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Texture-Mapping Progressive Meshes Pedro V. Sander Steven J. Gortler John Snyder Hugues Hoppe SIGGRAPH 2001 Harvard University Microsoft Research

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Texture-Mapping Progressive Meshes Pedro V. Sander Steven J. Gortler John Snyder Hugues Hoppe SIGGRAPH 2001 Harvard University Microsoft Research

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progressive mesh 69,000 faces 15,000 faces 600 faces

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progressive mesh 600 faces simplified mesh + normal map Conveys detail of original geometry 69,000 faces 15,000 faces 600 faces

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Texture mapping Authoring: map a texture image onto a surface

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Texture mapping Authoring: map a texture image onto a surface Our problem: sample an existing surface signal

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Our problem Sample the surface signal into a texture: (e.g. normal, displacement, BRDF, …) Goals: l single texture for entire PM sequence l quality metrics n minimize appearance changes over PM n efficiently distribute the texture samples Sample the surface signal into a texture: (e.g. normal, displacement, BRDF, …) Goals: l single texture for entire PM sequence l quality metrics n minimize appearance changes over PM n efficiently distribute the texture samples demo

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Our problem Sample the surface signal into a texture: (e.g. normal, displacement, BRDF, …) Goals: l single texture for entire PM sequence l quality metrics n minimize appearance changes over PM n efficiently distribute the texture samples Sample the surface signal into a texture: (e.g. normal, displacement, BRDF, …) Goals: l single texture for entire PM sequence l quality metrics n minimize appearance changes over PM n efficiently distribute the texture samples

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Simple approach: chart-per-face l Define texture for single-LOD mesh. l Cannot use texture for any simpler mesh! l Define texture for single-LOD mesh. l Cannot use texture for any simpler mesh! 500 faces atlas of 500 triangles [Soucy 96, Cignoni 98, Sander 00]

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Our approach: multi-face charts l Partition mesh into charts. l Simplify respecting chart topology. [Cohen 98] l Same texture still applicable. l Partition mesh into charts. l Simplify respecting chart topology. [Cohen 98] l Same texture still applicable.

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Chart constraint 1: Faces cannot span chart boundaries

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Chart constraint 2: Texture boundaries must be straight coarse mesh fine mesh texture map

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Our problem Sample the surface signal into a texture: (e.g. normal, displacement, BRDF, …) Goals: l single texture for entire PM sequence l quality metrics n minimize appearance changes over PM n efficiently distribute the texture samples Sample the surface signal into a texture: (e.g. normal, displacement, BRDF, …) Goals: l single texture for entire PM sequence l quality metrics n minimize appearance changes over PM n efficiently distribute the texture samples

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Parametrization quality metrics (1) Minimize texture deviation (stricter than geometric error) [Cohen et al 98] demo

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Parametrization quality metrics (2) Minimize texture stretch high stretch high stretch low stretch low stretch 2D texture undersampling

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Parametrization quality metrics (2) Minimize texture stretch high stretch high stretch low stretch low stretch blurring

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Contributions: Texture mapping PMs l Chartification algorithm (considers simplification quality) l Texture stretch metric (penalizes undersampling) l Parametrization algorithm (minimizes stretch) l PM optimization l Chartification algorithm (considers simplification quality) l Texture stretch metric (penalizes undersampling) l Parametrization algorithm (minimizes stretch) l PM optimization 470 faces 700 faces 1,200 faces 10,000 faces

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ApproachApproach (1) partition original mesh into charts (2) parametrize charts (3) resize chart polygons (4) simplify mesh (5) optimize parametrization (6) pack chart polygons (7) sample texture images

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ApproachApproach

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ApproachApproach

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ApproachApproach

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ApproachApproach

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ApproachApproach

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ApproachApproach

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ApproachApproach

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ApproachApproach

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Approach: Details (1) partition original mesh into charts (2) parametrize charts (3) resize chart polygons (4) simplify mesh (5) optimize parametrization (6) pack chart polygons (7) sample texture images

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Partition: chart merging l Assign each face to its own region. l Merge regions in greedy fashion based on n planarity distance 2 to best-fitting plane n compactness perimeter length 2 l Preserves mesh connectivity. [Maillot 93], [Eck 95], [Lee 98], [Garland 01] l Assign each face to its own region. l Merge regions in greedy fashion based on n planarity distance 2 to best-fitting plane n compactness perimeter length 2 l Preserves mesh connectivity. [Maillot 93], [Eck 95], [Lee 98], [Garland 01]

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Partition: boundary straightening l Improves parametrization (boundary will be straight in texture domain)

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Approach: Details (1) partition original mesh into charts (2) parametrize charts (3) resize chart polygons (4) simplify mesh (5) optimize parametrization (6) pack chart polygons (7) sample texture images

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ParametrizationParametrization 2D texture domain surface in 3D linear map singular values: γ, Γ

