A Wiener filter with circular-aperture-like point spread function to restore scanning tunneling microscopy (STM) images

Pattern Recognition Letters 11 (1990) 553-556 North-Holland

August 1990

A Wiener filter with circular-aperture-like point spread function to restore scanning tunneling microscopy (STM) images M. P A N C O R B O ,

E. A N G U I A N O ,

A. D I A S P R O * a n d M. A G U I L A R

lnstituto Ciencia de Materiales (C.S.L C.), Universidad Aut6noma de Madrid (C-Ill), 28049, Cantoblanco, EspaJqa

Received 4 January 1990 Revised 5 April 1990 Abstract: With a suitable noise analysis and a relative Wiener filter, utilizing a Point Spread Function in analogy with optical

cases, a good restoration of noisy fast-STM images can be achieved. Correlation between consecutive scans and integration along the scan direction could be the greatest disturbing facts. Key words: STM, tunneling, microscopy, noise, digital filter, scanning-tunneling-microscope.

Introduction Scanning Tunneling Microscopy is a well established technique to observe and to study surfaces and objects down to the atomic scale level (Binning and Rohrer, 1986). Upon approaching a very sharp tip, ideally ended by one atom, to the sample surface, the electron wave functions of the sample and of the tip overlap. Then a tunneling current can be established by applying a voltage of a few millivolts between tip and sample. For a separation of a few A, the current is about a few nanoamperes. A feedback circuitry keeps this current constant by taking its value in the input and sending, as response signal, a voltage to a piezoelectric rigidly joint to the tip. Now, if we get two other piezoelectrics parallel to the sample surface, we are able to perform a scanning. Then the feedback signal carries the information of the sample topography and can be digitized and stored on a computer as a matrix. The data are affected by noise stemming from different sources: the recording system, the feed* Permanent address: lstituto di BiofisicaUniversit~ di Genova, Sestri Ponente, Via Giotto 2, 16153, Genova, Italia.

back circuitry dynamic response, external unavoidable perturbations penetrating the system, strong irregularities in the specimen, etc. Moreover the data, picked point by point, are in a certain sense blurred by the image formation system Point Spread Function (PSF) owing to the finite lateral resolution of the STM detector, a very sharp tip.

Data analysis and processing When an image is taken, it is essential to choose the width of the measured zone in such a way that the increments A x and A y along and perpendicular to the scan direction, respectively (henceforth we assume that the scan direction is the x-axis), satisfy the W h i t t a k e r - K o t e l n i k o v - S h a n n o n theorem (Rosenfeld and Kak, 1982). This can be achieved if these increments are smaller than the lateral resolution of the system. We will follow the assumption of Stoll and Baratoff (1988), who consider that the PSF and the noise are instrumental properties which do not depend on the surface profile, i.e., they do not depend on the signal. This means that the noise is additive and uncorrelated with the signal. This

0167-8655/90/$03.50 © 1990 -- Elsevier Science Publishers B.V. (North-Holland)

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assumption does not imply that correlated, noadditive noise, could be present but we only take care of the additive noise, which anyway is the largest one. In the Fourier spatial-frequency domain we may write the following relation between the imaged (observed or recorded) profile G(q), the true surface profile P(q), the PSF Fourier transform H(q), also called transfer function, and the signal-independent noise N(q):

G(q) = P(q) H(q) + N(q).

(1)

To evaluate P(q) we utilize a Wiener filter, W(q), that produces an optimal least-squares-error estimate Pe(q).

Pe(q) = W(q) G(q).

