Scanning Microscopy

Volume 1992 Number 6 Signal and Image Processing in Article 16 Microscopy and Microanalysis

1992

Multi-Dimensional Data Analysis and Processing in Electron- Induced Microanalysis

N. Bonnet Université de Reims, France

P. Trebbia Université de Reims, France

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Recommended Citation Bonnet, N. and Trebbia, P. (1992) "Multi-Dimensional Data Analysis and Processing in Electron-Induced Microanalysis," Scanning Microscopy: Vol. 1992 : No. 6 , Article 16. Available at: https://digitalcommons.usu.edu/microscopy/vol1992/iss6/16

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MULTI-DIMENSIONAL DATA ANALYSIS AND PROCESSING IN ELECTRON-INDUCED MICROANALYSIS

N. Bonnet• and P. Trebbia 1

Unite INSEAM 314, Laboratoire de Microscopie Electronique (Universite de Aeims), 21 rue Clement Ader, 51100 Aeims, France 1LASSI (Universite de Aeims), B.P. 347, 51062 Aeims Cedex, France

Abstract Introduction

The new facilities offered by computer Due to the increasing facilities offered by controlled data capturing devices allow one to computer controlled data capturing devices, open the field of data acquisition from one or microanalysts and microscopists are now two-dimensional spaces to multi-dimensional allowed to open the field of data acquisition, ones. The methods used for analysing and moving from a one-dimensional space (e.g. processing such data sets have to move in spectrum l(E)) or a two-dimensional space parallel towards multi-dimensionality. Multi­ (e.g. image: l(x, y)) to higher dimensional variate Statistical Analysis is one of the tools spaces. This is the case, for example, for time­ which appear to be promising in : data analysis, dependent spectroscopy (l(E, t)) or for the data reduction, data processing (multivariate recording of image sequences either for noise filtering), data interpolation and studying dynamic processes (l(x, y, t)) or the extrapolation. Illustrations of these different location of a chemical species in a specimen possibilities are given in the fields of (l(x, y, E)). Even more sophisticated proce­ spatially resolved spectroscopy, time­ dures can be expected in the near future (time dependent spectroscopy and elemental mapping dependent energy filtered series, three­ from Electron Energy Loss Spectroscopy. dimensional chemical mapping). In conventional microscopy also, three-dimensional data may be recorded, either for performing 3D­ reconstruction (confocal microscopy, electron tomography, X-ray microtomography) or for studying dynamic events. Facing such a large amount of data, the question about the most appropriate tools for analysis and processing arises. Besides the extension of "standard" methods developed for mono-dimensional data, some tools speci­ fically developed for multi-dimensional data sets can be valuable. Key Words: Multivariate statistics, time and One such tool is Multivariate Statistical space series, elemental mapping, data Analysis (MSA). This technique was developed processing, microanalysis at the beginning of the century to help to interpret data sets in the fields of Sociology * Address for correspondence: and Econometry. It was introduced in the N. Bonnet seventies for applications in chemistry Unite INSEAM 314 (Universite de Aeims) (Malinowski and Howery (1980)) (mass 21, rue Clement Ader spectrometry, chromatography, spectro­ 51100 Aeims, FrancE;l photometry, etc.), and in the eighties for Telephone Number: 33-26-05-07-52 applications in Auger spectroscopy FAX Number: 33-26-05-19-00 (Garenstrom (1981, 1986), Browning (1987),

163 N. Bonnet and P. Trebbia

Prutton et al. (1987, 1990), Kenny et On the other hand, if any concentration al.(1992)), in X-Ray spectroscopic imaging variation between two probed locations exists, (King et al.(1989), Paque et al.(1990)) and in the two related spectra are not proportional high resolution electron microscopy (Van Heel and this situation is similar to two and Frank (1980, 1981 ), Frank and Van Heel independent vectors generating a vectorial (1982), Savoji and Burge(1983)) where it has space of dimension two. mainly been used for analysing a data set made MSA benefits from such a property. Each of similar (but not completely equivalent) measurement (i.e. a spectrum of 1024 views of a given object and for the automatic channels) is considered as a single vector partitioning of this complex data set into (with 1024 components}, the whole data set, several homogeneous subsets. It was later the N spectra, being a collection of N vectors. suggested that this technique could also be These N vectors are gathered in a single matrix well suited to the processing of energy filtered (N by 1024 in our example) and MSA merely image sequences used for elemental mapping finds the characteristics of the vectorial space (Hannequin and Bonnet (1988), Bonnet and generated by this matrix looking for the Hannequin (1989), Trebbia and Bonnet (1990), eigenvectors (the so-called factorial axes) Trebbia and Mory (1990)). enables one to find whether some information It was shown that MSA could be useful for is redundant (any couple of proportional performing several different tasks data spectra wou!d pertain to the same eigenvector, analysis, data processing (filtering) and also thus reducing the dimension of the vectorial data interpolation and extrapolation. These space generated by the whole data set) and to different possi-bilities are briefly described classify hierarchically the non-proportionality in the next subsection. between spectra (peak variations, noise MSA : Basics and Possibilities fluctuations, ... ) by looking at the eigenvalue associated to each eigenvector. Although it may appear to be some "magical The details of the calculations performed by speculation" to non-specialists, MSA is based MSA can be found elsewhere (Trebbia and on a solid mathematical background and relies Bonnet (1990)) and we would like to point out on a very simple idea which is summarized some comments here: hereafter. 1) MSA is an analysis procedure. Let us suppose that one has recorded several a) It gives the number N' of factorial axes. If measurements pertaining to a single N' < N, then some measurements are redundant experiment for example, a collection of N EDX and convey no special information, but the fact spectra obtained at different locations on a that some kind of stationarity has been given specimen. If no elemental concentration observed. variation between these different sampling b) Provided that the original matrix of data areas on the specimen exists, then all these has been properly set, the eigenvalue spectra should exhibit the same quantitative associated with each eigenvector is a direct features (peak locations, peak/background measure of the relative importance of the ratios). In other words, they should be specific information carried by this factorial proportional to each other (the only varying axis. parameter being the acquisition time) and, c) MSA gives the coefficient of the linear from a mathematical point of view, they are combination needed to build, from the original fully dependent from any given spectrum of data set, the eigenvector set. In other words, that collection, one can build any other one one gets the coordinates, also called weights, (within statistical error) provided that one on the factorial axes of each one of the original knows the arbitrary multiplicative coefficient spectra. These coordinates are nothing but the (integrated intensity ratio) to be used as a scaling factors discussed above. Note that, in scaling factor. The situation is identical to a the general case, these coordinates may be collection of parallel vectors with different positive or negative, depending on the location, norms all these vectors have the same basis, in the N' - dimensional vectorial space, of the that we shall refer to later as a factorial axis, point taken as origin. but different scaling factors containing the 2) Looking at these results (eigenvalue for information on the magnitude. Each one of each eigenvector and coordinates of each these scaling factors is the coordinate of the measurement), the user has to face the relevant vector on the factorial axis. interpretation of the results and to give a

