Geometric Applications to Computer Graphics and Imaging

(1)

Geometric applications to computer

graphics and imaging

Geometric applications to computer

graphics and imaging

Shing-Tung Yau Harvard University Shing

Shing--Tung YauTung Yau Harvard University

(2)

Outline

z Introduction z Motivation

z Global conformal parameterization z Future work z z IntroductionIntroduction z z MotivationMotivation z

z Global conformal parameterizationGlobal conformal parameterization

z

(3)

Motivation

z Texture Mapping z Geometry Remeshing z Geometry Morphing z Geometry Matching z Geometry Classification z Geometric Compression z Topological Computation z

z Texture MappingTexture Mapping

z

z Geometry RemeshingGeometry Remeshing

z

z Geometry MorphingGeometry Morphing

z

z Geometry MatchingGeometry Matching

z

z Geometry ClassificationGeometry Classification

z

z Geometric CompressionGeometric Compression

z

(4)

Introduction

z Surface Parametrization is a process to

map a surface to a region of the plane

z

z Surface Parametrization is a process to Surface Parametrization is a process to

map a surface to a region of the plane

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Texture Mapping - Non distortion

Texture Mapping

(6)

Texture Mapping - Cloth modeling

Texture Mapping

(7)

Geometry Remeshing

z Import image processing techniques on

geometry (e.g. wavelet compression,

signal processing, network transportation)

z Simplify graphics hardware architecture

Low end graphics engine

Mobile graphics hardware

z Improve geometric computation

z Geometric Media (e.g. 3D magazines) z

z Import image processing techniques on Import image processing techniques on

geometry (e.g. wavelet compression,

signal processing, network transportation)

z

z Simplify graphics hardware architectureSimplify graphics hardware architecture

Low end graphics engineLow end graphics engine

Mobile graphics hardwareMobile graphics hardware z

z Improve geometric computationImprove geometric computation

z

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Geometry Remeshing

z Change irregular connectivity to regular

one

z

z Change irregular connectivity to regular Change irregular connectivity to regular

one

(9)

Geometric Matching - brain mapping

Geometric Matching

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Geometric Matching - brain mapping

Geometric Matching

-

_-

brain mapping

_{brain mapping}

demo

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Geometric Matching - brain mapping

Geometric Matching

-

_-

brain mapping

_{brain mapping}

z Minimize L2 norm under Mobius

transformation

z Least square problem z

z Minimize L2 norm under Mobius Minimize L2 norm under Mobius

transformation

z

z Least square problemLeast square problem

demo

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Growth patterns in the developing human

brain

*

Growth patterns in the developing human

brain

**

*_{Thompson et.al Growth patterns in the developing brain detected by}

(13)

Brain Conformal Mapping

z Alzheimer’s Disease (AD) was as the 8th

leading cause of death in U.S. for 1999; it costs U.S. society $100 billion per year1

z Schizophrenia affects 0.2-2% of the worldwide

population; it costs $32.5 billion per year in U.S.2

z Brain conformal mapping can be used to the

diagnosis of AD and Schizophrenia, brain development study, and surgery mapping

z Our conformal mapping was demonstrated to

be a stable solution and superior than other

methods and is expected to become a standard in this field in the future

z

z AlzheimerAlzheimer’’s Disease (AD) was as the 8s Disease (AD) was as the 8thth

leading cause of death in U.S. for 1999; it costs

U.S. society $100 billion per year

U.S. society $100 billion per year11

z

z Schizophrenia affects 0.2Schizophrenia affects 0.2--2% of the worldwide 2% of the worldwide

population; it costs $32.5 billion per year in

U.S.

U.S.22

z

z Brain conformal mapping can be used to the Brain conformal mapping can be used to the

diagnosis of AD and Schizophrenia, brain

development study, and surgery mapping

z

z Our conformal mapping was demonstrated to Our conformal mapping was demonstrated to

be a stable solution and superior than other

methods and is expected to become a standard

in this field in the future

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Geometric Morphing

z Parameterize surfaces to a canonical

domain

z Match features by parameter z

z Parameterize surfaces to a canonical Parameterize surfaces to a canonical

domain

z

z Match features by parameterMatch features by parameter

demo

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Geometric Matching

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Geometric Matching

z Representation of genus zero surfaces as

functions on a sphere:

Conformal factor

Gauss map

z

z Representation of genus zero surfaces as Representation of genus zero surfaces as

functions on a sphere:

