random walker segmentation
DESCRIPTION
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Outline
• Problem Description and Basic Idea
• Solution of Random Walk Probability
• Properties and Results
• Relation to Other Methods
• Discussion
2
Problem Description• Interactive Segmentation
• Requirements:▫ fewer interactions
▫ fast computation, fast editing
Partially labeled image Segmented image
3
Previous Methods
• Interactive segmentation
▫ Intelligent Scissors (live wire)
fail in the presence of noise and weak boundary
too much interaction needed for high accuracy
▫ Active Contour
local minimum solution
inconvenient for editing
▫ Graph Cuts
small cut bias
no global optimal solution for multi-way cut
4
Basic Idea
A pixel p belongs to
which class?
A random walker starts
from p, which seeds he
will arrive first ?
Green Red Yellow Blue
The pixel intensity means: starting at this pixel the probability that a random walker will arrive the
target seed before arriving other three seeds
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Why formulate in this way?
1. Fast solution
▫ solving a sparse linear system
▫ ability of multiple segmentation
▫ fast editing
2. Nice properties
▫ location of weak/missing boundaries
▫ noise robustness
▫ avoidance of trivial solutions
6
Algorithm Overview
1. Allocate region seeds si for each region i
2. Calculate ui(x,y):
the probability of first arriving si for a
random walker starting from (x,y)
3. Assign (x,y) to Label k
if uk(x,y) is the largest among ui(x,y)
for i = 1…N
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Outline
• Problem Description and Basic Idea
• Solution of Random Walk Probability
• Properties and Results
• Relation to Other Methods
• Discussion
8
Definition of Image Graph
2 4 6 8 10 12 14 16
2
4
6
8
10
12
200 400 600 800 1000 1200
100
200
300
400
500
600
700
800
900
Graph nodes: pixels
Edge weights: similarity between neighboring pixels
Figures are borrowed from
http://www.cs.yale.edu/homes/spielman/sgta/
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2
2
( )i jI I
ijw e
Graph Terminology
• Adjacency Matrix
• Degree Matrix
• Laplacian Matrix
[ ] ij n nW w
1
D di ij
j
n n n
d
w
d
L D W
10
2
2
( )
0
i jI Iij
i j
w
e i j
i
ij
ij
d i jl
w i j
Random walk probability
Let pij be the the
probability of walking
from node i to node j:
Let xi be the probability,
starting at node i, of a
random walker reaching
the red node before
reaching the blue nodes
( , )
ij ij
ij
ik i
i k E
w wp
w d
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Random walk probability
• How to compute X = [x1 x2 … xn ]T?
• xi has following property
▫ xred = 1 for the red node
▫ xblue = 0 for the blue nodes
▫ for other nodesi ij j
j
x p x
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• If we define the transition matrix to be
• except for labeled nodes.
• Reorder the node vector X to make labeled nodes in
front of unlabeled :
1
( , )
ij
ij n nik
i k E n n
wP p D W
w
0
0
L L L
T
U U U
X D W BX D W
X D B W
13
PX X
1 1
1 1
L L L
T
U U U
D W D BP
D B D W
• The properties can be described by:
• XL are known and we need to calculate XU
• Finally we can compute XU by solving:
1 1
0L L
T
U U U U U
X I X
X D B D W X
1 1
1 1( )
T
U U L U U
T
U U U U L
X D B X D WX
I D W X D B X
( ) T
U U U LD W X B X
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• is positive definite for fully connected graph,
thus the linear system is nonsingular.