164 Multi-dimensional data processing in microanalysis physical meaning to the information conveyed one to find an appropriate model describing as by the factorial axes. It may be of use to look closely as possible the observed trends, and to at the projections of the whole data set on use this model for the estimation of the each factorial axis: a close inspection of which coordinates of an intermediate point. This channels in the spectra are linked to a specific "interpolation" facility, leading to the factorial axis is of great value for making a construction of a fictitious (i.e. not actually decision. recorded) measurement only requires two 3) Once the interpretation of MSA results has assumptions i) the validity of the model and been made, one may go to the next step : the ii) the continuity of the model in the processing of the original data. For example, unexplored area of the diagram. a) if one wants to discard a peculiar kind of e) the "extrapolation" facility, that is the information which has been interpreted as construction of a fictitious image beyond the noise or artefact, one has merely to reduce the limits of the actual diagram, is obviously a dimension of the vectorial space, removing any little bit more hazardous since the two unneeded factorial axis. Each measurement is previous assumptions, which must also be then restored in this sub-space. This process is fulfilled, are more speculative there is no analogous to applying a specific filter, the type way, with MSA, of confidently predicting that of filter being determined by the number, the there is no fatal break in the model at the importance and the physical meaning of the frontiers of the diagram. This point will be factorial axes which are taken into account. discussed in more detail in the section devoted Examples of such a process (reconstitution to the zero-dose extrapolation. with a reduced number of factorial axes) can be To conclude this subsection, we would like to found in (Bretaudiere and Frank (1986)). emphasize that the efficiency of MSA, when b) discarding a part of information which has compared to other monovariate procedures, been evaluated as being of less-importance can precisely relies on the fact that the analysis is also be seen as data compression. Let us performed on the complete data set, suppose, for example, that a collection of N constructed as a whole. The validity of the measurements gives rise to only N' analysis is therefore more established if many "significant" factorial axes, the word measurements are taken into account. The "significant" being determined through the higher the number of measurements (N in our interpretation step 2) above. Then, the example), the greater is our knowledge on the reconstruction of the whole data set with only possible links between them, and, for example, these N' "useful" factorial axes would lead to a the more accurate the model used for compressed data set from which any "useless" interpolation. information has been removed. But one must never forget that MSA is mainly c) one can build a scatter diagram an analysis tool, the validity of the processing (Jeanguillaume(1985), El Gomati et al.(1987)). tasks being highly dependent on the extra­ It consists in representing an element (pixel, assumptions needed by the process. energy channel) by a point in the N'­ dimensional vectorial space (generally, N' is MSA in Microanalysis: restricted to 2 or 3). The observation of this Some Possible Applications scatter diagram can be of great help in the interpretation of data the gathering of data In the present paper, we would like to within restricted space volumes would lead to investigate some possible applications in the automatic classification algorithms. MSA is field of electron induced microanalysis, i.e., thus able to discriminate among all the data electron induced spectroscopies (energy which pertain to a specific space volume, a dispersive X-ray microanalysis, EDX; Auger cloud, sharing the same specific properties, electron spectroscopy, AES; electron energy this specificity being determined by the loss spectroscopy, EELS; etc.) and elemental coordinates of the gravity center of the cloud. mapping. d) in the same way, one may look at this Spectroscopy: Classical spectroscopy scatter diagram and try to find some trends consists of recording the signal induced by a between selected data (linear evolution of physical event as a function of energy (or their coordinates along a specific axis, for wavelength). Improved spectroscopic example). Therefore, such an analysis enables techniques consist of repeating several such