Conformal factorConformal factor

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Geometric Matching

z Compute functions on surface

Gaussian curvature Height function

z Locate critical points of them z Match critical points first

z Minimize L2 norm under Euclidean transformation group z

z Compute functions on surface Compute functions on surface

Gaussian curvatureGaussian curvature

Height functionHeight function

z

z Locate critical points of themLocate critical points of them

z

z Match critical points firstMatch critical points first

z

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Geometric Matching

z Level set of Gaussian curvature z Gradient of Gaussian curvature z

z Level set of Gaussian curvatureLevel set of Gaussian curvature

z

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Geometric Parameterization

z Topological parameterization is too coarse z Isometric parameterization is too refined z Conformal parameterization is a good

candidate

z Applications:

Geometric Data base indexing

3D search engine

z

z Topological parameterization is too coarseTopological parameterization is too coarse

z

z Isometric parameterization is too refinedIsometric parameterization is too refined

z

z Conformal parameterization is a good Conformal parameterization is a good

candidate

z

z Applications:Applications:

Geometric Data base indexingGeometric Data base indexing

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Geometric Parameterization

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Conformal Invariants - Periods

Conformal Invariants

-

_-

Periods

_Periods

z Torus is conformally mapped to a

parallelogram

z The shape factors of the parallelogram are

conformal invariants

z

z Torus is conformally mapped to a Torus is conformally mapped to a

parallelogram

z

z The shape factors of the parallelogram are The shape factors of the parallelogram are

conformal invariants

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Conformal Invariants - Periods

Conformal Invariants

-

_-

Periods

_Periods

7500 7500 3750 3750 4.9928 4.9928 85.432 85.432 rocker rocker 11616 11616 5808 5808 31.150 31.150 85.1 85.1 knot knot 34048 34048 17024 17024 3.0264 3.0264 89.95 89.95 teapot teapot 2048 2048 1089 1089 2.2916 2.2916 89.987 89.987 torus torus Faces Faces Vertices Vertices Length Length ratio ratio Angle Angle Mesh Mesh Snap Snap shot shot

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Geometry Compression

z Map surface to a canonical domain

z Construct a functional orthonormal basis of

the domain

z Decompose surface signal as a spectrum z Fourier analysis, harmonic analysis,

Laplacian spectrum analysis

z

z Map surface to a canonical domainMap surface to a canonical domain

z

z Construct a functional orthonormal basis of Construct a functional orthonormal basis of

the domain

z

z Decompose surface signal as a spectrumDecompose surface signal as a spectrum

z

z Fourier analysis, harmonic analysis, Fourier analysis, harmonic analysis,

Laplacian spectrum analysis

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Geometry Compression

z Spherical harmonic functions z Spectrum compression

z

z Spherical harmonic functionsSpherical harmonic functions

z

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Topological computation

z Verify if two curves are topologically

equivalent

z Fundamental domain

z Universal covering space z

z Verify if two curves are topologically Verify if two curves are topologically

equivalent

z

z Fundamental domainFundamental domain

z

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Global Conformal Parameterization

(27)

Outline

z Introduction

z Genus zero closed surface

z Non zero genus closed surface z Surfaces with boundaries

z Summary

z Future research z

z IntroductionIntroduction

z

z Genus zero closed surfaceGenus zero closed surface

z

z Non zero genus closed surfaceNon zero genus closed surface

z

z Surfaces with boundariesSurfaces with boundaries

z

z SummarySummary

z

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Theoretic background

z There exists a unique harmonic map from

a topological disk to a planar convex

domain with fix boundary condition. The map is a diffeomorphism.

z Each cohomology class has a unique

harmonic one-form representative.

z

z There exists a unique harmonic map from There exists a unique harmonic map from

a topological disk to a planar convex

domain with fix boundary condition. The

map is a diffeomorphism.

z

z Each cohomology class has a unique Each cohomology class has a unique

harmonic one

(29)

Introduction: Structures on mesh

z Simplicial complex

Triangulation Homology Cohomology

z Geometric structure

Embedded in Euclidean Space, Riemann metric De Rham cohomology

z Conformal Structure

Holomorphic 1-forms Period matrices

z

z Simplicial complexSimplicial complex

TriangulationTriangulation Homology Homology CohomologyCohomology z

z Geometric structureGeometric structure

Embedded in Euclidean Space, Riemann metricEmbedded in Euclidean Space, Riemann metric

