# Assessment of sub-sampling schemes for compressive nano-FTIR imaging

Selma Metzner1, Bernd Kästner1, Manuel Marschall1, Gerd Wübbeler1, Stefan
Wundrack1, Andrey Bakin2, Arne Hoehl1, Eckart Rühl3 and Clemens Elster1
1Physikalisch-Technische Bundesanstalt, Braunschweig and Berlin, Germany
2Institut für Halbleitertechnik, Technische Universität Braunschweig, Hans-
Sommer Straße 66, 38106 Braunschweig, Germany
3Physikalische Chemie, Freie Universität Berlin, Arnimallee 22, 14195 Berlin,
Germany

###### Abstract

Nano-FTIR imaging is a powerful scanning-based technique at nanometer spatial
resolution which combines Fourier transform infrared spectroscopy (FTIR) and
scattering-type scanning near-field optical microscopy (s-SNOM). However,
recording large spatial areas with nano-FTIR is limited by long measurement
times due to its sequential data acquisition. Several mathematical approaches
have been proposed to tackle this problem. All of them have in common that
only a small fraction of randomly chosen measurements is required. However,
choosing the fraction of measurements in a random fashion poses practical
challenges for scanning procedures and does not lead to time savings as large
as desired. We consider different, practically relevant sub-sampling schemes
assuring a faster acquisition. It is demonstrated that results for almost all
considered sub-sampling schemes, namely original Lissajous, triangle
Lissajous, and random reflection sub-sampling, at a sub-sampling rate of 10%,
are comparable to results when using a random sub-sampling of 10%. This
implies that random sub-sampling is not required for efficient data
acquisition.

###### Index Terms:

nano-FTIR, sub-sampling, Lissajous, low-rank matrix reconstruction

##  I Introduction

Infrared (IR) spectroscopy is a non-destructive technique for material
characterization ranging from analytical chemistry [1, 2], materials sciences
[3, 4], life sciences [5, 6] to microelectronics [7]. While the spatial
resolution of conventional IR microscopy is limited to several micrometers due
to the diffraction limit [8], this limitation can be overcome by modern
scanning probe microscopy-based methods in applying, for instance, near-field
based techniques. Examples include scattering-type scanning near-field optical
microscopy (s-SNOM) [9, 10], atomic force microscopy-based IR spectroscopy
(AFM-IR) [11, 12], photoinduced force microscopy (PiFM) [13], tip-enhanced
photoluminescence spectroscopy (TEPL) [14], and tip-enhanced Raman
spectroscopy (TERS) [15].

Nano-FTIR [16, 17, 18] is another scanning-based technique which combines
Fourier transform infrared spectroscopy (FTIR) and s-SNOM. Here, FTIR
spectroscopy uses a broadband IR source such as tunable lasers, thermal, or
synchrotron radiation. The spatial resolution is enhanced in s-SNOM by
adapting the principle of atomic force microscopy. Nano-FTIR enables the
acquisition of the complete spectrum at each pixel of the scanned area with
nanoscale spatial resolution.

Usually, these scanning-based approaches take the spectra in a sequential
process, resulting in long acquisition times for extended 2D arrays. These
scans may take several hours to achieve nanometer spatial resolution over a
substantial field of view in the micron size regime. In addition, long
acquisition times may lead to sample and tip damage as well as drift
artifacts.

Several methods have been presented recently in order to overcome long
acquisition times. In [19] a compressed sensing approach was introduced. Its
potential benefit was demonstrated by reconstruction results achieved for
nano-FTIR measurements when using only 11% of the original data. A low-rank
matrix reconstruction was developed in [20] for FTIR measurements. An adequate
sample recovery was achieved for a sub-sampling rate of 5%. Both methods
require only a small fraction of all measurements which could in principle
reduce the experimental effort significantly. However, the fraction of
measurements was chosen randomly. Practically, this implies that each single
measurement position needs to be accessed individually, thereby also passing
along idle routes and positioning times during which no measurement is carried
out. Consequently, time savings are not as large as desired.

