Application of simulated poisson statistical processes to stem imaging

Signal Processing 8 (1985) 51-62 North-Holland

51

APPLICATION OF SIMULATED POISSON STATISTICAL PROCESSES TO STEM IMAGING Thierry PUN, Member EURASIP and James R . ELLIS Biomedical Engineering and Instrumentation Branch, Division of Research Services, Bldg. 13, Room 3W13 . National Institutes of Health, Bethesda, MD 20205, USA

Received 4 January 1984 Revised 9 July 1984

Abstract. The computer generation of Scanning Transmission Electron Microscope (STEM) images is one way of approximating controlled experimental conditions . These STEM images are assumed to be composed of signals derived from Poisson distributed variables ; three algorithms for Poisson deviate generation are examined . Relationships between the image acquisition parameters and resulting object contrast are given . An application to evaluation of the minimum detectable elemental concentration in Electron Energy Loss Spectrometry (EELS) images is presented . It is found that for typical experimental conditions, the required analysis time for the sample varies approximately as the inverse square of the concentration (when other factors are equal) .

Zosammenfassung. Die Computersimulation von Bildern, die mit einem Abtastelektronenmikroskop (STEM) gewonnen werden, erlauben die experimentellen Bedingungen zu reproduzieren . Diese Bilder werden aus Zufallsvariablen mit einer Poissonverteilung aufgebaut . Drei Algorithmen fuer die Synthese solcher Variablen werden untersucht . Der Zusammenhang zwischen den Parametern der Bildeaufnahme and dem optischen Kontrast des Ohjekts wird hergestellt . Als Anwendung wird die minimale chemische Konzentration bestimmt, die in einem EELS-Bild (Electron Energy Loss Spectrometry) noch gefunden werden kann . Es stellt sich heraus, dass die nbetige Untersuchungszeit einer Probe bei typischen experimentallen Bedingungen wie das Inverse des Quadrates der beobachteten Elementkonzentration variiert (wenn alle anderen Parameter konstant bleiben) .

Resume . La simulation par ordinateur d'images de microscope electronique a balayage (STEM) permet de reproduire approximativement des conditions experimentales donnees . Ces images sont composees de signaux derivant de variables a caractere poissonien ; trois algorithmes sont examines pour In synthese de telles variables . On detaille les relations liant les parametres d'acquisition de l'image avec le contraste visuel de I'objet . L'evaluation de la concentration chimique minimum detectable dans one image obtenue par spectroscopic a pertes d'energie (EELS) est presence a titre d'application . II en ressort que, pour des conditions experimentales typiques, le temps d'analyse requis pour on echantillon vane approximativement comme l'inverse du carve de la concentration de I'element observe (les autres parametres etant constants) . Keywords . STEM imaging, Poisson process, simulation, image processing, detectability limit .

1 . Introduction Computerized Scanning Transmission Electron Microscopes (STEM) are becoming operative in many analytical laboratories [1, 2] . Consequently, more and more interest is being devoted to understanding and modeling the physical processes involved in the microscope image formation [3, 4] . A problem of particular interest is the evaluation of the minimum detectable elemental (i .e . iron, nitrogen, etc .) concentration in a sample [5, 6, 7, 8] . This concentration threshold depends on the physics of the process, as well as on the shape parameters of the object as it appears to the observer . 0165-1684/85/$3 .30 <© 1985, Elsevier Science Publishers B .V . (North-Holland)



T Pun, J. Ellis / Simulation oJ'STEM images

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Visibility of objects in images is not a novel subject . Psychophysical experiments on vision have been performed extensively for many years (for example [9, 10, 11]), and are not the purpose of this paper . Results presented here concern the theoretical modeling of part of the STEM image formation, and the usage of simulation for reproducing laboratory conditions with controlled parameters . Visual contrast can be related to the chemical concentration . This provides a new and versatile tool, linking the image acquisition parameters with the minimum detectable elemental signal, and allowing optimization of the use of the STEM in a wide variety of situations . This paper is organized as follows . An elementary introduction to STEM imaging is presented (Section 2) . Then, for the sake of clarity, some well known facts about Poisson processes and various random number generators are reviewed (Section 3) . The relationships between acquisition parameters and image contrast are given . Since in STEM imaging "detectable" usually refers to the visually perceivable, simulation of simple patterns can provide an estimation of elemental detection limits (Section 4) .

2 . X-Ray and

EELS

imaging

Scanning Transmission Electron Microscopy (STEM) is based on the controlled scanning of the location of an electron beam which passes through a sample to be analyzed . The primary beam electrons undergo a rich variety of interactions, which are divided into two classes [1] . There are first elastic events, which affect the trajectories of the primary electrons without significantly altering their energy . Second, there are inelastic events, which result in a transfer of energy to the sample . These latter can lead to the generation of secondary electrons, Auger electrons, characteristic and continuum x-rays, electromagnetic radiation (visible, UV and IR), electron-hole pairs, phonons (lattice vibrations), and plasmons (electron oscillations) . The energy losses of the primary electrons also constitute a characteristic measure of the sample analyzed . These signals may carry morphological information, chemical information, or often both . They allow the formation of elemental chemical maps, i .e . images whose intensity at every pixel is related to the concentration of the particular element of interest . Two characteristic signals which are commonly analyzed are the x-rays and energy losses of electrons ; these techniques are known as x-ray microanalysis and Electron Energy Loss Spectrometry (EELS) . EELS and x-ray imaging provide different kinds of information and are rather complementary techniques . EELS is usually more efficient than x-rays, especially for light elements such as carbon, nitrogen, etc. Radiation damage is therefore decreased, and higher resolution can be achieved . The samples must however be thin, typically less than 500 Angstroms, to prevent multiple beam interactions . X-ray techniques are better suited for the study of heavy elements for which the signal-to-noise attainable with EELS measurements is very low . In STEM x-ray imaging, an x-ray spectrum is recorded at each point of the scanned sample . Transfer of energy between the primary electrons and the sample produces radiation of well known characteristic energies that depend on the structure of the atomic layers . The contribution of each element to one spectrum is a series of peaks at these known energies, of known relative amplitudes . Each of the peaks (for a given element) corresponds to a transition between two atomic shells or subshells . Fig. I presents an extremely simplified x-ray spectrum composed of only one peak, due, for example, to the transition between the K and L layers . An elemental (or chemical) map for this given element is then obtained by computing at each point in the sample a function of the variables p (peak, or characteristic), b (background, or continuum) and i (total) for the element of interest . Only t and b are directly measurable ; typical functions are [4] : p=t-b, Signal Processing

orp/b=(t-b)/b.

(1)



T Pun, J. Ellis /

Simulation

of STEM

images

53

counts

characteristic peak

total amplitude t = p + b

P

continuum background b energy E Fig. 1 . Idealized drawing of an x-ray spectrum composed of a single characteristic peak . Its amplitude p is the difference between the total measured amplitude t and the continuum background h. In STEM EELS imaging, the phenomenon analyzed is the loss of energy that primary electrons undergo when traversing the sample [3] . Each chemical element of interest is related to a particular loss of energy corresponding to the energy required for a transition between the atomic shells . The number of electrons losing this characteristic amount of energy is proportional to the elemental concentration . The transmitted beam is recorded at each point using a magnetic spectrometer, in which particles are deflected more or less according to their energies . It is therefore possible to count electrons having undergone specific energy losses . Fig . 2 presents a schematic EELS spectrum ; the characteristic core edge energy separates the estimation and integration regions (below/above edge) . The concentration of a given element, having

estimation region (n channels)

integration region (m channels)

core edge energy

ii

total area t = b + e

energy loss E Fig. 2 . Idealized drawing of an inner-shell ionization edge in the energy loss spectrum, showing the estimation region and the integration region (with n and m energy channels respectively) . In the integration region, the extrapolated background defines the background area b (below) and the edge area e (above) . Vol . 8, No. I, February 1985



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T Pun, J. Ellis / Simulation of STEM images

known characteristic core edge energy, is uniquely related to the edge value e. This value e is obtained by successively estimating the parameters of the background shape below edge (estimation region), computing an extrapolated background b above edge (integration region), and subtracting b from the recorded total number of counts t : (2) e=t-b. The elemental EELS image is the collection of edge values obtained at each point of the sample, for a given characteristic edge .

3 . Simulation Poisson processes in STEM imaging Counting processes, which include as typical examples the generation of x-rays or the arrival of electrons with a given energy loss, are Poisson processes . The Poisson distribution describes the frequency of occurrence of events in a given time interval . It presupposes that [12] (1) the probability of an event occurring in a time interval depends only on the length of this interval, and (2) the numbers of events occurring in disjoint time intervals are independent . Denoting the average number of events in a unit time interval by A, the probability p(N, AT) of having N events during a period of time T is * (AT)~ p( N`, N,

AT) =

NI

(3)

The parameter A is characteristic of the sample under study and of the type of process analyzed (x-rays, EELS, etc.) . Important factors in determining A are the element type, concentration and cross-section . For a given energy E;, AT becomes C;, which is the average number of counts about which the actual (recorded) c; fluctuates with a variance C ; . In x-ray imaging, the values t and b (eq . (1)) are sums of Poisson variables, but the functions defined by eq . (1) are not necessarily Poisson distributed [8] . In EELS imaging, only the value t (eq . (2)) is a sum of Poisson variables . Neither the distribution of b nor that of e are Poisson . The necessary estimation of variances for these quantities is not straightforward ; models as well as simplifying hypotheses have to be assumed [5, 6] . The simulation process involves three basic operations that have to be performed for each pixel in the STEM image (independence is assumed between neighbors) . First, uniformly distributed variables are generated . Second, Poisson deviates are obtained using the uniform ones . Third, several Poisson deviates are combined using an appropriate transformation into a final "experimental" value, such as t (eqs (1) or (2)) . Uniform deviate generation Computer generation of random 2D fields having uniform distribution can be difficult due to the large amount of data required, especially in the case of a 16 bit machine . The random sequence {Uk } given by the congruential (or modulo) algorithm is expressed by Uk „ Signal Processing

=(

g * Uk +h)Mod m, (g, h, m, k, Ua : integers)



T. Pun, J. Ellis / Simulation of STEM images

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with k rO, m>0, 0_g,h, Ua
(4)

where m is the modulus, g the multiplier and h the increment . In order to achieve the largest possible period (i .e . m), the following conditions must be satisfied [13] : 1) h is relatively prime to m ; 2) g - I is a multiple of every prime number dividing m : 3) g - I is a multiple of 4, if m is a multiple of 4 . Due to the 16 bit limitation, it is desired to have g * m smaller than 2" . Otherwise, an overflow would occur in addition to the modulo operation ; this could create perceivable patterns . To obtain the largest possible sequence, g has to be as small as possible ; therefore m must be divisible by as few primes as possible (condition 2 above) . Some optimal couples of values {g, m} are {5,8192=2"}, {4,6561=3 8 }, {7, 4374 = 2 * 3 7 } and {6, 3125 = 5 5 } . The first pair is not desirable when the 2D field has a side length which is a power of 2 . It has been observed that h, in addition to satisfying condition 1, should be not too small, and should be relatively prime to g and m . A uniform generator satisfying these conditions is Uk „=(4* U, +9695) Mod 6561 . However, when Poisson noise is generated, a large number of uniform deviates is required for a single Poisson value . Due to the rather small period (6561), visible patterns are produced . A value of h/m of approximately 21% yields a very low serial correlation [13] . Consequently, another generator could be Uk „ = ( 13077 * U,,+6925) Mod 32701 . The patterns obtained using this non-optimum congruential algorithm are broken by an exclusive-OR between the results of the modulo operation and a counter (incremented after each generation) : Uniform Deviate= Uk „ XOR Counter. Poisson deviate generation

Several algorithms can be found for generating Poisson distributed variables . Three of them are compared here . The most immediate one (ALGORITHM PI) is a transformation method based on the equivalence in probabilities p„ (v) dv=p„ (u) du,

with v=f(u)

(5)

where p„ (u) and p„( v) are the pdfs of the random variables U and V, and f is the transformation providing v knowing u . In the case of discrete bounded variables, eq . (5) becomes oa

uu

Y_ Pj(Q = E p . (j),

with

vo = f(uo)

(6)

where f is defined by the pairs of values (u a , v 0} satisfying eq . (6) . Here, p„ (j) is known (for example 1/256) and p„(i) is given by eq . (3) . A lookup table between u o and v o is computed since f has no usable analytical form . Another simple algorithm, adapted from [13], (ALGORITHM P2) is derived from the definition of a Poisson process . It consists of generating uniform deviates U,, U2 , - - - Um (in [0- I]) until the sum of their natural logarithms satisfies ln(U,)+In(U2)+ • . .+ln(Ur)-_-C where C is the mean value of the distribution . The Poisson deviate is V=m-1 . This method requires, on the average, the generation of C + I uniform variables . The third algorithm (ALGORITHM P3), due to J .H . Ahrens and U . Dieter [13], is of order log(C) . It is quite complex, and is not described here . Vul . 9, No. I, February 1985



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T Pun, J, Ellis / Simulation of STEM images

The pdfs (probability density functions) of the simulated 2D fields are evaluated by comparison with the ideal desired density histogram . An error measure s may be defined as follows : 1

nbins

(generated-ideal)' s

(7)

ns * nbins

where "nbins" is the number of bins of the density histogram . Other statistical tests for discrete distributions, such as the Chi-Square criterion, have been applied with similar results . Fig . 3 compares the three algorithms P1, P2 and P3 for a mean of 128 counts . Tables 1 and 2 give the values of s (eq . (7)), as well as the sample mean, standard deviation and range . The errors s obtained for different numbers of bins increase very clearly from algorithm P1 to P2 to P3 ; the distribution produced by PI is the closest to the desired one . For such a small mean (128), little storage is required for the lookup table, and therefore algorithm PI is a fast and effective way of generating Poisson deviates . Its primary disadvantage is that, due to the discretization of the variables in eq . (6), some values in the resulting distribution will never be generated .

Fig. 3 . Simulation of Poisson distributed 128 x 128 2D random fields, average 128 . 3(a) : Algorithm P1 ; 3(b) : Algorithm P2 ; 3(c) : Algorithm P3 . Range of values : [31-170].

Table I Poisson noise generation, figs 3 (average 128) and 4 (average 10 000). Error s as a function of the number of bins nbins

8

16

32

64

128

256

Fig . 3(a) Fig- 3(b) Fig- 3(c)

35 .6 283 .0 498 .1

34 .6 177.4 533 .6

25 .5 107 .4 297 .2

19 .9 68 .3 158 .9

19 .0 46.5 86 .5

13 .8 32 .7 57 .7

Fig . 4(a) Fig . 4(b)

1678 .9 52 .5

1180 .4 91 .0

660 .1 56 .3

335 .8 41 .9

173 .5 31 .8

90.1 23 .6

For large means, it is not practical to use a lookup table . Algorithm P2 becomes very slow, and furthermore very sensitive to round-off errors . Fig. 4 compares algorithms P2 and P3 with a mean of 10 000 counts. High count values are typical when simulating very efficient processes such as those occurring in EELS . Quantitative comparisons are shown in Tables 1 and 2 . Algorithm P2, in addition to being very slow, produces variables with considerably biased mean and variance ; furthermore the range of the values Signal Processing



T. Pun, J. Ellis / Simulation of STEM images

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Table 2 Poisson noise generation, Figs . 3 (average 128) and 4 (average 10000), Actual sample mean, standard deviation and range, and generation time per pixel (ms) on a DEC PDP 11/60. The "ideal" range is the one inside which distribution values are more than 1/4096 times the maximum one Measure Fig . 3(a) Fig . 3(b) Fig . 3(c) ideally Fig. 4(a) Fig. 4(b) ideally

Mean 128 .12 126.86 125.88 128 9924 .2 9998.2 10000

St . dev . 11 .19 11 .58 10.49 11 .31 79.1 101 .7 100

Range [99-166] [98-169] [93-168] [85-176] [9672-10137] [9677-10332] [9595-10410]

Time (ms) 1 .2 45 .0 10.1 3700 200 -

Fig . 4 . Simulation of Poisson distributed 128 x 128 2D random fields, average 10 000 .4(a) : Algorithm P2 ; 4(b) : Algorithm P3 . Range of values : [9000-10 400] .

is incorrectly distributed . Algorithm P3 is much faster, and shows results comparable to those obtained by algorithm PI when the average was 128 . For high mean values (>1000), algorithm P3 should be employed .

STEM image generation The respective collection efficiencies of either x-ray or EELS processes make the choice of a Poisson generator easy : x-rays yield typically small (-100) to very small (1-10) numbers of counts, while the opposite is true for EELS (--10000 counts per energy channel) . Algorithm PI should be employed in the first case, and algorithm P3 in the second . For x-ray images, a spectrum model is specified, and has to be simulated at each pixel . High-pass filtering [14] is a commonly used procedure to remove the background and evaluate the peak value . Simulating EELS images requires a large number of operations . A spectrum model is specified, where the relative edge concentration k is given by k=e/b.

(8)

This value is directly related to the actual elemental concentration . Random values are generated to simulate counts in the estimation region channels (Fig . 2) ; their mean values are given by the background Vol . 5, No. 1, Febmery 1985



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T Pun, J. Ellis / Simulation of STEM images

shape in this region :

C,=aE ;',

i=1, . . .,n

(9)

where n is the number of estimation channels . A log-log transformation is applied ln(c,) = ln(aE ; ')

(10)

y,=a-r(x;-x)

(11)

or

with y, - ln(c,),

x,

ln(E,),

x=(l/n)Y_x,

1nIV]jE,l

(12)

and a =1n (a)-rx.

(13)

The parameters a and r defined that way are statistically independent [6, 15] . They are obtained from the actual background counts using a least-squares estimation procedure . Then the background area b delimited by the extrapolated function (eq . (9)) in the integration region is evaluated : b=

I

exp(a-r(xj-x)),

m

j-l

where m is the number of integration channels and xi is In(E,') in this region . A second set of Poisson variables has to be generated for the total area t=b+e=(I +k) * b. The final value e is the difference e=t-b . Some simulation results are shown in Fig . 5 (discussed in the next section) . It would be interesting to compare quantitatively the statistical properties of these images with real data . However, due to the lack of real samples having precisely known characteristics, only qualitative comparisons can be made between those simulated pictures and actual microscopic images . They have been satisfactory . In addition, consistency checks between values provided by the analytical model and simulated values indicate good agreement .

4 . Minimum detectable concentration in EELS images Only recently have STEM x-ray or EELS systems coupled to and driven by computers become operative [2] . Such installations permit simultaneous and automated recording of pictures representing different kinds of information, without registration problems . Little processing of such elemental images has been attempted, and almost all the evaluation and quantitation work is performed off-line by human observers . It is therefore important to relate the minimum visually detectable signal in an EELS image with the corresponding limit elemental concentration (eq . (8)) . This determines the minimum number of counts to be recorded, and therefore the acquisition dwell time for each point of the sample . It can be shown that, under usual experimental conditions, the values a, r and b from eqs (11) and (14), as well as the edge area e, are approximately normally distributed with respective variances o,', it," Signal Processing



T Pun, I. Ellis / Simulation of STEM images

59

02 and oa [6] . The variances oa and oz are oa=

F=1

with a,=

a'

C,'

P r?

(15a) n

with r=-

(x' -x) (x, - x)2

(15b)

The values a ; and r, are known in advance, and depend on the estimation channels . The variance of the extrapolated background b is approximately o2=b 2 oa+I

-ab l , o,=b 2 oa+b,o,

`dr

(16)

and the variance of the edge area e is oa=t+o2 =(I+k) * b+oe .

(17)

The signal-to-noise ratio of the edge area e is snr=e/oe =

k*b

t-b

(18)

Jt+o J(1+k)b+b 2 oa+bao;

In this global measure of the goodness of the edge area estimation, some terms depend on the estimation region (oo , o,), some on the integration region (t or k), and some on both of them (b and b,) . The problem of how to maximize the snr is developed in [6] and [7] . The question addressed now is determination of the minimum detectable snr . Together with eq . (18), this yields the number of counts necessary to achieve detection of a given elemental concentration . What has been done here is by no means intended to be an exhaustive study of the very complex problem of visual discrimination . Display and object parameters are deliberately neglected . Influence of grouping individual pixels is not taken into account ; results obtained in this section refer only to the individual pixel snr . This oversimplification is done purposely ; the intent is rather to present some results on how to deal with the problem of detection limits . Caution has to be exercised when setting a visual observation limit . The important factor for such discrimination is indeed not the snr, but rather the image contrast . In general, this contrast is defined as the ratio of the interesting part IP(e) of the signal to its total variation TV(e) : C = IP(e)/TV(e)

(19)

It has to be as large as possible, and its relationship to the smaller detectable snr depends on the statistics of the noise . The "optimal contrast" is the ratio of the ideal signal value (here, the edge area e) to the dynamic range of actual signal perturbed by the noise . In the present case of Gaussian distribution, it may be assumed that this range extends 3a e below the reference value zero and 30, above the signal value, and therefore is a+6o, The optimal contrast is then Copt= e/(e+6oe )

=

snr/(snr+6)

(20a)

Inverting this relationship, the signal-to-noise ratio is snr -=6 * Copt/(1 - Copt)

(20b) Vnl . 8, No . I, Februarv 1985



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(If there is no noise, Copt is 1 .) This parameter is called optimal in the sense that the full dynamic range of the signal is mapped on the full dynamic range of the display . This may not always be the case ; when several pictures have to be compared, the largest of the dynamic ranges of these images is the one mapped on the display dynamic range (as in Fig. 5) .

Fig . 5 . Determination of the minimum visually detectable signal (64 x 128 elements) . The snr, the optimal and actual contrast ratios (Copt, C) are given . Figs . 5(a)-(e) are normalized so that the minimum and maximum values of each of them are mapped onto the full dynamic range of the display (0-255) . Figs . 5(f)-(j) are all normalized between the extremum values of Fig . 5(j) . Parameters : 10000 counts at 300 eV (core edge energy), r = 3, n = 5 estimation channels (250, 260, . . ., 290 eV), m = 5 integration channels (310, .0% ((c) and (h)), 2.0% ((d) and 320, . . ., 350 eV), relative edge concentrations 0 .30% (Figs . 5(a) and 5(f)), 0.55% ((b) and (g)), 1 (i)) and 5 .0% ((e) and (j)) .

Fig. 5 shows some of the patterns used in these experiments. An exponential transform is applied in order primarily to correct for the eye's response and therefore to linearize the grey scale represented on the images . They simulate various relative edge concentrations ; the actual contrast ratios (i .e . e/ (max-min)) are smaller than the optimal contrast Copt because, in eqs (20)), the actual dynamic range is larger than e+6ae . The visual detectability limit, for these EELS images, is for Copt close to 0 .05, i .e . for a snr of about 0 .3 . Figure 6 gives the number of counts at core edge energy necessary to be able to detect a given relative edge concentration with different snr's . For example, with the acquisition parameters given in the figure caption, about 5200 counts at an edge are necessary to "see" an elemental concentration of 1%, if a snr of 0.5 is taken as the detection limit . It can be shown from eq . (18) that for typical experimental conditions (such as those set for Figs . 5 or 6) and for small concentrations, the required number of counts increases approximately as fast as the square of the decrease in concentration . This appears clearly on Fig . 6 ; 21000 counts are required for a 0 .5% concentration, and about 515 000 counts for a 0 .1% concentration! Fig . 6 can also be used to estimate the minimum concentration which can be detected when the number of counts (i .e . the acquisition time) is fixed .

5. Conclusions Computer simulation of STEM microscopic imaging has been used to reproduce controlled experimental conditions consistent with physical principles of generation of x-ray and EELS chemical maps . Signal Processing



T. Pun, J_ Ellis / Simulation of STEM images

61

7

0 U

0 J 0 0

2 .5

5

RELATIVE EDGE CONCENTRATION (%) Fig . 6. Decimal logarithm of the total number of counts at a core edge necessary to reach a given signal-to-noise ratio, as a function of the relative edge concentration . Parameters : core edge energy 500 eV, r=4, estimation region starting at 450 eV (n = 5 channels : 450, 460, . . ., 490eV), integration region starting at 500eV (m=5 channels : 500, 510, . . ., 540eV) .

Two Poisson deviate generation methods suitable for the ranges of interest here have been selected on the basis of departure of their actual pdfs from the ideal one . The lookup method is effective for small parameters, although it has some limitations-primarily that it cannot generate all possible values within the range of the desired distribution . For large parameters, the algorithm labelled P3 is relatively fast and provides minimally biased results . Using these methods, images simulating EELS or x-ray chemical maps can be generated . This has been applied to evaluation of the minimum visually detectable signal in an EELS image . A measure of object contrast is introduced to relate visual response to signal . Optimization of this contrast measure leads to calculation of the minimum number of counts required to detect a given elemental concentration, providing therefore the minimum detectable signal . For typical experimental conditions, the required analysis time (proportional to this number of counts) decreases roughly as the square of the concentration of the object of interest .

Acknowledgements The authors thank Dr . R. D. Leapman and Dr . M . Eden of the National Institutes of Health, as well as Dr . M . Kocher of Bell Laboratories, for suggestions and remarks related to this work . The use of the image display system designed by computer scientists from the CSL laboratory at NIH is gratefully acknowledged . This research was supported in part by the NIH visiting program .

References [II

J .1 . Goldstein, D .E. Newbury, P . Echlin, D .CJoy, C . Fiari and E. Lifshin, Scanning Electron Microscopy and X-Ray Microanalysis, Plenum Press, New York, 1981 . VoL 8, Nn. I, February 1985



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[2] R .D . Leapman, C .E. Fiori, K .E. Gorlen, C.C . Gibson and C .R . Swyt, "Combined elemental and structural imaging in a computer controlled analytical electron microscope", Proceedings 41st Annual Meeting EMSA, G .W . Bailey ed ., San Francisco Press, 1983, pp. 10-13 . [3] D.C . Joy, "The basic principles of electron energy loss spectroscopy", Introduction to Analytical Electron Microscopy, J .J . Hren, 1 .] . Goldstein and D .C . Joy eds ., Plenum Press, New York, 1979, pp . 223-244 . [4] C .E . Fiori, C .R . Swyt and 1 .R . Ellis, "The theoretical characteristic to continuum ratio in energy dispersive analysis in the analytical electron microscope", Microbeam Analysis, K .F.J. Heinrich ed ., San Francisco Press, 1982, pp . 57-71 . [5] T. Pun and J .R . Ellis, "Statistics of edge areas in quantitative EELS imaging : signal-to-noise ratio and minimum detectable signal", Microbeam Analysis, R. Gooley ed ., San Francisco Press, 1983, pp . 156-162 . [6] T. Pun, J .R . Ellis and M . Eden, "Optimized acquisition parameters and statistical detection limit in quantitative EELS imaging", Journal of Microscopy, Vol . 135, pt . 3, Sept . [984, pp . 295-316 . [7] T. Pun, JR . Ellis and M . Eden, "Weighted least squares estimation of background in EELS imaging", Journal of Microscopy, Vol . 137, pt . 1, Jan . 1985 . [8] J .R. Ellis, T. Pun and G . Hook, "Statistical effects of background distribution on peak-to-background ratio in microanalytical X-ray spectroscopy", submitted for publication . [9] A . Rose, "Television pickup tubes and the problem of vision", Advances in Electronics, L. Marton ed ., Academic Press, New York, Vol . 1, 1948, pp. 131-166 . [10] F . Attneave, "Some informational aspects of visual perception", Psychological Review, Vol . 61, No. 3, 1954, pp . 183-193 . [11] T.N . Cornsweet, Visual Perception, Academic Press, New York, 1970 . [12] E . Parzen, Stochastic Processes, Holden Day, San Francisco, 1962, Chapter 4, pp . 117-159 . [13] D.E . Knuth, The Art of Computer Programming : Volume 2, Seminumerical Algorithms, Addison-Wesley Publishing Company, Reading, MA, 1981, chapter 3, pp . 1-177 . More particularly, section 3 .4.1 .E and exercise 3 .4 .1 .16 for gamma distribution . [14] C .E . Fiori, C .R . Swyt and K .E . Gorlen, "Application of the top-hat digital filter to a nonlinear spectral unraveling procedure in energy-dispersive x-ray microanalysis", Microbeam Analysis, R .H . Geiss ed ., San Francisco Press, 1981, pp . 320-324 . [15] A . Hale, Statistical Theory with Engineering Applications, John Wiley and Sons, New York, 1952, chapter 1S, pp . 522-584 .