De Rham cohomologyDe Rham cohomology

z

z Conformal StructureConformal Structure

Holomorphic 1Holomorphic 1--formsforms

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Introduction: Methodology

z Treat surfaces as Riemannian manifold z Conformal Structure z Holomorphic 1-forms z Conformal automorphism group z

z Treat surfaces as Treat surfaces as

Riemannian manifold

z

z Conformal StructureConformal Structure

z

z HolomorphicHolomorphic 11--formsforms

z

z Conformal Conformal

automorphism group

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Definition: Conformal Mapping

z Scaling first fundamental form z Angle preserving

z Similarities in the small z

z Scaling first fundamental formScaling first fundamental form

z

z Angle preservingAngle preserving

z

z Similarities in the smallSimilarities in the small

ij ij g

(32)

Methodology

z Genus zero closed surface: harmonic

energy minimization until tangential Laplacian is zero

z Non zero genus closed surface: compute

homology, cohomology, harmonic 1-form, holomorphic 1-forms

z Surfaces with boundaries: double cover

the surface to get a symmetric closed

surface, find symmetric holomorphic 1-form

z

z Genus zero closed surface: harmonic Genus zero closed surface: harmonic

energy minimization until tangential

Laplacian is zero

z

z Non zero genus closed surface: compute Non zero genus closed surface: compute

homology, cohomology, harmonic 1

homology, cohomology, harmonic 1--form, form, holomorphic 1

holomorphic 1--formsforms

z

z Surfaces with boundaries: double cover Surfaces with boundaries: double cover

the surface to get a symmetric closed

surface, find symmetric holomorphic 1

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Genus zero surfaces

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Genus 0 surfaces

z All conformally equivalent

z Harmonic is equivalent to conformal z Automorphism group Mobius group z

z All conformally equivalentAll conformally equivalent

z

z Harmonic is equivalent to conformalHarmonic is equivalent to conformal

z

z Automorphism group Mobius groupAutomorphism group Mobius group

demo

(35)

Global conformal parameterization

algorithm for genus zero surface

Global conformal parameterization

algorithm for genus zero surface

z Use Gauss map as the initial degree one

map

z Compute the gradient vector of harmonic

energy on each vertex

z Project the gradient vector to the tangent

space

z Update the image of each vertex along the

tangential gradient vector

z Normalize the mapping by shifting the

center of the mass to the sphere center

z

z Use Gauss map as the initial degree one Use Gauss map as the initial degree one

map

z

z Compute the gradient vector of harmonic Compute the gradient vector of harmonic

energy on each vertex

z

z Project the gradient vector to the tangent Project the gradient vector to the tangent

space

z

z Update the image of each vertex along the Update the image of each vertex along the

tangential gradient vector

z

z Normalize the mapping by shifting the Normalize the mapping by shifting the

center of the mass to the sphere center

(36)

Algorithm details

z Harmonic energy:

z Discrete harmonic energy

z Discrete Laplacian z

z Harmonic energy:Harmonic energy:

z

z Discrete harmonic energyDiscrete harmonic energy

z

z Discrete LaplacianDiscrete Laplacian

M M d f 2

σ

∫

∇

∑

∈ − = M v u uv f u f v k f E ] , [ 2 ) ( ) ( ) (

k

uv

=

cot

α

+

cot

β

∑

∈ − = Δ M v u uv f u f v k u f ] , [ )) ( ) ( ( ) (

(37)

Genus zero bunny example

z Highly non-uniform z

(38)

Mobius Transformation

z Linear rational group on complex plane z 6 dimensional group

z

z Linear rational group on complex planeLinear rational group on complex plane

z

(39)

Mobius Transformation

demo

(40)

Genus zero Gargoyle example

z Spherical barricentric embedding z Spherical conformal embedding z

z Spherical barricentric embeddingSpherical barricentric embedding

z

(41)

Non zero genus surface

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Basis idea

z Study the gradient fields of conformal

maps

z All such gradient fields form a linear space z The basis of such linear space is closely

related to the topology of the surface

z

z Study the gradient fields of conformal Study the gradient fields of conformal

maps

z

z All such gradient fields form a linear spaceAll such gradient fields form a linear space

z

z The basis of such linear space is closely The basis of such linear space is closely

related to the topology of the surface

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Definition: Homology group

z Curve space

z Homology Basis: A special set of curves

which can be deformed to any curve by merging, splitting, and duplicating

z

z Curve spaceCurve space

z

z Homology Basis: A special set of curves Homology Basis: A special set of curves

which can be deformed to any curve by

merging, splitting, and duplicating

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Cohomology group

z Tangential vector field space (1-form), we use

simplicial cohomology to approximate it

z Dual basis

z Conjugate 1-form z

z Tangential vector field space (1Tangential vector field space (1--form), we use form), we use

simplicial cohomology to approximate it

z

z Dual basis Dual basis

z

(45)

