interpolation via barycentric coordinates · • moving least squares coordinates [manson and...

Interpolation viaBarycentric Coordinates

Pierre Alliez

Inria

Some material from D. Anisimov

Outline

• Barycenter• Convexity• Barycentric coordinates

– For Simplices– For point sets

• Inverse distance (Shepard) • Delaunay / Voronoi• Natural neighbors (Sibson)

– For convex polyhedra– Generalized barycentric coordinates– Applications

BARYCENTER

Barycenter

• Law of lever:

Center of mass

Barycentric coordinates

Opposite problem?

v1 v2p

?

Opposite problem

p

l2/l l1/l unique!

v1 v2l2l1

CONVEXITY

Convexity

A set S is convex if any

pair of points p,q S

satisfy pq S.

p

q

non-convex

q

p

convex

p

q

non-convex

q

p

convex

CH(S)

Convex Hull

p

q

non-convex

q

p

convex

•The convex hull of a set S is:• The minimal convex set that

contains S, i.e. any convex set C

such that S C satisfies CH(S) C.

• The intersection of all convex sets

that contain S.

• The set of all convex combinations

of piS, i.e. all points of the form:S

p

q

non-convex

q

p

convex

1 1

, 0, 1n n

i i i i

i i

p

CH(S)

Convex Hulls

•The convex hull of a set is unique, up to

collinearities.

•The boundary of the convex hull of a point set

is a polygon on a subset of the points.

S

BARYCENTRIC COORDINATES

FOR SIMPLICES

Triangle barycentric coordinates

A. F. Möbius

[1790−1868]

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Properties:


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Data interpolation

Barycentric Coordinates for Simplices

…


FOR POINT SETS

Point Set

Barycentric Coordinates?

Weights such that it is a barycenter of the point set?

Barycentric Coordinates?

Weights such that it is a barycenter of the point set?

A. F. Möbius

[1790−1868]

Weights always exist if #points >= dimension

Inverse Distance [Shepard]

Natural Neighbor Coordinates [Sibson]

Voronoi Diagram

Voronoi Diagram• The collection of the non-empty Voronoi regions and their faces, together with their

incidence relations, constitute a cell complex called the Voronoi diagram of E.

• The locus of points which are equidistant to two sites pi and pj is called a bisector, all bisectors being affine subspaces of IRd (lines in 2D).

demo

Sibson Coordinates

Sibson Coordinates

• Well define over convex hull• Local support• Satisfy Lagrange property• C1 continuity, except at points pi (only C

0)

GENERALIZED


Quadrilateral?

Bilinear interpolation

Quadrilateral

Bilinear map on unit square

Quadrilateral

Image of bilinear map on unit square

Unified Formula! [Floater]

signed area

Generalized barycentric coordinates

Properties:

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Interpolation

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Different weights -> different coordinate functions

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• Wachspress coordinates [Wachspress, 1975]

• Discrete harmonic coordinates [Pinkall and Polthier, 1993]

• Mean value coordinates [Floater, 2003]

• Metric coordinates [Malsch et al., 2005]

• Harmonic coordinates [Joshi et al., 2007]

• Maximum entropy coordinates [Hormann and Sukumar, 2008]

• Complex barycentric coordinates [Weber et al., 2009]

• Moving least squares coordinates [Manson and Schaefer, 2010]

• Cubic mean value coordinates [Li and Hu, 2013]

• Poisson coordinates [Li et al., 2013]

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• Three-point coordinates [Floater et al., 2006]

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Three-point coordinates

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Wachspress Mean value Discrete harmonic

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Wachspress Mean value Discrete harmonic

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Discrete harmonic


Wachspress Mean value

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Mean value

Mean value coordinates

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Mean value

Mean value coordinates

15/36

Positive mean value coordinates [Lipman et al., 2007]












16/36

Metric coordinates

17/36

Metric coordinates

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Metric coordinates

19/36

Metric coordinates

20/36












21/36

Harmonic coordinates

No closed form!22/36

Biharmonic vs Harmonic

Biharmonic Harmonic

[Weber et al., 2012]

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28/36

Barycentric mapping

Source polygon Target polygon

29/36

Complex barycentric mapping

Source polygon Target polygon

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Complex barycentric coordinates

Induce conformal mappings

HarmonicGreen

Three-point coordinates are generalized to complex three-point coordinates

Green coordinates are members of complex three-point coordinates

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[Lipman et al., 2008]

APPLICATIONS

Editing

Image warping

Original image Warped imageMask

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Principal Component Analysis

Linear Least Squares Approximation

Pierre AlliezInria Sophia Antipolis

64

Definition (point set case)

Given a point set x1, x2, …, xn Rd, linear least squares

fitting amounts to find the linear sub-space of Rd

which

minimizes the sum of squared distances from the points

to their projection onto this linear sub-space.

65

Definition (point set case)

This problem is equivalent to search for the linear sub-

space which maximizes the variance of projected

points, the latter being obtained by eigen decomposition

of the covariance (scatter) matrix.

