Quantitation of Secondary Ion Mass Spectrometric Images by Microphotodensitometry and Digital Image Processing J. D. Fassett, J. R. Roth, and G. H. Morrison* Department of Chemistry, Cornell University, Ithaca, New York 14853
The quantitation of secondary ion mass spectrometric images produced by an ion microscope is accomplished by microphotodensitometry and digital image processing. The film negative of the ion image is converted into a matrix of point densities which maintains the spatial resolution of the original image. The image matrix in density space is converted into intensity space and concentration space with the use of standards. Image analysis techniques are demonstrated, including feature analysis and cross-correlation using the fast Fourier transform.
T h e ion microscope is a unique analytical tool combining ion sputtering, mass spectrometric filtering, and ion optics to give a spatially resolved mass analysis of the surface of a solid. As a microscope the instrument analyzes morphology in terms of elemental composition. T h e ion optics provide this direct microscopic imaging by retaining a one-to-one correspondence between the point of origin of an ion sputtered from the surface of a sample and its location in the final mass-filtered beam. T h e ion micrograph that is produced contains information about the spatial distribution of any selected element in a sample within a diameter of 250 pm or less and with a spatial point-to-point resolution of about a micrometer. T h e ion microprobe ( 1 , 2 ) ,electron microprobe ( 3 , 4 ) ,and Auger-ion sputtering spectrometer (5, 6) all have the proven capability to do compositional spatial analysis and elemental imaging of the surface of a sample. The imaging process in all these cases can be generalized as the sum of sequential point analyses with the probes operating in the scanning mode. This process is in contrast to the ion microscope which produces a micrograph by what can be considered as simultaneous multipoint detection. This contrast means that there is a substantial decrease of time for the ion microscope to produce the same type of information as these scanning microprobes or, conversely, the substantial increase in information produced in the same amount of time by the ion microscope. The compromise commonly made in microprobe imaging between time and total number of counts often results in strong statistical intensity variations which adversely affect the apparent resolution and quality of scanning microprobe images relative to ion images. T h e production of the compositional image is different in the scanning probe techniques from that of the ion microscope, and thus quantitation is achieved by differing paths. The image in the sequential point mode is created by the counting of appropriate radiation (ion, electron, or x-ray) for a particular point; storing the data; moving to the next point; and repeating until the desired field is covered. The final image is typically produced by modulation of the intensity on an oscilloscope with the photograph of the oscilloscope resulting in the permanent combination of all the point analyses. With the ion microscope, recording of all the data points is simultaneous; the image is directly produced on photographic film. Quantitation as described in this paper is accomplished by 2322
ANALYTICAL CHEMISTRY, VOL. 49, NO. 14, DECEMBER 1977
microphotodensitometry of the ion micrograph negative. As has been done in other techniques in chemical anal.ysis such as emission spectroscopy or spark source mass spectrometry which utilize photographic detection, quantitation in ion microscopy is achieved by the determination of the characteristic curve for t h e film and the conversion of darkening to intensity (7-9). Darkening a t any point on the photographic film can be correlated to the ion intensity produced from the analogous point on the surface of the sample. As opposed to these other analytical techniques, however, the determination of the characteristic curve is simplified by the fact that the photographic process is equivalent at all points on the image and film due to the conversion of the secondary ions into monoenergetic electrons. There are no corrections necessary for mass or wavelength differences, thus making the analysis and application of the characteristic curve to the image straightforward. Until this time ion micrographs have been utilized only for the qualitative data they contain ( 2 , 10-14). This paper describes how, by means of microphotodensitometry and computer analysis, image information is made quantitative. Image analysis through digitization of images is now widespread. Gaining its initial impetus from the work of Mariner (15) and subsequent extraterrestrial exploration, i t is now routinely used in light microscopy, x-ray diffraction, electron microscopy, astronomy, as well as many of the other photographic techniques that have large amounts of information to be processed (16-18). In this paper, procedures are developed to quantify ion micrographs in terms of ion intensity. Given standards, it is shown how this quantitation can be utilized to convert ion intensity into concentration. A simple bimetallic Ni-Cu composite system is used to illustrate the applicability of this procedure. Examples also are presented that illustrate two aspects of image analysis. The first, feature analysis, is used to