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Classification of SIMS Images Using a Kohonen Network M. Wolkenstein, H. Hutter,* Ch. Mittermayr, W. Schiesser, and M. Grasserbauer
Institute for Analytical Chemistry, Vienna University of Technology, Getreidemarkt 9/151, A-1060 Wien, Austria
The distributions of atomic and molecular ions of 12 different masses were measured by imaging SIMS at low mass resolution. Due to mass interferences, visual interpretation of the chemical phases represented in the different distributions is not possible. These single phases were extracted by classification using a Kohonen network. To demonstrate this technique, the behavior of the Kohonen map is compared with manual classification. For determination of the optimal dimension of the network (the number of nodes should be equal to the number of expected classes), and to reduce the artifacts due to noise and nonlinearities, principal component analysis was performed. Alternatively, the number of necessary classes can be determined by a second classification of the nodes of a Kohonen network that is sufficiently large with the help of dendrograms. Peak interferences between atomic and molecular ions are a general problem in mass spectrometry. By the use of a doublefocusing mass spectrometer, this problem can be avoided by adjusting to high mass resolution; however, this reduces the transmission of the instrument and, thereby, the detection power of the method. Distributions of elements in secondary ion mass spectrometry (SIMS) can be measured either in scanning mode or in imaging mode. In imaging mode, all ions emitted from the analyzed area are projected simultaneously to the channel plate, a laterally resolved single ion detector. In this mode, adjusting to high mass resolution results in an decreasing image quality. For optimal lateral resolution, the images are picked up at low mass resolution (M/∆M ) 600), which complicates the interpretation of distributions, especially of higher atomic and/or molecular masses. In the case of inhomogeneous samples, assignment of the overlapped distributions to chemical phases is often impossible. In scanning mode, energy filtering is possible (with some instruments) to overcome this problem; in imaging mode, this would lead to a significant deterioration of the image quality. Numerous chemometric approaches to this problem, like the deconvolution of low-resolution mass spectra14 or the minimum distance classificator,6,10 have been reported. This paper describes the use of Kohonen networks for assigning structures in secondary ion distributions to chemical phases. THEORETICAL SECTION Classification. Chemical and geometric features (like texture and shape) in digital images lead to pixel populations in coherent clusters and can, therefore, be further treated by multivariate statistical means to extract analytical information. A further step is the classification of the specific chemical features contained in the sample. To identify different objects in the feature space, it S0003-2700(96)00510-0 CCC: $14.00
© 1997 American Chemical Society
Figure 1. Manual classification of Cr-B. (Top left) Ion micrograph of B. (Top right) Ion micrograph of Cr. (Bottom left) Scatter diagram. (Bottom right) Classified image. Chemical correlation of secondary ion micrographs by means of image classification: ion micrographs of Cr and B, the scatter diagram indicating the correlation of these elements, and the spatial distribution of the chemical entities (phases) found.
is necessary to establish their frequency distribution. A scatter plot as a mode of display is calculated to represent the frequency distribution of gray levels.1-4,12,13 Figure 1 demonstrates the (1) Barkshire, I. R.; El Gomati, M. M.; Greenwood, J. C.; Kenny, P. G.; Prutton, M.; Roberts, R. H. Surf. Interface Anal. 1991, 17 (4), 203-208. (2) Boehmig, S. D.; Reichl, B. M. Fresenius J. Anal. Chem. 346:223-226, 1993, 346, 223-226. (3) Bright, D. S.; Newbury, D. E.; Marinenko, R. B. Concentration-Concentration Histograms: Scatter Diagrams Applied to Quantitative Compositional Maps; San Francisco Press Inc.: San Francisco, CA, 1988. (4) El Gomati, M. M.; Peackock, D. C.; Prutton, M.; Walker, C. G. J. Microsc. 1987, 147, 149-158. (5) Hutter, H.; Grasserbauer, M. Mikrochim. Acta 1992, 107, 137-148. (6) Hutter, H.; Graserbauer, M. Chemom. Intell. Lab. Syst. 1994, 24, 99-116. (7) Kohonen, T. Self-Organization and Associative Memory, 3rd ed.; Springer Verlag: Berlin, 1989. (8) Kohonen, T. The self-organizing map. Proc. IEEE 1990, 78, 1464-1480. (9) Kohonen, T. Self-Organizing Maps; Springer Verlag: Heidelberg, 1995. (10) Latkoczy, C.; Hutter, H.; Grasserbauer, M. Mikrochim. Acta 1995, 352, 537-543. (11) Manly, B. F. J. 1st ed.; Chapman and Hall: London, 1986; pp 101-113.