165 N. Bonnet and P. Trebbia experiments for different values of another context of elemental mapping. The first one parameter (time, space location, emergence consists of the several images required to angle, etc.) leading to the following (non build a single elemental map, when using exhaustive) classification : electron energy-filtered images for instance. Time dependent spectroscopy is a method It is well-known that one single image, which is beginning to be used in several recorded at the energy-loss characteristic of a laboratories where it benefits from the given element, cannot be considered as possibilities offered by parallel detection representative of the true concentration of spectrometers. Its goal is to study dynamic that element. The reason is that the phenomena such as the diffusion of ions under characteristic ionization signal (which is the electron beam, the nucleation process of proportional to the true concentration) is precipitates in materials (see for instance superimposed on a large background which is Ellis et al.(1985)). Such experiments have non characteristic but depends on several already been performed by Craven et al. (1989) phenomena (specimen thickness, presence of and Tence et al. (1990) for instance. Though the other species). It is also widely accepted that data sets produced by time-resolved even two images (one recorded below the spectroscopy can be processed by techniques characteristic energy-loss edge and one developed in classical spectroscopy recorded above the edge) are not sufficient. (background modelling and subtraction, peak Therefore, image sequences made of several area estimation), we believe that many tasks images (below and above the edge) are required can be solved much more conveniently by in order to compute an elemental map. The processing the whole data set at once with standard method used to perform this MSA. A preliminary investigation of the computation is based on a modelling of the possible applications of MSA to time­ background (below the edge) and a subtraction dependent spectroscopy was reported (Bonnet of the extrapolated background (above the edge) et al. (1991 )). A difficult but exciting (Jeanguillaume et al. (1978), Bonnet et challenge opened by MSA is the possible al.(1988)). Since this method sometimes extrapolation to zero-dose (Bonnet and suffers from drawbacks, the possible Hannequin (1988)). We will show that, in applications of MSA to these image sequences favorable situations, MSA seems to be one were investigated over the last three years. A appropriate method to perform this task. first method consisted of submitting the whole Spatially (or angular) resolved data set to MSA. A new alternative is to only spectroscopy is also beginning to be applied process the images below the edge and then, by in several labs. The space parameter can be the extrapolation in the factorial space, to depth in the specimen or a lateral dimension compute a "fictitious" background for energy (Disko et al.(1991) for instance). In the former losses above the edge. These possibilities will case, one speaks of depth profiling, a technique be described in the last section. which is sometimes used in Auger The other kind of image sequences spectroscopy, in addition to SIMS, in order to encountered in the field of elemental mapping get estimations of the concentration of a given consists of several maps (for different element as a function of depth. The latter case elements) of the same specimen region. This concerns studies of concentration variations situation can be found for any kind of along an interface for instance. Here again, the spectroscopy: X-Ray imaging, Auger mapping, spectrum sequence could probably be processed EELS mapping. In this case, the different more efficiently by multivariate methods than images (especially those of low concentration by a monovariate method applied sequentially elements) are often rather noisy. It is of to each individual spectrum. As an example, we course possible to improve the signal-to-noise will show how the concentration variations of ratio by applying standard processing one or several elements can be estimated techniques (Fourier filtering, etc.) to each without any need to model the background or image separately. But this procedure cannot the characteristic peaks, provided that some take the often important correlations which additional information is available concerning exist between the different maps into account. the spectra of pure elements. Here again, MSA is an appropriate method for Imaging : Image sequences are also used in performing such a task, but this point will not the field of microanalysis. Two different kinds be considered any more in this paper. of image sequences are in fact recorded in the

166 Multi-dimensional data processing in microanalysis

,------:::,,....0 10 8 9 .... ·x-"' ~ -; ·.: .B ~'"' C 0 "'., --; :aC.. 0 u0 0 Position within the interface

~ : Coordinates of the different spectra of the simulated series (see Fig. 1) on the first factorial axis, after MSA. In this favorable situation, this graph plots directly the law of evolution of the concentration parameter x as a function of the spatial parameter (here a cosinusoidal shape). Furthermore, the concentration in element A can be directly read Energy loss as its coordinate on axis 1 provided that this

~ Four of the spectra which compose a axis has been previously scaled from 0 simulated spatially resolved spectrum series. (spectrum without element A) to 1 (spectrum The concentrations in element A (lower corresponding to pure element A). For instance, characteristic energy loss) and in element B the coordinate of spectrum number 3 is 0. 79, are supposed to be anti-correlated, as it would which corresponds to the simulated value 0.5 be the case for a compound Ax B 1 -x for (1 +cos3n/10). instance, where x is a spatially-varying parameter. The four spectra are supposed to correspond to the compounds A1 Bo (n=0), quantifying the con-centration variations. For this purpose, we have built two simulated Ao.798021 (n=3), Ao_35Bo.65 (n=6) and AoB 1 spectrum sequences, each spectrum being (n=10). composed of a decreasing background, two overlapping Gaussian peaks and Poisson noise. Quantification of Concentration Variations First simulation perfectly correlated in Spatially Resolved Spectroscopy concentrations We suppose that we are facing a binary As stated in the introduction, this technique compound AxB _x• with x varying across an provides data sets composed of a series of 1 interface from x=1 to x=0, according to a spectra l(E) obtained at different locations in cosinusoidal law for instance. Eleven spectra the specimen. Generally, the specimen sites (n=0 to 10) are simulated and four of them explored are along a vertical direction (depth (corresponding to A B , A_ B_ , A_ B_ and profiling) or along a lateral direction, normal 1 0 79 21 35 65 A B ) are shown in Fig. 1. The MSA results can to an interface for instance. Quantifying the 0 1 concentrations of the different elements by be summarized as follows "standard" techniques would rely on background - only one factorial axis is significant, due to removal, peak area estimation ... The problem the fact that the concentration variation for becomes more difficult when several elements elements A and B are perfectly anticorrelated, give rise to overlapping peaks since in this - the coordinates of the spectra on the main situation, deconvolution or modelling must be factorial axis (number 1) display the shape of performed. In this section, we would like to the model which describes the concentration show that MSA applied to the whole spectrum variations, here a cosinusoidal model (see Fig. series could help in understanding and 2),

167 N. Bonnet and P. Trebbia

- the concentration in element A (or B) can be A B directly obtained from this figure, provided that concentrations are known for at least two experimental spectra. (In this situation, the acquisition of two spectra far away from the zone of variation gives these references): the factorial axis has only to be calibrated (in concentration units) and then the ordinate of any spectrum immediatly gives the co nee ntratio n. Second simulation partly correlated concentrations. If we suppose that the concentration variations of the two (or more) elements to be considered are no longer perfectly anti­ correlated, the situation becomes slightly Fig. 3 Eigen-spectra corresponding to a more complex because we have to work in a simulation of spatially resolved spectroscopy two-(or more) dimensional space. This is where the concentrations of two elements (A illustrated through another simulation. and B) are not exactly anti-correlated. In this Crossing an "interface", the concentration of case, the first eigen-spectrum (i.e. the element A is supposed to decrease exponen- component connected to the first eigen-value 0 5 tially as e· · n (n=0 to 10) and the in the matrix decom-position) corresponds to concentration of element B is supposed to the anti-correlated part of the concentration increase linearly as n/10. As in the previous variations and the second eigenspectrum simulation, the characteristic peaks related to corresponds to the correlated part. each element are supposed to be overlapping Gaussians with different widths. Now, MSA results can be summarized as - two factorial axes are significant because the concentration variations are no longer 10 exactly anti-correlated. These two main axes explain 99% of the total variance observed 9 between the eleven spectra. The nine remaining axes have been stated as non significant. 8 Factorial axis 1 is representative of the anti­ correlated part of the variations whereas factorial axis 2 represents the correlated part. Axis 1 This can be confirmed by the observation of the 6 factorial components, i.e. of the coordinates of 2 5 the energy channels on the factorial axes (see Fig. 3). - The coordinates of the spectra must be .Elg_,___1 Coordinates of the different spectra observed in a two-dimensional factorial space (second simulation) in the factorial subspace (axes 1 and 2) (see Fig. 4). generated by axes 1 and 2. - However, the coordinates of the different spectra in this representation space are not directly proportional to the concentrations in - The situation can be greatly improved if one elements A and B. The reason is that the is able to add to the data set submitted to MSA decomposition of the data set into orthogonal a spectrum corresponding to such a situation. components has no direct physical meaning (in In real world experiments, this should be done other words, axis 1 and 2 are not directly either by the acquisition of such a spectrum, if connected to elements A and B). Furthermore, there exists in the specimen regions where the point used as the origin of the factorial neither element A nor element B are present. space has no reason to correspond to a physical Or, more probably, this could be done by situation where the concentration in elements modelling the background in the experimental A and Bare nil (CA= C = 0). 8 spectrum sequence. In the present simulation,

