Image Processing Reference

In-Depth Information

(5)

Let
V
(
u
,
v
),
U
(
u
,
v
), and
F
(
u
,
v
) present the probability densities of
v
(
i
,
j
),
u
(
i
,
j
), and
f
(
i
,
j
), re-

spectively. Equation
(5)
can be expressed as

(6)

Making the approximation that ln
u
and ln
f
are uncorrelated random variables, Equation

(6)
is found by convolution as follows:

(7)

The multiplication corrupts the field and a division can undo the corruption. In the fre-

quency domain, multiplications and divisions convert to convolutions and deconvolutions as

follows:

(8)

In which
V
,
U
, and
F
are probability densities. After this stage, the uniformity distribution

(
F
) is modeled and viewed as blurring intensity distribution
U
that is the main stage for cor-

recting bias field.

2.4.2 Correction step

An straightforward technique for image bias field correction would be that if the spatial fre-

quencies of
u
(
i
,
j
) (the true image) and
f
(
i
,
j
) bias field are disjointed, the bias field can be re-

moved by filtering out the spatial frequencies illustrating the bias field. In some cases, use-

ful knowledge in MRI related to higher spatial frequencies than the intensity nonuniform-

ity. Thus, by removing low spatial frequencies, intensity nonuniformity can be suppressed.

The problem is that in most cases, spatial frequency information of the bias field and true im-

age are not perfectly separated, and they have some overlapped components. For example,

in MRI scans, when low spatial frequencies are removed, the spatial image will be consider-

ably changed, implying the existence of useful knowledge at low spatial frequency bands. In

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