Random Walks for Vector Field Denoising

João Paixão, Marcos Lage, Fabiano Petronetto, Alex Laier, Sinésio Pesco, Geovan Tavares,

Thomas Lewiner, Hélio LopesMatmidia Laboratory – Department of Mathematics

PUC–Rio – Rio de Janeiro, Brazil

MotivationVector Fields inScience and Engineering

Flow in an artificial heart

Flow patterns in a tubeUniversity of Cambridge (2009)

MotivationNoise 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

LimitationsFeature Destruction

Gaussian FilteringOriginal Original + Noise

LimitationsFeature Destruction

Random Walks on the Graph

Feature

Previous WorkSmolka et al. 2001 Random Walk for Image

Enhancement

Previous Work Sun et al. 2007 Mesh Denoising

Random Walks for Vector FieldsWhat we want-Meshless-Feature-preserving

What do we need- Graph- Probabilities that avoid crossing features

How to build the graph

Feature Functions

||||)(

)(

i

i

viF

iF

Direction

Magnitude

i

Feature Functions

||||)(

)(

i

i

viF

iF

Direction

Magnitude

Other feature functions in the paper!

i

Probabilities

is the neighborhood of vector i.

otherwise0

)( if22

2

21

2

2

)||()(||(2

)||()(||(

,iNjeCep V

jFiFjXiX

ji

)(iNV

3

4

2

14,1p

3,1p

2,1p

Probability from vector i to vector j

Time to walk

A

B

Time to walk

A

B

Time to walk

A

B

Time to walk

A

B

Time to walk

A

B

Time to walk

- the probability of going from node A to node B after n stepsnBAp ,

A

B

Random Walk Filtering

Weighted Average of Random Walk Probabilities

Fjj

njii vpv ,

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

Gaussian Random Walk

Particle Image Velocimetry

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