11 Electron Tomography
269
noise reduction are imperative. Besides, the problem requires the separation of the
contribution of each element to the spectra per pixel and their identification, which
can be regarded as a blind source separation (BSS) problem [20, 38–40].
First, the data (x, y, θ, ,E) needs to be treated to correct energy drift. Then,
weighted principal component analysis (wPCA) [38] is applied. The weighting of
the PCA is adapted to the dominant Poissonian noise. The PCA algorithm computes
a new spectral base for the spectrum image, where the base components are ordered
by the spectral variance in the original SI. Thus, the dimensionality is transformed
from intensity of each energy loss channel at a given point to weight of each new
base component in that given point. The method is based on three major assumptions:
(i) the problem to be solved is linear, (ii) the signal has higher variance than noise
and (iii) there is component orthogonality. The linearity and orthogonality mean
that the separated components can be treated as a basis for the energy loss spectra
space (i.e. each spectrum can be expressed as a weighted sum of the components).
The higher variance components resolved by the algorithm are generally related
with meaningful features of the sample (e.g. thickness and elemental composition),
whereas the components of lower variance are usually associated with pure noise and
do not offer further information pertaining to the spatial distribution of the elements.
The problem with PCA, and the reason it is not regarded as a valid BSS by itself,
is that it relies in second order statistics (variance) to separate components. This
is, the separation of components in PCA is not based on physical considerations,
so they may have no physical meaning [42]. Thereby, they are also, in principle,
unsuitable for tomographic reconstruction, since they may fail to fulfil the projection
requirement. Nonetheless, as a noise reduction step, wPCA plays an important role
since the BSS algorithms described ahead are not to be applied to noisy datasets. It
also plays the important role of reducing the number of components in the calculation,
thus reducing the computational time required.
Independent component analysis (ICA) is the first possible method for BSS. It
deals with the data as a mixture of independent, and therefore uncorrelated, components. Those components are found according to their non-Gaussian distribution and
they should unveil physically meaningful components of the dataset [39, 40].
A second possible method for BSS is the Bayesian linear unmixing (BLU) [42,
43] approximation, described by N. Dobigeon. One reason to choose BLU over ICA
is that the latter has been shown to fail performing endmember extraction precisely
when the spectral sources (components or endmembers) are not statistically independent, a strict condition for the implementation of ICA [42, 44]. Besides, this
Bayesian formulation allows the introduction of several constraints for the calculations, such as (1) sparsity, (2) non-negativity and (3) full additivity. The case of the
EELS-SV [36] calculation can clearly benefit from the introduction of the number of
components, non-negativity nature of the signal, and the proportions limitation (i.e.
full additivity for the proportions of the components expected in the sample) into
the data treatment, as a way of increasing the endmember identification accuracy.
Alongside, Dobigeon model [42] is characterized by estimating the parameters of the
new basis in a lower dimension space identified by a standard dimension reduction
269
noise reduction are imperative. Besides, the problem requires the separation of the
contribution of each element to the spectra per pixel and their identification, which
can be regarded as a blind source separation (BSS) problem [20, 38–40].
First, the data (x, y, θ, ,E) needs to be treated to correct energy drift. Then,
weighted principal component analysis (wPCA) [38] is applied. The weighting of
the PCA is adapted to the dominant Poissonian noise. The PCA algorithm computes
a new spectral base for the spectrum image, where the base components are ordered
by the spectral variance in the original SI. Thus, the dimensionality is transformed
from intensity of each energy loss channel at a given point to weight of each new
base component in that given point. The method is based on three major assumptions:
(i) the problem to be solved is linear, (ii) the signal has higher variance than noise
and (iii) there is component orthogonality. The linearity and orthogonality mean
that the separated components can be treated as a basis for the energy loss spectra
space (i.e. each spectrum can be expressed as a weighted sum of the components).
The higher variance components resolved by the algorithm are generally related
with meaningful features of the sample (e.g. thickness and elemental composition),
whereas the components of lower variance are usually associated with pure noise and
do not offer further information pertaining to the spatial distribution of the elements.
The problem with PCA, and the reason it is not regarded as a valid BSS by itself,
is that it relies in second order statistics (variance) to separate components. This
is, the separation of components in PCA is not based on physical considerations,
so they may have no physical meaning [42]. Thereby, they are also, in principle,
unsuitable for tomographic reconstruction, since they may fail to fulfil the projection
requirement. Nonetheless, as a noise reduction step, wPCA plays an important role
since the BSS algorithms described ahead are not to be applied to noisy datasets. It
also plays the important role of reducing the number of components in the calculation,
thus reducing the computational time required.
Independent component analysis (ICA) is the first possible method for BSS. It
deals with the data as a mixture of independent, and therefore uncorrelated, components. Those components are found according to their non-Gaussian distribution and
they should unveil physically meaningful components of the dataset [39, 40].
A second possible method for BSS is the Bayesian linear unmixing (BLU) [42,
43] approximation, described by N. Dobigeon. One reason to choose BLU over ICA
is that the latter has been shown to fail performing endmember extraction precisely
when the spectral sources (components or endmembers) are not statistically independent, a strict condition for the implementation of ICA [42, 44]. Besides, this
Bayesian formulation allows the introduction of several constraints for the calculations, such as (1) sparsity, (2) non-negativity and (3) full additivity. The case of the
EELS-SV [36] calculation can clearly benefit from the introduction of the number of
components, non-negativity nature of the signal, and the proportions limitation (i.e.
full additivity for the proportions of the components expected in the sample) into
the data treatment, as a way of increasing the endmember identification accuracy.
Alongside, Dobigeon model [42] is characterized by estimating the parameters of the
new basis in a lower dimension space identified by a standard dimension reduction