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11/15/2011
1
Level Sets Method
Introduction
Images
•
Curves as isolevels of an image, image as a surface

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2
Parametrized Curves
p
N
T
Arc Length, Unit Tengent, and Curvature
•
Arc length:
•
Unit tangent vector (verify:
)
•
Curvature tensor:
•
So
°±
°²
and
³
?
are parallel and (for some
´(?)
, the curvature)
? = 0
? = ?

11/15/2011
3
Two Important Formulas
•
Verify
•
and
Curves as Iso-Level of a Function
u
Differentiate (*) again:
Consider
(*)
(**)

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4
Formulas for Curvature of Level Curves
From |T’(s)|=1, we have
Thus
We can re-write the formula as
Boundary Detection Functional
We will apply calculus of variation to some image analysis problems. First let us
consider the problem of detecting the boundary of an object in an image. Recall
that boundary points are where the intensity changed most. This means, the
magnitude of the gradient of the intensity is very large on the boundary points.