(2)

The derivation of W(q), which depends on H(q) and on the spectral density of noise ¢(q), can be found in the referenced book by Rosenfeld and Kak (1982). Its expression is:

H*(q) W(q) = IH(q) l 2 + q~(q)

(3)

where the * denotes complex conjugation. Theoretical models (see for example (Hansma and Tersoff, 1987)) have been developed to predict at the same time what are the lateral resolution of a given tip and the current I from a given distance d from the tip to the surface of the sample. On this basis and on some considerations in analogy with optical defocused systems, such as the conical geometry of the tip-sample system, we have implemented a Wiener filter utilizing a transfer function as that used to correct defocus effects in optical systems with a circular aperture. We model, by this way, the interaction tip-sample utilizing a smearing function as that derived by Stokseth (1969) for a circular aperture with defocus (see e.g. the review of Castleman (1979) in order to obtain the complete formula) and described by this approximated form:

H(q) = (1 - 0 . 6 9 s + 0.0076 s2 + 0.043 s 3) × jinc [ 4kco ( 1 - 2 ) 2 ] where s=

2q --,

qc 554

q = Jql = q]/~x2 + q 2,

(4)

jinc(x) ---

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f

20 Jl(X) x

if x > 0 , otherwise.

qc is the cut-off frequency related with the resolution of the system. In the optical case k = 2n/;t (2 is the wave length of the employed light), o9 is the maximum defocus path length error and qc = 2a/2 where a is the numeric aperture (Castleman, 1979). In our case, the defocusing has two different origins. In the first place we have the electronic signal integration along the x-direction (related to the scan speed), that makes the finite-time sampling window become important. On the other hand, we have the effect of thermal drift and instabilities on the slowest movement of the tip along the y-axis. These processes produce a correlation between consecutive scans. Therefore we introduce two parameters, 0x and 0y, relative to these two effects, that will be treated as fitting parameters and have completely different physical meaning. The cut-off frequency is the inverse of the lateral resolution and this must be chosen according with physical criteria, i.e., it must reflect the STM lateral resolution. With these considerations we build up a transfer function, based on (4), which is:

H(q) = (1 - 0.69 s + 0.0076 s2 + 0.043 s 3) x jinc [ 2 r ( 1 - 2 ) ]

(5)

where r is a new variable which value is related to Ox and Oy values by the following expression: r-~

~/(qxOx) 2 + (qyOy) 2

qc Notice that (5) reduces to (4) when 0x = Oy = 2k09. In order to check the performance of our model we have compared it with the case of a gaussian transfer function,

H(q) = exp(-qZ/q2).

(6)

In this case the only parameter is qc which has the same meaning, in principle, that it has in equation (5). This is the usual function used in STM images restoration by Wiener filtering. The most characteristic noise in STM images is the 1/f&like one (Stoll and Marti, 1987), generated

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by the high input impedance amplifiers and the typical integrator-like form of the feedback circuitry, that affects the look of the image specially in the y-direction. These authors point out that at high frequencies, white noise becomes more important than 1/f~-like one. Then we assume for the noise-to-signal density power ratio a 1 / f B spectrum (that we call q)p(q)) for low frequencies, and a uniform one (white noise) for high frequencies. The general form of the ~0~(q) expression is picked from (Stoll and Baratoff, 1988), and it is:

O~/3(q) = [q2 + (qj2Ny)2]B/2

(7)

where Ny is the number of scans. Notice that q is essentially the same as f but in different units. The factor 2Ny dividing qy is something that depends on the mode of sampling and it is properly justified in the last reference. As we said, at high frequencies, white noise is higher than 1/f#-like one, so we introduce this fact in the final 0(q) expression as follows: 0~(q) = ~q~P(q) /._0%

if Op(q)>0~0, otherwise.

(8)

r / a n d q~0 are two fitting parameters which respectively represent the 1/f~-like noise quantity and the white noise one. The fl factor we have measured for our STM system is 1.0. Now we have all the tools that are needed to properly filter STM images. We use them as follows: after an image is stored, a two-dimensional Discrete Fourier Transform (DFT) is performed so we obtain G(q). Then we compute eq. (2) in order to obtain Pe(q). At this stage, the values of the H(q) and q~(q) parameters must be given. Finally Pe(q) is carried back to the real-space domain and displayed on a high resolution monitor.

Discussion and conclusion

We report, as an example, the grey-level pictures exhibited in Figure 1, where there is shown an image (256 × 256 12-bit pixels) of gold filtered with different H(q). Figure 1b shows the gaussian case and it is possible to see an improvement in the

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quality of the image but the 'defocusing' in xdirection is still present. Figure lc shows the circular aperture case with ~y>dx. The difference is clear between the image filtered with a gaussian transfer function and those filtered with a circularaperture like H(q): there is an enhancement of the quality in the last ones, though the estimated lateral resolution parameter is the same for the two transfer functions. The STM images presented here are very large ones, 70 A × 70 A, with atomic resolution. For this reason the blurring or noise is very large. The only objective measure to judge the quality of the image restoration is the quality of the details at atomic level that are clarified by the restoration. In fact, STM is the only technique that gives information of surfaces with atomic resolution and then it is not possible to compare the STM pictures of the surface with any picture or information obtained by other techniques. Thus, we present a zoom of a region of the image in the three cases that shows a dislocation. From the comparison of the different images it is clear that the method of the circular aperture is the best one, i.e., it gives the best image quality. We have repeated this analysis with several images. In all cases we have obtained the same result. This could mean, within our interpretation of what defocusing means in STM, that both the correlation

between consecutive scans and the integration along the scan direction (related to the scan velocity) are blurring factors. This is probably the most important result from our study. We believe that the results achieved and here presented demonstrate that our optical defocusing based model is correct. We also believe that the method for image restoration used here has general applicability and then, it may be useful for a correct application of digital image processing techniques to STM data. The approach, in this sense, is heuristic, but moves on the assumption that under an image formation viewpoint there is an analogy between the effect of a circular aperture and the fact that a STM tip, due to its conical aspect, picks up information from different surface points and not only from one.

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Figure 1. Profile of a polycrystalline gold surface taken with an electrochemically treated tungsten tip at atmospheric pressure and filtered as described in the text. Size of the area: 70 ,~ × 70 ,~. Size of the subarea: 19 A × 19 .&. Size of the image: 256 × 256 pixels (i.e. 256 scans with 256 points per scan). Time per scan: 50 msec. Tunnel current: 9 nA. Bias tension: - 0 . 4 V (sample referred). (a) Original (not treated). (b) Filtered with a gaussian H(q), given by (6). Estimated lateral resolution parameter: 1.4 A. (c) Filtered with H(q) given by (5). Defocusing factors: Jx = 0.5 and ~y = 10. Estimated lateral resolution parameter: 1.4 A. A 3 x 3 high emphasis convolution filter was performed in both figures (b) and (c) in order to improve the contrast of the image. The other parameters that are not mentioned have the same values in the two cases.

Acknowledgements

References

We wish to thank IBM Madrid Scientific Center for partial financial support, and in particular J.P. Secilla for his helpful 2D-DFT algorithms and other supporting software. We also thank the Comett project that has made possible the collaboration between our two groups.

Binnig, G. and H. Rohrer (1986). Scanning tunneling microscope. J. Res. Develop. 30 (4), 355-367. Castleman, K.R. (1979). Digital linage Processing. PrenticeHall, Englewood Cliffs, NJ. Hansma, P.K. and J. Tersoff (1987). Scanning tunneling microscope. J. Appl. Phys. 61 (2). Rosenfeld, A. and A.C. Kak (1982). Digital Picture Processing. Vol. 1. Academic Press, Orlando, FL. Stokseth, P.A. (1969). Properties of a defocused optical system. J. Opt. Soc. Amer. 59(10), 1314-1321. Stoll, E. and A. Baratoff (1988). Restoration and pictorial representation of STM data. Ultramicroscopy 25, 149. Stoll, E. and O. Marti (1987). Restoration of STM data blurred by limited resolution, and hampered by 1/f-like noise. Surface Science 181, 222.

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