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ParametrizationParametrization length-preserving (isometric) γ = Γ = 1 length-preserving (isometric) γ = Γ = 1 angle-preserving (conformal) γ = Γ angle-preserving (conformal) γ = Γ area-preserving γ Γ = 1 area-preserving γ Γ = 1 length-preserving (isometric) γ = Γ = 1 length-preserving (isometric) γ = Γ = 1 angle-preserving (conformal) γ = Γ angle-preserving (conformal) γ = Γ area-preserving γ Γ = 1 area-preserving γ Γ = 1 2D texture domain surface in 3D linear map TT singular values: γ, Γ

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Stretch-minimizing parametrization 2D texture domain surface in 3D linear map TT singular values: γ, Γ L (T) = Γ L 2 (T) = (γ 2 + Γ 2 )/2 L (M) = max T L (T) L 2 (M) = T (L 2 (T)) 2 A(T)

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Stretch-minimization algorithm l Start with uniform parametrization. l Perform several optimization iterations: n for each vertex, try random line searches. l Start with uniform parametrization. l Perform several optimization iterations: n for each vertex, try random line searches. demo

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Parametrization example

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Conformal parametrization ( MIPS, Floater) ( MIPS, Floater) L 2 = 2.28 L = 10.07 L 2 stretch minimization L 2 = 1.22 L = 2.13 ComparisonComparison

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ComparisonComparison Uniform parametrization L 2 = 2.60 L = 12.52 L 2 stretch minimization L 2 = 1.22 L = 2.13

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Area-preserving parametrization L 2 = 1.57 L = 4.19 L 2 stretch minimization L 2 = 1.22 L = 2.13 ComparisonComparison

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Example of stretch minimization ignoring stretch minimizing stretch demo

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Approach: Details (1) partition original mesh into charts (2) parametrize charts (3) resize chart polygons (4) simplify mesh (5) optimize parametrization (6) pack chart polygons (7) sample texture images l Half-edge collapses ordered by deviation l Constrained simplification l Half-edge collapses ordered by deviation l Constrained simplification

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Approach: Details (1) partition original mesh into charts (2) parametrize charts (3) resize chart polygons (4) simplify mesh (5) optimize parametrization (6) pack chart polygons (7) sample texture images

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Parametrization optimization l Min M in PM stretch(M) + deviation(M) l Improves deviation over entire range. l Improves stretch at coarser LODs (stretch was ignored during simplification). l Min M in PM stretch(M) + deviation(M) l Improves deviation over entire range. l Improves stretch at coarser LODs (stretch was ignored during simplification).

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Approach: Details (1) partition original mesh into charts (2) parametrize charts (3) resize chart polygons (4) simplify mesh (5) optimize parametrization (6) pack chart polygons (7) sample texture images

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Pack chart polygons l NP-Hard problem. l We designed a heuristic. l NP-Hard problem. l We designed a heuristic.

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Packing Heuristic l Calculate the minimum bounding rectangle. l Rotate chart to make rectangle vertical. l Calculate the minimum bounding rectangle. l Rotate chart to make rectangle vertical.

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Chart placement l Sort chart rectangles by height. l Sequentially place left-to-right and right-to-left. l Sort chart rectangles by height. l Sequentially place left-to-right and right-to-left. ~[Igarashi 01]

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Approach: Details (1) partition original mesh into charts (2) parametrize charts (3) resize chart polygons (4) simplify mesh (5) optimize parametrization (6) pack chart polygons (7) sample texture images mipmap artifacts!

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Results (Measurements) l Scale charts to meet low-stretch requirement. l Stretch efficiency 3D surface area / 2D chart area l Packing efficiency 2D chart area / texture domain area l Texture efficiency stretch efficiency * packing efficiency 3D area / texture domain area l Scale charts to meet low-stretch requirement. l Stretch efficiency 3D surface area / 2D chart area l Packing efficiency 2D chart area / texture domain area l Texture efficiency stretch efficiency * packing efficiency 3D area / texture domain area // // //

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ResultsResults Efficiencies on fine meshes: Modelsbunnyparasaurhorsehand # faces in M n 69,63043,86696,95660,856 # charts75 12060 uniform param. stretch efficiency 0.630.0030.610.11 our stretch efficiency0.840.630.800.68 packing efficiency0.670.630.700.62 texture efficiency0.560.400.560.42

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Results across PM uniformmin-stretchmin-stretch+optim demo stretchstretchdeviationdeviation

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DemosDemos

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SummarySummary l Automatic PM parametrization scheme. l Optimizes both deviation and stretch. l Novel stretch metric prevents undersampling at all locations and in all directions. l Robust parametrization algorithm. l Automatic PM parametrization scheme. l Optimizes both deviation and stretch. l Novel stretch metric prevents undersampling at all locations and in all directions. l Robust parametrization algorithm.

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Future work l Use hierarchical parametrization. l Constrain anisotropy. l Consider content of texture signal. l Address mip-mapping problems. l Use hierarchical parametrization. l Constrain anisotropy. l Consider content of texture signal. l Address mip-mapping problems.

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