evaluate heterogeneity in standards. The second, crosscorrelation is used to compare and combine information from two related images. Further, cross-correlation shows the use of image transforms and the possibilities that these procedures present. Both feature analysis and cross-correlation are illustrated by micrographs of NBS-660 steels. The above procedures and illustrative images represent only a brief introduction to digital image analysis and its potential application to ion microscopy. The broader application of image analysis techniques to more complex ion micrographs is being studied. EXPERIMENTAL Instrumentation. The instrument layout is schematicized in the block diagram, Figure 1. The CAMECA IMS-300 ion microscope used was developed by Castaing and Slodzian (19) and described in detail elsewhere (20). Instrument operating parameters are given in Table I. The camera assembly which makes the photographic detection is contained within the converter section of the instrument and is subject to the vacuum in the converter. Actual detection is of an electron image as the final mass filtered beam is projected onto the cathode of an ion-to-electron image converter and the secondary electrons emitted from the converter are accelerated to
Table 111. Microphotodensitometer Specifications Data format Raster setting Drum speed Positional accuracy Density range Time to acquire image
8 bits 100 pm x 1 0 0 pm
4 rps t 2 brn/cm on film 0.-3D or 0-2D 128 s
( 2 5 6 ~ 256)
Figure 1. Block
diagram of system layout
Table I. Instrument Operating Conditions Primary ion gases Primary ion energy Primary ion current Sputtered area of sample Secondary ion mass range Mass resolution Secondary ion polarity Image field of view Point-to-point resolution Image size to film Magnification Vacuum
Oxygen, argon Positive, 5 . 5 KeV Negative, 1 4 . 5 KeV to A/cm2 2.5 X cmz 1-250 300
Positive Negative 2 5 0 pm or less 1 pm 28 mm 110
lo-'
Torr
Table 11. Photographic Parameters Agfa Gavaert 37C50, 3 5 mm Film 40 images Camera capacity Developer Acufine ( 7 min) Rinse Running water (30 s) Kodak Rapid Fixer ( 2 min) Fixer Wash Running water (10 min) ~
20 keV and focused onto the film. Electron sensitive roll film
is used (Agfa Gavaert 37'250). Standard development procedures for the film are used and summarized in Table 11. A Photomation Mark I1 scanning microphotodensitometer interfaced to a PDP 11/20 computer is used to digitize the images. The film is placed in an opening in a cylindrical drum which rotates at high speed. Density is read by an incoherent optical system that is always on axis which guarantees linearity and fixed position. After each circumferential record has been scanned, the entire optical system (illumination and detection) is stepped along the axis of the drum in preparation for the next record. Zero density is electronicallycorrected with each revolution of the drum by the introduction of an open slit in the light path. The sampled density is amplified in a logarithmic amplifier and fed t o the analog-to-digital converter. Digital data are transferred to the computer through the interface and control circuits. Table I11 summarizes the densitometer specifications. The computer facilities consist of a PDP 11/20 with a 24K word memory; a 1.2-M word cartridge disk used for the RT-11 operating system, software development, and storage; a dual fixed disk assembly and a 9-track magnetic tape unit, which offer potentially unlimited data storage space; a 600-line-per-minute line printer for hard copies of program computations; an incremental plotter for contour plotting, calibration curves, and histographs; and a GT 40 graphics display and processor. The computer is used in this study entirely as a processor of image data matrices. However, through a computer-ion microscope interface, the computer also can be dedicated t o the operation of the ion microscope. Although the contemporaneous use of the computer is not possible, the potential exists for dual use and is being investigated. Computer Programs. Programming was done primarily in FORTRAS and stored on disk. The PDP 11/20 controls all acquisition and calculation programs. The acquisition programs
which create the primary image data files are designed to store the data in a minimum amount of space. Each image point has 8-bit resolution (0-255) and is packed into a single byte of 8 bits in the image data file. The total amount of disk space required by a 256 X 256 point image, the maximum size for which the system is designed, is 33K words. The calculation and manipulation of image data is done by a general program, IONPIX, consisting of a keyboard command interpreter which supervises calls to the main FORTRAN subroutines. IONPIX is much larger than the available memory space in the computer but is made operable by the overlay facility of the executive monitor, RT-11. The subroutines presently callable by IOXTX are classified in several broad functional categories which include mathematical, single file examining, image analysis, file shuffling, and ion micrograph specific. These subprograms represent a basic set of useful image file handling routines expanded by subprograms designed specificallyfor ion images. The structure of IONPIX is such that expansion and change is readily facilitated. Details concerning the operation and structure of IONPIX and its subroutines are available from the authors.
