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Ultramicroscopy 120 (2012) 2534
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Spectral mixture analysis of EELS spectrum-images
Nicolas Dobigeon a, Nathalie Brun b,n
a University of Toulouse, IRIT/INP-ENSEEIHT, 2 rue Camichel, 31071 Toulouse Cedex 7, Franceb University of Paris Sud, Laboratoire de Physique des Solides, CNRS, UMR 8502, 91405 Orsay Cedex, France
a r t i c l e i n f o
Received 10 March 2011
Received in revised form
18 May 2012
Accepted 23 May 2012Available online 1 June 2012
Electron energy-loss spectroscopy (EELS)
Multivariate statistical analysis
Spectral mixture analysis
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esponding author. Tel.: 33 1 69 15 53 80.ail address: email@example.com (N. Brun
a b s t r a c t
Recent advances in detectors and computer science have enabled the acquisition and the processing of
multidimensional datasets, in particular in the field of spectral imaging. Benefiting from these new
developments, Earth scientists try to recover the reflectance spectra of macroscopic materials (e.g., water,
grass, mineral typesy) present in an observed scene and to estimate their respective proportions in eachmixed pixel of the acquired image. This task is usually referred to as spectral mixture analysis or spectral
unmixing (SU). SU aims at decomposing the measured pixel spectrum into a collection of constituent
spectra, called endmembers, and a set of corresponding fractions (abundances) that indicate the proportion
of each endmember present in the pixel. Similarly, when processing spectrum-images, microscopists
usually try to map elemental, physical and chemical state information of a given material. This paper
reports how a SU algorithm dedicated to remote sensing hyperspectral images can be successfully applied
to analyze spectrum-image resulting from electron energy-loss spectroscopy (EELS). SU generally over-
comes standard limitations inherent to other multivariate statistical analysis methods, such as principal
component analysis (PCA) or independent component analysis (ICA), that have been previously used to
analyze EELS maps. Indeed, ICA and PCA may perform poorly for linear spectral mixture analysis due to the
strong dependence between the abundances of the different materials. One example is presented here to
demonstrate the potential of this technique for EELS analysis.
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Over the two last decades, scanning transmission electronmicroscopy (STEM) has benefit from important advances in elec-tron-based instrumentation and technology. These recent advanceshave enabled the development of electron energy-loss spectroscopy(EELS). EELS provide spectrum-images, that have been widely usedin various applications, including material science and chemicalanalysis [1,2]. The multidimensional data coming from EELS analysisexploit inherent spatial information to build elemental maps. Anelemental map is useful per se, however it does not exploitadditional crucial information present in the acquired spectrumimage. As EELS signal is sensitive to chemical changes and atomenvironment, building a map of the different materials would bemore much more relevant. Therefore, there is a real need forefficient techniques to process EELS spectrum-images, able toidentify and quantify the spectral components that represent thedifferent compounds present in the imaged sample.
Attempts to extract information from EELS spectra wereconducted in 1999 mainly based on multivariate data analysistechniques, specifically principal component analysis (PCA) .A PCA-based method was written for DigitalMicrograph and
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commercialized by Ishizuka in 2001  and is now rather widelyused for data filtering and dimensional reduction . However,such analysis faces the difficulty of extracting physically mean-ingful spectra from the computed eigenvalues.
Conversely, independent component analysis (ICA) aims atidentifying statistically independent components from multivari-ate data. In 2005, Bonnet and Nuzillard  applied the ICA-basedSOBI algorithm to process spectrum image data set. The authorsnoticed that, since EELS spectra are not composed of separatedpeaks, the independence hypothesis is not fulfilled. To overcomethis issue, successive derivatives of EELS-spectra are analyzed.From this analysis, it seems that first derivatives produce moreinterpretable results than second derivatives. Unfortunately, thisfinding was empirical and no theoretical argument was found tojustify this point. De La Pena proposed in  to use a kernelizedversion of ICA. This approach allows C, SnO2 and TiO2 signals to besuccessfully separated while analyzing a spinodally decomposedsolid solution. Satisfactory quantitative analysis was obtained butno fine structure analysis was performed. The authors noticedthat difficulties could be encountered because of multiple scatter-ing and energy instabilities introducing non linearity.
Recently a matrix factorization technique has been proposedto map plasmon modes on silver nanorods . The analysis,relying on the software AXSIA developed by Keenan  consists inlooking for a rotation matrix to be applied on orthogonal factorsto maximize the intrinsic simplicity of the decomposition.
Fig. 1. Geometrical formulation of spectral mixture analysis (SMA). The scatter-plot represents the data observed in a 2-D space. The mixed pixels (gray circles)
belong to the simplex (simplest geometric figure that is not degenerate in
n-dimensions), whose vertices are the 3 endmembers. SMA algorithms exploit
different properties of the simplex.
N. Dobigeon, N. Brun / Ultramicroscopy 120 (2012) 253426
Specifically, the optimal solution is defined by the sparsity of thespatial distribution of each individual material.
In a significantly different area namely remote sensing andgeoscience reflectance spectroscopy is widely used to charac-terize and discriminate materials on the Earth surface for variousapplications . Usually mounted on aircrafts, balloons orsatellites, spectral sensors collect electromagnetic radiations fromthe Earth surface. Most of the recorded signals are reflectancespectroscopic signals measured in the infra-red/visible range.The collection of these signals over an observed scene providesa multi-band image formed as a 3-dimensional data cube. Eachpixel of the atmospheric-corrected image is characterized by avector of reflectance measurements. Specifically, hyperspectralimages are composed of pixels with several hundreds of narrowand contiguous spectral bands.
Faced with this amount of data, the geophysicist community hasdeveloped analysis methods to extract physical information fromthese images. One of the main objectives of these methods is toidentify spectral properties corresponding to distinct materials in agiven scene and thus to get classification maps of the image pixels.However, because of the intrinsically limited spatial resolution ofthe hyperspectral sensors, several materials (e.g., water, grass,mineral typesy) usually contribute to the spectrum measured ata given single pixel. The resulting spectral measurement is acombination of the individual spectra that are characteristic of themacroscopic materials. Consequently, techniques to estimate theconstituent substance spectra and their respective proportions frommixed pixels are needed. Spectral unmixing is the procedure thataims at (i) decomposing the measured pixel spectrum into acollection of constituent spectra, or endmembers, and (ii) estimatingthe corresponding fractions, or abundances, that indicate the propor-tion of each endmember present in the pixel .
What is usually known as )spectrum image* in the microscopistcommunity corresponds very precisely to a )hyperspectral image* forthe geoscience-related applications. The analogy between these twofields of research is undeniable. However, at the present timemicroscopists are less advanced in their ability of conducting efficientmultivariate analysis of their data. In this work we describe how arecent spectral unmixing algorithm developed by Dobigeon et al. for analyzing hyperspectral images can be successfully applied tospectrum images resulting from EELS maps.
2. Methods and experimental setup
2.1. Spectral mixture analysis
This paragraph formulates the so-called spectral unmixing orspectral mixture analysis. Let Y denote the L by N observed datamatrix that gathers the whole set of N measured pixel spectra. Eachcolumn of the Y is a vector of size L which corresponds to thereflectances measured in the L spectral bands. The spectral mixtureanalysis (SMA) conducted on the spectrum image consists of decom-posing this matrix Y into a product matrix SA. In this decompositionscheme, each column of the L by R matrix S is the spectral signatureof a constituent (endmember). Conversely, each column of A is a setof R coefficients corresponding to the relative proportions of thesignatures in the pixels. Thus, like any factorization matrix method,SMA estimates the two latent variables S and A leading to theproduct SA that best approximates the observed matrix Y. Since