Mapping from one planar polygonal domain to another is a fundamental problem in computer graphics and geometric modelling. exploiting the properties of harmonic maps, we define smooth and bijective maps with prescribed behaviour along the domain boundary. these maps can be approximated in different ways, and we discuss the respective advantages and disadvantages. we further present a simple procedure for reducing their distortion and demonstrate the effectiveness of our approach by providing examples of applications in image warping and surface cross-parameterization. moreover, we briefly discuss the extension of our construction to 3d and its application to volumetric shape deformation.

1. Smooth Bijective Maps between
Arbitrary Planar Polygons
Teseo Schneider
University of Lugano
joint work with
Kai Hormann

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Introduction
! special bivariate interpolation problem
! bijective
! smooth
! linear along edges
! with low distortion

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Applications
notbijectivebijective
image warping

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Applications
shape deformation

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Applications
surface cross
parameterization
f
g = °1 ± f ± °0
¡1
°0
°1

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Harmonic map
! harmonic map ' : ­ ! £
! ¢' = 0
! '|@­ = b
! Radó–Kneser–Choquet theorem
! ' is bijective if £ is convex
'
£­

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Smooth Bijective Maps
­0
f = '1
¡1 ± '0
'0 '1
­1
£

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Solving the Laplace Equation
! FEM – Finite Element Method
! BEM – Boundary Element Method
! MFS – Method of Fundamental Solutions
[Strang and Fix 2008]
[Hall 1994]
[Fairweather and Karageorghis 1998]˜' ⇡
mX
i=1
wiGsi
+ A
˜' ⇡
1
!(x)
mX
i=1
di
Z
@⌦
GxBi ci
Z
@⌦
@Gx
@n
Bi
˜' ⇡
mX
i=1
ciBi

9. GMP 2015 – Lugano – 2 June 2015
Solving the Laplace Equation
smooth meshless
exact on the
boundary
precise near
the boundary
fast
FEM " " # # #
BEM # # " # "
MFS # # " " #
FEM MFSBEM

10. GMP 2015 – Lugano – 2 June 2015
Solving the Laplace Equation
smooth meshless
exact on the
boundary
precise near
the boundary
fast
FEM " " # # #
BEM # # " # "
MFS # # " " #
FEM MFSBEM

11. GMP 2015 – Lugano – 2 June 2015
Solving the Laplace Equation
smooth meshless
exact on the
boundary
precise near
the boundary
fast
FEM " " # # #
BEM # # " # "
MFS # # " " #
MFSFEM BEM

12. GMP 2015 – Lugano – 2 June 2015
Solving the Laplace Equation
smooth meshless
exact on the
boundary
precise near
the boundary
fast
FEM " " # # #
BEM # # " # "
MFS # # " " #
MFSFEM BEM

13. GMP 2015 – Lugano – 2 June 2015
Solving the Laplace Equation
smooth meshless
exact on the
boundary
precise near
the boundary
fast
FEM " " # # #
BEM # # " # "
MFS # # " " #
MFSFEM BEM
big sparse
linear system
boundary
integrals and
small dense
linear system
small dense
linear system

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Inverting '1 – Piecewise Linear
'1
'0
­0 ­1
£

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Inverting '1 – Piecewise Linear
'1
­0 ­1
£

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'1
Inverting '1 – Piecewise Linear
'0
­0 ­1
£

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Inverting '1 – Optimization
'1
'0
min
y2⌦1
kx '1(y)k2
x
­0 ­1
£

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Inverting '1
Piecewise linear Optimization

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Comparison
composite mean value mapssmooth bijective maps
source
[Schneider et al. 2013]

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Comparison
composite mean value maps, conformal distortion 203.77smooth bijective maps, conformal distortion 7.03

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Extensions
irregular intermediate polygon, isometric distortion 3.02optimized intermediate polygon, isometric distortion 2.49
£

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Extensions
linear boundary conditionsquadratic boundary conditions

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Extensions
Dirichlet BC Neumann BC
£

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Thanks for you attention