In this work, we introduce several, practically relevant sub-sampling schemes
which sub-sample the data according to specific routes. The acquisition of
these data avoids idle routes, is practically feasible and much faster than
taking the same number of measurements at randomly chosen positions. Using the
same low-rank matrix recovery method as in [20] we compare the low-rank
reconstructions of the fully sampled data set for different sub-sampling
schemes with results obtained from random sub-sampling. The comparison is
based on a data set for a sample corresponding to the edge of an ultrathin
gallium film beneath epitaxial graphene grown on 6H-SiC.

The paper is organized as follows. In Section 2, the measurement process, data
acquisition, and numerical low-rank approximation method are presented.
Furthermore, four sub-sampling schemes will be described. Section 3
demonstrates results on different sub-sampling schemes. Finally, Section 4
gives some conclusions and an outlook to promising future research.

##  II Methods

###  II-A Nano-FTIR

Nano-FTIR is a scanning probe technique which is based, on the one hand, on
Fourier-transform IR spectroscopy and, on the other hand, on atomic force
microscopy. A sharp metallic tip which is brought close to the sample scans
the sample surface and backscatters the incident IR radiation. The metallic
tip behaves as an optical antenna and strongly restricts the incident optical
field to the tip. The nearfield of the resulting nanoscale light source is
then suitable for high-resolution imaging and spectroscopy. In order to
separate the nearfield contribution from the farfield background the
backscattered signal is demodulated at the second harmonic of the tip-
modulation frequency. The interferogram is recorded by a Michelson
interferometer [18, 21] as a function of the optical path difference and the
corresponding spectrum can be obtained from a Fourier transform of the
interferogram.

###  II-B Sample and data acquisition

As a sample, we chose the edge of an ultrathin gallium film beneath epitaxial
graphene on 6H-SiC [22]. It results in a strongly resonant phononic spectrum
around 920 cm-1 as well as a broad metallic spectrum, exhibiting in addition a
transition region. Thus, we have a large spectral variation for benchmarking
the proposed reconstruction algorithm. Details about fabrication and further
sample properties can be found in the Appendix.

Data acquisition from the Michelson interferometer results in an interferogram
$I(z)$ with $N_{z}=1024$ equidistant points ($\Delta z\approx 1.39~{}\mu$m).
The interferogram is converted to a spectrum by Fourier transformation with a
spectral resolution of $\Delta\tilde{\nu}=1/N_{z}\Delta z\approx 7.03~{}$cm-1.
Altogether, 961 interferograms were recorded on a rectangular area of
$2\,\mu$m$\times 2\,\mu$m using a Cartesian grid with 31$\times$31 spatial
locations. The fully sampled nano-FTIR data set can thus be represented by a
3D data cube with dimensions of 31$\times$31$\times$1024, which was recorded
within a total measurement time of 170 minutes.

###  II-C Low-rank matrix reconstruction

The concept of low-rank matrices arises in many mathematical settings related
to, e.g., modeling and data compression. A variety of applications ranges from
signal processing [23] and image restoration [24] to machine learning [25].
The recovery of a data matrix derived from incomplete observations poses a
relevant example of tasks where low-rank techniques can be applied. In the
present case, the data matrix is approximated by a matrix product with each
factor having lower dimensionality, resulting in a recovery result of lower
rank. The main idea of low-rank matrix reconstruction is that a low-rank
approximation already captures the main characteristics of the data and less
informative dimensions will be removed. In [20], a low-rank matrix
reconstruction has been presented and successfully applied to sub-sampled FTIR
data. In more detail, for observations $X$, the task is to find matrices $U$
and $V$ by minimizing $\sum_{i,j}|X_{i,j}-\sum_{k=1}^{r}U_{i,k}V_{j,k}|^{2}$.
The so-called rank $r$ approximation $\hat{X}$ of $X$ is then given by
$\hat{X}=UV^{T}$. However, the solution to this decomposition is non-unique.
For this reason, and to stabilize the numerical reconstruction, an additional
Tikhonov regularization [26] has been utilized in [20] which also accounts for
the spatial smoothness of the focal images. We will use $r=20$ which was also
used in [20].