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Figure 2. Classification of Cr-B by Kohonen network. (Top) Decrease of the classification error if number of nodes is increased. (Bottom) Classified images (left to right): four, five, and seven nodes. Automatic classification of two elemental distributions using a Kohonen network. Increasing the number of nodes results in a decrease of the classification error.
construction principle. A lot of pixels in these diagrams tend to pile up at the same spots as they possess the same frequency distribution of gray levels in both images. A scatter diagram, therefore, allows the determination of pixel clusters (representing single sample phases), outliers, and gradients in terms of their density. The resulting clusters ideally represent the relationship of the constituents in the original sample, and they can be backprojected onto the original images to visualize the spatial distribution of the different phases. Self-Organizing Mapssthe Kohonen Network. A Kohonen network, called by its inventor a “self-organizing map” (SOM),7-9 maps objects from an n-dimensional input data space, Rn, to a twodimensional array of neurons. Every node is characterized by an n-dimensional weight vector, wj. The mapping is formed in an unsupervised learning process which applies competitive learning rules. The weights are initialized with random numbers within the minimum and maximum values of each variable. Then an input vector, xi, is selected and compared to every weight vector wj by calculating the Euclidean distance. The weight vector wc closest to the input vector is selected and becomes the center of the adaption process:
wc ) wj r min (||xi - wj||)
(1)
and its neighbor neurons are updated:
wrnew ) wrold + η(t)h(d,r,t)(xj - wrold)
(2)
η denotes the learning rate, d the distance of neuron j to the active neuron c, r the radius of the neighborhood function, and t the iteration/step number. The neighborhood function may have different forms: common ones are the “bubble”, Gaussian, and “Mexican hat” functions. One adaption step or iteration pulls the weight vectors in the direction of the actual input vector. This learning step is repeated with the whole data set several times. A decreasing learning rate η(t) during the learning process assures the convergence of the training process:
tmax - t tmax
η(t) ) η
(3)
For classification, a shrinking neighborhood radius, r(t), enhances the discriminatory power of the network, while for continuous mapping the learning should not be restricted to only one unit:
j)1...r
Depending on a neighborhood function h(d,r,t), the weights to the “winning” neuron (hence the term competitive learning) (12) Newbury, D. E.; Bright, D. S. Concentration Histogram Images: A Digital Imaging Method for Analysis of SIMS Compositional Maps; Wiley: New York, 1990; pp 929-933. (13) Prutton, M.; El Gomati, M. M.; Kenny, P. G. J. Electron Spectrosc. Relat. Phenom. 1990, 52, 197-219. (14) Ritter, M.; Hutter, H.; Grasserbauer, M. Fresenius J. Anal. Chem. 1994, 349, 186-190.
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r(t) ) 1 +
(rmax - 1)(tmax - t) tmax
(4)
By this iterative process, a nonlinear projection of the probability density function onto the two-dimensional array of neurons is performed, and topology of the input data space (neighborhood, density) is preserved.
