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Transcript of 1 Lecture #6 Variational Approaches and Image Segmentation Lecture #6 Hossam Abdelmunim 1 & Aly A....
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Variational Approaches and Image Segmentation
Lecture #6Lecture #6Hossam 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
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The curvature and The Implicit Function FormThe curvature and The Implicit Function Form
)0(0)( 1 CorC
The level set function has the following relation with the embedded curve C:
0)( sTC
Us the following derivative equation w.r.t. the arc-length s:
To prove that: (Assignment)
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Calculating Additional Quantities
||),2/())/cos(1()(
||,1)(
||)),/sin(1
1(5.0)(
H
HExample of a Level Set Function
iso-contours
H and Delta FunctionsApplying H FunctionApplying δ Function
,)( dxdyHA
,||)( dxdyL
• Enclosed Area
• Length of Interface
• Mainly used to track the Interface/contour:-
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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.
• 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.
Boundary Band Points.
Red line is the zero level set corresponding to
front.
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Level Set PDELevel Set PDE
0),).(,(
dt
dy
dt
dx
yxt0.
||||
Vt
0),,( tyx
Curve Contracts with time
0
dyy
dxx
dtt
Level Set Function changes with time
0||
Ft
Fundamental Level Set Equation
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.
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Speed FunctionSpeed Function
1F
Among several forms, the following speed function can be used:
Contour characteristics:
Forces the contour to evolve smoothly. The bending is quantized by ε.
Image data (force):
+1 for expansion
-1 for contraction
It will be a function of the image (I).
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Need for Re-initializationNeed for Re-initializationNeed for Re-initializationNeed for Re-initialization• Solving the PDE of level set evolution does not keep the definition.• Keeping the definition is very necessary to hold the front between
the positive and negative regions.
|)|1)(sgn( 0 t
• Solving this equation frequently often in parallel with the main equation keeps the function close to the signed distance definition.
1|| x
x0x
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Numerical Solution
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Upwind Scheme and Discontinuous Solutions
Upwind Scheme and Discontinuous Solutions
Consider the following PDE:
It is one dimensional in x and can have the following numerical solution for different values of the speed a (can be of course a function of x):
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Upwind Scheme and Discontinuous Solutions (Cont…)
Upwind Scheme and Discontinuous Solutions (Cont…)
So, we can define (as a notation):
To put the solution in the following general form:
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First Order Upwind Scheme and Discontinuous Solutions
First Order Upwind Scheme and Discontinuous Solutions
Consider the Solution of the re-initialization PDE:-
a+=max(a,0) and a-=min(a,0)
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First Order Upwind Scheme and Discontinuous Solutions whereFirst Order Upwind Scheme and Discontinuous Solutions where
)2
1)((2)( HS
And a smoothed version of the sign function is defined as follows:
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Numerical Algorithm for the Level Set Evolution Equation – Higher Order Scheme
Numerical Algorithm for the Level Set Evolution Equation – Higher Order Scheme
We consider the numerical solution of the equation:
0|| Ft
Note that it is very similar to the 1D equation we showed above. Without proof, this equation will have the following numerical solution:
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Numerical Algorithm for the Level Set Evolution Equation (Cont…) where
Numerical Algorithm for the Level Set Evolution Equation (Cont…) where
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Numerical Algorithm for the Level Set Evolution Equation (Cont…) and
Numerical Algorithm for the Level Set Evolution Equation (Cont…) and
The switching function m is given by:
The speed function is given by:
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Examples..1Examples..1Examples..1Examples..1• Curvature flow with a curvature speed:
F
Parts of the curve with different curvature signs, move in opposite directions.
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Examples..2Examples..2Examples..2Examples..2• Curvature flow with a positive curvature speed:
)0,max(F
Parts of the curve with -ve curvature do not move
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Examples..3Examples..3Examples..3Examples..3• Curvature flow with a negative speed vector:
Parts of the curve with +ve curvature do not move
)0,min(F