168 Multi-dimensional data processing in microanalysis

fulfilled for its use in experimental situations. These requirements are the need to define precisely the locations of the representative points O', A and 8 in the factorial space. Fixing points A and 8 implies that there are regions in the specimen where element A (respectively 8) can be found alone. Furthermore, the fact that the quantification can be performed without taking into account the background comes from the hypothesis that the background does not change from one spectrum to another, including the "reference" spectrum (number 11 in our simulation). There is no doubt that in many experimental situations, one or several of these requirements cannot be fulfilled and that the application of such a technique would be ~ : Coordinates of the different spectra in the new factorial subspace (axes 1 and 2) when more questionable than in these simulations. one spectrum (without any characteristic peak corresponding to element A or 8) has been Extrapolation to Zero-Dose added to the previous series. This additional in Time-Resolved Spectroscopy spectrum (number 11) defines the origin O' of coordinates in a new (oblique) factorial space In a previous paper (Bonnet et al. 1991 ), (axes 1' and 2') which can then be used, after we tried to define the possible applications scaling, to read directly the concentrations in of MSA to time-resolved spectroscopic data, elements A and 8. For instance, the relative i.e., to spectrum sequences recorded as a coordinates of spectrum number 1 are 0.09 and function of time: l(E,t). We have found that 0.63 on axes 1' and 2' respectively. These the first goal of applying a global multivariate values agree closely with the simulated values technique is to carefully analyse the data set. (0.10 and 0.61 respectively). Indeed, the more sophisticated the acquisition procedure, the more important the risks of artefacts. But we have shown that MSA is able it was of course easy to add a simulated to check the consistency of a data set and to background in the data set. depict artefacts such as primary current The weights (or coordinates) of the spectra modification (see also Barkshire et al. (1991 )), on the two new main factorial axes after spectrometer drift.. We have also shown that, introducting the "reference" spectrum 11 for consistent data, MSA is ab!e to plot the (concentrations of A and 8 nil) are displayed in evolution of the sequence, to find the number Fig. 5. of information components. The additional point O' in the factorial space After analyzing the content of the data set, can then be used as an origin for a new set of MSA can be used to process it by filtering the modified factorial axes (O'B, O'A), where A is information components and reconstituting the the point representative of CA = 1 ; CB = 0 and 8 data set with only the desired components. corresponds to CA = O ; CB = 1. (This procedure Besides reconstituting the data set with its is a particular realization of the so-called true weighting coefficients on the factorial "oblique analysis" described by Hannequin and axes, it is also possible to perform the Bonnet (1988)). reconstitution with other (interpolated or Now, it is possible to obtain the extrapolated) coefficients, thus leading to concentrations for any spectrum by "fictitious" spectra. This idea is at the origin determining the coordinates of the different of the extrapolation to zero-dose in representative points on the new factorial axes spectroscopy (Bonnet and Hannequin (1988)). (O'B, O'A). This is illustrated in Fig. 5. Note We must say that this idea did not lead to Though the procedure described above for the much experimental verification up to now, quantification of mixed elements spectra maybe because it is not completely mature. In appears very fascinating at first sight, it must the present paper, we would like to check by a be stressed that several requirements must be simulation an idea suggested by Cazaux (LASS!,

