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Mesh Parametrization and Its Applications

동의대학교멀티미디어공학과

김형석포항공과대학교 ( 이윤진 , 이승용 )

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Computer Graphics

Definition : all technologies related to producing pictures or images using a

computer Computer animation, VR(virtual reality), …

Goal : Reality and Real time

Reality Mapping(texture) / Rendering(light)

v

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Computer Graphics

v

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Computer Graphics

v

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Polygonal Objects(Mesh)

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Parametrization

Embedding 3D mesh to 2D parameter space

Requirements distortion minimization one-to-one mapping

s

t

x

yz

v(s,t)

triangular mesh in 3D parametrization in 2D

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Parameterization:[Levy]

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Previous Work

Energy functional minimization Green-Lagrange tensor [Maillot93] orthogonality and homogeneous spacing [Lévy98] Dirichlet energy [Hormann99]

Convex combination approach shape-preserving parametrization [Floater97] harmonic embedding [Eck95]

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Convex Combination Approach (1)

Convex combination and boundary condition determine shape of parameter space map boundary vertices onto a convex polygon determine coefficients for the inner vertices solve a linear system Ax = b

1-ring neighborhoodin parametric space

3D mesh parameter space

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Convex Combination Approach (2)

Benefit simple and fast, one-to-one embedding

Drawback high distortions near the boundary

parameterization with fixed boundary

3D mesh

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Reducing Distortion near Boundary

Floating boundary for the parameter space non-linear system [Maillot93] [Lévy98] [Hormann99] linear system [Lévy01] heavy computation and/or non-one-to-one mapping

parameterizationwith floating boundary

3D mesh

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Motivation

Extension of convex combination approach distortion minimization near the boundary simple and fast one-to-one mapping

floating boundary3D mesh fixed boundary

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Our Approach (1)

Virtual boundary virtual vertices attached to the real boundary virtual boundary is fixed but real boundary can

move to reduce the distortion in parameterization

virtual boundaryparametrization

with virtual boundary3D mesh

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Our Approach (2)

Parametrization process

Compute coefficients i,j

(inner vertices + boundary vertices)

Determine shape of parameter space(convex polygon)

Map virtual vertices to the polygon

Solve linear system

making virtual boundary

parametrization

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Virtual Boundary

Virtual vertices # of virtual vertices = 2 # of real boundary

vertices boundary vertex is adjacent to three virtual vertices no 3D positions are required for virtual vertices

real boundary

connectivity of virtual vertices

virtual boundary

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Coefficient Computation (1)

Shape-preserving parametrization [Floater97] conformal mapping of 1-ring neighborhood average of barycentric coordinates

conformal mappingonto 2D

1-ring neighborhoodin 3D

averagingbarycentric coord.

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Coefficient Computation (2)

Coefficients of real boundary vertices

1-ring neighborhoodin 3D

1-ring neighborhoodin 2D

1-ring neighborhood + virtual vertices

in 2D

map to 2Dwhile preserving

angles and lengths

place virtual vertices

in 2D

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2D Positions of Virtual Vertices

Mapping virtual vertices onto convex polygon using edge lengths between real boundary vertices

real boundary

mapping virtual boundary

virtual boundary

relation of real and virtual boundary

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Shape of Parameter Space

Strong influence on the parameterization simple choices such as circle and rectangle?

Convex hull of the projection of real boundary

circle rectangle convex polygonfrom projected boundary

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Extended Virtual Boundary (1)

More virtual vertices in multi-layered structure to reduce distortions near the real boundary

3D mesh parametrization

region far from the boundary

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Extended Virtual Boundary (2)

Structure each layer has the same # of virtual vertices

Coefficients for virtual vertices

real boundary

1st virtual layer

2nd virtual layer

…

connectivity an coefficientsof virtual vertices

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Extended Virtual Boundary (3)

Effect for concave real boundary

3D mesh one layerno virtual vertices

two layers three layers four layers

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Results (1)

rectangle circle projected polygon

3D mesh

mapthe

boundary

mapvirtual boundary

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Results (2)

Texture mapping

rectangle circle,virtual boundary

projected polygon,virtual boundary

3D mesh

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Applications(Texture):[Levy]

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Applications(Texture) ):[Levy]

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Applications(Texture) ):[Levy]

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Conclusion and Future Work

Extension of convex combination approach distortion minimization near the boundary

Virtual boundary fixed instead of the real boundary multi-layered structure

Future work connectivity and coefficients of virtual vertices speed up with multilevel approach