Holomorphic one-form

Holomorphic one

-

_-

form

_form

z A gradient field of a conformal map z A pair of tangential vector fields

z Curl is zero

z Both x and y gradient fields are harmonic

z x,y vector fields are orthogonal at every

point

z

z A gradient field of a conformal mapA gradient field of a conformal map

z

z A pair of tangential vector fields A pair of tangential vector fields

z

z Curl is zeroCurl is zero

z

z Both x and y gradient fields are harmonicBoth x and y gradient fields are harmonic

z

z x,y vector fields are orthogonal at every x,y vector fields are orthogonal at every

point point ) , (

ω

_x

ω

_y

0 )

(

,

0 )

(

_x

=

curl

_y

=

curl

ω

0 , 0 Δ = = Δ ω _x ω _y x y n ω ω = ×

(46)

Non Zero Genus Surface Algorithm

overview

Non Zero Genus Surface Algorithm

overview

z Compute a homology basis

z Compute a dual Cohomology basis z Diffuse to a harmonic 1-form basis

z Rotate each harmonic 1-form w to *w, pair

(w,*w) to a holomorphic 1-form

z Integrate a holomorphic 1-form on a

fundamental domain to get a conformal parameterization

z

z Compute a homology basisCompute a homology basis

z

z Compute a dual Cohomology basis Compute a dual Cohomology basis

z

z Diffuse to a harmonic 1Diffuse to a harmonic 1--form basis form basis

z

z Rotate each harmonic 1Rotate each harmonic 1--form w to *w, pair form w to *w, pair

(w,*w) to a holomorphic 1

(w,*w) to a holomorphic 1--form form

z

z Integrate a holomorphic 1Integrate a holomorphic 1--form on a form on a

fundamental domain to get a conformal

parameterization

(47)

Homology Group

z Simplicial complex structure

z Chain space (patch space, curve space,

point space)

z Boundary operator

z Smith normal form, kernel & image space

bases

z Multi-resolution z

z Simplicial complex structureSimplicial complex structure

z

z Chain space (patch space, curve space, Chain space (patch space, curve space,

point space)

z

z Boundary operatorBoundary operator

z

z Smith normal form, kernel & image space Smith normal form, kernel & image space

bases

z

(48)

Compute Cohomology Bases

z Select one handle

z Cut along homology bases (a,b) z Set the boundary to unit square

z Floater immerse the mesh to plane, dx,dy are the

desired dual cohomology bases

z

z Select one handleSelect one handle

z

z Cut along homology bases (a,b)Cut along homology bases (a,b)

z

z Set the boundary to unit squareSet the boundary to unit square

z

z Floater immerse the mesh to plane, dx,dy are the Floater immerse the mesh to plane, dx,dy are the

desired dual cohomology bases

demo

(49)

Harmonic Representative

z Define global function on mesh F z 1-form + derivative of F

z Minimize harmonic energy by adjusting F z

z Define global function on mesh FDefine global function on mesh F

z

z 11--form + derivative of Fform + derivative of F

z

z Minimize harmonic energy by adjusting FMinimize harmonic energy by adjusting F

∑

∈ = − + = + Δ M v u v F u F v u u F ] , [ 0 )) ( ) ( ]) , ([ ( ) )( (ω δ ω

(50)

Hodge Star Operator

z Local - rotate a vector field a right angle

z Global - linear transformation in

cohomology space

z Using wedge product of 1-forms z Solving linear systems

z

z Local Local -- rotate a vector field a right anglerotate a vector field a right angle

z

z Global Global -- linear transformation in linear transformation in

cohomology space

z

z Using wedge product of 1Using wedge product of 1--formsforms

z

z Solving linear systemsSolving linear systems

dx dy

dy

dx = ,* = −

(51)

