439
15 FtIR Imaging of tissues: techniques and methods of Analysis
=
+
(
)
′
1
1
2
σπ
Bk
(15.3)
where A is the absorbance, 2σ is the width at half height, k′ = (k − k 0 ) is the frequency
with respect to the band centre at k 0 , and B = 1/σ
2
. Substituting Eq. 3 into Eq. 1 gives
the second derivative.
A
B
Bk
Bk
′′
′
′
=
−
−
(
)
+
(
)
2 1 3
1
2
2
3
σπ
(15.4)
the band position for the second derivative is identical with the wavenumber value
of the original band centre k 0 . the half width of the second derivative, σ
II
is related
to the half width of the original line by
σ
σ
II
= / .
2 7
(15.5)
while the peak intensity A 0
′′
of the second derivative is related by the following
expression
A
A
0
0
2
2
′′
= −
/ σ
(15.6)
thus the peak height of the second derivative is proportional to the original peak
height and inversely proportional to the square of the original half-width. of course
for real spectra, where there are overlapping bands, which deviate from a true Lorentzian the situation is more complex but nonetheless these formulas provide a
reasonable approximation for second derivative interpretation [67].
In summary the second derivative corrects baseline shift, resolves overlapping
bands, reduces replicate variability and amplifies spectra variations without having
to model for spectral variation.
15.4.1.2 Normalization
Normalization is often employed to compensate for the differences in sample quantity (concentration and/or path length) in a spectral database or image hypercube.
Normalization pre-processing methods attempt to correct for these effects by identifying some aspect of each sample, which should be essentially constant from one
sample to the next, and correcting the scaling of all variables based on this characteristic. typically, normalization should be performed before any centering or scaling or other column-wise pre-processing steps and after baseline or offset removal.
the presence of baseline or offsets can impede correction of multiplicative effects.
the effect of normalization prior to background removal will, in these cases, not
improve model results and may deteriorate model performance.
15 FtIR Imaging of tissues: techniques and methods of Analysis
=
+
(
)
′
1
1
2
σπ
Bk
(15.3)
where A is the absorbance, 2σ is the width at half height, k′ = (k − k 0 ) is the frequency
with respect to the band centre at k 0 , and B = 1/σ
2
. Substituting Eq. 3 into Eq. 1 gives
the second derivative.
A
B
Bk
Bk
′′
′
′
=
−
−
(
)
+
(
)
2 1 3
1
2
2
3
σπ
(15.4)
the band position for the second derivative is identical with the wavenumber value
of the original band centre k 0 . the half width of the second derivative, σ
II
is related
to the half width of the original line by
σ
σ
II
= / .
2 7
(15.5)
while the peak intensity A 0
′′
of the second derivative is related by the following
expression
A
A
0
0
2
2
′′
= −
/ σ
(15.6)
thus the peak height of the second derivative is proportional to the original peak
height and inversely proportional to the square of the original half-width. of course
for real spectra, where there are overlapping bands, which deviate from a true Lorentzian the situation is more complex but nonetheless these formulas provide a
reasonable approximation for second derivative interpretation [67].
In summary the second derivative corrects baseline shift, resolves overlapping
bands, reduces replicate variability and amplifies spectra variations without having
to model for spectral variation.
15.4.1.2 Normalization
Normalization is often employed to compensate for the differences in sample quantity (concentration and/or path length) in a spectral database or image hypercube.
Normalization pre-processing methods attempt to correct for these effects by identifying some aspect of each sample, which should be essentially constant from one
sample to the next, and correcting the scaling of all variables based on this characteristic. typically, normalization should be performed before any centering or scaling or other column-wise pre-processing steps and after baseline or offset removal.
the presence of baseline or offsets can impede correction of multiplicative effects.
the effect of normalization prior to background removal will, in these cases, not
improve model results and may deteriorate model performance.