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Active Contour Model (Snake)
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Outline
Introduce Active Contour Model
Active Contour Model
First introduced by Kass, Witkin, and Terzopoulos[1987]
Why snake?
The shape of many objects is not easily represented by rigid primitives. Natural objects, such as bananas, have similar
recognizable shapes. But no two bananas are exactly the same.
In medical imaging, objects are similar but not exact. An exact representation of a vein's shape, for example, cannot be given.
Some objects, such as lips, change over time.
What’s snake?
Optimization technique Based on a spline Minimization the energies of the spline Deforms to fit local minima Local, not global, so initial location must be prov
ided
What’s snake?
Customizable Initialization Weights User constraints
Energy of Snake
A snake is defined as an energy function. To find the best fit between a snake and an object's shape, we minimize the energy.
Where the snake is parametrically defined as v(i) = (x(i),y(i)) Einternal : Internal spline energy caused by stretching and bending.
Eimage : Measure of the attraction of image features such as contours.
Econstraint : Measure of external constrains either from higher level shape information or user applied energy.
diivEivEivEE constraimageernalsnake ))()()(( intint10
Internal Energy
The internal energy provides a smoothness constraint.
α(i) is a measure of the elasticity of the snake. β(i) is a measure of the stiffness of the snake.
The first order term makes the snake act like a membrane The constant α controls the tension along the spine. The second order term makes the snake act like a thin plate The constant β controls the rigidity of the spine If β=0 then the function is discontinuous in its tangent, i.e. it
may develop a corner at that point. If α=β=0 then this also allows a break in the contour, a positi
onal discontinuity.
2
2
22
int )()(di
dvi
di
dviE ernal
Image Energy
The image energy is derived from the image data.
where ω is an appropriate weighting function. line functional is defined simply by the image function,
The edge functional is defined by
the termination functional allows terminations (i.e. free ends of lines) or corners to attract the snake.
termtermedgeedgelinelineimage EEEE
),( yxfEline
2),( yxfEedge
Constraint Energy
The constraint energy is determined by external constraints.
This energy may come in the form of a spring attached by the user. Or, the constraint energy may come from higher knowledge about the images in question.
