Noise Filtering Methods in MRI - Advance Image Processing in Magnetic Resonance Imaging - page 173

2.0

1.5

Φ

1.0

C

0.5

0.0

1K 2K 3K 4K

I

∇

FIGURE 6.2 Monotonic decrease of the diffusion coefficient c(x, t) and flow function

φ ( ∇ I) as a function of gradient ∇ I.

maximum flow is produced at the image location when I is below

K, the flow function reduces to zero because in almost homogeneous regions, the
flow is minimal. For

∇ I = K. When ∇

I larger than K, the flow function again decreases to zero,
halting diffusion at locations of high gradients. A proper setting of the K parameter
in the diffusion function not only preserves, but also enhances object edges. This
property will be exploited in the next section in an application dealing with
contour enhancement for myocardial image segmentation.
The performance of the anisotropic filter is also related to the choice of the
diffusion function. An alternative choice of the diffusion function is:

∇

1

ct () x = (6.13) ,

( ∇ It x ) k )

1 + || ( , || /

1 / α

or [14]:

1
2

ct ( x ) = [tanh( ( γ k − ∇ || I t ( , ) ||)) x + 1 ] (6.14) ,

Anisotropic diffusion, in its original form, is a well-accepted filtering tech-
nique because of its computational speed and algorithmic simplicity. It was
applied to 2-D and 3-D MRI data by Gerig [13]. Such a filter has shown maximum
performance in local filtering applications such as automatic MR cardiac image
segmentation [18].