Retrospective Evaluation and Correction of Intensity Inhomogeneity - Advance Image Processing in Magnetic Resonance Imaging - page 146

values must be chosen for each of these parameters. Wells et al. determined the
tissue-specific means and variances by manually selecting representative

voxels of each of the tissue types considered in a prototypical image and calcu-
lating sample means and variances. Once this training phase was performed, the
same model parameters were used to analyze thousands of images acquired with
the same MRI acquisition protocol. For the prior probabilities

µ σ 2

k

π , both uniform

k

and nonuniform distributions were used. The exact choice of the N × N covariance

is a difficult issue and will be discussed later.
Having obtained the posterior probability on the inhomogeneity field, Wells
et al. used the maximum a posteriori (MAP) principle to formulate an estimate
of the inhomogeneity field as the value of

matrix Σ

β

β having the largest posterior probability

ˆ

β = arg max py ( β | ) ,

β

which is equivalent to

ˆ

β = arg maxlog py ( β | ) .

β

to be zero for all
voxels, Wells et al. derived the following equations for the inhomogeneity field
estimation:

By requiring the gradient of log py ( β | ) with respect to β

i

ˆ

β = Hr (5.6)

with

∑

y −

−

y

w
w

µ

1

1

∑

ik k

r = y , =

k

, w = wwp , = / σ

2

y y

2 i ik ik ik k

2 i

i

k

where

ppl ==, ( ky | β )

ik i i i

py l (| =, k β )( pl = k )

=

i i i i

(5.7)

∑

py l (| = k ′, β )( pl = k )

i i i i

k ′

and H is a linear operator defined by

w …

w…

0

1

−− − 11 1

HI =+ W Σ β , W = 0 .

2