[PPT] - Generalized Barycentric Coordinates Kai Hormann Faculty of PowerPoint Presentation

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Generalized Barycentric Coordinates

Kai Hormann

Faculty of Informatics Università della Svizzera italiana Lugano Università della Svizzera italiana, Lugano

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Cartesian coordinates

point (2,2) with x-coordinate: 2 di 2

y

2 3 (2,2)

y-coordinate: 2 mathematically: (2 2) 2 (1 0)

x

1 2 3 –3 –2 –1 1 –1 (0,0) (–3,1)

(2,2) = 2 · (1,0) + 2 · (0,1) in general:

–3 –2 (1,–2)

(x,y) = x · (1,0) + y · (0,1)

René Descartes (1596–1650)

x- and y-coordinates w.r.t. base points (1,0) and (0,1)

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(1,0) and (0,1)

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Barycentric coordinates

(1 0 0)

point (a,b,c) with 3 coordinates w.r.t. base points A B C

(1,0,0) (0.5,0.5,0) (0,1,0)

base points A, B, C mathematically: (a b c) = a · A

(0.25,0.25,0.5)

(a,b,c) = a · A + b · B + c · C where

(0,0,1) (0.25,–0.25,1)

where A = (1,0,0) B = (0,1,0) ( )

August Ferdinand Möbius (1790–1868)

C = (0,0,1) and a + b + c = 1

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Barycentric coordinates

system of masses at positions position of the system’s barycentre:

position of the system s barycentre:

are the barycentric coordinates of

not unique at least

points needed to span

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Barycentric coordinates

Theorem [Möbius, 1827] :

The barycentric coordinates

f

with The barycentric coordinates of with respect to are unique up to a common factor

example:

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Barycentric coordinates for triangles

normalized barycentric coordinates properties

properties

partition of unity reproduction ep oduct o positivity Lagrange property g g p p y

application

linear interpolation of data

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linear interpolation of data

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Generalized barycentric coordinates

finite-element-method with polygonal elements

convex

[Wachspress 1975]

convex

[Wachspress 1975]

weakly convex

[Malsch & Dasgupta 2004]

arbitrary

[Sukumar & Malsch 2006]

arbitrary

[Sukumar & Malsch 2006]

interpolation of scattered data interpolation of scattered data

natural neighbour interpolants

[Sibson 1980]

f h h d

– " –
f higher order

[Hiyoshi & Sugihara 2000]

Dirichlet tessellations

[Farin 1990]

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Generalized barycentric coordinates

parameterization of piecewise linear surfaces

shape preserving coordinates

[Floater 1997]

shape preserving coordinates

[Floater 1997]

discrete harmonic (DH) coordinates

[Eck et al. 1995]

mean value (MV) coordinates

[Floater 2003]

mean value (MV) coordinates

[Floater 2003]

ther applications
ther applications

discrete minimal surfaces

[Pinkall & Polthier 1993]

l l colour interpolation

[Meyer et al. 2002]

boundary value problems

[Belyaev 2006]

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Arbitrary polygons

barycentric coordinates normalized coordinates normalized coordinates properties

linear precision

partition of unity reproduction

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for all

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Convex polygons

[Floater, H. & Kós 2006]

Theorem: If all , then

positivity

[ , ]

positivity Lagrange property linear along boundary linear along boundary

application application

interpolation of data given at the vertices d h h ll f h

inside the convex hull of the

direct and efficient evaluation

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Examples

Wachspress (WP) coordinates mean value (MV) coordinates discrete harmonic (DH) coordinates discrete harmonic (DH) coordinates

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Normal form

[Floater, H. & Kós 2006]

Theorem: All barycentric coordinates can be written as

[ , ]

with certain real functions

three-point coordinates

with

Theorem: Such a generating function

g g exists for all three point coordinates

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exists for all three-point coordinates

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Three-point coordinates

Theorem:

if and only if is

positive positive monotonic convex convex sub-linear

examples

WP coordinates MV coordinates DH coordinates

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Non-convex polygons

Wachspress mean value discrete harmonic

poles, if

, because

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Star-shaped polygons

Theorem:

if and only if is

positive positive super-linear

examples

MV coordinates DH coordinates

Th f if i

Theorem:

for some if is

strictly super-linear

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Mean value coordinates

[H. & Floater 2006]

Theorem: MV coordinates have no poles in

[ ]

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Mean value coordinates

properties

well-defined everywhere in Lagrange property linear along boundary g y linear precision for smoothness at , otherwise , similarity invariance for

application

direct interpolation of data

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Implementation

Mean Value coordinates

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Implementation

efficient and robust evaluation of the function

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Colour interpolation

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Vector fields

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Smooth shading

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Rendering of quadrilateral elements

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Transfinite interpolation

mean value coordinates radial basis functions

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Smooth distance function

Function approximates the distance function

and along the boundary

smooth, except at the vertices

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Mesh animation

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Image warping

riginal image

warped image mask

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Mesh warping

MV coordinates in 3D

[Ju et al. 2005] negative inside the domain

positive MV coordinates

[Lipman et al. 2007]

MVC PMVC

nly C0-continuous

no closed form

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MVC PMVC

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Harmonic coordinates

define normalized coordinate as solution of PDE

subject to subject to

Lagrange property

well-defined smooth

linear precision

positivity efficient

animation for Ratatouille

[Joshi et al. 2007]

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Positive barycentric coordinates

drawbacks so far … mean value coordinates mean value coordinates

negative

positive mean value coordinates

not smooth (only C0) ( y )

harmonic coordinates

rather expensive to compute not smooth in practice

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Maximum entropy coordinates

[H. & Sukumar 2008]

based on maximizing the Shannon-Jaynes entropy Lagrange property well defined smooth ()

[ ]

Lagrange property well-defined smooth () linear precision

positivity efficient ()

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