Random Walks for Image segmentation

RANDOM WALKS FOR IMAGE

SEGMENTATION

IEEE Transaction on pattern analysis and machine intelligence, November 2006

Leo Grady, Member, IEEE

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Outline Introduction Algorithm

Dirichlet Problem Behavioral Properties Result--Demo

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Introduction K-way image segmentation

User-defined seeds user-defined labels

How to labels and unseeded pixel ?

K objects

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Introduction Algorithm is resolving the question:

Given a random walker starting at this location, what is the probability that it first reaches each of the K seed points ?

Probability = ? A random walker first reach each of the K seed points.

K-tuple vector

A pixel Seed point

Probability?

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Introduction

The probability a random walker first reaches a seed point.

The solution to the Dirichelet problem with boundary

conditions at the locations of the seed points and the seed

point in question fixed to unity while the others are set to zero.

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Introduction Goal

1. location of weak (or missing) boundaries. 2. noise robustness. 3. ability to identify multiple objects

simultaneously. 4. fast computation (and editing). 5. avoidance of small/trivial solutions.

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Algorithm 1. generating the graph weights 2. establishing the system of equations

to solve the problem. 3. the practical details of

implementation.

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Defining a graph Graph G = ( V , E ) And edge, e, spanning two vertices, vi

and vj, is denoted by eij. The weight of an edge, eij, is denoted by

w(eij) or wij. Degree of a vertex is Assume this graph is connected and

undirected.

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Edge Weights Gaussian weighting function

The only free parameter

gi indicates the image intensity at pixel i

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Combinatorial Dirichlet Problem The Dirichlet integral

A harmonic function is a function that satisfies the Laplace equation

Dirichlet problem - finding a harmonic function subject to its boundary values.

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Combinatorial Dirichlet Problem(cont.)

Combinatorial Laplacian matrix

Where Lij is indexed by vertices vi and vj. The m x n edge-node incidence matrix

as

Incidence matrix is indexed by edge eij and node vk.

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Combinatorial Dirichlet Problem(cont.)

A combinatorial formulation of the Dirichlet integral

C is the mxm constitutive matrix (the diagonal matrix with the weights of each edge along the diagonal).

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Combinatorial Dirichlet Problem(cont.)

Partition the vertices into two sets, VM (marked/seed nodes) VU (unseeded nodes)

Finding the critical point yields

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Combinatorial Dirichlet Problem(cont.)

The probability (potential) assumed at node, vi, for each label, s, by .

Define the set of labels for the seed points as a function

Define the vector for each label, s, at node as

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Solving the combinatorial Dirichlet problem

For one label

For all labels

X has K columns taken by each and M has colums given by each

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Equivalences between random walks and electrical circuits

Three fundamental equations of circuit theory.

These three equations may be combined into the linear system

It is equivalent to with f = 0.

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Algorithm Summary1. Using , map the

image intensities to edge weights in the lattice.

2. Obtain a set, VM, of marked (labeled) pixels with K labels, either interactively or automatically.

3. Solve outright for the potentials or solve for each label except the final one, f. Set

4. Obtain a final segmentation by assigning to each node, vi, the label corresponding to

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Overview of segmentation computation

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Analogies Assigns an unseeded pixel to a label, given a

weighted graph: If a random walker leaving the pixel is most likely to

first reach a bearing label s, assign the pixel to label s. If the seeds are alternately replaced by grounds/unit

voltage sources, assign the pixel to the label for which its seeds being “on” produces the greatest electrical potential.

Assign the pixel to the label for which its seeds have the largest effective conductance.

If a 2-tree is drawn randomly from the graph, assign the pixel to the label for which the pixel is most likely to remain connected to.

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Effective Conductance Effective conductance

Dirichlet integral equals the effective conductance between nodes labeled “1” (“on”) and those labeled “0” (“off”).

x is intended to include both xM and xU

i j i jequals

Unit voltage

Current flow

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Effective Conductance (cont.) Effective conductance between two nodes, vi,

vj is given by

where T is a set of edges defining a connected tree

the sum is over all possible trees in the graph

Where TT(i, j) is used to represent the set of edges defining a 2-tree, such that node vi is in one component and vj is in another.

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2-tree A 2-tree is defined to be a tree with one

edge removed.

vi and vj are indifferent components and vt is in the same component as vj.

Note that

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2-tree (cont.) Then, the following expressions are

equivalent

The segmentation is computed from the potentials by assigning the pixel to the label for which it has greatest potential (probability).

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Behavioral Properties1. Weak Boundary detection2. Noise robustness3. Assignment of ambiguous regions

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Weak Boundaries

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Weak Boundaries - Comparison

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Noise Robustness

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Ambiguous Unseeded Regions

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Demo Videos http://www.cns.bu.edu/~lgrady/Random_

Walker_Image_Segmentation.html Brain Lung tumor Aorta-3D

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Definition of a harmonic function Any real function u(x, y) with continuous

second partial derivatives which satisfies Laplace's equation, is called a harmonic function.

Reference from Mathworld: http://mathworld.wolfram.com/HarmonicFunction.html

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