Biomedical Engineering Reference
In-Depth Information
quantify the time gained by the procedure as well as to decide on parameters
a priori, as the algorithm would likely have to obtain both from the data
quality and resolution enhancement. Another key step in the approach is the
use of an iterative scheme to estimate the likely structure, which involves a
careful selection of constraints on the solution (e.g., smoothness or positivity
of values).
An interesting observation reported was that the reconstruction was not es-
pecially sensitive to the instrument's point spread function. Hence, a detailed
knowledge of the optics of the instrument is not required for super-resolution.
Further, in principle, smoothing and noise rejection could be achieved in the
inversion process (though this idea was not reported in the chapter). As with
other inverse problems (e.g., filtered backprojection in computed tomogra-
phy) the choice of an appropriate filter may be made to minimize the effects
of noise and actually reconstruct both a high-resolution and low-noise image
set from constituent low-resolution, noisy data sets. Finally, caution must be
exercised in that results of both high-resolution and low-noise reconstructions
are dependent on the algorithm and do result in a loss of data. The results are
a best estimate of the actual structure/data quality under the constraints of
the algorithms employed. When used appropriately, though, both chemomet-
ric approaches help improve the trade-offs that a spectroscopist must consider
in the design and conduct of experiments.
8.2.5 Eigenvector-Based Multivariate Analysis
Spectral data are highly redundant (many vibrational modes of the same
molecules) and sparse (large spectral segments with no informative features).
Hence, before a full-scale chemometric treatment of the data is undertaken,
it is very instructive to understand the structure and variance in recorded
spectra. Hence, eigenvector-based analyses of spectra are common and a pri-
mary technique is principal components analysis (PCA). PCA is a linear trans-
formation of the data into a new coordinate system (axes) such that the largest
variance lies on the first axis and decreases thereafter for each successive axis.
PCA can also be considered to be a view of the data set with an aim to
explain all deviations from an average spectral property. Data are typically
mean centered prior to the transformation and the mean spectrum is used a
base comparator. The transformation to a new coordinate set is performed
via matrix multiplication as
Y
=
V
T
X
(8.1)
where
X
is the same data matrix shown in Fig. 8.1 and
Y
is the new data
matrix after rotation. The specific axes along which the data are rotated are
found by solving an eigenvalue equation and it can be shown that a rota-
tion matrix (
V
T
) always exists. The weights assigned to different axes are
representative of the spectral characteristics of separate sub-groups that may
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