Anal. Chem. 1995,67,4500-4507
Modeling the Degradation of Scanning Electrochemical Microscope Images Due to Surface Roughness Kevin A. Ellis, Mark D. Pritzker,* and Thomas 2. Fahidy Department of Chemical Engineering, Univecsiiy of Waterloo, Waterloo, Ontario, Canada N2L 3G 1
Visualization of surface roughness is widely used in electrochemical applications to provide a measure of the quality of a deposit or to determine the extent of localized corrosion. Generating high-resolutionthree-dimensional images of conducting or insulating surfaces in situ via scanning electrochemical microscopy (SECM) is one method. A three-dimensional image of a substrate is determined by measuring the variation in the tip current due to perturbations in the diffusion layer. Although surface roughness significantly distorts the resulting image, current models relating the tip current to the tipsubstrate spacing assume that the substrate is perfectly flat. The quality of SECM images is enhanced currently by heuristic techniques. In this paper, a method for rigorously modeling the blurring process is presented. The model can be used as a basis for restoring SECM images. In particular, it may bg feasible to resolve features smaller than the tip by scanning an image with a small sampling interval relative to the tip size and using knowledge of the blurring process to deconvolve the image. Scanning electrochemical microscopy (SECM) is a useful tool for reconstruction of high-resolution topographical maps of a conducting or insulating surface,'-* characterization of local surface chemistry,+14 and micr~fabrication.'~l~-~~ It is beyond the scope of this paper to give an overview of the basic theory of (1) Bard, A. J.; Denuault, G.; Lee, C.; Mandler, D.; Wipf. D. 0. Acc. Chem. Res. 1990,23, 357-363. (2) Bard, A J.; Fan, F.-R F.; Kwak, J.; Lev, 0. Anal. Chem. 1989,61,132138. (3) Bard, A J.; Denuault, G.; Friener, R A Anal. Chem. 1991,63, 1282-1288. (4) Bard, A. J.; Fan, F.-R F.; Pierce, D. T.; Unwin, P. R ; Wipf, D. 0.;Zhou, F. Science 1991,254,68-74. (5) Kwak, J.: Bard, A. J. Anal. Chem. 1989,61, 1794-1799. (6)Kwak, J.; Bard, A J. Anal. Chem. 1989,61,1221-1227. (7) Wipf, D. 0.; Bard, A. J. Anal. Chem. 1992,64,1362-1367. (8) Ludwig, M.; Krmz, C.; Schuhmann, W.; Gaub, H. Rev. Sci. Instrum. 1995, 66, 2857-2860. (9) Engstrom, R. C.; Small, B.; &Em, L. Anal. Chem. 1992,64, 241-244. (10) Lee, C.; Kwak, J.; Anson, F. C. Anal. Chem. 1991,63,1501-1504. (11) Wipf, D. 0.;Bard, A. J. J. Electrochem. Soc. 1991,138,U-L6. (12) Wipf, D. 0.; Bard, A. J. J. Electrochem. Soc. 1991,138,469-474. (13) Macpherson, J.; Unwin, P. J. Phys. Chem. 1994,98,1704-1713. (14) Casillas, M.; James, P.; Smyrl, W. J. Electrochem. SOC.1995,142,L16L18. (15) Mandler, D.; Bard, A J. J. Electrochem. Soc. 1990,137,1079-1086. (16) Sugimara, H.; Uchida, T.; Shimo, N.; Kitamura. N.; Masuhara. H. Ultramicroscopy 1992,12,168-171. (17) Kranz, C.; Ludwig, M.; Gaub, H.; Schuhman, W. Adu. Mater. 1995,7,3840.
4500 Analytical Chemistry, Vol. 67,No. 24, December 15, 1995
SECM operation; instead, interested readers can consult refs 5, 6, and 18. The currently accepted method of reconstructing a threedimensional image is to assume that the surface is flat at each point in the image and use a calibration curve to relate the measured current profile to the tip-substrate spacing. Surface roughness significantly distorts the resultant image, particularly near edges and other areas containing high-frequencyinformation. Several approaches have been used to increase the resolution of SECM images by using different modes of ~peration,~ decreasing the tip size,19*20 and using digital image processing.21 Lee et a1.21 improved the resolution of SECM images by implementing a Laplacian of a Gaussian (LOG) filter, which is commonly used to enhance images that have undergone optical blurring. It will be demonstrated in this paper that this filter is not optimal in the general case but can improve image resolution near surface protrusions. A short synopsis of signal processing theory is first given to clarify the assumptions and restrictions made in formulating the image degradation model. The theory is then applied to a microdisk electrode and used to formulate a model for the degradation of SECM images based on simulations of the feedback current for a series of circularly symmetric steps. Finally, a detailed description of the simulation procedure is provided, along with a series of step responses to illustrate the blurring process. IMAGE DEGRADATION An important class of operations in digital image processing
comprises linear shift invariant (LSD processes.22 A commonly used image degradation model that exhibits LSI is
where f k y ) and g(xy) are the original and blurred images, respectively,h @y)is the set of weighting factors commonly called the point spread function (PSF) or impulse response (IR), and n by) is the noise component. The coordinatesx and y correspond to the position withii the image or filter as appropriate. The asterisk denotes digital convolution. Thus, each point in the blurred image is a weighted local average of points in the original (18) Bard, A J. Electroanalytical Chemisty; Marcel Dekker, Inc.: New York, 1994; Vol. 18. (19) Lee, C.; Miller, C. J.; Bard, A J. Anal. Chem. 1991,63,78-83. (20) Mirkin, M. V.; Fan, F.-R. F.; Bard, A. J. J. Electroanal. Chem. 1992,328, 47-62. (21) Lee, C.; Wipf, D. 0.; Bard, A. J. Anal. Chem. 1991,63, 242-2447, (22) Gonzalez, R C.; Wintz, P. Digital Image Processing, 2nd ed.; Addison-Wesley Publishing Co., Inc.: Reading, MA, 1987. 0003-2700/95/0367-4500$9.00/0 0 1995 American Chemical Society
image, plus an additional noise term. The term LSI implies that h(x,y) does not depend on the magnitude off(x,y) or the position within the image. This result implies that if h (xy) is known, the response to any input can be calculated. The objective, then, is to calculate the PSF for a microdisk electrode used in SECM experiments. The PSF may be used in an image restoration procedure to estimate f(xy), thereby improving the resolution of SECM images. MICRODISK ELECTRODE
Since a microdisk electrode is circularly symmetric, its PSF will also be symmetric. Therefore, the dimensionality of the system may be reduced by replacing Cartesian coordinates with cylindrical coordinates and h(xy) with h(r). Assuming a LSI system, the input signalf(r) can be arbitrarily chosen for the purpose of finding h(r). For convenience, let the input signal be a circularly symmetric step about the origin, i.e.,
where f i and fi are constants, and the edge of the step occurs at an arbitrary radius S. Let g(.9 be the measured signal at the origin as a function of the step position. Then,
In the next section, details of the computer simulations and results from the step tests are presented, with a brief discussion of the implementation of the results into an image restoration procedure. SIMULATIONS
Simulations of the tip current over a wide range of tipsubstrate spacings and substrate geometries were performed using a finite difference scheme. Under conditions of normal SECM usage, migration and convection of the electroactive mediator species can be ignored. The governing transport equation is then given by the two-dimensional Fick’s law, i.e.,
By normalizing the concentration (c) with respect to the bulk concentration ( c * ) , the radial and axial distances (r and z) with respect to the tip radius (a), and the tip current with respect to the tip current for semi-infhite diffusion, eq 6 can be expressed in terms of dimensionless parameters. Then, at steady state and under diffusion-limited conditions,
(7)
where C = c/c*, R = r/a, Z = z/a, and the boundary conditions are To prevent bias in the process, a constraint is placed on the PSF
=O
C(R,O) = 0 (R < 1) This constraint preserves the mean of the signal between the degraded and the original signals. In the limitng case where the substrate is flat, the constraint ensures that the estimated distance matches the true distance determined through tip calibration. Equation 2 then becomes
&(S) =hV(s> +fin - V(S)l
(3)
where
V(S)= ZnKrh(r) dr Rearranging eq 3 yields
&(a-fi
V(S) = -
fi -fi
(4)
Thus, by simulating the SECM response for a series of steps with different radii, the cumulative distribution of the PSF can be obtained. Finally, the PSF is found by differentiating eq 4, i.e.,
h(r) = -2 5
(5)
C ( R B = 1(RJ-
w)
The solution using a finite difference scheme is cumbersome because of the discontinuity in the boundary condition at R = 1, where it changes from zero concentration to zero flux. This problem can be resolved by using an orthogonal coordinate transformation which folds the solution space at the discontinuity. Table 1 lists some of the transformations, the resultant steadystate differential equations, boundary conditions, and integral equations for the tip current that have been sed.^^-^^ In each case, the discontinuity in the boundary condition disappears. The transformed differential equations have variable coefficients, which does not cause a convergence problem but marginally slows down each iteration of the Gauss-Siedel update. However, this is more than compensated by the fact that all three transformations lead to a much smaller grid for a given level of accuracy, so that the overall computation time is greatly reduced. Prior to calculation of step responses, calibration simulations were run to compare the results obtained via finite differencing schemes to the model proposed by Kwak and Bard,6 based on finite element methods (Figure 1). The predicted normalized currents approach unity as the tip-substrate spacing approaches inhity, except in the case of model 111, due to the poor convergence rate for large tip-substrate spacings (Le., small a/d). (23) Newman, J. S.J. Electrochem. SOC.1966,113, 501-502. (24) Amatore, C.; Fosset, B. J. Electround. Chem. 1992,328, 21-32. (25) Michael, A.; Wightman, R. J. Electroanul. Chem. 1989,267, 33-45.
Analytical Chemistry, Vol. 67, No. 24, December 15, 1995
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Table 1. Coordinate Transformations Suitable for the Simuiatlon of Diffusion at a Microdisk Electrode
model In
model IIb
model IIIc
Amatore and F o s ~ e t . Michael ~~ and Wightma11.~5N e ~ m a n . ~ ~
Table 2. Comparison of Models Obtained for SECM Calibration
finite
finite
parameter Kwak et al. difference parameter Kwak et al. difference ki k2
0
0.5
1
1.5
2
2.5
3
ala
Figure 1. Calibration curve relating the normalized current as a function of reciprocal distance for the SECM. (-) Model reported by Kwak et al.; 0,+, and 0 correspond to simulations based on models I, II, and 111, respectively. (- - -) Nonlinear regression based on finite difference simulations.
Increasing the maximum number of iterations reduces the discrepancy between the models in this region. When the tip-substrate spacing becomes very small (a/d >> l), the current becomes proportional to the gap width, with a proportionality constant close to x/4, as predicted for thin-layer cells.6 The models assume the same functional form,
ZT/ZT,== k, + k , d d
+ k, exp(-k,a/d)
(8)
with parameters for model I shown in Table 2, indicating fairly close numerical agreement. Figure 2 shows typical concentration contours predicted by models I and I1 (Table 1) and the underlying grid for a flat substrate. The surface of the microdisk electrode lies at 2 = 0. The conductive tip lies between R = 0 and R = 1, and the insulating\sheath extends from R = 1outward. The vertical axis (R = 0) corresponds to the axis of circular symmetry. In Figure 2, parts a and c, the substrate is located far from the tip (d/a = 10) and is off the scale of the plot. The contours in this case approximately correspond to those for semiinfinite diffusion. In Figure 2, parts b and d, the substrate is located at 2 = 1. At large tip-substrate spacing, the concentration profile closely follows the grid lines. As the tip approaches the substrate, the 4502 Analytical Chemisfry, Vol. 67, No. 24, December 15, 7995
0.6800
0.6353
k3
0.7838
0.8171
k4
0.3315 1.0672
0.3647 0.9349
concentration profile flattens near the axis of symmetry. However, in the critical region near the edge of the microdisk, the concentration profile closely follows the grid l i e s in both cases. Thus, the coordinate transformations are well suited to estimate the feedback current in SECM applications and to treat semiin6nite diffusion problems. Table 3 shows the variation in tip current with grid size. The estimated current for a coarse grid is typically within 1%of the value obtained from model I using the finest grid and typically within 6%for model 11. The grid sue becomes more important as the tip-substrate spacing decreases. Both models were found to be very stable, further reflecting the suitability of the transformations. It is interesting to note that with the exception of the case of d / a = 0.2, the solution obtained using model I increases monotonically with increasinggrid size, while the solution derived from model I1 decreases monotonically with increasing grid size. This finding suggests that the true solution is bounded by the solutions obtained using models I and 11. Model 111has poorer convergence characteristicsfor large tipsubstrate spacings if an exponentially increasing grid is used in the 6 direction and poorer stability for small tip-substrate spacings if a linear grid is used. Thus, only models I and I1 were used for the step tests. The finite difference scheme can be further enhanced by implementing a multigrid A common problem with iterative solutions is the rate of convergence. If a coarse grid is used, convergence is fast, but the accuracy is poor. Accuracy improves as the grid size is reduced, but the convergence rate decreases by the order of Nz, where N is the number of grid points. This difficulty can be partially eliminated by solving the problem on a coarse grid and then interpolatingthe solution to a 6ner grid with an appropriate smoothing function and estimating the "defect" in the fine grid solution. This defect is used to ~~
(26) Press, W.; Flannery, B.; Teukolsky, S.; Vetterling, W. Numerical Recipes in C: ne At? of Scieafific Computing, 2nd ed.; Cambridge University Press: Cambridge, MA, 1992.
0.8
0.8
0.6
0.6
N
N
0.4
0.4
0.2
0.2
n 0
1
2
r
3
0
5
4
0
1
2
3
4
5
r
1
0.8
0.8
,..‘
...I
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I* j,./
...* ,,...” ; ..’ ! ./ ...‘
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0.4
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0.2
0.2
a 0
1
2
3
4
5
.... _....’
0
0
1
3
2
4
5
r
Figure 2. Effect of tip-substrate spacing on the shape of the concentration contours (-) for two grid structures (b) model I, d a = 1; (c) model 11, d a = 10; and (d) model 11, d a = 1. Table 3. Comparison of Normalized Current as a Function of Normalized Tip-Substrate Spaclng at Three Grld Resolutions normalized model I model I1 tip-substrate spacing 10 x 10 20 x 20 40 x 40 10 x 10 20 x 20 40 0.200 00 0.400 00 0.600 00 0.800 00 1.000 00 1.200 00 1.400 00 1.600 00 1.800 00 2.Ooo 00
5.181 81 2.697 60 2.065 14 1.760 69 1.580 28 1.471 21 1.388 58 1.334 35 1.291 30 1.254 30
4.651 39 2.709 10 2.076 39 1.769 88 1.591 07 1.476 93 1.397 51 1.340 32 1.296 81 1.262 33
4.652 66 2.712 20 2.079 43 1.772 46 1.594 31 1.479 61 1.400 20 1.342 44 1.298 66 1.264 52
5.140 11 3.491 50 2.365 93 1.909 80 1.695 84 1.576 13 1.492 20 1.426 33 1.380 10 1.344 09
5.823 05 2.774 26 2.124 96 1.811 18 1.628 27 1.510 44 1.429 32 1.369 39 1.324 88 1.289 27
(..e):
(a) model I, d a = 10;
Table 4. Comparison of Direct Solution with Multlgrld Solution normalized model I model I1 tip-substrate spacing x
40
4.683 04 2.730 29 2.094 56 1.785 51 1.606 11 1.490 21 1.409 99 1.351 58 1.307 30 1.272 70
compute a correction term to the finite difference equations on the coarse grid. The process of prediction and correction is repeated until the solution converges at each resolution level. The multigrid approach allows fine grid accuracy at a greatly accelerated rate of convergence and gives a measure of the stability of the solution as the concentration profile is solved simultaneously at each resolution. Table 4 shows a comparison of calibration results for models I and I1 using threq multigrid levels. The “1”columns are direct solutions on 10 x 10 grids, in exact agreement with Table 3. Each additional level effectively doubles the resolution of the grid in
0.200 00 0.400 00 0.600 00 0.800 00 1.000 00 1.200 00 1.400 00 1.600 00 1.800 00 2.OOo 00
5.181 81 2.697 60 2.065 14 1.760 69 1.580 28 1.471 21 1.388 58 1.334 35 1.291 30 1.254 30
4.651 39 2.709 21 2.076 61 1.770 05 1.590 63 1.476 57 1.396 11 1.339 04 1.295 32 1.259 53
4.653 95 2.714 14 2.081 00 1.773 37 1.593 72 1.478 67 1.398 13 1.340 49 1.296 54 1.261 12
5.140 11 3.491 50 2.365 93 1.909 80 1.695 84 1.576 13 1.492 20 1.426 33 1.380 10 1.344 09
5.823 04 2.774 26 2.124 96 1.811 18 1.628 24 1.510 44 1.429 39 1.369 35 1.324 89 1.289 38
4.683 05 2.730 53 2.095 09 1.786 23 1.606 81 1.490 94 1.410 91 1.352 15 1.308 19 1.273 31
each dimension. For example, the “2” columns correspond to an exact 10 x 10 solution with iterpolation and correction for a 20 x 20 grid, and the “3” columns have two interpolation and correction steps, giving a solution comparable to direct solution on a 40 x 40 grid (“able 3). Computational savings become pronounced at higher levels. For example, the data in Table 3 were generated in 5 min, whereas the data in Table 4 required only 1 min on an IBM Rs6OOO Model 350. Four multigrid levels, equivalent to a direct solution on an 80 x 80 grid, were used to generate the calibration curve in Figure 1and do the step tests. The multigrid solution requires only about 10 s per point, whereas the direct solution requires about 125 s. The solutions produced by the two Analytical Chemistry, Vol. 67, No. 24, December 15, 7995
4503
1 ' 2 r - L q
..... .......................................
0.8
0.8 N
Substrate
...........................................................................
1
0.6
N
0.6
0.4
0.2
0.2 0
0
0.5
1
1.5
2
2.5
0
3
0.5
1
1.5
2
2.5
3
1.5
2
2.5
3
r
I
1.2 I
............................................
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
n
0 0
0.5
1
1.5 r
2
2.5
3
0
0.5
1
r
Figure 3. Typical concentration profiles for a pit or a protrusion with height z = 0.5 and a spacing of z = 1 between the tip and the base of
the feature. methods are in close agreement, with an absolute difference less than 0.001. STEP TEST RESULTS When the substrate profile contains a step, the overall shape of the concentration profile changes (Figure 3) relative to the "standard" flat substrate profiles. The effect on the tip current may be substantial in the case of a protrusion (Figure 3a,b) or relatively small when the substrate has a narrow pit (Figure 3 4 . A narrow protrusion still has an effect on the surrounding solution since its edge is conductive. As shown in Figure 3a, the isoconcentration lines are compressed near the axis of symmetry, which increases the concentration gradient near the tip (positive feedback) and increases the tip current. On the other hand, a narrow pit has a small effect on the feedback current since the species reacting at the tip is regenerated along the edge before it diffuses to the base of the pit, thereby teducing the sensitivity of the tip current to resolve the pit. This difference in the limiting cases suggests that the process cannot be described by a single PSF. However, the method previously discussed for identifying the PSF of an LSI process is still useful for establishing the degree of nonlinearity and for finding a family of F'SFs to be used for an image restoration technique based on piecewise linearization. A method for constructing a restoration filter is outlined in the next section. 4504 Analytical Chemistry, Vol. 67, No. 24, December 15, 1995
0.2
1 0
" "
x : I
0,s
-
0
1
1.5 2 Radius of Pit (S)
2.5
3
Figure 4. Estimated distance as a function of step position for pits of the form d a = 1.O - mu(r - s),where (0)m = 0.2;(+) m = 0.4; (0) m = 0.6; and ( x ) m = 0.8. u(3 is the Heaviside step function.
Figure 4 shows simulations for a series of pits, in which the distance determined from the calibration curve for tip current (Figure 1) is plotted against the pit radius for four pit depths (m = 0.2, 0.4, 0.6, and 0.8). When its radius is less than half of the tip radius, the pit cannot be resolved, and the distance estimated from the calibration curve is the distance to the surrounding substrate level. As the pit radius increases, the distance estimate
1 e
1 I
0.8
.X
I
Table 5. Regression Parameters Parameters for Various Step Heights and Tip-Substrate Spacings
step tip height posihon
1
-0.50 -0.50 -0.25 -0.25 0.25 0.25 0.50 0.50 0
0.5
1.5
1
2
2.5
3
Step Position (S)
Figure 5. Cumulative PSF for the data shown in Figure 4.
2 1 2
1 2 1 2
a 2.2213 3.3280 2.9508 3.9956 6.0981 5.3283 7.8112 5.9590
P
c1
CZ
2.0876 3.3456 0.2989 0.3562 1.7883 2.8074 0.2613 4.3672 3.8270 0.3163 2.6514 3.1616 0.1822 36.2096 5.4885 0.2838 3.2834 3.5236 0.1544 100.2907 6.4767 3.3957 3.6887 0.2711
c3
72
0.2213 1.3280 0.9508 1.9956 4.0981 3.3283 5.8112 3.9590
0.997 0.996 0.999 0.993 1.000 0.997 0.999 0.998
1 0.9
0.8
increases until the bottom of the pit is fully resolved. For a twodimensional scan of a surface, a cylindrical pit with a radius half that of the tip would have only a marginal effect on the feedback current. With a further increase in its radius, the pit would appear as a depression in the substrate, growing to twice the tip radius, when the base of the pit would appear to flatten in agreement with the actual surface. Figure 5 shows the cumulative PSF computed for the data presented in Figure 4. If the blumng process exhibits LSI behavior, the cumulative PSF should collapse to a single curve, irrespective of the dimensions of a feature. The results show that there is a weak nonlinearity in the case of a pit. The curves for both pits and protrusions were found to have the general shape of an incomplete r function:
Substituting eq 9 into eq 5 yields
where
Nonlinear regression analysis confirms a good fit (Table 5), with 12 correlationsranging from 0.993 to 1.OOO. For a protrusion (step height < 0), the value of q is the most critical for determining the shape of the PSF. For a pit, the most sensitive parameter is C1.
If c3 5 0, eq 10 takes the form of an exponentiallow-pass filter (LPF). The inverse of an exponential LPF is a Laplacian highpass filter (HPF), with a scaling factor dependent on the severity of the blur. This special case suggests that the widely used LOG filtefi is an appropriate filter for improving the resolution of dendrites very close to the tip (cg < 1).
0.7 0.6
0.5 0.4
0.3 0.2
0.1
0 0
0.5
1
2
1.5
2.5
3
L
Figure 6. Estimates of the PSF as a function of step height for substrates of the form d a = 1 - mu(r - s),where (-) m = -0.50; (- -) m = -0.25; (- -) m = 0.25; and m = 0.50.
-
(e-)
However, when the surface has cavities, or when the tip is scanned farther away from the substrate, blurring deviates strongly from an exponentialLPF. Under these conditions,c3 >> 0, and the PSF has a skewed bell shape. Although the image will still be sharpened by a LOG filter, it may also be distorted. This distortion arises from the fact that the current density is greatest at the edge of the tip for a bell-shaped PSF. Thus, the tip current is relatively insensitive to small features centered under the tip and highly sensitive to features located near the perimeter of the tip. The PSF for a protrusion is far more sensitive than the PSF for a pit (Figures 6 and 7). When the spacing between the tip and the base of the substrate is kept constant and the size of the step varies, the PSF for the protrusion changes from an exponential decay to a skewed bell shape (Figure 6). The PSF for the pits varies only slightly under these conditions,but the increased width indicates that the blumng is more severe for a pit. When the imaging plane is retracted from the substrate, the degree of blurring increases. Again, the increase is more apparent for a protrusion than for a pit (Figure 7 ) , and the nonlinearity in the PSF for a pit becomes signi!icant. These results suggest that the tip-substrate spacing should be minimized and that optimal restoration requires an adaptive approach while accommodates the variation in the PSF. IMAQE RESTORATION
Since the blurring process is nonlinear, there are several options for restoring microscope images. Linear image restoration is based on the best linear approximation to the nonlinear process. Analytical Chemistry, Vol. 67, No. 24,December 15, 1995
4505
problem, several approaches may be used. One common ap proach is to multiply the Laplacian term in eq 13 by a Gaussian LPF. With the Gaussian term, eq 13 takes on the form of a LOG filter,21which may be expressed in the spatial domain as
1 0.9 0.8
0.7
0.6 0.5 0.4
0.3 0.2
0.1
0
0.5
1
2
1.5 r
3
2.5
Figure 7. Estimates of the PSF as a function of tip position for substrates of the form d a = t - mu(r - s),where (-) t = 1, m = -0.50; (- -) t = 2, m = -0.50; (- - -) t = 1, m = 0.50; and t= 2. m = 0.50. (e.-)
Although the image restoration will be suboptimal, the technique outlined previously can be used to obtain the best linear a p proximation. Nonlinear adaptive restoration procedures would be a more sophisticated approach. A means of generating a linear restoration filter is presented below (the development and implementation of a nonlinear filter is beyond the scope of this paper). To restore an image, a filter is applied which is the inverse of the blurring process. In many cases, the true inverse cannot be applied due to its instability, and a pseudoinverse is used which is a stabilized approximation to the true inverse. The filter may be applied to the blurred image in spatial domain (i.e., Cartesian coordinates) using a convolution operation or in frequency domain using a Fourier transform. Two special cases will be considered-c3 x 0 and c3 > 0. In the first case, the PSF has the form of an exponential LPF. Converting eq 10 back to Cartesian coordinates, the PSF has the form
where Ah is the sampling interval relative to the tip radius, assumed to be equal in both the x and y directions. Thus, x and y are integers corresponding to sampling points in the image. Fourier transformation of h(xy) yields
which can be inverted to yield
where t = ( ~ 2 A h ) - ~h&y) , is the LOG filter, and u is an empirical tuning parameter related to the cutoff frequency of the restoration filter, which in turn depends on the degree of noise in the image. Comparing eq 14 to the results obtained by Lee et al.,2l the advantage of incorporating knowledge of the blurring process becomes apparent. The number of empirical tuning parameters has been reduced from two to one, which simplifies the amount of fine tuning required to achieve good restoration. The relationship between t and Ah indicates that the degree of filtering depends largely on the sampling interval, If the sampling interval is large (CZT >> l),then little improvement in the image is possible. As the sampling interval decreases, the contribution of the filter increases. There is a lower limit to the size of the sampling interval, since reducing the interval increases the sensitivity to high-frequency noise, and u will have to be increased to compensate. Nonetheless, resolution of features smaller than the tip should be possible. Figure 8b shows the result of applying the LOG filter to restore a circularly symmetric step (Figure Sa), degraded with an almost exponential LPF (CI = 2.09, c2 = 3.35, c3 = 0.22). The resultant image differs from the original because some high-frequency information in the blurred image is too severely attenuated and the Gaussian component of the LOG filter prevents amplification of high-frequency noise. When c3 >> 0, improvement in the resolution is poor (Figure Sc). In this case, the blurring (c1 = 2.65, c2 = 3.16, c3 = 2.00) is more severe than that for an exponential LPF. The "restored image is still closer to the true image, but there is less improvement than is possible with a more appropriate filter. The optimal filter in a least-mean-squares sense is the Wiener filter, which has the formz2
where H*(u,v) is the complex conjugate of the PSF in the frequency domain and F(u,v) and N(u,v) are the Fourier transforms of the input signal f(xy) and noise n (xy), respectively. In many cases, the forms of fky) and n(xy) are unknown, and the last term in the denominator of eq 15 is replaced with a tuning parameter E , Le., (16)
where V is the Laplacian operator. Equation 13 is unstable because H-'(u,v) is unbounded as (u,v) -. To overcome this
-
4506 Analytical Chemistry, Vol. 67, No. 24, December 15, 1995
When H(u,v) is approximately an exponential LPF, eq 16 generates a restoration filter that is similar to the LOG filter. The main difference is that the Wiener filter acts as a LPF when the signal-to-noise ratio approaches zero, thus combining debluning and noise reduction. Since both eqs 14 and 16 have one tuning
Fisure 8. Restoration of blurred protrusion: (a, top len) original image; (b, bonom ien) restoration of blurred image (c, = 2.09. c2 = 3.35,03 = 0.22) using a LOG filter (Ah = 0.2, r = 1.95. (I = 0.5);(c, top right) restoration 01 blurred image (c, = 2.65, c2 = 3.16. C2 = 2.00) Using the LOG filter in part b: and (d, bonom right) restoration of image in part c with Wiener restoration filter (c = 1). The grid corresponds to the blurred image, and the transparent surface corresponds to the restored image.
parameter, the amount of tuning required is comparable for the two approaches. Figure 8d shows the results of applying a Wiener restoration filter to the blurred image in Figure 8c. It is clear that the restored image is much closer to the true image when the deviation in the PSF from an exponential LPF is taken into account in the restoration filter (Figure 8ac,d). Thus. the LOG filter is only suitable under limited circumstances, and in general a more suitable filter can be found based on Wiener restoration. But even the Wiener filter cannot optimally filter a nonlinear blur, and the filtering is optimal only in a portion of the image, where the blurring is accurately described by the PSF used to generate the filter. Current research in the authors' laboratory indicates that the approach may, nevertheless, he extended to accommodate nonlinear behavior. CONCLUSIONS
The blurring of SECM images due to surface roughness cannot be described by a single LSI process. Tne nonlinearity in the PSF can be reduced by reducing the gap between the tip and the substrate, and it appears that the constant current feedback mode, in which the vertical position of the tip is varied over the course of a scan. permitting closer tracking of the substrate features, may give more consistent blurring and facilitate image restoration. The
variation in the PSF estimate with substrate geometry suggests the use of a nonlinear adaptive restoration technique for best results. An adaptive restoration technique based on the Wiener filter may be feasible. The use of a LOG filter provides some improvement in the image, particularly where there are large protrusions; however, the LOG filter does not adequately filter the blur observed in a substrate cavity. Work is underway in the authors' laboratory to verify the PSF models by using them to restore experimentally obtained images. The approach should further enhance the SECM technique to enable resolution of features smaller than the tip. ACKNOWLEDGMENT
This work was funded by the Natural Sciencesand Engineering Research Council of Canada (NSERC) and the Ontario Graduate Scholarship (OGS) program. Received for review March 28,1995. Accepted October 3, 1995." AC9503015 Abstnrt published in Advance ACS Abstmch, November 1.
1995.
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