RESULTS AND DISCUSSION A typical ion image has a 250-pm diameter circular field of view with a point-to-point resolution of 1 pm. Thus, there are 50 000 information bits in a typical ion image with each information bit representing the total number of ions of a particular mass being extracted from a 1-pm2 area of the surface of a sample. The analysis of the information bits in an ion image is readily accomplished via computerization. The ion micrograph is 28 mm in diameter and is digitized into a 256 X 256 square point matrix using the 100-wm2 aperture of the microphotodensitometer. If the image is centered, digitization results in the loss of about 6% of the total image with no loss of point-to-point resolution. One of the results of digitizing a circular image into a square data file is the inclusion of non-ion image points in the file. About 10% of the digitized points in the image file are background points to the ion image field. This background is used in making background corrections on image points. Each datum in the primary image is acquired in the range of 0-2D (100% to 1% transmittance) or 0-3D (100% to 0.1 70 transmittance). In both cases, the number of steps measureable is 256. The point density resolution is therefore inversely proportional to the range. T h e experimenter has the choice of increasing the resolution on images that do not exceed a maximum density of two. However, the increased dynamic range in an image that has a maximum density that approaches three is more important than higher point resolution. The experimenter aims to satisfy this condition in the determination of the exposure times in the images that are taken. The pivotal step in quantitation of the ion image is the determination of the characteristic curve of the film and the conversion of film darkening per information bit to ions extracted per information bit. The characteristic curve is determined by the taking of a series of standard images of known intensity and of graded exposures from background to saturation for each roll of film to be quantized. These standard images define the curve of density vs. intensity. The intensity of image points of experimental images are extrapolated from this curve. ANALYTICAL CHEMISTRY, VOL. 49, NO. 14, DECEMBER 1977
2323
L C 5 1NTENSI-Y X T I V E i
Flgure 2. Characteristic curve for ion microscope film: (0)density vs. logarithm of the exposure; ( 0 ) Seidel density vs. logarithm of exposure. The exposure is in arbitrary units of intensity-time
As previously described, the recorded ion image is produced on film by a n electron beam, with the electrons accelerated to 20 keV. T h e response of photographic emulsions to electrons is well understood and can be described as a single hit process whereas the exposure to light requires the cooperative action of several photons (21). One result of this single hit process is that density is proportional to exposure in the low density range D < D,/4, where D, is the saturation density of the film. Experimentally, i t was found that the useful density range where this linearity manifested itself was extremely limiting and large deviations occurred for densities greater than 1. I t was decided, therefore, to analyze the characteristic curve in the more familiar manner of density vs. the logarithm of the exposure (Hurter and Driffield curve). This curve for electrons is similar to that of light with a toe region at low densities. Although not as large as that for light, the central region of the curve is linear. Empirically, it was found that the linear region of the D vs. loglo E curve extended through the uppermost digitized density of 3. I t was found that the linear range could be extended significantly in the toe region by the modification of densities according to Seidel (22):
D, = log ( l / T , - 1)
Flgure 3. Standard image. (a) Contour plot with equidistant isodensity levels from maximum to minimum densities of the image. (b) Three-dimensional representation where the Z-axis represents density and X and Y-axes, position. Size of image file: 64 X 64 I
-
(1)
where D , is the Seidel density and T , is the background corrected transmittance. This correction has little effect on t h e higher density values but succeeds in straightening the toe of the curve so that essentially the entire range of densities is linearly calibrated. Figure 2 illustrates a typical D vs. log,, E curve for a roll of ion microscope film, as well as the results of the Seidel correction. Experimentation was done also on the modification of t h e recorded densities according to the procedure of Arrak (23),where the 1 in the Seidel equation is a n adjustable parameter. I t was found that significantly better results were not obtained by changing the parameter from being equal to 1. T h e standard images are made of an aluminum disk in which the field of view is limited by a 35-pm diameter circular field aperture. T h e aluminum disk is defocused to ensure a homogeneous intensity across the field of view. The intensity level can either be read in terms of current off the photomultiplier or converted into counts per second. Since the total number of information bits is fixed and equivalent for each individual bit, the ion intensity per information bit is readily calculated. T h e field of view is limited to 35 fim for two reasons. The first is that it guarantees complete measurement (integration) of t h e signal by the photomultiplier which nominally sees a 2324
