Biomedical images are often affected and corrupted by various types of noise and artifact. Any ...

418 Pages·2008·6.43 MB·English

Biomedical Image Analysis
3
Removal of Artifacts
Biomedical images are often affected and corrupted by
various types of noise and artifact.
Any image, pattern, or signal other than that of interest
could be termed as interference, artifact, or simply noise.
–345– c R.M. Rangayyan, CRC Press
(cid:13)
Biomedical Image Analysis 3.1. CHARACTERIZATIONOFARTIFACTS
3.1 Characterization of Artifacts
3.1.1 Random noise
Random noise: interference from a random process such as
thermal noise in electronic devices and the counting of photons.
Random process characterized by its PDF: probabilities of
occurrence of all possible values of a random variable.
–346– c R.M. Rangayyan, CRC Press
(cid:13)
Biomedical Image Analysis 3.1. CHARACTERIZATIONOFARTIFACTS
Consider a random process η characterized by the PDF p (η).
η
The process could be a function of time as η(t), or of space in
1D, 2D, or 3D as η(x), η(x, y), or η(x, y, z);
it could also be a 4D spatio-temporal function as η(x, y, z, t).
Mean µ : first-order moment of the PDF.
η
µ = E[η] = η p (η) dη, (3.1)
η ∞ η
Z
−∞
where E[ ] represents the statistical expectation operator.
Common to assume mean of a random noise process = zero.
–347– c R.M. Rangayyan, CRC Press
(cid:13)
Biomedical Image Analysis 3.1. CHARACTERIZATIONOFARTIFACTS
Mean-squared (MS) value: second-order moment.
2 2
E[η ] = η p (η) dη. (3.2)
∞ η
Z
−∞
2
Variance σ : second central moment.
η
2 2 2
σ = E[(η µ ) ] = (η µ ) p (η) dη. (3.3)
η η ∞ η η
− −
Z
−∞
Square root of variance = standard deviation (SD) σ .
η
–348– c R.M. Rangayyan, CRC Press
(cid:13)
Biomedical Image Analysis 3.1. CHARACTERIZATIONOFARTIFACTS
2 2 2
Note that σ = E[η ] µ .
η η
−
2 2
If the mean is zero, it follows that σ = E[η ]:
η
variance = MS value.
–349– c R.M. Rangayyan, CRC Press
(cid:13)
Biomedical Image Analysis 3.1. CHARACTERIZATIONOFARTIFACTS
0.25
0.2
0.15
0.1
0.05
e
u
al 0
v
e
s
oi
n −0.05
−0.1
−0.15
−0.2
−0.25
50 100 150 200 250
sample number
2
Figure 3.1: A time series composed of random noise samples with a Gaussian PDF having µ = 0 and σ = 0.01.
MS value = 0.01; RMS = 0.1. See also Figures 3.2 and 3.3.
–350– c R.M. Rangayyan, CRC Press
(cid:13)
Biomedical Image Analysis 3.1. CHARACTERIZATIONOFARTIFACTS
2
Figure 3.2: An image composed of random noise samples with a Gaussian PDF having µ = 0 and σ = 0.01.
MS value = 0.01; RMS = 0.1. The normalized pixel values in the range [ 0.5,0.5] were linearly mapped to the
−
display range [0,255]. See also Figure 3.3.
–351– c R.M. Rangayyan, CRC Press
(cid:13)
Biomedical Image Analysis 3.1. CHARACTERIZATIONOFARTIFACTS
0.016
0.014
0.012
e
c
n 0.01
e
urr
c
c
o
of 0.008
y
bilit
a
b
Pro 0.006
0.004
0.002
0
−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5
Normalized gray level
Figure 3.3: Normalized histogram of the image in Figure 3.2. The samples were generated using a Gaussian
2
process with µ = 0 and σ = 0.01. MS value = 0.01; RMS = 0.1. See also Figures 3.1 and 3.2.
–352– c R.M. Rangayyan, CRC Press
(cid:13)
Biomedical Image Analysis 3.1. CHARACTERIZATIONOFARTIFACTS
A biomedical image f(x, y) may also, for the sake of generality,
be considered to be a realization of a random process f.
This allows for the statistical characterization of
sample-to-sample or person-to-person variations in
a collection of images of the same organ, system, or type.
–353– c R.M. Rangayyan, CRC Press
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Biomedical Image Analysis 3.1. CHARACTERIZATIONOFARTIFACTS
Statistical averages representing populations of images of a
certain type are useful in
designing filters,
data compression techniques, and
pattern classification procedures
that are optimal for the specific type of images.
However, in diagnostic applications, it is the
deviation from the normal or the average
in the image on hand that is of critical importance.
–354– c R.M. Rangayyan, CRC Press
(cid:13)