The Savitzky-Golay function is mostly used as low-pass filter to render
visible the relative widths and heights of spectral lines in noisy data
without major loss of intensity. This procedure is called smoothing.
Each data point of a 2D or 3D data object will be evaluated under consideration
of a number of neighboring data points and the overall resulting slope
of the data. The algorithm can also be used to calculate derivatives.
Each data point value fi of the 2D or
3D data object is therefore replaced by a linear combination gi of itself and
some number of neighboring data points,
Where nL
is the number of data points used relative to the left of a ith data point
and nR
the number of points used to the right. In the software nL and nR are equivalent
to compute gi
as the weighted average of data points around the ith data point.
The weighting factor cn
for each data point i is derived
from a polynomial least squares fit using a polynomial of the degree M. The polynomial is defined as
Required polynomials for the least squares fit for each data point of
a 2D or 3D data object can be easily obtained using a previously designed
matrix to produce required linear combinations of the polynomials. The
matrix AT
looks like this:
The weighting factors or Savitzky-Golay coefficients cn for each data
point will be derived from the vectors aj in terms of
the vectors fi
of the matrix Aij
with the specific forms
and
where f is replaced by the
unit vector en if the coefficient cn is the component
a0.
cn
is then calculated as:
For smoothing purposes only the coefficients a0 are interesting,
because they represent the smoothed data point intensity values.
This value indicates the order of the polynomial fit function applied
for smoothing of 2D or 3D data objects. A positive integer value (greater
than 0) must be entered into this text field.
An odd number of data points around each smoothed data point of the
spectrum will be taken into account for calculation of the smoothing polynomial.
For details, please refer to the Savtizky-Golay
documentation. The number of data points can be selected from the
drop down combo box by clicking the icon at the right side of the parameter field.
Tip: With increasing polynomial order
the number of window points must be increased accordingly. If the number
of window points is too small, a math error message is displayed on calculation.
References
Savitzky A., and Golay, M.J.E. 1964, Analytical Chemistry, vol. 36,
pp. 1627â1639.
icon at the right side of the parameter field.
Tip: With increasing polynomial order
the number of window points must be increased accordingly. If the number
of window points is too small, a math error message is displayed on calculation.