169 N. Bonnet and P. Trebbia

Reims; personal communication) in order to 5 increase the chances of success of such an 4 extrapolation to zero-dose. The problem is that, even if MSA may help in determining a model for the evolution of an element in a time - resolved experiment, the extrapolation of the model outside the experimental domain (in forecasting as well as exploring towards the past) requires the introduction of hypotheses, the weakest of which being the hypothesis of continuity. In order to avoid the use of strong (and maybe unjustified) hypotheses, it would be interes­ ting to get not only one evolution model but several models supposed to converge towards a ~ : Coordinates on the factorial axis 1 of unique origin. the 15 spectra simulating an experiment in To perform time-resolved spectroscopy time-dependent spectroscopy. The horizontal means the realization of several experiments axis represents the time parameter. The three (on several equivalent specimen regions) at spectrum sets (a,b,c) correspond to three different dose rates and the submission of the different dose rates: a) the amplitude whole data set to the analysis. If the different decreases following the model (1-n/10) 2 , dose rates lead to different evolution models, where n is the spectrum number in the then these models could have a common origin sequence, b) model (1-n/10), c) model (1- at a point which represents the zero-dose n/10)112_ In this simulation, the dose effects situation. were supposed to be proportional to the dose In order to illustrate this possibility and the (without any threshold effects introducing ability of MSA to process such data, we discontinuities). Therefore, the three data sets performed the following simulation : converge towards one single point at zero­ Three groups (a,b,c) of five spectra are dose. The coordinate of this unique point on computed. Each one is made of a decreasing axis 1 can be used to compute the fictitious background and a Gaussian characteristic peak. zero-dose spectrum. Poisson noise is also simulated. The amplitude of the peak decreases as a function of time but the law which describes the amplitude decrease is different for the three spectrum reconstitution is very close to the corresponding simulated spectrum at n O (see sequences (1-n/10) , (1-n/10) 2 and (1- Fig. 6c in Bonnet et al. (1991) for a similar n/10)112 , where n is the spectrum number in result). the series (n=1 to 5). Here again, the actual realization of such an The fifteen spectra are submitted together to experiment could obviously be more MSA. Only one factorial axis is significant (77% disappointing than this crude simulation. There of the total variance). The weights of the is no guarantee of obtaining traces with different spectra on this axis are displayed in different slopes, nor of obtaining several Fig. 6. One can see that the three groups traces intersecting at the same location. But corresponding to three different dose rates are the idea is that, even if such an experiment characterized by three different traces in the does not allow attainment of the ultimate goal, (time - axis 1) space. These three traces can be which is to get information about the state of mathematically modelled and the the specimen at zero-dose, at least such an representative point corresponding to the zero­ analysis seems to be a useful tool to dose can be obtained at the intersection of the understand the effects of the dose. For three traces, giving the weight of the instance, if the intersecting point (even fictitious spectrum on axis 1. This weighting roughly approximated) seems to be far away coefficient can then be used to build the from the experimental points in the factorial fictitious spectrum, taking into account the space, this means that large effects occurred eigenfactors resulting from the analysis of the before the first measurement, thus rendering initial data set. The result of this hazardous any extrapolation to zero-dose. If

170 Multi-dimensional data processing in microanalysis instead of the continuous traces observed in data and to cross-correlate the different this simulation, one obtains trace results obtained. discontinuities, this militates against any Since MSA seems particularly suitable for extrapolation but, at the same time, brings new processing image sequences as well as single insight into largely unknown phenomena. data sequences or spectrum sequences, its potential possibilities were investigated. Application in Electron Energy Loss First method investigated. Elemental Mapping This consists of submitting the whole image sequence (energy-filtered images below and As described in the introduction, image above the characteristic energy-loss) to some sequences are necessary to perform elemental variant of MSA (Correspondence Analysis or mapping by electron energy loss and Auger Principal Component Analysis). The possibili­ spectroscopy. They are made of several ties of the procedure are data analysis, data energy-filtered images below the compression, data processing. characteristic edge of the element that one Data analysis: the consistency of the data wants to map and one or several energy­ set can be checked through the visualization filtered images above this edge. In many of the weights of the different images and situations, a pixel to pixel image processing of the weights of the different pixels in the fac­ the whole sequence allows one to obtain a torial space. The interpretation of these pro­ meaningful result. Such a process consists, for jections can help in understanding the data each pixel of the sequence, of a modelling of set precisely, depicting eventual acquisition the signal variation below the energy-loss artefacts. Examples of this possibility have edge, an extrapolation of the model to energy­ been described by Trebbia and Mory (1990). losses above the edge and a subtraction of Data compression : at any time, the original these extrapolated values from the expe­ image sequences can be reconstituted from its rimental signal intensities recorded for these decomposition into factorial images. Since a energy-loss values, the final result being small number of these factors contain considered as the net signal produced by the significant information, a large number of atomic species. images can be replaced by a small number of But, in some situations, this procedure is not factorial images, thus leading to an important convenient, for several reasons : data reduction. Though this point is not - one has to define an a-priori model (I = AE fundamental at the present state of the routine acquisition procedure, where the number of R is the most commonly used, where E is the images in the sequence is less than ten, it energy-loss and A,R two adjustable could become really useful with new parameters). When the chosen model does not acquisition procedures: the "spectrum imaging" match the experimental variation, the procedure for instance, where one complete modelling, and thus the extrapolation, will be energy spectrum is recorded for every pixel defective. (Jeanguillaume and Colliex (1989), Thomas et - the computation being done for each pixel, al. (1990), Balossier et al.(1991 )). the statistical confidence in the resulting Data processing Since one or several factorial parameters is rather weak, especially for images can be discarded during the situations where one must take care of the reconstitution process, there is a way to filter electron dose received by the specimen, the data set. Keeping or rejecting one of the - the apparatus and acquisition procedures different factors is, of course, the for recording the image sequences involved in responsability of the user and must rely on a this kind of experiment are very sophisticated careful interpretation of the meaning of this (see for instance Tence (1987)). Therefore, factor (or axis). One interesting possibility is there is no guarantee that the data sets are to improve the signal-to-noise ratio of the free from artefacts, such as primary electron image sequence (in low dose experiments). This beam current or spectrometer drifts, specimen is possible because an important part of the shifts, etc. Therefore, it is of primary noise is rejected in the low order factorial importance to check the consistency of the axes, especially when the noise components are whole data set submitted to the process. orthogonal to the useful information For these different reasons, it seems useful components. The hypothesis that a factorial to use several methods for the processing of