Table 1. Measurement Conditions primary ions primary beam intensity primary beam energy scanned area analyzed area diameter
O2+ 2 µA 5.5 keV 500 × 500 µm2 400 µm
EXPERIMENTAL SECTION Sample Description and Data Acquisition. The measured specimen is a solder alloy used to join steel and chromium. The solder material was a nickel-based alloy (Cr, 7.0%; Fe, 3.0%; B, 3.0%; Ni, 82.5%; Si, 4.5%; B, 3%) in the form of a foil. The previous homogeneous material develops cracks and brittle phases during soldering. After the solder, the soldering joint was polished. The instrument we used is the double-focusing secondary ion microscope CAMECA IMS3f (for measurement parameters, see Table 1) with a typical lateral resolution of 1-3 µm in imaging mode. To obtain elemental distribution images, an intensive primary beam homogeneously illuminates the sample by scanning rapidly (100 frames/s) over an area of typically 500 × 500 µm2. The ion optical system of the mass spectrometer produces a mass-filtered secondary ion image of the surface registered using a CCD camera system (Pulnix TM 760) in combination with a double microchannel plate fluorescent screen assembly (Galileo HOT). The camera signal is digitized by an ITI 151 image processor and stored on the controlling computer.5 Depending on signal intensity, the secondary ion distributions were summed for 5-20 s. Micrographs were taken at 12 different masses. Computing. The software used is based on the public domain package SOM written by Teuvo Kohonen and co-workers. The source code of the package is available for the anonymous ftpuser at Internet site cochlea.hut.fi (130.233.168.48). For handling image data, the input and output routines were rewritten. The program was compiled and linked on a 150 MHz clock rate ALPHA-PC from Digital Equipment Cooperation under Windows NT. Due to the high speed of this computer, processing time for large image data sets (the full data base of all 12 images consists of 160 000 vectors in 12 dimensions) was only in the range of a few minutes. RESULTS Classification of Two Images. (1) Manual Classification. A simple classification is obtained if all image pixels forming different features in the scatter diagram are assigned to a class (Figure 1). In the scatter diagram, a region with high Cr and B signal is visible, and the corresponding chemical phase is CrB. Additionally, features with high B and low Cr signal, and vice versa, are assigned to one class. Other phases, which are obviously present, can be detected only by classifying more than two distributions, which can hardly be realized by manual classification (n-dimensional scatter diagram). (2) Classification by the Kohonen Network. To demonstrate the technique, we compared the Kohonen network with manual classification. The greatest problem when using a Kohonen network is setting the dimensions of the network (i.e., the number of nodes). The number of classes is identical to the number of nodes, but we do not know a priori how many chemical phases are present in the images. Increasing the number of nodes results in a lower classification error because the average
Figure 3. Classification of the nodes. (Left) Scatter diagram of 5 × 5 nodes (b, position of the nodes; (, new class center). (Right) Resulting classified image (four nodes). By clustering a larger number of nodes with the help of a dendrogram, the number of resulting clusters represents the number of classes.
Figure 4. Classification of two images. (Left) Classified image. (Right) Scatter diagram. The table gives the vector elements of the nodes. All elements of the vector of a node correspond to a secondary ion intensity. By comparing these values, the nodes (classes) can be assigned to chemical phases.
Figure 5. Classification of four images. Automatic classification of four elemental distributions.
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Figure 6. The 12 original images (a, top) and the 12 principal components (b, bottom). 780
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Table 2. Measured Masses and Possible Interferences measd mass 11 28 39 52 56 58 63 67 69 80 84 86
assigned element or molecule
possible interferences
11B 28Si
12C16O, 56Fe2+
28Si11B
(39K)
52Cr
(40Ca12C) 28Si, (40Ca16O) 58Fe, 29Si , 28Si30Si 2 (63Cu)
56Fe 58Ni 52Cr11B 56Fe11B 58Ni11B
58Fe11B
52Cr28Si
58Ni11B
56Fe28Si 58Ni28Si
2 52Cr16O 2 58Fe28Si
Euclidean distance of the vectors to the next node decreases (Figure 2). If there are more nodes than existing clusters, a second classification is able to calculate the necessary number of classes and, therefore, the optimal dimension of the map. Figure 3 displays the position of the nodes from a 5 × 5 network in the scatter diagram. By clustering these nodes with the help of a dendrogram,11 the number of resulting clusters represents the number of classes. Both elements of the vector of a node correspond to a secondary ion intensity. By comparing these values, the nodes (classes) can be assigned to chemical phases, e.g., a class showing high intensities of boron, chromium, and chromium boride can be labeled as chromium boride (Figure 4). Classification of a Higher Number of Images. To identify further phases, additional elements have to be classified. The complex task of n-dimensional classification can be achieved only automatically because of the impossibility