b
ANALYTICAL CHEMISTRY, VOL. 49, NO. 14,DECEMBER 1977
ti
I
;EYS
I
I
2
3
I TV
Flgure 4. Histogram of standard image in Figure 3
125-pm diameter field of view and which has a slight loss of light collection efficiency around the edges due to the optics that carry the light to the photomultiplier. The second reason is that i t produces an image on the film of 3.9 mm and this size can be conveniently digitized into a 64 X 64 image disk file. An iso-density contour plot and a three-dimensional representation of a standard image are illustrated by Figure 3. It should be noted that there is present a large amount of background as well as foreground area in each standard image file. A computer routine is used to extract the average background and foreground densities and estimate the standard deviations for these values from the standard images. Figure 4 illustrates a histogram of a typical standard image. T h e background and foreground are assigned to the maxima of the
low and high density peaks, respectively. T h e standard deviations are estimated from the points to the low density side of the background and to the high density side of the foreground. The points in between the two peaks are those that lie in the foreground-background edge region and thus would be expected to cause deviation from Gaussian character for the peaks. The data file consisting of the background and foreground densities and exposure values for the standard images is input to the routine that determines the characteristic curve. The computer calculates the best linear fit for the logarithm of exposure vs. the background corrected, Seidel modified density values, and outputs the coefficients for the line and the deviation from this line that each point suffers. T h e experimental image is converted from density space to ion intensity space by application of the characteristic curve on the background corrected and exposure-time corrected densities. The process involves the determination of the center of t h e ion image circle which delineates the ion image field in the entire image file. This determination is accomplished by a n iterative computer routine that finds the center point of the image by minimizing points over the background threshold in the background. The center point is typically within 5 pixels (picture elements) of the center of the square image file and varies according to where the photomultiplier head was manually stationed a t digitization time. Each ion image file consists of the image field and background to the image field in a ratio of digitized points of about 9 to 1. The film background, which is a function of the history of the film, is in the density range 0.07 to 0.15 (85% to 70% transmittance). The determination of the ion image field and consequently the background field allows the determination of the background in the immediate vicinity of the image and background correction of image field points. T h e effective range of the analytical curve is limited a t the upper end by saturation of the microphotodensitometer. The saturation density of the microphotodensitometer is sufficiently less than the saturation density of the film so that all t h e upper points lie on the straight line region of the characteristic curve. It is found that for the low end of the density range the deviations from the characteristic curve begin to become significant a t approximately twice the background level. T h e lowest density quantitative point of the characteristic curve is designated the low limit to the analytical range, and it is typically twice the background level or slightly higher. Thus, the analytical range in this quantitation process extends over a density range of greater than 2.5 with more than 200 levels. Since density is related exponentially to intensity with a slope approximately of 1, the analytical intensities extend over a range between 100 and 1000. Precision and Accuracy. A discussion of the precision and accuracy involved in ion image quantization encompasses the digitization process, the film darkening and development, and the determination and application of the characteristic curve. Integral to this discussion is the question of what minimum change in ion intensity produces a detectible change in density (contrast) for the process. Also important to the discussion is the effective analytical range of the process, as well as the ramifications of the conversion from a logarithmic to linear quantized image. The accuracy of the digitization process can be characterized by the quantization error, the error due to the quantized steps into which the density range is divided. This error is constant a t each step and is equal to 0.289 X DRIQL, where DR is the density range and QL are the number of quantized levels (24). Therefore, the relative error is inversely proportional to the density and only a t low density levels does it become truly significant. As illustrated in Figure 5 , it is less than 5% for
Figure 5.