171 N. Bonnet and P. Trebbia axis is mainly representative of noise can be ascertained by observing that the factorial image does not contain organized information but displays a random contrast. Therefore, discarding these factors which mainly contain noise results in a reconstituted sequence with .;,, 1b• a higher signal-to-noise ratio. Applying the a• standard processing procedure to such a noise­ filtered sequence provides better quality resu Its. • • Second method investigated The method previously described can only be ... Iii- considered as a pre-processing method since it • jd• does not lead directly to the desired elemental c• map. It only performs data analysis, checking and data processing (noise filtering). A more • ambitious task would be to obtain the map "" without returning to the standard procedure, which needs to define an a-priori model for the .,. ~.~ ....,; •" . """; 1,-, pixel intensity evolution as a function of the E~ -F'♦ energy-loss. For this purpose, a second method has .E.iCJ..a_Z: Gallery of two uranium maps computed undergone preliminary investigation (Bonnet et from a series of 10 electron energy filtered al. (1992)). It consists of submitting to MSA images. These 10 images where recorded at 5 only the images below the characteristic edge. different energy losses and duplicated. a, b) Then, after data analysis and check for elemental map produced with the standard consistency, the extrapolation procedure in the procedure consisting in background modelling factorial space, described in the previous and subtraction from energy filtered images sections, is applied in order to get the above the characteristic edge. c, d) In each "fictitious" images of the background for the case, the 5 experimental images were energy-losses of interest (above the edge). submitted to MSA and then reconstituted after Then, these "fictitious" images of the keeping only the first factorial component, background are subtracted from the thus discarding a large part of noise and other experimental images recorded at these energy artefacts (50 Hz ripples). The 5 reconstituted losses, providing the desired elemental map. images were then submitted to the standard Though this procedure was sucessfully procedure. e, f) In each case, the 4 applied to an experimental sequence of a metal experimental images below the characteristic cobalt catalyst on a Ce0 2 layer (Bonnet et al. edge were submitted to MSA. From the (1992)), numerous other tests have to be coordinates of these images in the factorial performed before we can claim that it is really space, the coordinate of the fictitious (i.e. not superior to the standard procedure. Among attainable experimentally) background image them, we studied once again the image was then computed and subtracted from the sequence of uranium clusters (decorating a DNA fifth experimental image above the edge. on a thin carbon film) previously investigated by Trebbia and Mory (1990). This sequence has the advantage that the different images are duplicated (i.e. twice recorded), giving an obtained by the multivariate procedure opportunity to study the consistency of the (standard procedure applied to a reconstituted different processing steps (see Mory et al. sequence obtained by retaining only axis 1, (1988)). which represents 70% of the total variance for Fig. 7 displays the results obtained with the odd as well as even images). Fig. 7 e,f display various procedures described here above for the results obtained with the second odd numbered and even numbered images. Fig. 7 multivariate procedure. The weights of the a,b display the results obtained by the standard different images on axis 1 (which carries 43 % procedure. In Fig. 7 c,d are shown the results and 45 % of the total variance ,or odd and even

172 Multi-dimensional data processing in microanalysis

5 recorded on both sides of the characteristic / edge (see Fig. 4 in Colliex (1986)). / / / / Conclusions / / / The goal of th is paper is to take stock of the 4 / / possibilities offered by Multivariate Statistical Analysis for the analysis and processing of various multivariate data sets which can be recorded in the field of electron­ induced microanalysis. The basic idea is that Energy loss applying repetitevely a monovariate procedure on individual data seems less satisfactory than applying a mutivariate procedure on the whole .Eig_._.6.: Coordinates of the odd numbered images data set since it allows one to understand and on the factorial axis 1 (as a function of the process all the correlations between these energy loss) and the extrapolated value for the data at once. MSA is not the only multivariate fictitious background at 112 eV (above the procedure available. Geostatistics for instance uranium 045 excitation threshold). (Daly et al.(1992)) offers another option. However, we concentrated on this technique images) are shown on Fig. 8. From these which can be implemented easily and already weights, the weight of the fictitious offers several possibilites data analysis, data background image is extrapolated (extrapolated reduction, data processing, data interpolation value is shown) and the corresponding images and extrapolation. are built. We have considered three possible The results obtained from the three applications. In the context of spatially procedures are consistent from a qualitative resolved spectroscopy, we have shown that point of view. However, there are some points concentration variations can be quantitatively which deserve further attention estimated from some simple computations in - the different methods provide slightly the factorial space, without performing any different quantitative results, data processing (such as background - though they are supposed to be equivalent subtraction, peak area estimation) in the space (apart from statistical noise), the image pairs of the experiment (i.e. l(E)). have not exactly the same coordinates on the In applications related to time-resolved main factorial axis , thus leading to two spectroscopy, MSA can probably be useful to different extrapolated back-ground images. We understand the evolution (as a function of time, have not been able to explain this difference, or dose) of several element features. In this either by looking at the odd and even factorial paper, we have focused our attention on the images, or by looking at some intermediate perspective of extrapolating such experimental parameters (such as R) in the standard data towards the zero dose limit. Though we procedure (modelling). are fully aware that this task is quite In conclusion, it seems to us that the new difficult, we tried to show that specific procedure merits further studies before being experiments and multivariate processing of the adopted as a safe technique. It is clear that its recorded data could help in understanding the application would be rendered easier by phenomena which take place at the very recording more thc.n three or four images below beginning of the irradiation. the edge, as it was the case in the In the field of elemental mapping by energy­ experimental sequences used so far. This will loss spectroscopic technique, we have shown evidently be the case with the new spectral that two procedures relying on MSA can be used imaging procedure. in conjunction with the standard technique It should also be stressed that if the based on a modelling of the intensity of every extrapolation process could be replaced by an pixel. These complementary methods can help interpolation process, the task would be much in understanding the data set, in depicting more easy. This can be done in the favorable eventual acquisition artefacts, in performing situation where background images can be data reduction when very large sequences are

