Anal. Chem. 1983, 55, 557-564 --___
Registry No. In, 7440-74-6; K, 7440-09-7.
Table I. Origin of Transient LEI Pulse Derived from Steady-State Data ( 2 0)
LITERATURE CITED
time after aspiration (arbitrary units) concn of K (pg/mL) concnof In(ng/mL) %signal recovery LEI signala a
1
2
0.05 1
0.5
99 1
3
4
2.5 50
3.5
10
86 8.6
60 30.0
25 17.5
One signal unit/concentration unit.
_____I___
70
557
5 5 100 0 0
-_
signal for indium should reach a maximum and then return to a value which corresponds to the steady-state signal for indium with a potassiurn matrix as the experiment suggested. The maximum amplitude of the transient LEI signal was linear with indium concentration. Since it was also linear with potassium concentration, a standard addition technique would be necessary to quantify the indium. This approach permits the determination of an analyte in a matrix that would otherwise completely suppress the analyte signal. Observation of the transient LEI signal for the analyte is the instrumental analogue of sample dilution to reduce matrix interferences.
Green, R. 8.; Keller, R. A.; Schenck, P. K.; Travis, J. C.; Luther, G. C. Appl. Phys. Lett. 1978, 29, 727-729. Green, R. B.; Kell'er, R. A.; Schenck, P. K.; Travis, J. C.; Luther, G,C. J . Am. Chem. SOC. 1978, 98,8517-6518. Turk, G. C.; Travis, J. C.; DeVoe, J. R.; O'Haver, T. C. Anal. Chem. 1978, 50,817-820. Travis, J. C.; Turk, G. C.; Green, R. B. ACS Symp. Ser. 1978, No. 85,91-101. Turk, G. C.; Travis, J. C.; DeVoe, J. R.; O'Haver, T. C. Anal. Chem. 1979, 57,1890-1696. Turk, G. C.; Mallard, W. G.; Schenck, P. K.; Smyth, K. C. Anal. C b m . 1979, 51, 2408-2410. Gonchakov, A. S ; Zorov, N. B.; Uuzyakov, Yu. Ya.; Maveev, 0.I. Anal. Lett. 1979, 72,1037-1048. Turk, G. C.; DeVoe, J. R.; Travis, J. C. Anal. Chem. 1982, 54, 643-645. Travis, J. C.; Schenck, P. K.; Turk, G. C.; Mallard, W. C. Anal. Chem. 1979, 51, 1516-'1520. Green, R. B.; Havrilla, G. J.; Trask, T. 0. Appl. Spectrosc. 1980, 34, 561-569. Havrilla, G. J.; Gruen, R. B. Anal. Chem. 1980, 52,2376-2383. Green, R. B. Anal. Chem. 1981, 53,320-324. Trask, T. 0.; Turk, G. C. Anal. Chem. 1981, 53, 1187-1190. Smith, B. W.; Parsons, M. L. J . Chem. Educ. 1973, 5 0 , 679-681. Corson, D. R.; Lorraln, P. "Introduction to Electromagnetic Fields and Waves"; W. H. Frtseman: San Francisco, CA, 1962; Chapter 4. Smith, K. C.; Mallard, W. G. Combust. Sci. Techno/. 1981, 26, 35-41. Mallard, W. G.; Smyth, K. C. Combust. Name 1982, 44,61-70.
ACKNOWLEDGMENT The authors thank Ken McElveen, David Paul, and Ted Beeler for their help with the electrical aspects of this project. The authors also acknowledge the helpful comments provided by E. H. Piepmeier.
RECEIVED for review July 27, 1982. Accepted December 6, 1982. This research was supported by the National Science Foundation under the Arkansas EPSCOR Grant and NSF Grants CHE-79186126 and CHE-810500.
nhancement and Restoration of Chemical Images from Secondary Ion Mass Spectrometry and Ion Scattering Spectrometry Bernard G. M. Vandeginstel and Bruce R. Kowalski" Laboratory for Chomometrcs, Department of Chemistry BG- 10, University of Washington, Seattle, Washington 98 195
Digital image processing methods are applied to chemical images of surfaces obtained by a computer-controlled Ion beam spectrometer. Various image restoratlon and image enhancement methods are tested according to their abllltles to remove the blurring e4fect imposed on the ion scattering spectrometry or secondary Ion mass spectrometry images by using a scannlng Ion beam wlth a finite beam dlameter. These methods provlde llmproved spatial resolutlon allowing much finer detail to be observed whlle, In some cases, also improving the slgnal to noise ratio. The methods tested are part of a large image proccesslng system applicable to images produced by a number of surface analytical Instruments. They are seen as the first step In complete Image processhg.
Resolution in the time (chromatography), wavelengtb (spectroscopy), and spatial domains (surface analysis) is an important .topicin analytical chemistry as it limits the amount 'Permanent address: llepartment of Analytical Chemistry, University of Nijmegen, Toernooiveld, 6525ED Nijmegen, The Netherlands. 0003-2700/83/0355-0557$0 1.50/0
of information that can be obtained during analysis. In many cases the search for improved resolution has been the motive for both progress in instrumentation and the introduction and development of novel mathematical tools, such as fact(or analysis (I,Z),multiple linear regression analysis (3, 4 ) and the Fourier convolution theorem (5). Most of all present-dtiy chemical applications of these mathematical methods have been on one-dimensional analytical methods, where a signal is recorded as a function of one variable (e.g., spectra, chrlomatograms). A new and logical development initiated in analytical chemistry is the augmentation of the dimensionality of the analytical data, realized by linking two one-dimension,sl methods, e.g., GC/MS, HPLC/UV, etc. Here, traditional univariate statistical methods fail to extract the full amount of analytical information and should be replaced by multivariate methods. A recent example (6) is the application of multivariate curve resolution to GC/MS data to determine the number of compoinents in a complex mixture and estimate the spectra of the pure compounds even when the chromatographic peaks are only partially resolved. Recently. multidimensional analytical methods have been developed in the area (of surface analysis that produce chemical 0 1983 American Chemical Society
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ANALYTICAL CHEMISTRY, VOL. 55, NO. 3, MARCH 1983
data in two spatial axes, calibrated in units of length. Important methods in that field are secondary ion mass spectrometry (SIMS) and ion scattering spectrometry (ISS). By representing scattered or sputtered ion intensities on a gray scale, we obtain an image of the surface that can be considered a chemical image (7). The necessary development of the mathematical tools to fully exploit the analytical capabilities of these methods has just recently begun. Specifically, digital image processing techniques, mainly developed to analyze medical and satellite images, have been applied to display and analyze data from an ion microscope. Morrison and coworkers (8-12) were the first to demonstrate the potential of digitizing chemical images and submitting them to various digital image processing techniques. They interfaced a direct imaging SIMS instrument with a minicomputer, using a TV camera as an interface between the SIMS instrument and an image display system (13). Direct imaging SIMS instruments (ion microscope) bombard the entire sample surface with a fixed ion beam of a sufficiently large size, followed by a simultaneous multipoint detection. As a result the system has a very high resolution. However, some complicated procedures are necessary to calibrate the TV camera. While the ion microscope uses a large diameter beam and ion optics similar to those of an electron microscope, its sister instrument, the ion microprobe, uses a very narrow ion beam to analyze the surface of a sample. Ion microprobes are not normally used to obtain chemical images. Rather, these instruments only examine small select areas over the surface of the sample. Both types of instruments have been used extensively in geology, medicine, semiconductor research, and material science to better understand the chemistry at the surface of samples. They have given new life to the field known as surface science. In the present paper, a fully computer-controlled digitizing system for an ion microprobe is presented and used to obtain chemical images comparable to an ion microscope. Here, a small ion beam raster scans across a given sample area and the emitted ion current is detected simultaneously. A pulse counting detector converts the measured ion intensities into pulses that are readily counted with digital counters, interfaced with an LSI 11/23 microcomputer. In addition to this data-acquisition system, the LSI 11/23 controls the scanning process of the ion beam over the sample surface. Maximal flexibility of the digitization system has been acquired by a full control of the count time per pixel (a point in the image), the number of collected pixels, and scanned area. In many respects, the quality of the acquired images is much superior to the images displayed on the storage scope of the SIMS instrument. Better resolution, better dynamic range, and better reproducibility are important benefits made possible by our system. Microelectronics is an example where the need to analyze smaller and smaller microstructures on the surface and to various depths has great benefits. If the ion beam diameter of a microprobe or the spatial resolution of an ion microscope is larger than the microstructure, its structure will be blurred and important detail lost. Other studies where maximum resolution is highly desirable include contact metalligation problems, tissue examination, and the detection of precipitates in semiconductors. This paper presents algorithms for image enhancement and restoration to improve the spatial resolution of these instruments and is a first step toward the application of more sophisticated image-processing algorithms.
EXPERIMENTAL SECTION The heart of the image-SIMS combination (Figure 1)is the interface between a LSI 11/23 microprocessor and a 3M (now Kratos) SIMS/ISS instrument. This interface has three functions:
Figure 1. Diagram of the SIMS/ISS Digital Imaging System.
Table I. Performance Characteristics of the SIMS/ISS Data Acquisition System size counts count time raster time
64 X 64 to 256 X 256 0-lOOO/pix reduced to 8 bit levels 1-30 ms/pixel 4096 X 4096 pixels over max 10 X 10 mm beam deflection
sampling distance time
min. 2.5 pm 6 s-32 min
default value
128 X 128 X 8 20 ms/pixel 40 pm 5 min
(i) control of the position of the ion beam, (ii) control of the cylindrical mirror analyzer (ISS) and the quadrupole mass analyzer (SIMS), and (iii) data acquisition. The interface consists mainly of a D/A converter, which translates the desired beam positions into voltages applied to the ion beam deflection plates and two 16 bit counters for counting the pulses generated by the electron multiplier and setting the count time per pixel. The D/A converter allows 4096 different positions of the ion beam between full deflections in both directions. Besides a Color Graphic Display system (512 X 512 X 12 bit), a Tektronix 4012 graphical display (64 X 64 X 5) and Printronix (128 X 128 X 5) image printer are used for image display. All routines for (i) beam positioning, (ii) count time setting, (iii) pulse counting, and (iv) beam switching are written in MACRO-11 and are called from a FORTRAN program that controls the entire data acquisition process. A complete picture up to 256 X 256 pixels can be stored in the RAM memory. Afterward, the image is stored on disk or sent t o the VAX 11/780 for image processing. The performance of the data acquisition system (Table I) is mainly determined by the characteristics of the 3M SIMS/ISS instrument. The ion beam can scan about 10 mm across the sample surface (dependent on the distance between the sample and the detector), in 4096 discrete steps, resulting in a minimal sampling distance of 2.5 pm. Unfortunately, the beam diameter (about 125 fim for a small beam) constitutes a severe limitation in resolution. The ion counting rate dictates the count time per pixel that is necessary t o obtain a reliable digitized signal. Although the images are reduced to 8-bit gray levels afterward, the number of counts (N,) per pixel should be as high as possible in order to obtain a better signal to noise ratio. From the measurement of 36 consecutive frames, we observed that the detector follcws the Poisson counting statistics very well, with VAR(N,)/N, = 1. Therefore, a count time was selected that yielded at least 1000 counts at the high signal levels in the image (3.3% relative standard deviation). As a consequence, for average cost rates, the exposure time for a 128 X 128 image was -5 min. Although
ANALYTICAL CHEMISTRY, VOL. 55, NO. 3, MARCH 1983
Iimage nput
I
P r e p roc: e s s I n g SCALING
559
Te c h n I q u e s
I
, ROTATION, SHIFT,
M A G N I FY
E--
- - -
_ I -
--
Image Restoration
I N V E R S E and WIENER
CONSTRAINED
Image Processing FILTER,
LEAST SQUARES
L
MEDIAN FILTERING,NOTCH FILTER
C O N T O U R , 3-DIM ,
3
S M O O T H I N G( S P A T I A L )
THRESHOLDING
HISTOGRAM MODIFICATION
--
-
__-
--
-
Figure 2. Diagram of the main routines of the digital image processing (IPS) software package (contact author for program) I l l R G E DEGRRDRTION
4 7
Fourier theory stater4 that a convolution in the spatial domain becomes a multiplication in the Fouier domain. Thus, i n the
absence of noise G(u,u) = F(u,u) X H(u,u)
where G(u,u),F(u,u), and H(u,u) are the Fourier transforms of G(x,y), F(x,y), and H(r,y), respectively. Including the noise N ( x , y )in the model as an independent randomly distributed additive function, eq 2 becomes G(u,u) = F(u,u) X H(u,u)
Figure 3. Image degradation by beam geometry. SIMS is known to alter the surface structure during analysis, the amount of time the beam Eipends at one spot during imaging with our system is low keeping damage to a minimum. A software package for interactive image processing (IPS) has been developed to apply the most common image processing operations on chemical images. In Figure 2 a survey is given of all operations that are available.
RESULTS AND DISCUSSION Sources of Image Degradation in SIMS. Unfortunately, scanning SIMS instrumeints do not make an exception on the general rule that measurements are not a perfect reflection of reality. Noise and other blurring sources degrade images. Our major source of degradation was the blur caused by the fairly large diameter of the ion beam. Since the ion beam bombards an area around the target (Figure 3) sharp edges in the image become more or less fuzzy. The degree of blur depends on the radial distribution of the ion beam current density. In terms of image degradation, this is called pointspread-function (psf). As a demonstration of the effect of the beam size, two Au images are shown of a 150 pm diameter Au wire, one taken with a 12!5 pm diameter beam, and the other with 600 pm beam (Figure 4). The degradation can be expressed mathematically as a convolution(*) of the true image, F ( x , y ) , with a pointspread-function, H ( x , y ) ,where
( 2)
+ N(u,u)
(3)
Here, it is clear that the potential for correcting for blur and noise depends on a prior knowledge of H and N . If models for H and N are used to enhance the resolution, and thus the quality of the image. then in the jargon of image processing, the operation is called “image restoration”. On the other hand, operations without recourse to knowledge of the degrading phenomena are termed “image enhancement”. Image Enhancement. The purpose of image enhancement is to improve the signal-to-noise ratio and to enhance the details in the image by sharpening the blurred edges. However, both goals are somewhat contradictory: a deblurririg operation on one hand enhances the edges and thus emphasizes the high frequencies leading to a poorer signal to noise ratio. On the other hand, noise reduction methods tend to blur the image. Generally speaking, image operations can be done in both spatial and frequency (e.g., Fourier) domains. The latter requires a Fourier transformation of the two-dimensional data, that can be calculated by consecutive onedimensional Fourier transforms of the rows and columns of the image matrix, using a fast Fourier transform algorithm (14). A visual representation of the Fourier transform can be obtained by displaying the logarithm (to increase the dynamic range) of the magnitude of the transform coefficients. Figure 5 represents a SIMS A1 image of a 3 X 3 mm circwit and its Fourier transform, with the origin shifted to the cenkr of the image (128 X 128 X 8 bit). Enhancement in the frcbquency domain is usually obtained by applying a transfer function, T(u,u),to the Fourier transformed images, whereafter the result is back-transformed to the spatial domain, e.g.
G’(u,u) = G(u,v) X T(u,v)
(4)
where, e.g., T(u,u) = 1 for u 5 uo and u -i uo and T(u,u) = c) otherwise. Depending on the shape of the transfer function
560
ANALYTICAL CHEMISTRY. VOC. 55. NO. 3. MARCH 1983
Figure 4. Degradation as a function of the beam diameter, SlMS Au image (128 X 128 X 4) of 150 Om thick Au wire: (top) beam size. 125 pm: (bottom) beam size, 600 pm.
many types of filters can he designed (e.g., low-pass or high emphasis filters). However, since Fourier transforms are relatively time-consuming operations, and the above-mentioned filters perform relatively poorly, enhancement techniques in the spatial domain are preferable. Since our aim is to enhance detail, we implemented and investigated two candidate filters; a Laplacian high-emphasis spatial frequency filter (15)and a notch filter (16). For a long time, spectroscopists have improved the resolution of their spectra by subtracting the second derivative of a spectrum from the original. The idea behind this procedure is that the second derivative of a peak is negative between the two inflection points (where the second derivative is zero) and positive on the wings. The result of the subtraction then is increased peak intensity, sharper peaks, and
Figure 5. SlMS AI image (128 X 128 X 4) of a 3 X 3 mm circuitry. small beam (150 pml, field of view (3 mm X 3 mm), exposure time
(30 mslpixel): (top] original image: (bonom)Fourier transform (origin
in the center).
better resolution. This effect can he obtained on two-dimensional data as well (14). Thus, a two-dimensional linear second derivative operator
Pf a y - - + - = A z 2 f ( i j )+ A y 2 f ( i j ) - 6x2 Sy2
(5)
should he applied to the images which in discrete form (14) are
vy =
(f(i + 1j)+ f ( i - 1 j ) + f ( i j + 1)
+ f ( i j - 1)) - 4 f ( i j ) (6)
This operation is analogous to a convolution of the image with the mask
ANALYTICAL CHEMISTRY, VOL. 55. NO. 3, MARCH 1983
561
This is a very typical kind of operator, which is used in didtal image proceasing because of its easy implementation on digital computers. Memory requirements are very small since only three image lines need be stored in memory a t a time. For example other image processing techniques such as edge d e tection, line detection, etc., are all possible by the convolution of the original image with an appropriate mask (17). The calculation and subtraction of the second derivative also can be combined in one operator f-VZf= 5f(ij) - (f(i + l j ) + f(i
- 1:) + f ( i j + 1) + f ( i j - 1)) (7)
and the mask becomes 0 -1 0 -1 5 -1 0 -1 0
An example of the effect of this type of filtering is shown in Figure 6. I t represents an SIMS Fe image of a razor blade edge partially covering a hole in the sample holder (stainless steel). Clearly the edges have been sharpened and more detail is visible. The low frequency notch filter is a well-known nonlinear filter, that is very simple to implement. It is a modification of the subtractive box filter, where the average gray value is computed over a given region (s = filter size) surrounding the pixel, which in turn is subtracted from the pixel value itself: f ( i d = (1 - k)(f(ij)
- fijm) + k f ( i j )
(8)
where k represents the fraction of the original image that is retained in the result Varioua modifications of that filter have been proposed (16) claiming a reduction in unwanted side effects. Basically all variations of the box filter constitute a modification of the defnition of the set s in eq 6 as: (i) variable zonal notch filter, the new set contains all pixels in s whose gray levels lie within a given threshold value; (ii) symmetric variable zonal notch filter, a modification of the variable zonal notch filter, hut the new set contains only the symmetrical pairs of pixels which meet the threshold condition. We have implemented both of the above filters developed by Towfiq and Sklansky (18) who have shown that these algorithms reduce the production of false edges in the image. Figure 5(top) is an unfiltered digitized SIMS AI-image of a 5 X 5 mm transistor. Clearly, the image contains some strong shading which obscures the presence of the AI-free divider. After applying a symmetrical zonal notch filter (Figure 7), shades are corrected and details become better visible. Image Restoration. If the blurring source is known and ita effect can be expressed mathematically as a convolution of the true image with a pointspread-function (psf~,then an estimate of the unblurred image F ( x , y ) can he obtained by the inverse operation of a convolution or
This reaults in the expression of the well-known inverse filter, with a filter operator M(u,u) = l / H ( u , u ) . In the ideal case where the image is noise free, the estimated image F(u,u) is equal to the true image F(u,u). However, if the image is degraded by noise, the problem becomes more complicated. Noise may introduce a considerable power a t high frequencies, as shown in Figure 5(bottom). As a result N(u,u)/H(u,u)>>
,
1.
-b’
Flgum 6. H W emphasis spatial filtering. SIMS Fe Image (128 X 128 X 4) of a 5 X 5 mm Fe blade, small beam (150 pm), field of view (5 mm X 5 mm). exposure time (30 mstpixel): (top) original image; (bottom) flltered version.
F(u,u), and the estimated image &,u) is not even vaguely recognizable. The application of inverse filtering on our SIMS/ISS images does not lead to acceptable results in estimating F(u,u). Various modifications of the inverse filter have been proposed, but all bear some arbitrariness in use. A restoration filter that minimizes the mean squared error between the estimated and true image has been developed by Wiener (14). The feasibility of estimating the ratio of the power spectra of the noise and the true image determines the applicability of the filter to SIMS/ISS images. We have observed that our pulse counting detector has Poisson statistics which has two consequences: (i) The relative standard deviation of the measurements of the pixel intensities is inversely proportional to the square root of the signal. (ii) The spectral density of the noise is constant, as the noise is spectrally white (Poisson noise is uncorrelated). In ISS/SIMS, the noise is uncorrelated, but multiplicative rather than additive. Therefore, one of the conditions for applying the Wiener filter is not fulfilled. Therefore, we implemented the very elegant alternative method proposed by Hunt (191, where an estimate F ( x y ) of the true image
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ANALYTICAL CHEMISTRY. VOL. 55, NO. 3. MARCH 1983
R h l e 11. Half-Height Width of the Point-Spread-Function (pm)-HorieonW/Verticd Cross Sectiolu of . I IISS h S g e of a Au Grid Horizontal Cross Sections beam position: mm V H: -2.40 -1.62 -1.83 -0.44 +0.74 + 1.53 + 2.36 -1*."* a9
-1.05 -0.17 +0.70 +I% +2.45
H
291 245 ~
v:
-2.18
-2.36 -1.48 -0.61 +0.26 +1.13
262 218 183 210
-1.35
297 349
131 114 157 183 218
175 157 175 210 245
201 183 197 271
Vertical Cros Section8 beam position,' mm -0.52 +0.21 -1.05
+ 1.83
+ 2.62
192 122 139 192 201
175 140 140 192
25.1 ___ 179
279 218 218
140 183
1_ RR ..
170 148 131
210 __. 148 148 122
116 148 157
10.01 is the center of the imane.
filter, however are shifted to the determination of the value to noise ratio in the image. of 7 . which dewnds on the sSi& that valie cannot be &essed in a direct way, we used an iterative procedure proposed by-Hunt where y is iterated until the condition xmixmj(G - C)z = m2 VAR(N) a is fulfilled (G is a rn X m image). It is clear that for the calculation of the Hunt filter only the variance of the noise (besides the nature of the degradation) need be known. Because of the signal dependency of the noise in SIMS/ISS images, we developed a modification of the Hunt filter where the residuals between the measured and estimated images are weighted aa well as the s u m of squares of the noise (lln112).The weighted sum of squares of the noise equals
d
*
or
If the measured image w88 scaled down with a factore K (51) to a smaller number of gray levels, then an analogoua derivation gives
npure 7. Symmetrical notch finering. SIMS AI Image (128 X 128 X 4) of a 3 X 3 m m circuilry. fikered version: same condnions as In Figure 5.
F ( x y ) is calculated in such a way that after reconvolution of *e result with the pointspread-function H ( x y ) ,an estimate C(xy)of the measured image G(xJ) in obtained, that satisfies the criterion that the sum of the residuals ~ " ' i - , ~ " ' j - l (-G &ecluals the sauare sum of the noise: mVAR(N). Since the solution of that problem is not unique, Hunt included a constraint that the original undegraded image is relatively smooth. In a mathematical sense, this means that the sum of squares of the second derivative of the image should be minimal. After an elaborate derivation he obtained the following filter equation in the Fourier domain: 1
IH(u,v)12
M(u,o) = (lo) H(u.u) IH(u,u)l2 + y P ( u , ~ ) ~ where P(u,u)is the Fourier transform of the Laplace operator, defined by eq 6. The problems encountered with the Wiener
The weighted s u m of squarea of the differences between and Gij equals
me problem is to find a
for which
IlrIlZ _ -- -lln1I2* a mz
Gij
rnz
(15)
Our experiments have confvmed that ll#/mz is a m o n o t i d y decreasing function of y. Consequently, y can be calculated iteratively, using a Newton-Raphson algorithm. The effect of the application of the Hunt algorithm on SIMS/ISS h a g is 2-fold: (i) an enhancement of the signal to noise ratio as a result of the applied smoothing constraint; (ii) a correction for the blurring effect of the point-spread-function of the ion
ANALYTICAL CHEMISTRY. VOL. 55, NO. 3. MARCH 1983 503
I S - hl GRID (t3p
mp
nen8.
smulatfon Of me rk3termMhl Of me half* widm (w,n) of a Oa&n p o h t v b h m c t b n by measwingme h p & e respmw of a system: bandwidths of the impulse. 160 pm and 190 pm; w,,? (psf). actual half-height wldm of the psf; wl,? (slg). halfheight w!dth of the slgnal. c
rc
."
L'
'.
Flgura IO. Determlnation of half-heighi wldm of the p s f same condnlons as Figure 5: uods section of an ISS Au Image (128 x 128 X 4) of a 5 X 5 Au mm grld.
Determination of thee half-heioht width of the osf: 1SS Au image (128X 128 X 4)of a 5 X 5 rnm Au grid (30 p k thick wires on 800 pm distance). Flour0 9.
beam. However, the ratio between the smoothing and deblurring effect depends distinctly on the signal to noise ratio in the input image. For high noise levels (small count times per pixel) the optimal y value remains relatively high, causing the filter to act like a smoothing algorithm rather than a dehlurring algorithm. However, our results suggest that the Hunt operation introduces almost no blur, as opposed to the conventional smoothing operators. Obviously, the NewtonRaphson iterative procedure converges to the inverse filter (y = O), in the absence of noise. Of crucial importance for the application of this type of filter is the feasibility of measuring the point-spread-function. For SIMS/ISS, that function depends on both the position (distance, angle) of the sample surface relatively to the ion gun and the properties of the ion beam itself (diameter and ion beam current). Hence, the psf must be determined every time one of these parameters has been changed. In our opinion, the most practical way to determine the psf experimentally is by measuring an impulse or step response of the system. The measurement of impulse
It I
i
Flpur 11. Constrained leest-squares mering (&nil: same m a n ~ o n s as F @ m 5: SlMS Fe h g e (128X 128 X 41 of a 5 X 5 mm Fe osde.
responses is preferred as the psf is measured directly, if the bandwidth of the impulse is much smaller than the bandwidth of the psf (Figure 8). In order to determine the impulse response in both directions and over the entire sample surfam, an Au grid consisting of 40 pm thick wires separated by 800 urn was used. Expected half-height widths of the psf larger than 150 pm are then accurately measured. An ISS image of the Au grid is shown in Figure 9. The drop of intensity toward the ends of the image is a well.known and easily corrected phenomenon, caused by the changing acceptance anEle of the detector. Ry taking cross-sections in two directions, one obtains a plot of the psf as shown in Figure 10. These cross sections can be described with Gaussian functions from which half-height widths are determined as a function of the spatial cwrdinatps.
584 * ANALYTICAL CHEMISTRY. VOL. 55. NO. 3. MARCH 1983
f
results from our instrument (Figures 11 and 12) are very promising. The signal to noise ratio has been improved, while the edges are sharpened. The overall result is a considerable improvement of the quality of the images. By removing the blurring effect with a Hunt filter, we estimate that microfeatures on the order of 5 pm can be analyzed by our instrument. Comparison of these results with those obtained with the spatial high emphasis filter shows that the restoration sharpens and smooths the image in a single step, while the high emphasis filter introduces noise while sharpening. The Hunt filter is much more complicated to implement compared to the high emphasis filter. During the iteration steps several two-dimensional Fourier transforms must he calculated. Moreover, the picture matrix, the Laplacian o p erator, and the 2-D point-spread-function as well have to be extended with zeros to avoid wrap-around errors, causing a considerable augmentation of the matrix size (sometimes a doubling, in order to be able to calculate the FFT on the base of 2), and, consequently, the calculation time. A detailed mathematical outline of the Hunt filter can be found in the original paper of Hunt (19) and the textbooks of Rosenfeld ( 1 4 ) and Gonzales (15).
ACKNOWLEDGMENT The authors owe a great debt to Margaret Grifiith for many hours of instrument operation and Clemens Jochum and Mark Champion for construction of the instrument control and data acquisition system. LITERATURE CITED (1) (2) (3) (4)
Howery. D. 0. Am. Lab. ( F a W . Con".) 1978. 8 . 14. MalInowSLl. E. R. Anal. Chem. 1977. 4 9 , 806. Saxberg. B. E. H.: Kowakki. 8. R. Anal. Chem. 1979. 51. 1031. Warner. 1. M.; DBVMSOII. E. R.: Christaln. G. 0. Anal. Chem. 1977. 40 . a,
*.CI
