Correction of intensity nonuniformity in breast MR images - Emerging Trends in Image Processing, Computer Vision, and Pattern Recognition - page 180

(3)
where N p is the normalized pixel value, p is the pixel value, P m is the minimum pixel value in
the image, P M is the maximum pixel value in the image, and R M is the maximum value of pixel
bit depth.
In this step, a locally normalization via a resizable sliding window is performed to com-
pensate intensity nonuniformity in MR images.
2.4 Hybrid Method for Bias Field Correction
Nonparametric intensity nonuniformity normalization (N3) was proposed by Sled et al. [ 3 ] for
solving the problem of artifacts in MRI images. It is an iterative method, which estimates mul-
tiplicative bias field and true tissue intensity distribution. This method is an intensity model-
based or histogram-based approach. Unlike some other methods such as EM, N3 does not rely
on tissue classiication.
One of the main advantages of nonparametric methods is that they do not make any as-
sumptions about the patient anatomy. In addition, it is fully automatic, accurate, and robust
method. In this step, a combination of N3 algorithm and singularity function analysis (SFA)
model is used in breast MR images.
2.4.1 Bias field model
The following equation is the basis of the N3 method [ 23 ] . Bias field is often modeled as a mul-
tiplicative field:
(4)
In which, f is an unknown bias field or intensity inhomogeneity, v is the observed image, x
designates the spatial position or voxel, u is the true image, and n is the noise which assumed
to be independent of u .
The main stage for correcting bias field is estimating its distribution ( f ). The mixture of mul-
tiplicative and additive model makes this stage difficult
The challenging problem of bias field correction consists of recovering u ( x ) from informa-
tion about the multiplicative factor f ( x ) and the additive term n ( x ). Due to the simultaneous
presence of n ( x ) and f ( x ), it is difficult to solve the problem. Thus, a common solution is to neg-
lect the n ( x ) that is an additive noise. For the two-dimensional discrete image case and using a
log transform, the bias field is made additive.
Consider a case without noise in which u and f are independently distributed random vari-
ables. Instead of v , u , f , we deal with log v , log u , log f , then the formation model becomes ad-
ditive: