Variational Approaches and Image
Segmentation
Lecture #7
Hossam Abdelmunim1 & Aly A. Farag2
1Computer
& Systems Engineering Department, Ain
Shams University, Cairo, Egypt
2Electerical
and Computer Engineering Department,
University of Louisville, Louisville, KY, USA
ECE 643 – Fall 2010
1
The curvature and The Implicit Function Form
The level set function has the following relation with the
embedded curve C:
 (C)  0
or
C   1 (0)
Us the following derivative equation w.r.t. the arc-length
s:
( )T Cs  0
To prove that: (Assignment)
Calculating Additional Quantities
•
A   H ()dxdy,
Enclosed Area

•
L    () |  | dxdy,
Length of Interface

•
Mainly used to track the Interface/contour:-
 ()  (1  cos( /  )) /(2 ),|  | 
H ()  0.5(1 


H ()  1, |  | 

1

sin( /  )),|  | 
Applying δ Function
H and Delta Functions
Example of a Level Set Function
iso-contours
Applying H Function
Narrow Banding
•
Points of the interface/front/contour are only the
points of interest.
•
The points (highlighted) are called the narrow band.
•
The change of the level set function at these points
only are considered.
Boundary Band Points.
•
Other points (outside the narrow band) are called far
away points and take large positive or large
negative values.
•
This will expedite the processing later on.
Red line is the zero level set
corresponding to front.
Level Set PDE
Curve Contracts with time
( x, y, t )  0
Level Set Function changes with time



dt 
dx 
dy  0
t
x
y
   dx dy


(
,
).( , )  0
 |  |
.V  0
t
x y dt dt
t
|  |
Fundamental Level Set Equation

 |  | F  0
t
The velocity vector V has a component F in the
normal direction. The other tangential component
has no effect because the gradient works in the
normal direction.
Speed Function
Among several forms, the following speed function is used:
F  1  k
Image data (force):
Contour characteristics:
+1 for expansion
Smoothes the evolution
and the bending is
quantized by ε
-1 for contraction
It will be a function
of the image (I).
Variational Edge-based Segmentation
Edge map
1
g (I ) 
1 | (G * I ) |
Where g is an indicator function of the image gradient:
Variational Edge-based Segmentation
(Cont…)
Energy = Arc-Length + Enclosed Area:
E()    g ( I ) () |  | d   g ( I ) H ()d


By calculus of variation:

 t   ()[div( g
)  g ]
|  |
The amount of bending is controlled by λ>0.
The sign of ‫ ע‬depends on the position of the contour w.r.t.
the object.
Variational Segmentation without Edges
Chan-Vese Model
c1
H () Id


 H ()d
c2
H () Id


 H ()d
Object
Mean
Background
Mean
Ecv   [( I  c1 ) 2 H ()  ( I  c2 ) 2 H ()   | H () |]d

Maximizes the distance between c1 and c2
Only one level set function is used
Variational Segmentation without Edges
Chan-Vese Model (Cont…)
The PDE will be:

 t   ()[div(
)  ( I  c1 ) 2  ( I  c2 ) 2 ]
|  |
For computational issues:
 t   ()[div(
where:

) ]
|  |
 1 if  ( I  c1 ) 2  ( I  c 2 ) 2  0

 1 if  ( I  c1 ) 2  ( I  c 2 ) 2  0
Chan & Vese--Examples
Multi-phase Evolution
Chan & Vese
C3
In this example 2 functions are used.
Then 22=4 regions are considered.
The energy will be:
Ф2>0
Ф1<0
Ф1>0
Ф2>0
Ф2<0
Ecv    ( I  c3 ) 2 H (1 ) H ( 2 )

 ( I  c 4 ) 2 H (   1 ) H (  2 )
  (| H (1 ) |  | H ( 2 ) |)]d
C4
Ф1<0
[(I  c1 ) 2 H (1 ) H ( 2 )
 ( I  c 2 ) 2 H ( 1 ) H (  2 )
Ф1>0
Ф2<0
C2
C1
Multi-phase Evolution
Chan & Vese (Cont…)
Using calculus of variations will result in:
1
  (1 )[(I  c1 ) 2 H ( 2 )  ( I  c2 ) 2 H ( 2 )  ( I  c3 ) 2 H ( 2 )
t
 ( I  c4 ) 2 H ( 2 )  1 ]
 2
  ( 2 )[(I  c1 ) 2 H (1 )  ( I  c2 ) 2 H (1 )  ( I  c3 ) 2 H (1 )
t
 ( I  c4 ) 2 H (1 )   2 ]
Multi-phase Evolution
Chan & Vese (Example)
The given image contains 4 regions. Three different color boxes are represented in the
foreground. The background is considered the fourth region.
Multi-phase Evolution
8 Regions-3 Level sets
2
5
6
1
4
3
7
8
Chan & Vese (Cont…)
The curvature is included with a coefficient μ which helps
in segmenting images with noise but when the noise level
is high, the weight needs to be increased. This affects the
boundaries of the object and also increases the
convergence time.
Number of regions are always 2n depending on the
number of level set functions n.
No vacuum pixels appear because if any point does not
belong to a certain region, it will go to another one.
Unless the region can be described by only its mean, the
segmentation will fail.
Thank You
&
Questions