Eigenvectors corresponding to large eigenvalues are the

directions in which the data has strong component, or

equivalently large variance. If eigenvalues are the same

there is no preferable sub-space.

66

PCA finds an orthogonal basis that best represents given data set.

The sum of distances2 from the x’ axis is minimized.

PCA – the general idea

x

y

x’

y’

67

PCA – the general idea

PCA finds an orthogonal basis that best represents given data set.

PCA finds a best approximating plane (in terms of distances2)

3D point set

x y

z

68

Notations

Denote our data points by x1, x2, …, xn Rd

11 1

1 2

22 2

1 2

1 2

, , ,

n

n

dd d

n

xx x

xx x

xx x

1 2 dx x xK

MM M

69

The origin is zero-order approximation of our data set (a

point)

It will be the center of mass:

It can be shown that:

The origin of the new axes

1

1

n

in

i

m x

2

1

argminn

i

i

x

m x - x

70

Scatter matrix

Denote yi = xi – m, i = 1, 2, …, n

where Y is dn matrix with yk as columns (k = 1, 2, …, n)

TS YY

1 1 1 1 2

1 2 1 1 1

2 2 2 1 2

1 2 2 2 2

1 2

1 2

d

n

d

n

d d d d

n n n n

y y y y y y

y y y y y yS

y y y y y y

L L

L

M M M M M

L L

Y YT

71

Variance of projected points

In a way, S measures variance (= scatterness) of the data in different directions.

Let’s look at a line L through the center of mass m, and project our points xi onto it.

The variance of the projected points x’i is:

Original set Small variance Large variance

21

1

var( ) || ||n

in

i

L

x m

L L L L

72


Given a direction v, ||v|| = 1, the projection of xi onto L = m + vt

is:

|| || , || || ,T

i

i i ix m v x m v v y v y

vm

xi

x’iL

73


So,

2 2 21 1 1

1 1

21 1 1 1 1

var( ) || || ( ) || ||

|| || , ,

n n

n n n

i i

T T T

n n n n n

L Y

Y Y Y YY S S

T T

i i

T T T

x - m v y v

v v v v v v v v v

2 21 1 1 1

1 2

2 2 2 2

2 1 2 1 2 21 2

1 1

1 2

( ) || ||

i n

n nT d di n

i i

d d d d

i n

y y y y

y y y yv v v v v v Y

y y y y

T

iv y v

L

L LM M M M

L

74

Directions of maximal variance

So, we have: var(L) =

Theorem:

Let f : {v Rd

| ||v|| = 1} R,

f (v) = (and S is a symmetric matrix).

Then, the extrema of f are attained at the eigenvectors of S.

So, eigenvectors of S are directions of maximal/minimal variance.

75

Summary so far

We take the centered data points y1, y2, …, yn Rd

Construct the scatter matrix

S measures the variance of the data points

Eigenvectors of S are directions of max/min variance.

TS YY

76

Scatter matrix - eigendecomposition

S is symmetric

S has eigendecomposition: S = VVT

S = v2v1 vd

12

d

v2

v1

vd

The eigenvectors form

orthogonal basis

77

Principal components

Eigenvectors that correspond to big eigenvalues are the

directions in which the data has strong components (=

large variance).

If the eigenvalues are more or less the same – there is no

preferable direction.

78

Principal components

There’s no preferable

direction

S looks like this:

Any vector is an

eigenvector

There is a clear preferable

direction

S looks like this:

is close to zero, much

smaller than .

TVV

79

For approximation

x

y

v1

v2

x

y

This line segment approximates

the original data set

The projected data set

approximates the original data

set

x

y

80

For approximation

In general dimension d, the eigenvalues are sorted in

descending order:

1 2 … d The eigenvectors are sorted accordingly.

To get an approximation of dimension d’ < d, we take the d’

first eigenvectors and look at the subspace they span (d’ =

1 is a line, d’ = 2 is a plane…)

81

For approximation

To get an approximating set, we project the original data

points onto the chosen subspace:

xi = m + 1v1 + 2v2 +…+ d’vd’ +…+dvdProjection:

xi’ = m + 1v1 + 2v2 +…+ d’vd’ +0vd’+1+…+ 0 vd

82

Optimality of approximation

The approximation is optimal in least-squares sense. It

gives the minimal of:

The projected points have maximal variance.

2

1

n

k k

k

x x

Original set projection on arbitrary line projection on v1 axis

PCA on Point Sets

83

1 1 1 1 2

1 2 1 1 1

2 2 2 1 2

1 2 2 2 2

1 2

1 2

d

n

d

n

d d d d

n n n n

y y y y y y

y y y y y yS

y y y y y y

L L

L

M M M M M

L L

demo

demos/pca_2d.exe

PCA on Geometric Primitives?

84

Coordinate relative to center of mass

interpolation via barycentric coordinates · • moving least squares coordinates [manson and...

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