretch efficiency80% stretch efficiency

Our methodOur method99% stretch efficiency99% stretch efficiency

ParameterizationParameterizationParameterizationParameterization

Goal: Penalizes undersamplingGoal: Penalizes undersampling LL22 geometric stretch of Sander et. al. 2001 geometric stretch of Sander et. al. 2001 Hierarchical algorithm for solving minimizationHierarchical algorithm for solving minimization

Goal: Penalizes undersamplingGoal: Penalizes undersampling LL22 geometric stretch of Sander et. al. 2001 geometric stretch of Sander et. al. 2001 Hierarchical algorithm for solving minimizationHierarchical algorithm for solving minimization

ParameterizationParameterizationParameterizationParameterization

Goal: Penalizes undersamplingGoal: Penalizes undersampling LL22 geometric stretch of Sander et. al. 2001 geometric stretch of Sander et. al. 2001 Hierarchical algorithm for solving minimizationHierarchical algorithm for solving minimization

Goal: Penalizes undersamplingGoal: Penalizes undersampling LL22 geometric stretch of Sander et. al. 2001 geometric stretch of Sander et. al. 2001 Hierarchical algorithm for solving minimizationHierarchical algorithm for solving minimization

Angle-preserving metricAngle-preserving metric

(Floater)(Floater)

Angle-preserving metricAngle-preserving metric

(Floater)(Floater)

Chart packingChart packingChart packingChart packing

Goal: minimize wasted spaceGoal: minimize wasted space Based on Levy et al. 2002Based on Levy et al. 2002 Place a chart at a time Place a chart at a time

(from largest to smallest) (from largest to smallest) Pick best position Pick best position and rotationand rotation

(minimize wasted space) (minimize wasted space) Repeat above for multiple MCGIM rectangle shapesRepeat above for multiple MCGIM rectangle shapes

pick bestpick best

Goal: minimize wasted spaceGoal: minimize wasted space Based on Levy et al. 2002Based on Levy et al. 2002 Place a chart at a time Place a chart at a time

(from largest to smallest) (from largest to smallest) Pick best position Pick best position and rotationand rotation

(minimize wasted space) (minimize wasted space) Repeat above for multiple MCGIM rectangle shapesRepeat above for multiple MCGIM rectangle shapes

pick bestpick best

Packing ResultsPacking ResultsPacking ResultsPacking ResultsLevy packing Levy packing

efficiency 58.0%efficiency 58.0%

Our packing Our packing efficiency 75.6%efficiency 75.6%

Sampling into a MCGIMSampling into a MCGIMSampling into a MCGIMSampling into a MCGIM

Goal: discrete sampling of parameterized charts Goal: discrete sampling of parameterized charts into topological discsinto topological discs Rasterize triangles with scan conversionRasterize triangles with scan conversion Store geometryStore geometry

Goal: discrete sampling of parameterized charts Goal: discrete sampling of parameterized charts into topological discsinto topological discs Rasterize triangles with scan conversionRasterize triangles with scan conversion Store geometryStore geometry

Sampling into a MCGIMSampling into a MCGIMSampling into a MCGIMSampling into a MCGIM

Boundary Boundary rasterizationrasterization

Non-manifold dilationNon-manifold dilation

Zippering the MCGIMZippering the MCGIMZippering the MCGIMZippering the MCGIM

Goal: to form a watertight reconstructionGoal: to form a watertight reconstruction Goal: to form a watertight reconstructionGoal: to form a watertight reconstruction

Zippering the MCGIMZippering the MCGIMZippering the MCGIMZippering the MCGIM

Algorithm: Algorithm: Greedy (but robust) approach Greedy (but robust) approach

Identify cut-nodes and Identify cut-nodes and cut-path samples. cut-path samples.

Unify cut-nodes.Unify cut-nodes.

Snap cut-path samples Snap cut-path samples to geometric cut-path. to geometric cut-path.

Unify cut-path samples.Unify cut-path samples.

Algorithm: Algorithm: Greedy (but robust) approach Greedy (but robust) approach

Identify cut-nodes and Identify cut-nodes and cut-path samples. cut-path samples.

Unify cut-nodes.Unify cut-nodes.

Snap cut-path samples Snap cut-path samples to geometric cut-path. to geometric cut-path.

Unify cut-path samples.Unify cut-path samples.

