Random Walks for
Vector Field Denoising
João Paixão, Marcos Lage, Fabiano Petronetto,
Alex Laier, Sinésio Pesco, Geovan Tavares,
Thomas Lewiner, Hélio Lopes
Matmidia Laboratory – Department of Mathematics
PUC–Rio – Rio de Janeiro, Brazil
Motivation
Vector Fields in
Science and Engineering
Flow in an artificial heart
Flow patterns in a tube
University of Cambridge (2009)
Motivation
Noise in vector data-acquisition
Flow around a live swimming fish (Yoshida et al 2004)
Problem
Problem:Noise
Denoising
Gaussian Filtering
E.g. 5x5 Gaussian Filter
Limitations
Feature Destruction
Original
Original + Noise
Gaussian Filtering
Limitations
Feature Destruction
Random Walks on the Graph
Feature
Previous Work
Smolka et al. 2001 Random Walk for Image
Enhancement
Previous Work
Sun et al. 2007 Mesh Denoising
Random Walks for Vector Fields
What we want
-Meshless
-Feature-preserving
What do we need
- Graph
- Probabilities that avoid crossing features
How to build the graph
Feature Functions
F (i)   i
Direction
F (i) || vi ||
Magnitude
i
Feature Functions
F (i)   i
Direction
F (i) || vi ||
Magnitude
Other feature functions in the paper!
i
Probabilities
Probability from vector i to vector j
||( F ( i )  F ( j )|| 2
 ||( X (i )2X ( j )||
2 1
Ce
2 22
e
pi , j  
0

2
NV (i)
is the neighborhood of vector i.
2
p1, 2
4
p1, 4
1
p1,3
3
if j  NV (i)
otherwise
Time to walk
B
A
Time to walk
B
A
Time to walk
B
A
Time to walk
B
A
Time to walk
B
A
Time to walk
B
A
pAn ,B
- the probability of going from node A to node B after n steps
Random Walk Filtering


n
vi    pi , j v j 
 jF

Weighted Average of Random Walk Probabilities
Feature-preserving
Discontinuity
Simple Example
Original
Original + Noise
Simple Example
Gaussian
Random Walk
Granular Flow
Granular Flow
Gaussian Filtering
Random Walk Filtering
Particle Image Velocimetry
Particle Image Velocimetry
Gaussian
Random Walk
Landslide
Landslide
Landslide
Landslide
Landslide
Summary
-Feature Preserving
-Meshless
-Interpretative
-Flexible
-Easy to implement
Limitations
-Number of parameters
-Dependency in them
Future Works
- 3D vector field denoising algorithm
Thank you for your attention