Retrospective Evaluation and Correction of Intensity Inhomogeneity - Advance Image Processing in Magnetic Resonance Imaging - page 157

most violations of the class model do not result in incorrect solutions. Failure
of the class model was only observed in very low resolution MR datasets,
where as much as 30% of partial volume voxels can be present. In this case,
an additional preprocessing step identifies partial voluming voxels using a
dilated edge filter. The identified partial volume voxels are then excluded from
further computations.

x

2

fx () = (5.16)

k

x

2

+ 3 σ

2

k

∑∑ = .

e = e y () = f ( y − βµ − ) (5.17)

tot i ki i k

kK 1

y y

i i

Both additive and multiplicative inhomogeneity fields are modeled as a para-
metric inhomogeneity field using Legendre polynomials ψ L as basis functions in

x , y , and z . In case of a multiplicative inhomogeneity field, all computations are
performed in log-space. For Legendre polynomials up to the degree l , the size m
of the parameter vector c ijk is given by For instance, Legendre
polynomials up to the third degree would therefore require 20 coefficients. The
choice of the maximal degree of Legendre polynomials largely depends on prior
knowledge of the coil and the expected type and smoothness of the inhomogeneity
field. The inhomogeneity field estimate

( l + 2

2

) ( l + 3 )

ml =+ ( 1 ) .

3

β i is determined as follows:

l ( li − ) ( li j −− )

∑∑ ∑
0

β = c ψ () j x ψ () y ψ , () z (5.18)

i ijk L i , i L , i Lki

i = j = 0 k = 0

ψ L,i (.) denoting a Legendre polynomial of degree i .
Finding the parameter vector c ijk with minimum energy e tot is a nonlinear
optimization problem, independent of the type of inhomogeneity field and energy
function. In principle, any nonlinear optimization method could be applied.
PABIC uses an adapted version of the (1

with

+ 1)-evolution strategy (ES), which

belongs to the family of evolutionary algorithms (for an introduction see [ 50 ]).

This method adjusts locally the search direction and step size, and provides a
mechanism to step out of nonoptimal minima. Furthermore, the method is fast
enough to cope with the large data sets and overcomes the problem of parameters
with different scaling.
The optimal polynomial parameters correct the original MR image so that the
intensity statistics of the corrected image fit the given class model optimally. This
property of PABIC can also be employed for normalizing the intensity statistics for
different images. A slice-by-slice intensity normalization is necessary when vene-

tian blind artifacts are present in the image (see Section 5.1 ). A volume-by-volume

intensity normalization is necessary when analyzing the absolute intensities of MRI