Transcript Random Walks for Image Segmentation

RANDOM WALKS FOR
IMAGE SEGMENTATION
Leo Grady, Member, IEEE
IEEE Transaction on pattern analysis and machine
intelligence, November 2006
Outline
2


Introduction
Algorithm
 Dirichlet


Problem
Behavioral Properties
Result--Demo
Introduction
3

K-way image segmentation
 User-defined
seeds
user-defined labels
K objects
 How
to labels and unseeded pixel ?
Introduction
4

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 ?
K-tuple vector
Probability?
A pixel
 Probability
Seed point
= ? A random walker first reach each of
the K seed points.
Introduction
5
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.
Introduction
6

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.
Algorithm
7



1. generating the graph weights
2. establishing the system of equations to solve the
problem.
3. the practical details of implementation.
Defining a graph
8





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.
Edge Weights
9

Gaussian weighting function
The only free parameter
gi indicates the image
intensity at pixel i
Combinatorial Dirichlet Problem
10



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.
Combinatorial Dirichlet Problem(cont.)
11

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.
Combinatorial Dirichlet Problem(cont.)
12

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).
Combinatorial Dirichlet Problem(cont.)
13

Partition the vertices into two sets,
 VM
(marked/seed nodes)
 VU (unseeded nodes)

Finding the critical point yields
Combinatorial Dirichlet Problem(cont.)
14



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
Solving the combinatorial Dirichlet
problem
15

For one label

For all labels
X
has K columns taken by each
given by each
and M has colums
Equivalences between random walks
and electrical circuits
16


Three fundamental equations of circuit theory.
These three equations may be combined into the
linear system
 It
is equivalent to
with f = 0.
Algorithm Summary
17
1. 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
Overview of segmentation computation
18
Analogies
19

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.

Effective Conductance
20

Effective conductance
i
j
equals
Unit voltage
i
j
Current flow


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
Effective Conductance (cont.)
21

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.
2-tree
22

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
2-tree (cont.)
23


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).
Behavioral Properties
24
1. Weak Boundary detection
2. Noise robustness
3. Assignment of ambiguous regions
Weak Boundaries
25
Weak Boundaries - Comparison
26
Noise Robustness
27
Ambiguous Unseeded Regions
28
Demo Videos
29

http://www.cns.bu.edu/~lgrady/Random_Walker_I
mage_Segmentation.html
 Brain
 Lung
tumor
 Aorta-3D
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Definition of a harmonic function
37

Any real function u(x, y) with continuous second
partial derivatives which satisfies Laplace's
equation,
is called a harmonic
function.
Back
Reference from Mathworld: http://mathworld.wolfram.com/HarmonicFunction.html