Quantization error for digitization range 0-2D with 256 steps
0
-
I
v3.
a
CEYS!TV
Flgure 6 . Histogram peak resolution as measured by normalized peak width at half maximum vs. density
densities greater than 0.1 and less than 1% for densities greater than 0.5. The precision of the digitization process was estimated by the repetitive scanning of images with no adjustments to the densitometer. The standard deviation of the process was determined by averaging the standard deviations for corresponding points on different image files. This average standard deviation was equal to 0.65 of one density step for both low and high average density images. The precision associated with the image formation process can be estimated conveniently from the standard images used to determine the characteristic curve. The histogram of the foreground field produces a symmetrical Gaussian-like peak whose width is relatively independent of the exposure except a t maximum density levels. Figure 6 shows the effect of measurement precision on the resolution of histogram peaks. The major sources of imprecision in the imaging process are probably nonhomogeneity in the standard source for the image and statistical variations in the film. There is no way to distinguish the relative contribution of these effects to the overall spreading of densities. The fact that the eye is the final judge of the homogeneity of the image being photographed, and the variation in the measured densities would be indistinguishable to the eye, suggests that the primary component to measurement imprecision could be from the s a m d e itself. The characteristic curve least-squares linear fit shows typically an average deviation of points of 5 % . The accuracy of conversion of the experimental density points from density space to ion intensity space by the application of this characteristic curve must be considered to have a similar accuracy. T h e fact that the conversion is made from the APrlALYTICAL CHEMISTRY, VOL. 49, NO. 14, DECEMBER 1977
2325
Table IV. Calculated Concentrations of Composite Features Average Exposure density Image Isotope time, s Feature 1
58
Ni
1
5
0.625 0.647 0.780 0.759 0.814
2 2 3
6 T U
60Ni
20 20
1 2
a Cu characteristic curve; D , for isotopic abundance.
= 0.661
+
0 . 9 2 1 log
r C , Si,,G - 2 where i is the illumination, ions collected per unit area of =
surface times time; 7 is the practical ion yield, number of M+ ions collected per number of M atoms removed from sample; C , is the atomic concentration corrected for isotopic abundance; S is the sputtering yield, number of atoms removed per incoming particle; i, is the primary ion density; and G is the degree of magnification. If the practical ion yields and sputtering yields of a sample and standard are equal, the ion intensity or illumination is directly proportional to concentration when the analyses of both are made under similar conditions. When the assumptions that sputtering yields and practical ion yields are equal for sample and standard are made, the characteristic curve can be calibrated in terms of concentration of the standard element rather than in ion intensity. The characteristic curve as previously analyzed exhibits a linear relationship between Seidel densities and the logarithm of the product of ion intensity and time, or of d&!3iPG-* and time. For images exposed under similar conditions and of similar materials the logarithm of 7C,SiPG-' is equal to the logarithm of C,t plus the logarithm of 7 S i P G 2which is constant. Thus, given the characteristic curve for the film and the density of t h e image exposed for a time, t , of a standard material, the curve can be recalibrated to the logarithm of the product of concentration and time vs. Seidel density. Alternatively, the 2326
Concna
0.094
0.939 1.046 1.099 0.960 1.138
0.103 0.099
(Ct). Ni characteristic curve; D , = 0 . 2 6 4
logarithm of the exposure to the time-corrected intensity results in a larger relative error and a nonsymmetrical error bar in intensity space. Furthermore, conversion of a Gaussian distribution of densities into intensity space results in a skewed peak in intensity space by the nonequivalent widths of the quantized steps. The result of this conversion, then, is the disproportionate weighting of higher intensities which causes a n overestimate for the average intensity value for a feature with a Gaussian distribution of densities. The error is dependent upon the slope of the characteristic curve, which determines the intensity range, the width of the peak, and the number of quantized levels which is constant. This error was measured on a series of test images and was always less than 8% for a typical characteristic curve of slope 1.0. This error, however, is potentially correctable by the recognition and smoothing of features prior to conversion into intensity space. Applications. I . Concentration Mapping. The fundamental limitation to quantitative analysis by SIMS is the variability of the secondary ion yield which is dependent upon the state of the surface, the chemical environment (matrix effect), and effects induced by the primary ion sputtering. Thus, a compositional micrograph in ion intensity space is not necessarily directly convertible into concentration space. However, it has been shown that by using closely matched standards quantitative accuracies of 10% and better can be achieved (25,26). The limitation that exists here is the general nonavailability of suitable standards. From Morrison and Slodzian (20) the relationship between concentration and ion intensity is:
i
Background
ANALYTICAL CHEMISTRY, VOL. 49, NO. 14, DECEMBER 1977
+ 0.938
Relative error, % - 3.1 4.6 9.9
- 4.0 13.8
log (Ct), corrected
characteristic curve can be constructed from a series of graded exposure images of the standard in the same manner as prescribed for ion intensity. As a case in point of how standards can be used in the determination of elemental concentrations of micrographic features, a sample was prepared from nickel and copper foils consisting of two alternating layers of the foils, each with a nominal thickness of 100 ,um, clad together by heat and pressure. Images were taken of this composite and of the individual standard foils, all under similar conditions. The characteristic curves for both elements were determined from graded exposures of 0.5 to 10 s. The curves are linear with correlation coefficients of 0.9969 for copper and 0.9991 for nickel. The slopes of the curves are equal within experimental error. From these curves the images of the composite were converted into concentration space by the extrapolation of the background and time-corrected Seidel densities to concentrations. The results of this procedure as illustrated by the three related ion images of 58Ni+,63Cu+,and 60Ni+of the composite in Figure 7 are summarized in Table IV. I t is seen that the average concentrations of the features are equal to 1.00, the nominal concentration, within experimental precision. T h e average relative error in accuracy is 7%. The average precision equals 17 % . Much of the imprecision can be ascribed to the materials; for instance, obvious crystal grain structure for the copper and topography induced by ion sputtering a t the interfaces of the composite and the mounting material. T h e contour plots of the images in concentration space also are illustrated in Figure 7. Although simply designed, this example shows the applicability of concentration mapping. Furthermore, the interchangeability of isotopic images is also shown. I I . Sampling Constants. As previously stated, a major limitation to quantitative analysis with the ion microprobe is the lack of suitable standards. I t is common practice to calibrate signals with standards that have been certified for bulk concentrations using techniques that are insensitive to microscale variations in concentration. As opposed to bulk techniques in which homogenization of large amounts of material will eliminate sampling problems, probe techniques are constrained by their intrinsic nature to microsampling of materials that may or may not be homogeneous on the micro-scale. As a result of this constraint, sampling error can assume the preeminent fraction of the total error in the microprobe measurement. The level and range of heterogeneity of a sample can be estimated by the application of statistical techniques to the microprobe data (27). The procedure involves the compilation of single point analyses and the determination of the coefficient of variance for these points. The result of this analysis is that for a given probe diameter, a level of confidence can be assigned that sampling is of a homogenous area. Whereas the previous method of analysis gives an indirect indication of homogeneity, an ion micrograph (which can be considered a compilation of one ,urn2point analyses) gives all the information necessary to directly determine the feasibility
Figure 7. Bimetallic composite sample: (a)”Nit ion micrograph, 5-s exposure. (b) 6 3 C ~ ion f micrograph, 204 exposure. Fields of view: 225 p m . The 60Ni image is similar to ’*Ni+ and is not shown. (c) and (d) Isoconcentration contour plots of (a) and (b), respectively. Contour levels: 1. Calculated ion image field: 2-5. Concentration levels 25%, so%, 75%, and 100%. Size of image files 256 X 256
of probe analyses to the level of resolution of the image by showing the presence or absence of inclusions or secondary phases. Scilla and Morrison (28) have illustrated how images can be so utilized. Here, it is illustrated how the formulas they have derived can be combined with digital image analysis to determine the sampling constant for any material. This sampling constant, as they have shown, is related to the degree of uncertainty expected (sampling error) at a given confidence level for a probe measurement of a given area of the sample. The sampling constant of Scilla and Morrison is applicable when the element of interest is randomly dispersed as inclusions in the material under study. The greater the density of the elemental inclusions in number per volume, the greater is the homogeneity of the element in the material, and the smaller is this element’s sampling constant. In their determination of the sampling constant, Scilla and Morrison utilized absolute feature counting, making the assumption that all inclusions are of relatively constant size and intensity. With digital processing of the image, a more exact determination of the sampling constant can be made by utilizing the intensity information in the image. It can be shown that:
(3) where K , is the sampling constant in micrometers; i, is the intensity of inclusion i; N is the total number of inclusions; IT is the total intensity of the ion image; AT is the total area of the image, in Fm2; and AI is the total area of inclusions.