As described above, the nano-FTIR data can be represented by a 3D data cube.
The low-rank matrix reconstruction applies to a 2D object which will be
handled in the following way: if $X$ denotes the matrix of measurements for
the low-rank matrix reconstruction, then the rows of $X$ indicate the spatial
position of the measurements and the columns of $X$ the interferometer
position. Note that through a mapping from 3D to 2D (and vice versa) randomly
chosen elements of the 2D matrix imply a specified selection of the 3D data
cube (and vice versa).

The resulting minimization problem is solved using an alternating algorithm
which generates linear problems in each iteration. An implementation of the
employed Python code has been made publicly available [27].

###  II-D Sub-sampling schemes

In the following, the considered sub-sampling schemes are presented.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 1: Visualization of different sub-sampling schemes. Top row: an xy-view
of the first 1000 sampling points, bottom row: the sum of the sampled points
along the z-axis for (a)+(d) triangle Lissajous (b)+(e) original Lissajous
(c)+(f) random reflection. The colors in (d), (e) and (f) indicate the number
of samples in the interferometer direction for each pixel in the $xy$-plane.
Note that (e) has a different color scale than (d) and (f).

#### Random

According to [28], samples in a data matrix are chosen randomly to avoid
information loss when recovering a low-rank matrix. The most obvious solution
of sub-sampling is therefore a randomly selected subset from the full data
set. We refer to random if the samples were selected uniformly along the
interferometer axis and within the image plane.

#### Original Lissajous

The usage of random sub-sampling does not necessarily lead to significant
savings in scan time. For randomly selected nano-FTIR measurements each sample
would need to be approached by nanopositioning systems separately which is
time-consuming. In addition, transit times between the samples are not used
for measurement. We therefore consider routes along which the stages move,
which usually occurs in a continuous manner. First, we consider an original
Lissajous sub-sampling. A 2D Lissajous curve [29] is the trajectory of a
moving point whose coordinates are simple harmonic motions. The parametric
equation is given by

$t\mapsto\begin{pmatrix}A_{0}+A\,\text{sin}(\omega_{1}t+\varphi_{1})\\\
B_{0}+B\,\text{sin}(\omega_{2}t+\varphi_{2})\end{pmatrix}\,,$

for $t\in[0,\infty)$. 2D Lissajous curves have been applied to several
scanning techniques, e.g., in the fields of medical imaging [30], atomic force
microscopy [29] and magnetic particle imaging [31]. We will expand 2D
Lissajous to 3D Lissajous by simply expanding the parametric equation as
follows

$t\mapsto\begin{pmatrix}A_{0}+A\,\text{sin}(\omega_{1}t+\varphi_{1})\\\
B_{0}+B\,\text{sin}(\omega_{2}t+\varphi_{2})\\\
C_{0}+C\,\text{sin}(\omega_{3}t+\varphi_{3})\end{pmatrix}\,.$

Figure 1b shows an $xy$-view of the first 1000 sampling points along the 3D
Lissajous curve. More details of the employed scheme can be found in the
Appendix.

#### Triangle Lissajous

Although the samples of original Lissajous scatter across the entire image
stack, sinusoidal waves fail to create a uniform sampling pattern. In Figure
1e the sum along the $z$-axis of all sample points is shown. Clearly, more
samples are located at the border of the image which potentially could lead to
an undesirable biased image reconstruction. In [32] a 3D triangle Lissajous
sub-sampling scheme has been introduced for spectroscopic laser-scanning
imaging. In Figure 1a an $xy$-view of the first 1000 sampling points along the
triangular Lissajous curve is shown. It can be seen (cf. Figure 1d) that the
triangular wave design yields a much more uniform sampling density than
original Lissajous sub-sampling.

