Retrospective Evaluation and Correction of Intensity Inhomogeneity - Advance Image Processing in Magnetic Resonance Imaging - page 155

More recently, a number of authors have also used a single inhomogeneity field
model that is continuous over tissue boundaries within the K-means/fuzzy C-means
framework. Assuming that tissue-specific means and covariances are determined
in advance, Rajapakse and Kruggel iteratively alternated between a crisp classifi-
cation and an inhomogeneity field estimation step, the latter being performed by
averaging local inhomogeneity field estimates with a sliding window [45]. Pham
and Prince extended the fuzzy C-means algorithm with an explicit inhomogeneity
field model, by optimizing the objective function

∑∑

Q ( µβ ,, u | y ) = ( u ) (

m

y − βµ )

2

+ Q ( β ) (5.15)

AFCM ik i i k

i k

where ) is a regularization term that ensures

that the inhomogeneity field is spatially smooth and slowly varying [46].
Optimizing Equation 5.15 with respect to the membership values u , tissue
means

β is the inhomogeneity field and Q ( β

yields a 3-step algorithm that iteratively
alternates between a fuzzy tissue classification, estimation of the mean inten-
sities, and inhomogeneity field estimation. Whereas the mathematical aspects
differ significantly from the EM-based method proposed by Van Leemput et al.,
described earlier, both techniques are conceptually very similar. A similar 3-
step approach, based on an extension of the fuzzy C-means algorithm that takes
interactions between neighboring voxels into account, was described by Ahmed
et al. [47].

µ and inhomogeneity field β

5.4 CORRECTION BASED ON EVALUATION

OF THE HISTOGRAM

In contrast to the methods discussed in the previous section, the methods dis-
cussed in this section do not perform a segmentation of the image but rather base
their computations mainly on the histogram of the image. All of these methods
formulate the intensity inhomogeneity correction as an optimization problem

based on a metric computed directly from the histogram of the corrected image

(see Figure 5.5 ).

5.4.1 P ARAMETRIC B IAS C ORRECTION

We first discuss the parametric bias correction (PABIC) method [10,48] because this
method can be regarded as a hybrid between the combined segmentation/correction
methods and the other histogram-based methods.* While no explicit segmentation
is computed, a prior parametric model for all tissue classes in the image is a necessary
parameter of the histogram metric.

* PABIC is part of the National Library of Medicine Insight Segmentation and Registration Toolkit

(ITK), an open-source software system available at http://www.itk.org .