Retrospective Evaluation and Correction of Intensity Inhomogeneity - Advance Image Processing in Magnetic Resonance Imaging - page 144

tissue class in the MR image as an input to the estimation of the parametric
inhomogeneity field. Tincher et al. [16] and Meyer et al. [17] present automatic
techniques that fit polynomial functions to presegmented regional patches in the
image. The individual fits are then combined to find an estimate for a global
inhomogeneity field.
Some of the early methods assumed an additive inhomogeneity effect (e.g.,
[15]) whereas other methods proposed a multiplicative inhomogeneity effect (e.g.,
[11,16]). Nowadays, researchers agree that inhomogeneity effects in MRI are
better modeled as a multiplicative effect. While a smooth multiplicative inhomo-
geneity effect is consistent with most characteristics of the underlying acquisition
principles, it is noteworthy that minor sources of intensity inhomogeneity cannot
be fully incorporated by this model [18]. Additive inhomogeneity effects are
rarely observed in MRI, but can be seen in images acquired with other imaging
means, such as confocal microscopy.

5.3 COMBINED SEGMENTATION AND

INHOMOGENEITY CORRECTION METHODS

Whereas inhomogeneity correction methods are often needed to obtain good
segmentations, the early approaches of Dawant, Tincher, and Meyer indicate
that a good segmentation in turn facilitates inhomogeneity estimation. Observ-
ing this dependency between segmentation and inhomogeneity correction, the
idea emerged to solve both problems simultaneously using an iterative approach
in which increasingly accurate inhomogeneity corrections yield increasingly
accurate segmentations and vice versa. Although other techniques exist [19,20],
a landmark paper in this respect was published by Wells et al. [7], described
in the following text.
Assuming a multiplicative inhomogeneity field model, Wells et al. logarithmi-
cally transformed the MR intensity data in order to make the inhomogeneity an
additive artifact. Let y = ln ( z ) denote the log-transformed intensity at voxel i ,where

i i

z is the voxel's measured MRI signal intensity.* Assuming that there are K

i

different tissue types present in the image, and that the log-transformed intensity
distribution of each of these tissues can be modeled by a normal distribution after
taking the inhomogeneity field model into account, we have

py l (| , β ) = G ( y −− µ β ) (5.1)

i i i σ i l i

l i i

where l ∈ {1, 2, … , K } and β denote the tissue type and the inhomogeneity field

i i

at the i th voxel, respectively, µ denotes the mean intensity of tissue type k , σ 2

k

* For the sake of simplicity, only a single intensity value per voxel is assumed, although the technique
readily applies to multispectral MRI data.