Biomedical Engineering Reference
In-Depth Information
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APPENDIX A: DIFFUSION METHODS
In Eq. (13), if the function F depends on the curvature, we may have a geo-
metric flow in which high curvatures are diffused.
In image processing, diffusion schemes for scalar and vector fields have also
been applied. Gaussian blurring is the most commonly known one. Other ap-
proaches include anisotropic diffusion and Gradient Vector Flow. We will sum-
marize these methods and conjecture their unification.
Anisotropic diffusion is defined by the following general equation:
∂I ( x, y, t )
∂t
= div ( c ( x, y, t )
I ) ,
(16)
where I is a graylevel image [32].
In this method, the blurring on parts with high gradient can be made much
smaller than in the rest of the image. To show this property, we follow Perona-
Malik [32]. First, we suppose that the edge points are oriented in the x direction.
Thus, Eq. (16) becomes:
∂I ( x, y, t )
∂t
∂x ( c ( x, y, t ) I x ( x, y, t )) .
=
(17)
If c is a function of the image gradient: c ( x, y, t )= g ( I x ( x, y, t )), we can
define φ ( I x )
g ( I x ) ·
I x and then write Eq. (16) as
∂I
∂t =
∂x ( φ ( I x )) = φ ( I x ) ·
I t =
I xx .
(18)
∂I x
∂t .If c ( x, y, t ) > 0,we
can change the order of differentiation and with a simple algebra demonstrate that
We are interested in the time variation of the slope:
∂I x
∂t =
∂I t
∂x = φ ·
I xx + φ ·
I xxx .
At edge points we have I xx =0and I xxx << 0, as these points are local
maxima of the image gradient intensity. Thus, there is a neighborhood of the edge
 
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