Both original Lissajous and triangle Lissajous trajectories are sampled in
equidistant time steps. For each time step the interferogram value of the
nearest point on the $xyz$-grid is recorded.

#### Random reflection

Similar to triangle Lissajous we consider a so-called random reflection sub-
sampling. From a given starting point the samples are acquired on a 3D line
until one of the borders is reached. Then, specified by a random angle, the
line is reflected and new samples can be measured until the next border is
reached. This is repeated until the total number of samples is reached. In
Figure 1c an example of the $xy$-view of the first 1000 sampling points is
shown and in Figure 1f the sum along the $z$-axis of all sample points.
Contrary to original Lissajous more samples are now in the center area.

#### White light

Finally, we want to introduce the so-called white light sub-sampling. The idea
here is to measure full $xy$-planes but only for some randomly chosen points
along the interferometer axis. This means for some points along the $z$-axis
no points will be measured in the image plane, while for others the entire
$xy$-plane is recorded.

##  III Results and Discussion

To test the influence of the different sub-sampling schemes, five subsets of
the whole data set were created. The points belonging to each subset are
chosen accordingly to one of the five sub-sampling schemes presented above
using a sub-sampling rate of 10%.

In the following, we will assess the quality of the low-rank matrix
reconstructions in terms of the available full data set. Furthermore, we want
to particularly compare the results when using a random sub-sampling scheme to
the results for the other schemes.

In Figure 2 the $xy$-planes at the interferogram value of 1021.6 $\mu$m are
shown for the full data set, the low-rank reconstructions, the differences
between them, and the (sub-sampled) data set for each sub-sampling scheme.

Figure 2: Reconstruction results for the different sub-sampling schemes at
interferogram value of 1021.6 $\mu$m using 10% of the full data. First column:
triangle Lissajous, second column: original Lissajous, third column: random
reflection, fourth column: white light and fifth column: random sub-sampling.
The number above the reconstructed data indicates the root-mean square error
at the interferogram value of 1021.6 $\mu$m.

A summary of different similarity measures between the full data set and low-
rank reconstructions can be found in Table I. The relative errors are
calculated as $||\hat{X}-X||_{F}/||X||_{F}$, where $||\cdot||_{F}$ denotes the
Frobenius norm, and where $\hat{X}$ and $X$ denote the reconstructed and the
full data set, respectively. Note that the correlation coefficient cannot be
calculated for white light sub-sampling since the reconstruction is 0
everywhere for any $z$-plane where no sample was measured.

TABLE I: Similarity measures for the five different low-rank reconstructions. Each value represents the average over the respective measure for the reconstructed planes from 415.6 $\mu m$ to 1166.2 $\mu m$.  Sub-sampling | Root-mean | Relative | Correlation
---|---|---|---
scheme | square error | error | coefficient
Triangle Lissajous | 0.30 | 0.48 | 0.68
Original Lissajous | 0.32 | 0.50 | 0.66
Random reflection | 0.30 | 0.47 | 0.68
White light | 1.18 | 0.93 | -
Random | 0.29 | 0.46 | 0.69

While the results of triangle Lissajous, original Lissajous and random
reflection deliver similar results compared to a random sub-sampling, the
results from the white light reconstruction are extremely poor.

Note that Figure 2 relates to an $xy$-plane that included no samples for the
white light sub-sampling scheme. When looking at results of the white light
sub-sampling for $xy$-planes that were fully measured, the reconstruction is
almost perfect there.

In Figure 3 the resulting spectra of the five reconstructions are shown for
pixel (4, 25) and from 731.1 cm-1 to 1188.1 cm-1 together with the spectrum of
the full data set. Similar to the above results, the white light
reconstruction is extremely poor whereas the other sub-sampling methods yield
comparable results. Especially when comparing the random sub-sampling to
triangle Lissajous and original Lissajous, the spectra are similar to each
other.

