Bayesian equivalent of Fourier-ratio deconvolution

R.F. EGERTON; F. WANG; M. MALAC; M.S. MORENO; F. HOFER

Edmonton, Canada

Congreso; Annual meeting of the Microscopical Society of Canada; 2007

Fourier-ratio deconvolution is a technique used in electron energy-loss spectroscopy (EELS) to remove the effect of plural scattering that involves two distinct processes (plasmon and inner-shell excitation) where one of these (the plasmon excitation) can be measured separately, as a low-loss spectrum. It involves treating the latter as if it were an energy-broadening function, describing the energy resolution of the spectrometer system. The measured spectrum is then a convolution of this function with an ideal spectrum, recorded using a spectrometer with perfect energy resolution and from a vanishingly thin specimen, in which plural scattering would be negligible. Transforming to Fourier space, the convolution becomes a product and the ideal spectrum is obtained by dividing Fourier coefficients, followed by an inverse transform. The procedure is effective in removing plural scattering but is of limited use for improving the energy resolution because high-frequency spectral noise becomes amplified unacceptably for improvement factors greater than about 2.  By adding constraints such as the requirement for positive electron intensity, Bayesian procedures (maximum-entropy or maximum-likelihood deconvolution) promise greater resolution enhancement and have been used successfully in astronomy and more recently in EELS [1-5].  However, these are iterative procedures and as the number of iterations increases, so does the noise and the occurrence of spectral artifacts [5,6]. The zero-loss peak can be used as the resolution function (or kernel) but it has to be recorded separately (with specimen displaced out of the electron beam) or extracted from the low-loss spectrum, and both of these procedures involve approximations. However, if we use the entire low-loss spectrum as a resolution function, as in Fourier-ratio deconvolution, these procedures are avoided. Moreover, it should be possible to correct core-loss data for both instrumental broadening and plural scattering simultaneously. The success of this procedure is illustrated in Fig.1, where it is shown that Fourier-ratio and Bayesian deconvolution (using the same low-loss kernel) provide the same result, except for details of the noise. This particular example involves removal of plural scattering from a high-energy edge (Ti-K, threshold ~ 5keV) recorded from a thick sample, where plural scattering greatly distorts the edge shape (black curve in Fig. 2) and makes conventional quantification procedures difficult. An additional test is to re-convolve the derived single-scattering profile (blue curve) with the low-loss region, in which case the broadened edge is successfully recovered (red curve in Fig. 2).