Assuming constant intensity for each inclusion, this equation reduces to the equation of Scilla and Morrison:
(4) where R is the number of inclusions per unit area. I I / I Tis the fraction of the signal produced by the inclusions. If there is a background signal, that is, if the element studied is present in two phases, one of which is homogeneous, the sampling constant is reduced proportionately. The term (1- A I / A T ) is a correction due to the surface coverage by inclusions. Assuming that the fraction occupied by inclusions is small eliminates this term. I t should be pointed out that although homogeneity encompasses the three-dimensional distribution of phases, the ion microprobe is essentially a two-dimensional analyzer of an elemental surface. Thus, the two-dimensional description of homogeneity, inclusions per unit area, is more appropriate to the ion microprobe measurement. It is true that in making the image, exposure time represents integration over depth. However, the depth of field vs. the breadth of field is insignificant (Angstroms vs. micrometers) justifying the assumption that the image represents an elemental projection of the surface. The determination of sampling constants is exemplified using the two images of titanium, Figure 8, in NBS-662 and 664 steels. Titanium is one of the most notoriously inhomogeneous elements in the NBS steels. These images were digitized, the image field was determined, and the image density points were converted to ion intensities. The inANALYTICAL CHEMISTRY, VOL. 49, NO. 14, DECEMBER 1977 * 2327
Table V. Determination of Sampling Constant Element/ standard Titanium/ NBS-664 Titanium/ NBS-662
Total intensity of image,
Total area,
IT
AT
Total intensity of inclusions I1
I1 /IT
AIIAT
Ks(Pm)
9420
37 100
8400
0.894
0.083
56.6
20998
36200
17585
0.838
0.105
46.8
lustrated how the discrete Fourier transform can be applied to ion images t o cross-correlate similar images of different elements from the same spot on a sample. Combining information in two cross-correlated images produces a fourdimensional array of information, containing two spatial dimensions and two elemental ion intensity dimensions. T h e cross-correlation function is a traditional method of matching two images. T h e basis for the operation of this function is that for aligned images of similar object fields, the cross-correlation is the sum of large values multiplied by large values and small values multiplied by small values. When the images are of improperly oriented similar fields, or of different object fields, the value of the cross-correlation function is the sum of large values multiplied by small values and small values multiplied by large values. The former case will always yield a higher number than the latter case, and therefore the cross-correlation function will reach a maximum for properly oriented, related pictures. T h e formula for the discrete cross-correlation function (CCF) of two images, I1 and 12,is:
CCF(MX,MY) = C C I,(JX,JY)I,(JX JY JX
MX, J Y
+ MY)
+ (5)
T o get a matrix of N X N CCF values for two image arrays that are each N X N, a total of N4 operations must be executed. I t can be shown, however, that:
FT(CC F ) = FT( 11)FT * ( 1 2 )
Figure 8. 4aTi+ ion images from: (a) NBS-662 steel, 2-s exposure; (b) NBS-664 steel, 20-s exposure. Fields of view: 250 p m
clusions are identified in the image field by thresholding. The threshold value is arbitrarily chosen as the average value of the image. At this intensity, the partial derivative of integrated intensity equals the partial derivative of integrated area. Once the inclusions are isolated, K , as defined in Equation 3 is straightforwardly found as all variables are calculable. The results of this calculation are listed in Table V. The application of sampling constants is described in the paper by Scilla and Morrison. In brief, it can be said that to achieve less than 25% sampling error at the 95% confidence limit using a 100-Fm diameter circular probe there are required 26 probe analyses for titanium in NBS-664 steel and 18 probe analyses of titanium in NBS-662 steel. I I I . Image Cross-Correlation. The theory of digital image processing is rapidly expanding, as well as the application of this theory in all fields. The ability to compress, enhance, and amend the information contained in images is made possible through the use of image transforms and complex mathematical procedures. One of the most commonly used image transforms is t h e discrete Fourier transform. Here. i t is il2328
ANALYTICAL CHEMISTRY, VOL. 49, NO. 14, DECEMBER 1977
(6)