Zippering: SnapZippering: SnapZippering: SnapZippering: Snap

SnapSnap Snap discrete cut-path samples to Snap discrete cut-path samples to

geometrically closest point on cut-pathgeometrically closest point on cut-path

SnapSnap Snap discrete cut-path samples to Snap discrete cut-path samples to

geometrically closest point on cut-pathgeometrically closest point on cut-path

Zippering: UnifyZippering: UnifyZippering: UnifyZippering: Unify

UnifyUnify Greedily unify neighboring samplesGreedily unify neighboring samples

UnifyUnify Greedily unify neighboring samplesGreedily unify neighboring samples

How unification worksHow unification worksHow unification worksHow unification works

UnifyUnify Test the distance of the next 3 movesTest the distance of the next 3 moves Pick smallest to unify then advancePick smallest to unify then advance

UnifyUnify Test the distance of the next 3 movesTest the distance of the next 3 moves Pick smallest to unify then advancePick smallest to unify then advance

How unification worksHow unification worksHow unification worksHow unification works

UnifyUnify Test the distance of the next 3 movesTest the distance of the next 3 moves Pick smallest to unify then advancePick smallest to unify then advance

UnifyUnify Test the distance of the next 3 movesTest the distance of the next 3 moves Pick smallest to unify then advancePick smallest to unify then advance

How unification worksHow unification worksHow unification worksHow unification works

UnifyUnify Test the distance of the next 3 movesTest the distance of the next 3 moves Pick smallest to unify then advancePick smallest to unify then advance

UnifyUnify Test the distance of the next 3 movesTest the distance of the next 3 moves Pick smallest to unify then advancePick smallest to unify then advance

Geometry image optimizationGeometry image optimizationGeometry image optimizationGeometry image optimization

Goal: align discrete samples with mesh featuresGoal: align discrete samples with mesh features Hoppe et. al. 1993Hoppe et. al. 1993 Reposition vertices to minimize distance toReposition vertices to minimize distance to

the original surface the original surface Constrain connectivityConstrain connectivity

Goal: align discrete samples with mesh featuresGoal: align discrete samples with mesh features Hoppe et. al. 1993Hoppe et. al. 1993 Reposition vertices to minimize distance toReposition vertices to minimize distance to

the original surface the original surface Constrain connectivityConstrain connectivity

Multi-chart resultsMulti-chart resultsMulti-chart resultsMulti-chart results

genus 2; 50 chartsgenus 2; 50 charts 478x133478x133 RenderingRenderingPSNR 79.5PSNR 79.5

Multi-chart resultsMulti-chart resultsMulti-chart resultsMulti-chart results

genus 1; 40 chartsgenus 1; 40 charts174x369174x369

RenderingRenderingPSNR 75.6 PSNR 75.6

Multi-chart resultsMulti-chart resultsMulti-chart resultsMulti-chart results

genus 0; 25 chartsgenus 0; 25 charts 281X228281X228 RenderingRenderingPSNR 84.6 PSNR 84.6

Multi-chart resultsMulti-chart resultsMulti-chart resultsMulti-chart results

genus 0; 15 chartsgenus 0; 15 charts466x138466x138

RenderingRenderingPSNR 83.8 PSNR 83.8

478x133478x133

irregularirregularoriginaloriginal

singlesinglechartchart

PSNR 68.0PSNR 68.0

multi-multi-chartchart

PSNR 79.5PSNR 79.5demodemodemodemo

Multi-chart resultsMulti-chart resultsMulti-chart resultsMulti-chart results

Comparison to semi-regularComparison to semi-regularComparison to semi-regularComparison to semi-regular

Original irregularOriginal irregular Semi-regularSemi-regular MCGIMMCGIM

Comparison to semi-regularComparison to semi-regularComparison to semi-regularComparison to semi-regular

Original irregular meshOriginal irregular mesh

Semi-regular meshSemi-regular meshPSNR 87.8PSNR 87.8

MCGIM meshMCGIM meshPSNR 90.2PSNR 90.2

SummarySummarySummarySummary Contributions:Contributions:

Overall: MCGIM representationOverall: MCGIM representation– Rendering simplicityRendering simplicity

Major: zippering and optimizationMajor: zippering and optimization Minor: packing and chartificationMinor: packing and chartification

Contributions:Contributions: Overall: MCGIM representationOverall: MCGIM representation

– Rendering simplicityRendering simplicity Major: zippering and optimizationMajor: zippering and optimization Minor: packing and chartificationMinor: packing and chartification

Future workFuture workFuture workFuture work

Provide:Provide: CompressionCompression Level-of-detail rendering controlLevel-of-detail rendering control

Exploit rendering simplicity in hardwareExploit rendering simplicity in hardware

Improve zipperingImprove zippering

Provide:Provide: CompressionCompression Level-of-detail rendering controlLevel-of-detail rendering control

Exploit rendering simplicity in hardwareExploit rendering simplicity in hardware

Improve zipperingImprove zippering