
Deconvolution and fit of EEL spectra are widely used for removing spectrum background, separating the spectra generated from different elements, and removing artifacts in EELS signal:
i) Numerous background subtraction techniques have been
proposed to extract EELS signals. However, their use depends on the shape of the EELSspectrum
and/or the energy of the edge under consideration. Powerlaw fit is applicable to most ionization edges.
ii) Deconvolution consists of estimating O_{i}(E)for each element i in the materials. In general, we can view the observed EEL spectrum I(E) as
 [3936]
where,
O(E)  The single scattering EELS spectrum (SSD),
P(E)  The point spread function (PSF) including multiple scattering and instru
mental broadening,
N(E)  The additive noise.
Tables 3936a and 3936b show common methods for such deconvolution and background fit functions. Some are ‘smart’ algorithms to address some particular problems in specific cases in background fit and edge deconvolution.
Table 3936a. Techniques used for EEL spectrum background fit.
Techniques 
Characteristics 
Powerlaw fit 
Most common and simplest background fit technique. Remove EELS background by fitting in a region preceding the excitation edge. Applicable to most ionization edges (note: does not work on low energy edges) 
Exponential fit

Fit to low energy loss regions at moderate thicknesses, e.g. for background subtraction under
the NbM_{2,3} edges which is complicated by the intense tail of the
preceding NbM_{4,5} edges.[12] Works without the assurance that the energy dependency of the background does not vary under the
edge. 
Experimental background fit

Use an experimental spectrum as a reference to extract the background of other similar spectra, [13] it is used when no other better choice is available. 
Four window method 
To remove background for elemental mapping with overlapping or closely spaced ionization edges, e.g. to map Cr with the L_{2,3}edge in oxidized specimens [10] 
Linear fit 
Fits over small energy window when edges
overlap 
(Log)polynomial
function 
Fit background, e.g. for background subtraction under
the NbM_{2,3} edges which is complicated by the intense tail of the
preceding NbM_{4,5} edges.[12] It is used when no other better choice is available. Works without the assurance that the energy dependency of the background does not vary under the
edge. 
Differentiation 
Record two spectra with small relative shift along the energy axis, and then obtain the difference of the two spectra 
Multivariate statistics
approaches 

Thirdorder polynomial 
Remove EELS background: work on edges below the
Fe M_{2,3}edge (49 eV) [9] 
Table 3936b. Techniques used for EEL spectrum deconvolution.
Techniques 
Characteristics 
Fourier ratio method

Remove plural inelastic scattering contributions [11], and it is a highfrequency enhancement technique, then highfrequency noise in the coreloss spectrum is substantially amplified 
Fourierlog deconvolution 
Remove multiplescattering artifacts 
Fourier techniques 
Multiple scattering deconvolution 
Multiple linear leastsquares (MLLS) fitting 
Deconvolute overlap of edges, e.g. O K and Cr L_{2,3 }[8] 
MLLS fitting and kmeans clustering 

Straightforward deconvolution 
Very limited application [11] 
Gaussian modifier 
Deconvolution by damping the higher frequencies with a Gaussian function 
Wiener filter 

Maximumentropy (ME) 
Not significantly affected by highfrequency noise 
Richardson–Lucy (RL) algorithms 
Not significantly affected by highfrequency noise 
Singular
value decomposition (SVD) 
E.g. used for the MLLS fit routine
implemented in the DM software 
In order to improve the energy resolution of EEL spectrum, before spectrum deconvolution, scripts (e.g. applied in Gatan Digital Micrograph) can be used to automatically acquire and store each spectrum separately, and then to evaluate and correct the energy drift in each acquisition. [1, 2] After the drift correction, the EEL spectrum can be deconvoluted using software. For instance, some deconvolution techniques are Fourier ratio method, maximumentropy (ME) [3,4] and Richardson–Lucy (RL) algorithms [5]. In the Fourier ratio method, the "ideal" coreloss spectrum is obtained by inverse Fourier transform. Since the Fourier ratio deconvolution is a highfrequency enhancement technique, highfrequency noise in the coreloss spectrum is substantially amplified. The ME and RL algorithms estimate a predicted spectrum by convoluting it with an observed lowloss spectrum, and thus they are not so significantly affected by highfrequency noise as compared with the Fourier ratio deconvolution technique.
For thick TEM specimens, an incident electron that has undergone innershell scattering can also cause outershell excitation with very high probability. This mixed inelastic scatterings is the sum of both innershell and outershell scatterings, resulting in a mixed energy loss and a peak broadening (plasmon behavior) above the ionization threshold. If necessary, this outershell scattering intensity can be removed from the spectrum by deconvolution.
In general, if the TEM specimen is too thick (t/λ > 0.4), a deconvolution process must be employed to remove the effect of plural scattering, since the increase of plural scattering intensity in the higher energy region of an ionization edge can cause some artifacts:
i) Mask the fine structure;
ii) Make the background signal on subsequent edges deviate significantly from the power law model.
In this case, in order to deconvolute the coreloss spectrum, the Fourierratio method is applied. The deconvolution procedures then are:
i) Collect both the low and coreloss spectra from the same region of the specimen under the same conditions (including eV/change, convergence and collection semiangles).
ii) Isolate the edge of interest and remove the background intensity.
iii) Fouriertransfer the lowloss spectrum and backgroundsubtracted edge.
iv) Divide the coreloss spectrum Fourier transform by the lowloss Fourier transform.
v) Inverse the Fourier transform to yield the desired deconvolved spectrum.
However, in most cases, recording lowloss and coreloss spectra under the same conditions is extremely challenging, since the acquisiton time required for a good SNR (signal to noise ratio) in the coreloss spectrum is usually not short enough to avoid saturation of the signal from the ZLP. Therefore, in practice, it is necessary to sacrifice the SNR in the coreloss signal, or utilize a spectrometer system that has an ultrafast electrostatic shutter installed.
The low loss region in Figure 3936 shows the overlap between FeM edge and LiK edge in an EEL spectrum taken from discharged FeOF materials in a Liion battery. The extracted LiK edge was obtained by deconvolution technique.
Figure 3936. EEL spectrum taken from discharged FeOF materials in a Liion battery. Adapted from [6] 
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