where F T is the Fourier transform function. Whereas the number of operations required by the discrete CCF is prohibitive on large image arrays, the Fourier transform reduces the number of operations by two to three orders of magnitude making the calculation feasible. The application of Fourier transforms and their subsequent use in cross-correlation is accomplished by specific subroutines of IONPIX. I t is impossible to perform a continuous Fourier transform with a digital computer forcing the use of the discrete Fourier transform. T h e discrete Fourier transform of an N X N array, A(IX,IY) is: ;\--1
B(IKX,IKY)= C
N-1
C
I X = o IY = u
(7) for IKX, IKY from zero to n - 1. An advantage of the discrete Fourier transform is that the Cooley-Tukey or “fast Fourier transform” algorithm allows this function to be calculated relatively quickly (29). There is required for the Fourier transform a conversion of the image array from real to complex numbers, as each datum in the Fourier space has a real and imaginary part. This conversion results in a doubling of computer memory required to contain the image. The maximum image size is then reduced to 128 X 128. The procedural steps in the cross-correlation of two real image files are the conversion of each image file from real to complex space; the Fourier transform of each complex file; the complex multiplication of the two Fourier transforms; and finally the inverse Fourier transform of the product. T h e cross-corre-
A
-
C
E
:CF
FiLE
RESULTS
NB OFFSET
0
I
( n dimensions) information about a sample. As a result, ion microscopy is a very powerful but also a very complex technique. The fact that the power of the technique has not been fully exploited is probably due to its complexities. The use of the digital computer with ion imaging challenges the complexities with its multidimensional information handling. The applications in this article illustrate differing aspects of ion image quantitation but only scratch the surface. The exercise of the technique as an analytical tool to problems in such disparate fields as biology, metallurgy, and semiconductor technology is extremely promising.
ACKNOWLEDGMENT The authors are grateful to B. F. Addis for preparation of the composite sample and to B. F. Siegel, E. Kirkland, and K. Welles for their helpful discussions in the course of this project.
LITERATURE CITED
Figure 9. Cross-correlation of ion images. (a) and (b) are contour plots of 48Ti+ and 93Nb+ in NBS-662 (2-s and 100-s exposures, respectively). Size of files 64 X 64; subsections of full field 256 X 256 images. (c) Contour plot of the CCF of (a) and (b). Peak a t (X,Y) = (5,60) indicates offset (-4,+5) for image (b). (d) 93Nb+ image subsection with indicated offset
lation f i e is then searched for the maximum point and position of the maximum point. The height or magnitude of the maximum point is a measure of absolute correlation. The offset position indicates whether the files are properly oriented relative to each other. If the maximum is at point (1,l)either the images are best matched relative to each other or there is no better match calculable. The images must be within an absolute distance apart for a meaningful offset to be calculated, if they are related. This distance, known as the capture radius, is about N / 3 , where N is the number of picture elements to a side. For a precise determination of the offset, the original images should be within N/4 picture elements. If the indicated offset is made to the original picture and another cross-correlation to the reference image made, either a perfect or very close to perfect match can be expected with an increased height of the correlation peak. Images can be expected to be positioned within one-half picture element relative to each other. T o illustrate the application of cross-correlation, two images of NBS-662 steel of titanium and niobium were used. It is well known that Ti and Nb are strong carbide formers in steels, and that both elements are strongly heterogeneous (30). Therefore, there should exist a strong correlation between the images. Correlation was made on sub-images of the two ion images of size 64 X 64, or 1/16 the total image field. Contour plots of the sub-images, the resultant cross-correlation file, and the offset image indicated by the correlation peak are illustrated in Figure 9. The height of the correlation peak is indicative of the strong correlation between the images.
CONCLUSION Ion microscopy is a multidimensional analytical technique producing simultaneous spatial (3 dimensions) and elemental
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RECEIVED for review August 22, 19'77. Accepted September 27, 1977. Financial support was provided by the National Science Foundation under Grant No. CHE-7608533 and through the Cornel1 Materials Science Center. One of the authors (J.D.F.) was the recipient of the Edwin Dowzard Summer Fellowship of the Analytical Division, American Chemical Society.
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