Figure 3: Reconstructed spectra from 731.1 cm-1 to 1188.1 cm-1 at pixel (4,25)
for the different sub-sampling schemes. Note that pixel (4,25) corresponds to
point (a) in Figure 4.

In Figure 4 the amplitudes of the spectra at 920.9 cm-1 for fully sampled data
and triangle Lissajous sub-sampling are shown and appear to be in agreement.

Figure 4: Amplitude of the spectra at peak position 920.9 cm-1 for fully
sampled data and sub-sampled data using a triangle Lissajous scheme. Four
different pixel locations (a)-(d) are marked and evaluated in Figure 5.

We choose four different points each characterizing a special part of the
spectrum and compare the Fourier amplitude and the Fourier phase of the
spectra from 731.1 cm-1 to 1188.1 cm-1 (cf. Figure 5). The quality of the low-
rank reconstruction for the triangle Lissajous sub-sampling leads to an
agreement to the fully sampled data set.

Figure 5: Amplitude and phase of the reconstructed spectra at different pixel
locations for triangle Lissajous sub-sampling and fully sampled data. The
pixel locations in the $xy$-plane are (a) (4,25), (b) (13,22), (c) (10,8), and
(d) (24,25).

In order to verify that the results for sampling schemes with random elements
(i.e. random and random reflection sub-sampling) are representative, two
further, randomly drawn subsets were created for each scheme and used for the
reconstruction of the full data set. It turned out that the above results do
not change significantly. In particular, we obtained the same numbers as in
Table I.

We also created five data sets with a sub-sampling rate of 5%. Results of the
low-rank reconstructions can be seen in Figure 6. As expected, reducing the
amount of data leads to a deterioration of the reconstruction quality. This is
also indicated by the increased root-mean square errors of the
reconstructions. Nevertheless, for triangle Lissajous, original Lissajous, and
random reflection the main features are still clearly visible and comparable
to a random sub-sampling. The white light sub-sampling is still delivering
poor results.

Figure 6: Reconstruction results for the different sub-sampling schemes at the
interferogram value of 1021.6 $\mu$m using 5% of the full data. First column:
triangle Lissajous, second column: original Lissajous, third column: random
reflection, fourth column: white light and fifth column: random sub-sampling.

##  IV Conclusion & Outlook

Different sub sampling schemes have been compared for compressive nano-FTIR
measurement. Triangle Lissajous, original Lissajous, and random reflection
have been shown to lead to a suitable data reconstruction for a sub-sampling
rate of 10%. Even with a sub-sampling rate of 5% reasonable results could be
achieved. All three sub-sampling schemes yield similar results. Furthermore,
these results are in agreement with results of random sub-sampling. We
therefore have presented three practically relevant sub-sampling schemes which
could lead to tremendous time savings for nano-FTIR measurements.

Future work could apply the sub-sampling schemes to other nano-FTIR
measurements and even other scanning probe techniques. Additionally, a
hardware implementation of these schemes is necessary to compare the results
regarding their acquisition times. We expect that due to the absence of idle
times and the possibility of continuous stage movement the reduction in
acquisition times approaches the sub-sampling rate.

## Acknowledgments

We gratefully acknowledge fruitful discussions with Alexander Govyadinov and
Markus Raschke. We acknowledge financial support by Deutsche
Forschungsgemeinschaft (Grants EL 492/1-1, RU 420/13-1). Bernd Kästner has
received funding within the project 20IND09 PowerElec from the EMPIR programme
co-financed by the Participating States and from the European Union’s Horizon
2020 research and innovation programme.

