Retrospective Evaluation and Correction of Intensity Inhomogeneity - Advance Image Processing in Magnetic Resonance Imaging - page 154

1, the fuzzy C-means algorithm devolves to the K-means algorithm.
For m > 1, the fuzzy C-means algorithm alternates between a fuzzy segmentation

For m =

( )

21 1

−

/ ( m − )

| y

y

− | µ

µ

∑

i k

u = (5.13)

ik

| − |

i j

j

and an update of the tissue means

∑
∑

() uy

m

ik i

µ k =

i

. (5.14)

() u

m

ik

i

Whereas the classification step in the K-means algorithm uniquely assigns
each voxel to one single tissue type, the fuzzy C-means algorithm calculates for
each voxel fuzzy membership values in each of the tissue types. The parameter
m is a weighting exponent that regulates how fuzzy those membership values
are; typically, m

2 is used.
In [39], Pappas extended the K-means algorithm to account for local intensity
variations by letting the tissue means gradually vary over the image area.* Rather
than using one single mean intensity

=

µ k for every tissue type throughout the image,
a different mean is used in every voxel i , which is calculated by performing
the averaging operation in Equations 5.11 over a sliding window rather than over
the whole image area:

µ i

∑
∑

uy

ik i

iW ∈

µ i =

i

u

ik

iW ∈

i

with W i , a window centered at voxel i . This approach, and variations upon it, was
applied to MR images of the brain by a number of authors [40 -44]. Variations
include describing the spatially varying mean intensities by B-splines [41], using
the fuzzy C-means algorithm instead of the K-means algorithm [42], calculating
spatially varying tissue covariances as well [43], and replacing the crisp tissue
classification by a partial volume estimation [44]. One notable aspect of these
approaches is that the mean intensities of the tissue types are allowed to vary
independently of one another, i.e., no continuity of the inhomogeneity field over
the tissue boundaries is assumed. As a result, in places where the number of
voxels assigned to class k is small, the mean

µ k cannot be reliably estimated, and

appropriate precautions must be taken.

* Pappas also took an MRF model into account, but that is outside the scope of this text.