• Proof:
15
U UD W
2 2
,
( )
( )
( )
( ) ( )
U
U U
T
U U U U
T s s
U U U U
T T s s
U U U U U
ij i ij i j
i U j L i j U
x D W x
x D D W x
x D x x D W x
w x w x x
U ij
j L i U
D diag w
U
s
ij
j U i U
D diag w
Degree Matrix
for the subgraph
containing
unlabeled nodes
s
U UW W
Adjacency Matrix
for the subgraph
containing
unlabeled nodes
Minimization Point of View
2
( , )
1 1( )
2 2
. . 1, 0
T
ij i j
i j E
red blue
X LX w x x
s t x x
• It can be shown that the random walk
solution also minimizes following energy:
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Continuous case
• In continuous case, it is called the Dirichlet integral
• The Eular-Lagrange equation of the above functional
is the Laplace equation. Thus the solution of random
walk probability in the continuous case is a harmonic
function with predefined boundary conditions
21( ) | |
2D u u d
2 0 ( ) 1 ( ) 0red blueu u s u s
17
Outline
• Problem Description and Basic Idea
• Solution of Random Walk Probability
• Properties and Results
• Relation to Other Methods
• Discussion
18
The secretes of nice properties
• Property of Harmonic function
▫ Maximum Principle:
the maximum and minimum are only attained
on the boundary.
▫ Mean value property:
the function value is an average(or weighted
average) of neighboring values.
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Outline
• Problem Description and Basic Idea
• Solution of Random Walk Probability
• Properties and Results
• Relation to Other Methods
• Discussion
22
Graph Cut / Min-Cut
• The energy for Graph Cut segmentation is:
• It can be proved that the constraint is slack,
which means the Min-cut problem is a linear
programming problem which minimize the energy
with L1 penalty.
( , )
| |
. . {0,1}, 1, 0
ij i j
i j E
red blue
w x x
s t x x x
{0,1}x
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Normalized Cut (I)
• Revisit the Normalized Cut
• Relax y to be real numbers, the problem becomes:
( , ) ( , )( , )
deg( ) deg( )
( )... with {1, }, 1 0
TT
iT
cut A B cut A BNcut A B
A B
y D W yy b y D
y Dy
1
( )min
T
y D T
y D W y
y Dy
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Normalized Cut (II)
• Let , finally we want to solve
• is called normalized Laplace matrix.
• The energy minimized in the Normalized Cut is
similar to the random walk energy. They both have
quadratic form of a Laplace matrix, while the
constraints are different.
1
2z D y
1/2
1/2
* 1/2 1/2
1
1/2 1/2
1
arg min ( )
arg min
T
z D
T
z D
z z D D W D z
z D LD z
1/2 1/2D LD
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Random Walk Graph Cut Normalized Cut
Objective
function
Constraints
Solution. (k=2) Linear system Maximal FlowEigenvector of the
normalized Laplace
Solution. (k>2)Linear system Alpha-expansion
(approximated)Spectral clustering
Uniqueness Unique Not unique Unique
Result biasShift related to
seed placementSmall-cut bias
No bias
Comparison
( , )
| |ij i j
i j E
w x x
TX LX1/2 1/2Tz D LD z
1, 0red bluex x 1/2
1z D1, 0red bluex x
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Outline
• Problem Description and Basic Idea
• Solution of Random Walk Probability
• Properties and Results
• Relation to Other Methods
• Discussion
27
Conclusions
• The advantages of random walker
segmentation are mainly due to
1. It describe the segmentation problem with a
simple physical process, whose solution has
well-studied probabilistic interpretation and
nice mathematical properties.
2. The discrete calculus formulation of the
problem can be solved analytically and
efficiently.
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Why Use Graph?
• Capture the discrete nature of digital images
naturally model the spatial coherence of pixels
can be easily extended to high dimension
• Continuous differential operators can be replaced by
combinatorial operators, which can be directly
implemented.
• The linear algebra representation of the energy often
has analytical solution.
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Key References
[1] L. Grady, "Random walks for image segmentation," PAMI, pp.
1768-1783, 2006.
[2] A. K. Sinop and L. Grady, "A seeded image segmentation
framework unifying graph cuts and random walker which yields a
new algorithm," ICCV 2007, pp. 1-8.
[3] J. Shi and J. Malik, "Normalized cuts and image
segmentation," PAMI, vol. 22, pp. 888-905, 2002.
[4] U. Von Luxburg, "A tutorial on spectral clustering," Statistics and
Computing, vol. 17, pp. 395-416, 2007.
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