## References

  * [1] R. Salzer and H. W. Siesler, Infrared and Raman spectroscopic imaging.  John Wiley & Sons, 2014.
  * [2] S. Morsch, S. Lyon, S. Edmondson, and S. Gibbon, “Reflectance in AFM-IR: implications for interpretation and remote analysis of the buried interface,” Analytical Chemistry, vol. 92, no. 12, pp. 8117–8124, 2020\.
  * [3] F. Ruggeri, G. Longo, S. Faggiano, E. Lipiec, A. Pastore, and G. Dietler, “Infrared nanospectroscopy characterization of oligomeric and fibrillar aggregates during amyloid formation,” Nature Communications, vol. 6, no. 1, pp. 1–9, 2015.
  * [4] J. Chae, Q. Dong, J. Huang, and A. Centrone, “Chloride incorporation process in $CH_{3}NH_{3}PbI_{3-x}Cl_{x}$ perovskites via nanoscale bandgap maps,” Nano Letters, vol. 15, no. 12, pp. 8114–8121, 2015.
  * [5] M. Pilling and P. Gardner, “Fundamental developments in infrared spectroscopic imaging for biomedical applications,” Chemical Society Reviews, vol. 45, no. 7, pp. 1935–1957, 2016.
  * [6] C. Paluszkiewicz, N. Piergies, P. Chaniecki, M. Rękas, J. Miszczyk, and W. Kwiatek, “Differentiation of protein secondary structure in clear and opaque human lenses: AFM–IR studies,” Journal of Pharmaceutical and Biomedical Analysis, vol. 139, pp. 125–132, 2017.
  * [7] W. S. Lau, Infrared characterization for microelectronics.  World Scientific, 1999.
  * [8] M. Born and E. Wolf, Principles of optics: electromagnetic theory of propagation, interference and diffraction of light.  Elsevier, 2013.
  * [9] B. Knoll and F. Keilmann, “Near-field probing of vibrational absorption for chemical microscopy,” Nature, vol. 399, no. 6732, pp. 134–137, 1999.
  * [10] F. Keilmann, R. Hillenbrand, A. Zayats, and D. Richard, “Near-field nanoscopy by elastic light scattering from a tip,” Nano-Optics and Near-Field Optical Microscopy, pp. 235–265, 2009.
  * [11] A. Dazzi, R. Prazeres, F. Glotin, and J. Ortega, “Local infrared microspectroscopy with subwavelength spatial resolution with an atomic force microscope tip used as a photothermal sensor,” Optics Letters, vol. 30, no. 18, pp. 2388–2390, 2005.
  * [12] A. Dazzi and C. B. Prater, “AFM-IR: Technology and applications in nanoscale infrared spectroscopy and chemical imaging,” Chemical Reviews, vol. 117, no. 7, pp. 5146–5173, 2017.
  * [13] I. Rajapaksa, K. Uenal, and H. K. Wickramasinghe, “Image force microscopy of molecular resonance: A microscope principle,” Applied Physics Letters, vol. 97, no. 7, p. 073121, 2010.
  * [14] C.-F. Wang, M. Zamkov, and P. Z. El-Khoury, “Ambient tip-enhanced photoluminescence with 5 nm spatial resolution,” The Journal of Physical Chemistry C, vol. 125, no. 22, pp. 12251–12255, 2021.
  * [15] R. M. Stöckle, Y. D. Suh, V. Deckert, and R. Zenobi, “Nanoscale chemical analysis by tip-enhanced raman spectroscopy,” Chemical Physics Letters, vol. 318, no. 1-3, pp. 131–136, 2000.
  * [16] F. Huth, A. Govyadinov, S. Amarie, W. Nuansing, F. Keilmann, and R. Hillenbrand, “Nano-FTIR absorption spectroscopy of molecular fingerprints at 20 nm spatial resolution,” Nano Letters, vol. 12, no. 8, pp. 3973–3978, 2012.