173 N. Bonnet and P. Trebbia recorded, in filtering noise and (for the latter Colliex C (1986). Electron Energy-Loss method) in producing a fictitious background Spectroscopy analysis and imaging of image. This image can then be subtracted from biological specimens. Annals New-York Acad. the experimental images recorded at an energy Sci. 483, 311-325. above the characteristic energy loss in order to Craven A, Cluckie M, Duckworth S and Baker T produce the element map. (1989) Analysis of small vanadium carbide The techniques described in this paper were precipitates using electron energy loss mainly illustrated through numerical spectroscopy. Ultramicroscopy 28, 330-334. simulations. When applied on real data given by Daly C, Jeulin D and Benoit D (1992) Non experiments which are now in progress, they linear statistical filtering and applications to will show whether these assumptions are segregation in steels from microprobe imaging. reasonable or not. Scanning Microsc .. Suppl. 6, 137-'145. Disko M, Luton M, and Shuman H (1991) References Energy-loss near-edge fine structure and compositional profiles of cryomilled oxide Balossier G, Thomas X, Michel J, Wagner D, dispersion strengthened aluminium. Ultra­ Bonhomme P, Puchelle E, Ploton D, Bonhomme A microscopy 37, 202-209. and Pinon J.M (1991) Parallel EELS elemental El Gomati M, Peacock D, Prutton M and Walker mapping in scanning transmission electron C (1987) Scatter diagrams in energy analysis microscopy : use of the difference methods. digital imaging: application to scanning Auger Microsc. Microanal. Microstruct. 2, 531-546. microscopy. ,1. Microsc. 147, 149-158. Barkshire I, Greenwood J, Kenny P, Prutton M, Ellis T, Dubois L, Kevan S and Cardillo M Roberts R and El Gomati M (1991) Image (1985) Time-resolved electron energy loss correlation: application to correction of beam spectroscopy. Science 230, 256-261. current fluctuations in quantitative surface Frank J and Van Heel M (1982) Correspondence microscopy. Surface and Interface Analysis. 17, analysis of aligned images of biological 209-212. particles. J. Mol. Biol. 161, 134-137. Bonnet N, Colliex C, Mory C and Tence M Garenstrom S (1981) Principal component (1988). Developments in processing image analysis of Auger shape lines at solid-solid sequences for elemental mapping. Scanning interfaces. Appl. Sur. Sci. 7, 7-18. Microsc. Suppl. 2, 351-364. Garenstrom S (1986) Application of factor Bonnet N and Hannequin P (1988). Zero-dose analysis to elemental detection limits in extrapolation in Spectroscopy (Prospect). Inst. sputter dep1h profiling. Appl. Surf. Sci. 26, Phys. Conf. Ser. 93, 179-180. 561-574. Bonnet N and Hannequin P (1989) Chemical Hannequin P and Bonnet N (1988) Application mapping and multivariate statistical analysis of multivariate statistical analysis to (prospect.). Ultramicroscopy 28, 248-251. energetic image series. Optik 81, 6-11. Bonnet N, Simova E and Thomas X (1991) Hannequin P, Liehn JC and Valeyre J (1989) Application of multivariate statistical The determination of the number of analysis to time dependent spectroscopy. statistically significant factors in factor Microsc. Microanal. Microstruct. 2, 129-142. analysis of dynamic structures. Phys. Med. Biol. Bonnet N, Simova E, Lebonvallet S and Kaplan 34, 1 21 3-1227. H (1992) New applications of multivariate Harauz G and Chiu D (1991) Covering events in statistical analysis in spectroscopy and eigenimages of biomolecules. Ultramicroscopy. microscopy. Ultra-microscopy 40, 1-11. 38, 305-317. Bretaudiere JP and Frank J (1986) Jeanguillaume C (1985) Multi-parameter Reconstitution of molecule images analysed by statistical analysis of STEM micrographs. J. correspondence analysis: a tool for structural Microsc. Spectrosc. Electron. 10, 409-415. interpretation. J. Microsc. 144, 1-14. Jeanguillaume C, Trebbia P and Colliex C Browning R (1987) Materials analysis by (1978) About the use of electron energy loss scanning Auger microscopy: why the informa­ spectroscopy for chemical mapping of thin tion crunch is needed. In: Analytical Electron foils with high spatial resolution. Microscopy. Joy DC (ed.). San Francisco Ultramicroscopy . 3, 138-142. Press. 31 1 -31 6.

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Jeanguillaume C and Colliex C (1989). representation. Inst. Phys. Conf. Ser. 98, 311- Spectrum-Image the next step in EELS digital 314. acquisition and processing. Ultramicroscopy Thomas X, Balossier G, Wagner D, Martin P and 28, 252-257. Bonhomme P (1990) Interface between a Gatan Kenny P, Prutton M, Roberts R, Barkshire I, PEELS and an EDAX 9900 computer for Greenwood J, Hadley M and Tear S (1992) The simultaneous energy loss analysis and application of multispectral techniques to chemical mapping. Proc. Xllth Int. Congress E.M. analytical electron microscopy. Scanning (San Francisco Press) 11,30-31. Microsc. Suppl. 6, 361-367. Trebbia P and Bonnet N (1990). EELS King P, Browning R, Pianetta P, Lindau I, elemental mapping with unconventional Keenlyside M and Knapp G (1989) Image methods. I- Theoretical basis image analysis processing of multispectral X-ray photo­ with multivariate statistics and entropy electron spectroscopy images. J. Vac. Sci. concepts. Ultramicroscopy 34, 165-178. Technol. A. 7, 3301-3304. Trebbia P and Mory C (1990). EELS elemental . Malinowski E (1977) Determination of the mapping with unconventional methods. II­ number of factors and the experimental error Applications to biological specimens. in a data matrix. Analytical Chemistry. 49, Ultramicroscopy 34, 179-203. 612-617. Van Heel M and Frank J ( 1980) Classifica­ Malinowski E and Howery D (1980) Factor tion of particles in noisy electron micrographs analysis in chemistry, Wiley-lnterscience, using correspondence analysis. In: Pattern New-York, 1-239. Recognition in Practice. Gelsema E and Kanai Mory C, Bonnet N, Colliex C, Kohl H and Tence L (eds.). North-Holland, 235-243. M (1988) Evaluation and optimization of the Van Heel M and Frank J (1981) Use of performance of elastic and inelastic scanning multivariate Statistics in analysing the images transmission electron microscope imaging by of biological macromolecules. Ultramicroscopy. correlation analysis. Scanning Microsc. Suppl. 6, 187-194. 2, 329-342. Paque J, Browning R, King P and Pianetta P ( 1990) Quantitative information from X-ray Discussion with Reviewers images of geological materials. Microbeam Analysis. San Francisco Press. 195-198. R. Browning: I'm not sure that (1-n/10) 1 12 Prutton M, El Gomati M and Walker C (1987). can be a physically realistic process. Doesn't Quantitative imaging in the scanning Auger n 112 imply the surface has pre-knowledge of microscope. Inst. Phys. Conf. Ser. 90, 1-8. the coming events. It is the same for any Prutton M, El Gomati M and Kenny P (1990). sublinear dependence. I would expect the linear Scatter diagram and Hotelling transforms : dependence of a decaying feature to be : application to surface analytical microscopy. J. Rt= Roe-kt Electron Spectroscopy and Related Phenomena Authors: We agree with the comment that the 52, 197-219. models used for the amplitude dependence as a Savoji M and Burge R (1983) Elemental function of dose could be physically classification in multi-detector STEM images unrealistic. However, to our knowledge, there using image analysis clustering techniques. is no theoretical or experimental evidence for Ultramicroscopy. 12, 1-8. any model which can describe the effect of Tence M ( 1987). Un systeme informatique dose on microanalytical data of diffusible pour !'acquisition et le traitement des elements. We are not even sure that this effect donnees en Microscopie electronique: can be described by continuous models. realisation, mise au point et applications (A Therefore, we chose these models for testing computer system for data acquisition and a data processing method rather than for processing in electron microscopy: design, checking a physical hypothesis. However, as constructions and applications). These you suggested, we have undergone new Docteur lngenieur. Universite Paris-Sud simultations with a model of the type Rn = Ro (Orsay, France). e-kn, with different values of k (0.1, 0.2 and Tence M, Walls M, Jeanguillaume C, Colliex C, 0.3). The qualitative results are comparable to Thomas X, Jbara O and Cazaux J (1990) Electron those of Fig. 6 (results available from the irradiation effects a time-energy authors), indicating that the principle

