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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

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Motivation Vector Fields in Science and Engineering Flow in an artificial heart Flow patterns in a tube University of Cambridge (2009)

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Motivation Noise in vector data-acquisition Flow around a live swimming fish (Yoshida et al 2004)

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Problem

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Problem:Noise Denoising

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Gaussian Filtering E.g. 5x5 Gaussian Filter

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Limitations Feature Destruction

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Limitations Feature Destruction

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Random Walks on the Graph Feature

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Previous Work Smolka et al. 2001 Random Walk for Image Enhancement

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Previous Work Sun et al. 2007 Mesh Denoising

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Random Walks for Vector Fields What we want -Meshless -Feature-preserving What do we need -Graph -Probabilities that avoid crossing features

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How to build the graph

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Feature Functions Direction Magnitude

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Feature Functions Direction Magnitude Other feature functions in the paper!

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Probabilities is the neighborhood of vector i. 3 3 4 4 2 2 1 Probability from vector i to vector j

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Time to walk A B

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A B

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A B

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A B

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A B

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

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Random Walk Filtering Weighted Average of Random Walk Probabilities

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Feature-preserving Discontinuity

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Simple Example

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Granular Flow

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Gaussian FilteringRandom Walk Filtering

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Particle Image Velocimetry

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GaussianRandom Walk Particle Image Velocimetry

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Landslide

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Summary -Feature Preserving -Meshless -Interpretative -Flexible -Easy to implement

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Limitations -Number of parameters -Dependency in them

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Future Works - 3D vector field denoising algorithm

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Thank you for your attention

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