Wiener Subdivision

Presented by Koray KAVUKCUOGLU

Geometric Modeling

Spring 2004

May 05, 2004 2

• Introduction– Concepts

• Wiener Filtering– Theory

• Wiener Subdivision– Midpoint Subdivision– Application of Filter– Parameters

• Results

Outline

May 05, 2004 3

aim

– Derive and Implement a subdivison scheme

Based on Marc Alexa’s Wiener Filtering of Meshes

methodology

– Midpoint Linear Subdivision

– Create refined mesh

– Wiener Filtering

– Relocate vertices to obtain a smooth surface

Introduction

May 05, 2004 4

– Filtering of Irregular Meshes using Wiener Filter

– Recovering original smooth geometry from noisy data

Wiener Filtering

May 05, 2004 5

Mesh

– Triangular domain (K,V)

connectivity info vertices in R3

–Topological Distance ()

0

( , ) 1 { , }

min( ( , ) 1) { , }

i j

i j i j K

i k k j K

Wiener Filtering - Theory

May 05, 2004 6

– Neighborhood Definition

m-ring neighborhood

( ) { | ( , ) }mN i j i j m

Collection of rings, with radius up to m

– Expectation

linear operator

( ) ( ) ( )E a b E a E b

– Correlation

( ( , )) ( )C d a b E ab

Distance between two vertices

Wiener Filtering - Theory

May 05, 2004 7

Representation of Vertex Locations

i i iv v r

vertex position in noisy mesh

true vertex position random noise contribution

Estimate each point as a linear sum of given noisy points

i ij j ij

v a v

Find coefficients that minimize square of discrepancy

Wiener Filtering - Theory

May 05, 2004 8

Wiener Filtering - Theory

Linear System

i i iC a b

{ ( ( , ))}i j kC C d v v { ( ( , ))}i i jb C d v v

Solution of this

system gives,

coefficients aij

Need to define distance and correlation functions

i

1

2

dd

d

May 05, 2004 9

Wiener Subdivision

development environment

– Language C++– Mesh format GTS– Windows XP– Cygwin

external libs / tools

– TNT (template numerical toolkit)

Supersedes Lapack++– Jama/C++ (uses TNT - linear system solution)– Mesh Viewer for visualization

May 05, 2004 10

Wiener Subdivision

mesh data structure

– Tree each triangle

divided into 4 childs

– Triangles

– Edge Sharing

May 05, 2004 11

Wiener Subdivisionmesh refinement

– Linear midpoint subdivision

May 05, 2004 12

Wiener Subdivision

filtering– computing Topology

– compute m-ring neighborhood

BFS over vertices– compute distance and correlation

1xe

a bv v

x is parameterized

for smoothness control

May 05, 2004 13

filtering– solve linear system– LU decomposition method

– Jama/C++

Wiener Subdivision

May 05, 2004 14

Wiener Subdivision

parameters– size of m-ring neighborhood (1, 2, …) <-m>– smoothness parameter <-sp>– fraction of old vertex location in new location <-p>

May 05, 2004 15

results

-m1 / -n3 / -sp2

May 05, 2004 16

results

-m1 / -n3 / -sp0 -m2 / -n3 / -sp2

May 05, 2004 17

results

-m1 / -n3 / -sp0

-m2 / -n3 / -sp2

-m1 / -n3 / -sp2

May 05, 2004 18

results

-m1 / -n3 / -sp2 -m1 / -n3 / -sp2 / -p0.3

May 05, 2004 19

results

-m1 / -n3 / -sp2 / -p0.3-m2 / -n3 / -sp2

May 05, 2004 20

results

May 05, 2004 21

results

May 05, 2004 22

Questions?