175 N. Bonnet and P. Trebbia

underlined in section 2 holds for any model, P. Statham: Whereas random "high fre-quency" provided the hypothesis of continuity is spatial intensity variations are instantly fulfilled. recognisable as "noise", statistical fluctuations in the original data will also have P. Statham: The essence of MSA is to reduce "low frequency" components which could in the dimensionality of a problem to something principle be misinterpreted as undulations in comprehensible. A key stage in this process is the background. If the data set is not very the elimination of "insignificant" eigenvectors. large, is it possible that this could introduce In Fourier transform spectroscopy, it is often the sort of variation in results that the authors possible to estimate the expected variance in have noticed in the Uranium maps example (Fig. magnitude of Fourier components caused by 7) ? statistical noise in the raw data. Are the Authors: There are two aspects to your authors aware of any techniques in MSA for question which we would like to address estimating the expected variance in separately. The first one concerns the "high eigenvalues caused by statistics so that frequency/low frequency" considerations. We objective significance tests could be used to think that this concept cannot be handled by eliminate eigenvectors ? MSA, which considers every pixel of the series Authors: We agree with the idea that the independantly of its neighbors. From this point question concerning the number of "useful" of view, filtering by MSA is rather different eigenvectors (i.e. the number of factors which from frequency filtering. really represent information and the number of The second aspect concerns the size of the factors which are essentially due to noise) is data set. We are aware of the fact that MSA often very important for the different was developed in order to process large data applications of MSA. This question was often sets (many individuals and many variables) and considered in the different fields of that we are applying it to data sets which are application of MSA. Here is a list of different large in one dimension (pixels or energy answers reported in the literature channels) but small in the other dimension - drop of the eigenvalues : there is sometimes (images or spectra). This may have some a gap between the eigenvalues associated with consequences concerning the noise rejection. the "useful" eigenvectors and those associated For instance, if we have only three images, we with noise. can only define two factorial axes in - this qualitative observation was quantified Correspondence Analysis. If these three images by Malinowski ( 1977) through several indexes contain two real information components which can be used to estimate the real number ("background" and "object" for instance), these of factors. Some of these indexes need an two components are going to define (more or estimate of the root mean square error in the less) the two factorial axes and there is no data submitted to factor analysis. more factorial axis available for the noise. In - likelihood ratio test a reconstitution of the this situation (which we encounter in series is undergone by including an increasing processing color - RGB - images for instance), number of factors. When there is no longer a the noise is constrained to occupy also the two change (this being asserted by a statistical available factorial axes and no noise rejection test), the new factor is "useless" (Hannequin et is possible. al., 1989). Returning to your question, it is possible - methods for choosing factors useful for a that these kinds of effects are responsible for given purpose appeared recently in the field of the (small) variations observed in the results pattern recognition (for an application in reported in Fig. 7. electron microscopy and relevant references, see Harauz and Chiu, 1991). P. Statham: Since the authors are making a A systematic comparison of these case for MSA as a more efficient and different methods in different situations convenient approach than conventional would be necessary before drawing conclusions spectrum modelling techniques, I would have concerning their actual usefulness. liked to see more of an objective appraisal of the various approaches to the Uranium map example in Fig. 7. Without the authors' help, I cannot see why the reader should automatically

176 Multi-dimensional data processing in microanalysis see any way of ranking the solutions given by M. Prutton: In the second simulation Fig. 7. considering quantification of concentration Authors: Our purpose is not to make an variations in spatially resolved spectroscopy efficiency rank between different methods of the components A and B have different data analysis. We agree that it would be functional variations with respect to position difficu It for the reader to decide from Fig. 7 across an interface. A and B are which process is best suited for extracting the concentrations and cannot add up to 100% as useful information. This can only be done with the simulation is set up. This means, simulations or by computing some objective presumably, that at least one element is criteria, such as the signal to noise ratio for present or some property of the details of the instance. measurement technique cause A and B not to be The purpose of Fig. 7 is only to show that perfectly anti-correlated in an apparently different methods can lead to "a result", not to binary system. If the analyst has other decide which one is the best. evidence that a real piece of material is binary Obviously, if one has good reasons for then such absence of perfect anti-correlation assuming that a given model is well adapted is teaching him that there may an artefact for the description of background and/or noise, introduced by the measurement technique. If then such an a priori model must be used for there is no such evidence then perhaps the data reduction. analyst should seek elements other than A and But in the case where no a priori B wich may be localised at the interface. Would information is available, then, provided that you agree with this kind of interpretation? the data set is large enough, MSA is even an Authors: Yes, we totally agree with this ultimate resort for extracting the different interpretation. components of this data set, filtering and exploiting them. Therefore, we consider that MSA is not a more efficient and convenient approach than modelling techniques, but an alternative method when no model is available.

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