  * [17] A. C. Jones and M. B. Raschke, “Thermal infrared near-field spectroscopy,” Nano Letters, vol. 12, no. 3, pp. 1475–1481, 2012.
  * [18] P. Hermann, A. Hoehl, P. Patoka, F. Huth, E. Rühl, and G. Ulm, “Near-field imaging and nano-fourier-transform infrared spectroscopy using broadband synchrotron radiation,” Optics Express, vol. 21, no. 3, pp. 2913–2919, 2013.
  * [19] B. Kästner, F. Schmähling, A. Hornemann, G. Ulrich, A. Hoehl, M. Kruskopf, K. Pierz, M. B. Raschke, G. Wübbeler, and C. Elster, “Compressed sensing ftir nano-spectroscopy and nano-imaging,” Optics Express, vol. 26, no. 14, pp. 18115–18124, 2018.
  * [20] M. Marschall, A. Hornemann, G. Wübbeler, A. Hoehl, E. Rühl, B. Kästner, and C. Elster, “Compressed FTIR spectroscopy using low-rank matrix reconstruction,” Optics Express, vol. 28, p. 38762, dec 2020.
  * [21] P. Hermann, B. Kästner, A. Hoehl, V. Kashcheyevs, P. Patoka, G. Ulrich, J. Feikes, M. Ries, T. Tydecks, B. Beckhoff, et al., “Enhancing the sensitivity of nano-FTIR spectroscopy,” Optics Express, vol. 25, no. 14, pp. 16574–16588, 2017.
  * [22] S. Wundrack, D. Momeni, W. Dempwolf, N. Schmidt, K. Pierz, L. Michaliszyn, H. Spende, A. Schmidt, H. Schumacher, R. Stosch, et al., “Liquid metal intercalation of epitaxial graphene: large-area gallenene layer fabrication through gallium self-propagation at ambient conditions,” Physical Review Materials, vol. 5, no. 2, p. 024006, 2021.
  * [23] Z. Weng and X. Wang, “Low-rank matrix completion for array signal processing,” in 2012 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp. 2697–2700, IEEE, 2012.
  * [24] Y. Peng, J. Suo, Q. Dai, and W. Xu, “Reweighted low-rank matrix recovery and its application in image restoration,” IEEE Transactions on Cybernetics, vol. 44, no. 12, pp. 2418–2430, 2014.
  * [25] Q. Yao, J. T. Kwok, and W. Zhong, “Fast low-rank matrix learning with nonconvex regularization,” in 2015 IEEE International Conference on Data Mining, pp. 539–548, IEEE, 2015.
  * [26] H. W. Engl, M. Hanke, and A. Neubauer, Regularization of inverse problems, vol. 375.  Springer Science & Business Media, 1996.
  * [27] M. Marschall, G. Wübbeler, and C. Elster, “Regression–working group 8.42.” https://www.ptb.de/cms/nc/en/ptb/fachabteilungen/abt8/fb-84/ag-842/regression-842.html, 2021\.  Accessed: 2022-01-12.
  * [28] E. J. Candes and Y. Plan, “Matrix completion with noise,” Proceedings of the IEEE, vol. 98, no. 6, pp. 925–936, 2010.
  * [29] T. Tuma, J. Lygeros, V. Kartik, A. Sebastian, and A. Pantazi, “High-speed multiresolution scanning probe microscopy based on lissajous scan trajectories,” Nanotechnology, vol. 23, no. 18, p. 185501, 2012.
  * [30] H. Feng, H. Gu, D. Silbersweig, E. Stern, and Y. Yang, “Single-shot mr imaging using trapezoidal-gradient-based lissajous trajectories,” IEEE Transactions on Medical Imaging, vol. 22, no. 8, pp. 925–932, 2003.
  * [31] F. Werner, N. Gdaniec, and T. Knopp, “First experimental comparison between the cartesian and the lissajous trajectory for magnetic particle imaging,” Physics in Medicine & Biology, vol. 62, no. 9, p. 3407, 2017.
  * [32] H. Lin, C.-S. Liao, P. Wang, N. