Restoration of AFM Images that were Produced using the Estimated
AFM Tip at Three Different Scanning Speeds
AHMED AHTAIBA
Electrical and Electronic Engineering Department,
Sirte University,
Sirte,
LIBYA
Abstract: - The contact between the sample and the AFM tip causes distortions in all atomic force microscope
(AFM) pictures. With the three-dimensional tip form in hand, the distorted picture may be straightened out and
the surface structure's original state "restored" using deconvolution methods. Compared to the initial distorted
image, the restored image provides a more realistic portrayal of the sample's true 3D surface. In order to
estimate the impulse response of the AFM, this work presents a new method that uses contact mode AFM to
measure the dimensions of a micro-cylinder. Subsequent AFM pictures are subsequently restored using the
predicted impulse response.
Key-Words: - AFM, image restoration, scanning speed, pillar sample, AFM tip.
Received: March 24, 2024. Revised: October 9, 2024. Accepted: November 13, 2024. Published: December 23, 2024.
1 Introduction
Atomic force microscopes (AFMs) are highly
specialized tools capable of analyzing the surfaces
of metals, semiconductors, and insulators. They
operate effectively in various environments,
including vacuum, liquid, or air, and are extremely
sensitive to atomic forces. These microscopes
provide ultra-high resolution, allowing them to
image, measure, manipulate, and probe objects at
the micro and nanoscale. AFMs have a significant
advantage over other high-resolution microscopes
like the Scanning Tunneling Microscope (STM) in
that they can examine both conducting and non-
conducting materials.
Deformities in AFM images are caused by the
sample's interaction with the impulse response of
the AFM, despite their capabilities. By obtaining the
impulse response, one may rectify the distortions
and restore the original surface structure, usually by
employing deconvolution techniques. This process
results in a more accurate representation of the
sample’s actual two-dimensional surface compared
to the initial, deformed picture. The difficulty of
accurately re-creating surface topography from
AFM samples has been the subject of several
research efforts. Notably, researchers such as
Pingali and Jain employed mathematical
morphological operators to successfully restore
AFM images.
[1], after Keller and Franke used photos of
known samples to rebuild the AFM tip shape, they
used that shape to recover AFM images, [2].
Villarrubia developed a method for blind tip
reconstruction, which is also grounded in
mathematical morphology, [3]. Dongmo applied this
algorithm to reconstruct the tip of a stylus
profilometer, afterwards, we'll compare the SEM
picture with the rebuilt tip form [4]. Subsequently,
Todd showed that AFM picture noise skews the tip
shape estimate, and he suggested a better way to
improve the method, [5]. The practical application
of the algorithm also considered factors like
sampling intervals and instrumental noise, [6].
Following this, they laid up some ground rules and
set up several suitable experiments to test the blind
estimation process. In this work, a technique is
offered for approximating 3 dimensions AFM tip
shape from the measurement of a micro-cylinder
with well-known and independently measured
dimensions. If the same tip is used under identical
measurement condition, the predicted tip shape can
be utilized to reconstruct AFM pictures taken at a
later time. The ability to deduce the impulses
responses of the AFM is a notable feature of this
approach. The efficacy of this novel method in
repairing AFM pictures has been confirmed by
means of both virtual and physical experimental
AFM pictures.
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2 The Impulse Response of an AFM
is Estimated by Utilizing a
Cylindrical Pillar Sample
The HS-100MG standard sample, with its defined
dimensions, silicon construction, and 2D array of
tiny cylindrical columns, was measured using the
contact mode AFM. After taking this AFM
measurement, we chose one pillar out of the picture.
As seen in Figure 1(a), setting a threshold allowed
us to calculate the size of the top of the column in
the AFM picture, which is nearly a complete circle.
In Figure 1(b), we can see the outcome of applying
the Canny edge detection technique to determine the
pillar's outside border. Figure 1(c) depicts the steps
used to estimate the data on tip distortion around the
column's perimeter by increasing the circle's outside
edge. In this case, d represents the cylinder's
diameter, the size expansion was between around
1.2d and 2d. The data for the cylinder's tips, which
had been originally placed around its outside edge,
were then transferred radially inwards to the center
of the cylinder that had been removed using
Breshnam's line method. In Figure 1(d), we can see
the result of using the suggested method, which is
the predicted form of the 3-dimensional AFM
impulse response.
A priori knowledge of non- negativity and flux
conservation is incorporated into the widely-used
Lucy-Richardson algorithm. Through an iterative
approach, it is able to recover AFM pictures. The
fundamental premise is that there is a connection
between the ideal image of the AFM and its impulse
response. Lucy-Richardson optimizes the picture's
likelihood function using Poisson statistics as a
model, [7], [8]. Typically, the hazy image is used as
the initial estimate. Some examples of iterative
algorithms include the Lucy-Richardson method,
[9], [10], [11].
󰇛 󰇜
󰇛 󰇜󰇛󰇜󰇛󰇜
󰇟󰇛󰇜󰇛󰇜󰇠󰇛󰇜 (1)
Where ,, represents the blurred AFM picture, -
(−,−) denotes the transpose of the system's
impulse response, and ,-.(,) represents the prior
estimate of the AFM image, ℎ(,) denotes the
impulse response of the AFM system, and ,-
+1.(,) signifies the current estimate of the AFM
picture.
Fig. 1: Standard operating procedure (SOP) for
determining the AFM's impulse response: (a) first, a
standard sample with a cylinder of predetermined
dimensions is subjected to a threshold. Then, the
cylinder's position in the image is determined by
drawing its outer boundary. To remove the
cylinder's pixels from the image, the outer boundary
is enlarged. Finally, the AFM's impulse response is
extracted.
3 Experimental Results
3.1 AFM Image Restoration at a Scanning
Rate of 1 Hz
Two genuine samples were analyzed at a scanning
speed of 1 H-z using contact mode atomic force
microscopy (AFM). One of the samples was a grid
of elevated square pillars, and the other was a true
specimen with an array of raised cylindrical pillars.
The identical methods described before and
illustrated in Figure 1 were utilized while scanning
at this pace. After the AFM impulse response was
estimated at this speed, the raw AFM image, which
had been captured at the same detecting speed as the
AFM impulse response, and the derived impulse
response were subjected to a Lucy-Richardson
deconvolution procedure. As shown in Figure 2(c),
the recovered AFM picture showed qualitative
improvements in terms of fidelity after using the
Lucy-Richardson deconvolution approach to an
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AFM image of a genuine material with cylindrical
pillars.
The numerical findings for the original AFM
picture (Figure 2(a)) and the recovered image
(Figure 2(c)) at a detecting speed of 1 Hz are
provided in Table 1. We compared the AFM image's
first row of pillar measurements to the repaired
AFM image's corresponding row. In the AFM
picture, the height (H1) of the first pillar (P1,1) is
91.38 nm, while in the restored image, the
equivalent height (H2) is 91.68 nm. This pillar is
situated near the row center, at (X = 2.647 μm, Y =
2.441 μm. There is a 0.3 nm height disparity
between H1 and H2, which is equal to a 0.327%
percentage difference (D%).
H1 measures 88.59 nm for the second pillar (P1,2)
in the same row of the AFM picture, which is
located at (X = 7.617 μm, Y = 2.441 μm). In the
corrected picture, the matching height (H2) is 88.89
nanometers. With a height disparity of 0.3 nm, or
0.337%, between H1 and H2 is the result.
According to the AFM picture, the third pillar (P1,3)
stands at 86.24 nm in height (H1), with coordinates
(X = 12.62 μm, Y = 2.441 μm). In the reconstructed
picture, the comparable height (H2) is 86.52 nm,
which is different from the original by 0.28 nm, or
0.323%.
When we compare the height disparities
between H-1 and H-2 at a scanning speed of 1 Hz,
we find that the percentage differences are minor,
with the lowest being 0.323%.
The 1st row of pillars in the AFM picture (Figure
2(a)) and their corresponding row in the
reconstructed image (Figure 2(c)) are compared
quantitatively in Table 1.
Reconstructing future AFM pictures is possible
with the help of the Lucy-Richardson deconvolution
process, which uses the raw AFM picture and the
AFM's impulse response (already stated and shown
in Figure 1).
Table 1. A comparison of quantitative values of the
first row of pillars in the AFM image (Figure 2(a))
with the values of the corresponding row in the
restored image (Figure 2(c))
Pillar
position
P(1,1)
P(1,2)
P(1,3)
X[µm]
2.647
7.617
12.62
Y[µm]
2.441
2.441
2.441
H1[nm]
91.38
88.59
86.24
H2[nm]
91.68
88.89
86.52
D[nm]
0.3
0.3
0.28
D%
0.327%
0.337%
0.323%
Fig. 2: A juxtaposition of the original experimental
picture of the AFM and the recovered picture of the
AFM acquired using the suggested method with a
scanning speed of 1 Hz: (a) a picture of the real
sample made with an AFM tip, showing cylindrical
pillars; (b) a two-dimensional depiction of the
AFM's impulse response; (c) a three-dimensional
illustration of the AFM's impulse response; and (d)
an image of the restored The AFM picture and the
estimated impulse response were used in a Lucy-
Richardson deconvolution procedure to get the
AFM representation
Fig. 3: With a scanning speed of 1 Hz, the
experimental raw AFM picture and the recovered
AFM image, which was created using the suggested
approach. (a) A picture of the actual sample with
square pillars as captured by an AFM tip; (b) A
picture of the cleaned-up AFM output made from
the AFM picture plus the estimated impulse
response of the AFM using Lucy-Richardson
deconvolution.
Figure 3(a) and Figure 3(b) illustrate a
comparison for the original AFM topography image
acquired at a 1 Hz scanning speed and the recovered
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AFM image produced by the Lucy-Richardson
deconvolution method, respectively. The
unprocessed picture illustrates a series of raised
square columns. Figure 3(b) demonstrates that the
Lucy- Richardson deconvolution approach is
successful, resulting in a more accurate restored
image.
3.2 Bringing Back AFM Pictures while
Scanning at 2 Hz
All previously measured samples at 1 Hz are
employed here. Subsequently, the experimental
images of these specific materials made use of an
AFM scanning rate of 2 Hz. Images of an
experimental sample with a grid of raised square
pillars (Figure 4(a)) and a configuration of raised
cylindrical pillars (Figure 5(a)) are presented,
respectively. Using a Lucy-Richardson
deconvolution process on the raw AFM picture and
the estimated AFM impulse response, as described
in Section 2 and shown in Figure 4(c), the
experimental images captured by the AFM at a 2 Hz
scanning speed may be recovered. Consequently, in
comparison to the unprocessed experimental AFM
images in Figure 4(d) and Figure 5(b), the restored
images exhibit a marked enhancement in fidelity.
Table 2 compares the matching quantitative values
in the AFM picture (Figure 4(a)) and the restored
image (Figure 4(d)) for the first row of pillars.
Using a scanning speed of 2 Hz, the AFM image
was measured. Pillar P(1,1), situated at coordinates
(X = 2.647 μm, Y = 2.441 μm), has a height (H1) of
91.59 nm. In the corrected picture, the matching
height (H2) is 92.14 nm. The dissimilarity (D)
between the two hydrogen bonds, H1 and H2, is
0.55 nm, or 0.596%.
The first pillar, P(1,2), stands at a height of 94.2
nm, with coordinates (X = 7.617 μm, Y = 2.441
μm). In the corrected picture, the matching height
(H2) is 94.76 nm. A 0.56 nm discrepancy, or
0.590% percentage difference, separates H1 and H2
for P(1,2).
The first pillar, P(1,3), stands at 86.65 nm in
height, with coordinates (X = 12.62 μm, Y = 2.441
μm). In the corrected picture, the matching height
(H2) is 87.17 nm. For P(1,3), there is a 0.52 nm
discrepancy between H1 and H2, which translates to
a 0.596% percentage difference.
At 0.590 percent, P(1,2) has the narrowest
percentage discrepancy between H1 and H2 in
Table 2.
Table 2 shows the numbers for the first row of
pillars in both the AFM picture (Figure 4(a)) and the
reconstructed image (Figure 4(c)).
Table 2. A comparison of quantitative values of the
first row of pillars in the AFM image (Figure 4(a))
with the values of the corresponding row in the
restored image (Figure 4(c))
Pillar
position
P(1,1)
P(1,2)
P(1,3)
X[µm]
2.647
7.617
12.62
Y[µm]
2.441
2.441
2.441
H1[nm]
91.59
94.2
86.65
H2[nm]
92.14
94.76
87.17
D[nm]
0.55
0.56
0.52
D%
0.596%
0.590%
0.596%
Fig. 4: Result of applying the suggested method at a
scanning speed of 2 Hz to an AFM picture,
compared to the original experimental AFM image:
(a) a picture of the real sample with cylindrical
pillars taken by the tip of the AFM; (b) a two-
dimensional depiction of the impulse response of
the AFM; (c) the three-dimensional depiction of the
AFM's impulse response; and (d) the reconstructed
image of the AFM subsequent to the application of
an AFM picture and its predicted impulse response
by a Lucy-Richardson deconvolution method.
Figure 5 shows the raw AFM picture and the
restored AFM picture, both generated using the
suggested approach at a scanning speed of 2 Hz.
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Fig. 5: The raw AFM picture and the restored AFM
picture both generated using the suggested approach
at a scanning speed of 2 H-z. (a) the original AFM
picture of the sample with square pillars measured
by the tip; (b) the recovered AFM image created by
merging the original image with the estimated
impulse response from the AFM using the Lucy-
Richardson deconvolution method.
3.3 AFM Image Restoration at a Scanning
Rate of 2.5 Hz
The experimental pictures of an actual sample with
a grid of raised square pillars and an array of raised
cylindrical pillars are shown in Figure 6(a) and
Figure 7(a), respectively. Nevertheless, these
pictures have been assessed using the AFM at a 2.5
Hz scanning rate in the subsequent set of data.
Using a Lucy-Richardson deconvolution technique,
which was described in Section 2, on the original
AFM picture and the impulse response obtained as
shown in Figure 6(c), the AFM images of the two
actual samples may be restored. Figures 6(d) and
7(b) display the recovered AFM pictures that were
created by implementing the deconvolution
procedure, correspondingly. By eliminating the
impacts of the AFM impulse response convolution
that was present in the initial hazy raw AFM photos,
the deconvolution technique significantly enhanced
the quality of the recovered AFM images.
Table 3 compares numerical data from the first row
of pillars in the AFM picture (Figure 6(a)) with
numerical data from the corresponding row in the
restored image (Figure 6(d)). The scanning speed
used to measure this AFM picture was 2.5 Hz. At
coordinates (X = 2.647 μm, Y = 2.441 μm), the
pillar P(1,1) stands at a height of 93.45 nm, also
known as H1. In the corrected picture, the matching
height (H2) is 94.3 nm. For P(1,1), there is a
0.901% discrepancy between H1 and H2, which is
0.85 nm.
In the restored picture, the equivalent height
(H2) is 92.57 nm, while for the second pillar in the
first row, P(1,2), at a location (X = 7.617 μm, Y =
2.441 μm), the height (H1) is 91.73 nm. In terms of
percentage, the 0.84 nm gap between H1 and H2 is
equivalent to 0.907%.
In the first row, the third pillar (P(1,3)) has a
height (H1) of 88.67 nm. H2 stands for 89.49
nanometers in the corrected picture. With a
percentage difference of 0.916% and a difference of
0.82 nm, H1 and H2 are not identical.
With respect to P(1,1), the table shows that the
minimal percentage difference between H1 and H2
is 0.901%.
The first row of pillars in the AFM picture
(Figure 6(a)) and the corresponding row in the
restored image (Figure 6(d)) were compared
quantitatively in Table 3.
Table 3. A comparison of quantitative values of the
first row of pillars in the AFM image (Figure 6(a))
with the values of the corresponding row in the
restored image (Figure 6(d))
Pillar
position
P(1,1)
P(1,2)
P(1,3)
X[µm]
2.647
7.617
12.62
Y[µm]
2.441
2.441
2.441
H1[nm]
93.45
91.73
88.67
H2[nm]
94.3
92.57
89.49
D[nm]
0.85
0.84
0.82
D%
0.901%
0.907%
0.916%
Fig. 6: The unprocessed experimental AFM image is
contrasted with the recovered AFM image, which
was generated using the recommended procedure
using a 2.5 H-z scanning rate. (a) One view shows
the sample with cylindrical pillars as measured by
an AFM tip; (b) another shows the AFM impulse
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response in two dimensions; (c) a third shows the
AFM impulse response in three dimensions; and (d)
the last view is the recovered AFM image, which is
made by applying a Lucy-Richardson deconvolution
technique to both the AFM image and the expected
impulse response of the AFM.
Fig. 7: The original experimental AFM picture and
the recovered AFM image, which were generated
using the recommended method at a scanning speed
of 2.5 H-z. (a) an AFM-captured picture of the
actual sample with square pillars; (b) a recovered
AFM picture created by merging the original and
predicted AFM impulse responses using the Lucy-
Richardson deconvolution technique.
4 Conclusion
In order to restore AFM pictures and approximate
the AFM tip, this research presents a new method.
The study's experimental findings are shown by
a number of instances. The two authentic examples
that were scrutinized were a grid of raised square
pillars and an array of elevated cylindrical pillars.
The actual samples were measured using three
distinct scanning rates: 1 Hz, 2 H-z, and 2.5 H-z.
The unprocessed AFM pictures and the impulse
response determined for each scanning speed were
combined using a Lucy-Richardson deconvolution
technique after the AFM impulse response was
estimated. The measured AFM height pictures were
rendered more faithfully after undergoing this
deconvolution procedure.
You can see the comparison between the AFM
picture and the reconstructed image at 1 H-z, 2 H-z,
and 2.5 H-z scanning rates for the first row of pillars
quantitative indicators in Table 1, Table 2 and Table
3, respectively.
References:
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Deconvolution Algorithm using a New
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Shape from the Square Sample, WSEAS
Transactions on Signal Processing, Vol. 19,
2023, pp.103-107,
https://doi.org/10.37394/232014.2023.19.11.
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Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
The authors equally contributed in the present
research, at all stages from the formulation of the
problem to the final findings and solution.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The authors have no conflicts of interest to declare.
Creative Commons Attribution License 4.0
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