Smooth Bijective Maps between Arbitrary Planar Polygons Teseo Schneider University of Lugano joint work with

Kai Hormann

GMP 2015 – Lugano – 2 June 2015

Introduction

! special bivariate interpolation problem !  bijective

!  smooth

!  linear along edges

!  with low distortion

GMP 2015 – Lugano – 2 June 2015

Applications

not b

iject

ive

bije

ctiv

e

image warping

GMP 2015 – Lugano – 2 June 2015

Applications

shape deformation

GMP 2015 – Lugano – 2 June 2015

Applications

surface cross

parameterization

f

g = °1 ± f ± °0¡1

°0 °1

GMP 2015 – Lugano – 2 June 2015

Harmonic map

! harmonic map ' : ­ ! £

!  ¢' = 0

!  '|@­ = b

! Radó–Kneser–Choquet theorem !  ' is bijective if £ is convex

'

£ ­

GMP 2015 – Lugano – 2 June 2015

Smooth Bijective Maps

­0

f = '1¡1 ± '0

'0 '1

­1

£

GMP 2015 – Lugano – 2 June 2015

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

�d

i

Z

@⌦G

x

B

i

� c

i

Z

@⌦

@G

x

@n

B

i

�

'̃ ⇡mX

i=1

ciBi

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 MFS BEM

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 MFS BEM

GMP 2015 – Lugano – 2 June 2015

Solving the Laplace Equation

smooth meshless exact on the boundary

precise near the boundary fast

FEM " " # # #

BEM # # " # "

MFS # # " " #

MFS FEM BEM

GMP 2015 – Lugano – 2 June 2015

Solving the Laplace Equation

smooth meshless exact on the boundary

precise near the boundary fast

FEM " " # # #

BEM # # " # "

MFS # # " " #

MFS FEM BEM

GMP 2015 – Lugano – 2 June 2015

Solving the Laplace Equation

smooth meshless exact on the boundary

precise near the boundary fast

FEM " " # # #

BEM # # " # "

MFS # # " " #

MFS FEM BEM

big sparse linear system

boundary integrals and small dense linear system

small dense linear system

GMP 2015 – Lugano – 2 June 2015

Inverting '1 – Piecewise Linear

'1

'0

­0 ­1

£

GMP 2015 – Lugano – 2 June 2015

Inverting '1 – Piecewise Linear

'1

­0 ­1

£

GMP 2015 – Lugano – 2 June 2015

'1

Inverting '1 – Piecewise Linear

'0

­0 ­1

£

GMP 2015 – Lugano – 2 June 2015

Inverting '1 – Optimization

'1

'0

miny2⌦1

kx� '1(y)k2

x

­0 ­1

£

GMP 2015 – Lugano – 2 June 2015

Inverting '1

Piecewise linear Optimization

GMP 2015 – Lugano – 2 June 2015

Comparison

composite mean value maps smooth bijective maps

source

[Schneider et al. 2013]

GMP 2015 – Lugano – 2 June 2015

Comparison

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

GMP 2015 – Lugano – 2 June 2015

Extensions

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

£

GMP 2015 – Lugano – 2 June 2015

Extensions

linear boundary conditions quadratic boundary conditions

GMP 2015 – Lugano – 2 June 2015

Extensions

Dirichlet BC Neumann BC

£

GMP 2015 – Lugano – 2 June 2015

Thanks for you attention