Wedge Product

z Wedge Product z Algebraic computation z

z Wedge ProductWedge Product

z

z Algebraic computationAlgebraic computation

∫

ω ∧τ =

∫

ω ×τ ⋅n 1 1 1 ) ( ) ( ) ( ) ( ) ( ) ( 6 1 2 1 0 2 1 0 e e e e e e T τ τ τ ω ω ω τ ω ∧ =

∫

(52)

Conjugate Wedge Product

z Conjugate Wedge Product z

z Conjugate Wedge ProductConjugate Wedge Product

) ( 2 ) ( 2 | ) ( 2 24 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 b a a c b b c a a c b a c c b a b c a c b a c b S T + − + − + − + + − + − + − + + = ∧

∫

ω τ

∫

∧

=

T

pMq

τ

ω

*

)) ( ), ( ), ( ( )) ( ), ( ), ( ( 2 1 0 2 1 0 e e e q e e e p τ τ τ ω ω ω = =

(53)

Linear System for conjugate one-form

Linear System for conjugate one

-

_-

form

_form

z The conjugate of a harmonic 1-form is

harmonic 1-form

z Given a harmonic 1-form basis

z The conjugate 1-form can be represented

as a linear combination of the bases

z Linear system z

z The conjugate of a harmonic 1The conjugate of a harmonic 1--form is form is

harmonic 1

harmonic 1--formform

z

z Given a harmonic 1Given a harmonic 1--form basis form basis

z

z The conjugate 1The conjugate 1--form can be represented form can be represented

as a linear combination of the bases

z

z Linear systemLinear system

} ,..., , {ω₁ ω₂ ω₂_g

∑

= = g k k k c 2 1 *ω ω

∫

∑

∫

∧ = ∧ = k i g i i k ω c ω ω ω 2 1 *

(54)

Integrate on a fundamental domain

z Fix a base point, map it to the origin

z For any vertex, find an arbitrary path to the

base point, integrate 1-form along the path

z The integration is path independent z

z Fix a base point, map it to the originFix a base point, map it to the origin

z

z For any vertex, find an arbitrary path to the For any vertex, find an arbitrary path to the

base point, integrate 1

base point, integrate 1--form along the pathform along the path

z

(55)

Holomorphic one-form basis

Holomorphic one

-

_-

form basis

_{form basis}

z Holomorphic 1-forms z

z Holomorphic 1Holomorphic 1--forms forms

demo

(56)

Holomorphic one-form basis

Holomorphic one

-

_-

form basis

_{form basis}

z Genus 2 surface z

z Genus 2 surfaceGenus 2 surface

demo

(57)

Holomorphic one-form basis

Holomorphic one

-

_-

form basis

_{form basis}

z 2g real dimension z Dual to homology z

z 2g real dimension2g real dimension

z

(58)

Linear combination

z Linearly combine holomorphic 1-form

bases

z Different holomorphic one-form, different

properties (conformal factor, zero points)

z

z Linearly combine holomorphic 1Linearly combine holomorphic 1--form form

bases

z

z Different holomorphic oneDifferent holomorphic one--form, different form, different

properties (conformal factor, zero points)

(59)

Global Conformal Parametrization

z Genus g surface

Global conformal, no boundaries

2g – 2 zero points

Each handle maps to the plane periodically

z

z Genus g surfaceGenus g surface

Global conformal, no boundariesGlobal conformal, no boundaries

2g 2g –– 2 zero points2 zero points

(60)

Zero points

z Zero points of the tangential vector fields z

(61)

Zero points

z 2g – 2 zero points z

z 2g 2g –– 2 zero points2 zero points

demo

(62)

Zero points

demo

(63)

Global Conformal Atlas

z Each handle is conformally mapped to a

modular space (represented as a parallelogram)

z Handle separators z

z Each handle is conformally mapped to a Each handle is conformally mapped to a

modular space (represented as a

parallelogram)

z

(64)

Global Conformal Atlas

demo

(65)

Handle Separation

demo

demo demodemo

z Locate zero points

z In parameter domain, find connecting

curves

z Lift the planar curve to the original surface z

z Locate zero pointsLocate zero points

z

z In parameter domain, find connecting In parameter domain, find connecting

curves

z

(66)

Properties: Homology Basis Independent

(67)

Properties: Homology Basis Independent

z Boundary independent z

z Boundary independentBoundary independent

demo

(68)

Properties: Triangulation & Resolution

Independent

Properties: Triangulation & Resolution

Independent

(69)

Conformal Mapping Properties

z Intrinsic

z Depends on metric continuously z

z Intrinsic Intrinsic

z

z Depends on metric continuouslyDepends on metric continuously

demo

(70)

