Random Walk with Restart (RWR) for Image Segmentation

Sungsu Lim

AALAB, KAIST

Image Segmentation Computer vision: make machine to see or to under-

stand/ interpret the scenes (images & videos) like human do.

Image segmentation is one of the most challenging issues in computer vision.

Two major difficulties of conventional algorithms: weak boundary problem & texture problem.

Semi-supervised segmentation approaches are pre-ferred since user inputs can reduce the ambiguity in images.2011-02-08 RWR for image segmentation 2

Random Walks for Image Segmenta-tion RW (L. Grady, PAMI2006): In image segmentation,

random walks are used to determine the labels (i.e., “object” or “background”) to associate with each pixel.

K-way image segmentation: given user-defined seeds indicating regions of the image belonging to k ob-jects. Each seed specifies a location with a user-de-fined label.

We can use hitting time or commute time as rele-vance score between two nodes (seed and unlabeled pixel). By assigning each pixel to the label for which the best value is calculated.2011-02-08 RWR for image segmentation 3

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Random walk with restart

Example of RW

What if we start at a different node?

Start node

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RWR for Classification

RW with start nodes being

labeled points in class A

RW with start nodes being

labeled points in class B

Nodes frequented more by RW(A) belongs to class A,

otherwise they belong to B

Simple idea: use RW for classification

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RWR for Image Segmentation Limitation of RW: only considers the local rela-

tionship between the pixel and that border. (more prone to hit popular nodes)

RWR (Kim, Lee and Lee, ECCV2008): a new generative image segmentation algorithm based on Random Walks with Restart (Pan,Yang and Faloutsos, KDD2004)

Most previous semi-supervised image seg-mentation algorithms focus on the inter-label discrimination, but it introduce a generative model for image segmentation.2011-02-08 RWR for image segmentation 7

Generative Model

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Random Walk with Restart

Imagine a network, and starting at a specific node, you follow the edges randomly.

But (perhaps you’re afraid of wondering too far) with some probability, you “jump” back to the starting node (restart!).

If you record the number of times you land on each

node, what would that dis-tribution look like?

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Random Walk with Restart The walk distribution r satisfies a simple equa-

tion:

ePππ cc )1(

Transition ma-trix (relevance

vector)

Seed vec-tor

(start nodes)

“Keep-go-ing” proba-

bility (damp-ing factor)

Restart probabil-

ity

Equivalent to the well-known Google PageR-ank if all nodes

are start nodes! (e is

uniform)Rank-

ing vector

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0.13 0 1/3 1/3 1/3 0 0 0 0 0 0 0 0

0.10 1/3 0 1/3 0 0 0 0 1/4 0 0 0

0.13

0.22

0.13

0.050.9

0.05

0.08

0.04

0.03

0.04

0.02

0

1/3 1/3 0 1/3 0 0 0 0 0 0 0 0

1/3 0 1/3 0 1/4 0 0 0 0 0 0 0

0 0 0 1/3 0 1/2 1/2 1/4 0 0 0 0

0 0 0 0 1/4 0 1/2 0 0 0 0 0

0 0 0 0 1/4 1/2 0 0 0 0 0 0

0 1/3 0 0 1/4 0 0 0 1/2 0 1/3 0

0 0 0 0 0 0 0 1/4 0 1/3 0 0

0 0 0 0 0 0 0 0 1/2 0 1/3 1/2

0 0 0 0 0 0 0 1/4 0 1/3 0 1/2

0 0 0 0 0 0 0 0 0 1/3 1/3 0

0.13 0

0.10 0

0.13 0

0.22

0.13 0

0.05 00.1

0.05 0

0.08 0

0.04 0

0.03 0

0.04 0

2 0

1

0.0

n x n n x 1n x 1

Example of RWRIterative update until convergence

ePππ cc tt 1)1(

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Use of RWR Linear solution:

It can be reformulated as

( )

It considers all relations at all scales in the image.

ePπ 1))1(( cIc

0

)1(t

ttcc ePπ 10 c

As t increases weight becomes

smaller.

Weighted av-erage of all probability

Restart probabil-

ity

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Use of RWR

As restarting probability c decreases, coarser scale is more emphasized in likelihood term.

0

)1(t

ttcc ePπ

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Energy minimization framework Quadratic energy (cost) minimization:

similar to the formulation of RWR

( )

)(minarg* πππ

E

i

iji

jiijPE 2

,

2 )()()( eππππ

0eπPπππ

)(E

ePππ cc )1(

1

c

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Experimental Results

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Applications 1. Data-Driven RWR (Kim, Lee and Lee,

ICIP2009)It use the restart probability matrix. The restarting probability of each pixel depend on its edgeness, generated by Canny edge detector.

2. High-order RWR (multi-layer graph model)

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