to visualize an ndimensional scatter diagram. Figure 5 shows the automatic classification of four masses (B, Cr, Fe, and Ni). Six nodes can be associated with chemical phases corresponding to their vectors (three different NiB phases with growing B content, Fe, Cr, and CrB). The fourth node cannot be identified; for this phase, a classification of all images would be necessary, but the interpretation of the results of a classification of 12 distributions (Figure 6a) is rather difficult due to image artifacts. Small “unimportant” features and noise result in several additional classes. So we have looked for a method to separate the noise from the real features. Classification of the Principal Components. Various chemical phases of the sample are represented in different secondary ion distributions (e.g., chromium boride (CrB) can be observed in the Cr+, B+, and CrB+ ion distribution). All distributions of atomic and cluster ions are linear combinations of the phases in the sample. Because of the complexity of SIMS spectra, there are more secondary ion distributions than chemical phases (e.g., for one phase CrB distributions of Cr+, B+, CrB+, and CrB2+ are observed). To achieve high lateral resolution and high transmission, the distributions were acquired at low mass resolution (Figure 6a). Due to this, micrographs registered at a certain mass can be a combination of several secondary ions (see Table 2), which makes difficult both the visual assessment of individual pictures and the classification. To reduce the complexity of the distributions, we used principal component analysis (PCA). The PCA images are linearly independent and ordered by the variance of their gray levels (Figure 6b). Because the number of measured distributions is higher than the number of chemical phases in the sample, the principal components (PCs) with higher numbers
Table 3. Variance of the Gray Level of the Principal Components PC no.
sum of squares (%)
cumulative sum of squares (%)
1 2 3 4 5 6 7 8 9 10 11 12
48.802 15.279 8.758 6.557 6.372 4.257 3.154 2.329 1.573 1.168 1.053 0.699
48.8 64.1 72.8 79.4 85.8 90.0 93.2 95.2 97.1 98.2 99.3 100
Figure 7. Classification of the first three PCs. The classified image is shown, and the table gives the vector elements of the nodes. Chemical correlation of secondary ion micrographs by means of principal component analysis (PCA): automatic classification of the first three principal components leading to the identification of six phases.
present only the noise and nonlinearities, while the first few PCs represent nearly the whole image information. This makes possible the use of only the first few principal components for the classification to find the number of chemical phases in the sample. A chemical phase results in the presence of features in several distributions; this fingerprint is linearly independent of other phases. Because of this, the number of linear independent distributions is equal to the number of chemical phases represented in the images. Table 3 shows the decrease of the variance of the gray level of the PCs. The first three PCs represent 72% of all information, and the first six represent 90%. The six residual PCs represent the rest of the information. In fact, there are only six important PCs, so only six chemical phases are present. These six chemical phases can be detected by a classification of the first three PCs (Figure 7). CONCLUSION This work demonstrates the use of a Kohonen network for classification in image analysis for feature detection. The behavior of the Kohonen map is compared with manual classification, and Analytical Chemistry, Vol. 69, No. 4, February 15, 1997
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the optima dimension of the network was determined using either PCA or a second classification of the nodes of a sufficiently large Kohonen network with the help of dendrograms. The study is illustrated for but not restricted to SIMS images, and the acquired results may be applied to any analytical images that exhibit features similar to those of the test examples. The exact stoichiometry of the given phases in the sample could not be determined; therefore, their true identity remains open. Neighboring submicrometer phases could not be resolved and might have influenced the identification. Furthermore, mixed MeSiB phases have not been considered in this work. This example of PCA with subsequent classification is not restricted to imaging SIMS measurements alone but can be applied to any analytical imaging method. The use of specialized classification algorithms in the future promises further developments in this field.
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The determination of the number of classes (the dimension of the map) can be done either by a second classification of enough nodes by classical techniques (dendogram) or by calculation of the number of the important principal components. ACKNOWLEDGMENT This work was financially supported by the Austrian Scientific Research Council (Project S6205) and the Austrian Industrial Research Council (Project 2/293). The authors would like to thank Prof. Kohonen for providing the SOM software package to the scientific community. Received for review May 21, 1996. Accepted October 9, 1996.X AC9605105 X
Abstract published in Advance ACS Abstracts, December 1, 1996.