Results: Knot

(71)

Example: Knot

(72)

Example: Rocker

(73)

Example: Teri Surface

(74)

Example: Sculpture

(75)

Surfaces with boundaries

(76)

Surfaces with boundaries

z Copy the surface, invert the orientation z Glue two copies together along the

boundaries

z Treat the double covering as a closed

surface

z Keep symmetry z

z Copy the surface, invert the orientationCopy the surface, invert the orientation

z

z Glue two copies together along the Glue two copies together along the

boundaries

z

z Treat the double covering as a closed Treat the double covering as a closed

surface

z

(77)

Neumann Boundary Condition

z Surface boundary can slide on the

parameter domain boundary

z Tangential Laplacian is zero

z Fixed boundary vs. Neumann boundary z

z Surface boundary can slide on the Surface boundary can slide on the

parameter domain boundary

z

z Tangential Laplacian is zeroTangential Laplacian is zero

z

(78)

Example: Minimal Surface

z Genus one, 3 boundaries z Genus four

z

z Genus one, 3 boundariesGenus one, 3 boundaries

z

(79)

Conformal Factor

z Level Sets of conformal factor function z

(80)

Conformal Factor

(81)

Non Uniformity

z Extruding parts are with high conformal

factors

z Non uniform z

z Extruding parts are with high conformal Extruding parts are with high conformal

factors

z

(82)

Topology Modification

z Punch small holes at the end of the

extruding region

z

z Punch small holes at the end of the Punch small holes at the end of the

extruding region

demo

(83)

Summary of the whole process

z Double cover surfaces with boundaries z Compute a basis of its holomorphic

1-forms

z Select a holomorphic 1-form, integrate it on

a fundamental domain

z Construct conformal atlas

z Constructing geometry images z

z Double cover surfaces with boundariesDouble cover surfaces with boundaries

z

z Compute a basis of its holomorphic 1Compute a basis of its holomorphic 1-

-forms

forms

z

z Select a holomorphic 1Select a holomorphic 1--form, integrate it on form, integrate it on

a fundamental domain

z

z Construct conformal atlasConstruct conformal atlas

z

(84)

Step one: punch holes and double cover

z In order to improve the uniformity, three

holes are punched

z Double cover z

z In order to improve the uniformity, three In order to improve the uniformity, three

holes are punched

z

(85)

Step two: Compute Homology Basis

(86)

Step three: compute holomorphic 1-form

Step three: compute holomorphic 1

-

_-

form

_form

z Compute holomorphic 1-form basis

z Linearly combine to get a desired 1-form z

z Compute holomorphic 1Compute holomorphic 1--form basisform basis

z

(87)

Step three: construct conformal atlas

z Integrate the holomorphic 1-form on a

fundamental domain

z Locate zero points

z Find handle separators z

z Integrate the holomorphic 1Integrate the holomorphic 1--form on a form on a

fundamental domain

z

z Locate zero pointsLocate zero points

z

(88)

Step four: Construct geometry image

z Use regular grids to sample each chart z

(89)

Regular Connectivity

z Angle preserving z Connectivity is regular z

z Angle preservingAngle preserving

z

(90)

Geometry Image Comparison

z More accurate reconstructed normal,

comparing with geometric stretch method (gu geometry images siggraph 02)

z

z More accurate reconstructed normal, More accurate reconstructed normal,

comparing with geometric stretch method

(

(91)

David

(92)

David

demo

(93)

Summary

z Global conformal parameterization for

general surfaces with arbitrary topologies

z Conformal invariants

z Canonical Conformal atlas

z Intrinsic parameterization, independent of

triangulations, insensitive to resolutions

z

z Global conformal parameterization for Global conformal parameterization for

general surfaces with arbitrary topologies

z

z Conformal invariantsConformal invariants

z

z Canonical Conformal atlasCanonical Conformal atlas

z

z Intrinsic parameterization, independent of Intrinsic parameterization, independent of

triangulations, insensitive to resolutions

(94)

Future work

z Improve efficiency

z Canonical parameterization, which is

independent of the choice of the holomorphic one-form

z Improve uniformity of conformal factors z

z Improve efficiencyImprove efficiency

z

z Canonical parameterization, which is Canonical parameterization, which is

independent of the choice of the

holomorphic one

holomorphic one--formform

z

(95)

Thank you !

(96)