Anisotropic Mesh Filtering

Anisotropic Mesh Filtering
Mesh Processing

November 19, 2012

Table of Contents

Motivation

Signal Processing on Meshes

Linear Anisotropic Mesh Filtering

Bilateral Mesh Smoothing

Table of Contents

Motivation

Signal Processing on Meshes

Linear Anisotropic Mesh Filtering

Bilateral Mesh Smoothing

Main Idea
Polygonal Mesh
Connectivity Vertex positions Optionalproperties

Main Idea
Polygonal Mesh
Connectivity Vertex positions Optional properties

Figure: Convolution filtering with a user-defined transfer function.

Smoothing

Figure: Noise removed by low-pass filtering of the surface geometry.

Compression

Figure: Horse model reconstructed using a few spectral coefficients derived from the graph Laplacian.

Others

ResamplingWatermarking

Table of Contents

Motivation

Signal Processing on Meshes

Linear Anisotropic Mesh Filtering

Bilateral Mesh Smoothing

Introduction

Figure: Seahorse shape depicted by a closed contour (left). Smoothed version (right)

Laplacian Smoothing

Midpoint smoothing:

ˆ vi =

1 1 2 2

(vi −1 + vi ) +

1 1 2 2

ˆ (vi + vi +1 ) = vi = vi −1 + vi + vi +1
4 2 4

11

1

Laplacian Smoothing

Midpoint smoothing:

ˆ vi =

1 1 2 2

(vi −1 + vi ) +

1 1 2 2

ˆ (vi + vi +1 ) = vi = vi −1 + vi + vi +1
4 2 4

1

1

1

1D discrete Laplacian:

δ(vi ) = (vi −1 + vi +1 ) − vi
2

1

Signal Representation
V : coordinate vertices. n × 2 vi : coordinate of a vertex. X : x component of V. Treated as a 1D signal.

Signal RepresentationV : coordinate vertices. n × 2 vi : coordinate of a vertex. X : x component of V. Treated as a 1D signal.

Figure: The x-component of the contour viewed as a 1D periodic signal (left). 1D plot for the seahorse (right).

Laplacian Smoothing in Matrix Form

ˆ New contour X from Laplacian smoothing:  1 1    1 ˆ x1 x1 0 ... ... 0 2 4 4 1  x2   1 1 0 . . . . . . 0   x2  2   ˆ  42   .  .  .  . . . . . ˆ = . =. .  = SX . . . . . X  . . . . . .   .  .  1 1 1 xn−1   0 . . . . . . 0 ˆ xn−1  4 2 2 1 1 1 ˆ xn 0 ... ... 0 xn 4 4 2

Laplacian Smoothing in Matrix Form
Discrete 1D Laplacian for vertices:  1 −1 0 ... 2 − 1 1 − 1 0 2  2  . . . . . . . δ(X ) = LX =  . . . .  .  0 ... ... 0 1 −2 0 ... ...

... ...
. . .
1 −2

0

−1 2

...
. . . 1

0

−1 2

 X  1 −
2

0  

1

Laplacian Smoothing in Matrix Form
Discrete 1D Laplacian for vertices:  1 −1 0 ... 2 − 1 1 − 1 0 2  2  . . . . . . . δ(X ) = LX =  . . . .  .  0 ... ... 0 1 −2 0 ... ...

... ...
. . .
1 −2

0

−1 2



...
. . . 1

0

−1 2

 X  1 −
2

0  

1

Relation between the smoothing operator S and Laplaceoperator L:
1 S =I− L 2

Spectral Analysis
L is symmetric: real eigenvalues and a set of real and orthogonal set of eigenvectors. Any vector of size n can be expressed as a linear sum of these basis.

Spectral Analysis
L is symmetric: real eigenvalues and a set of real and orthogonal set of eigenvectors. Any vector of size n can be expressed as a linear sum of these basis.

X can beexpressed as linear sum of the eigenvectors:

E11 E1n E11 n E21  E2n  E11      ˜ ˜ ˜ X= ei xi =  .  x1 + · · · +  .  xn =  .  .   .   . . . . i =1 En1 Enn E11











... ...
. . .

...

˜  x1 E1n  x2  ˜  E1n    .  ˜ .   .  = EX .  .  . x  ˜n−1 E1n ˜ xn  

˜ X = ET X ˜ xi = eT .X i

DFT-like spectral transform
˜ X →X

Plot ofeigenvectors

Figure: First 8 eigenvectors of the 1D Laplacian: More oscillation as eigenvalues (frequencies) increase

Laplacian Smoothing as Filtering

S = I − 1L 2

Laplacian Smoothing as Filtering

S = I − 1L 2 Repeated application of S:
X (m ) = S m X =
n m

1 I− L 2
n

X=
m

i =1

1 I− L 2

m

˜ ei xi =
i =1

1 e 1 − λi 2

˜ xi

Laplacian Smoothing as...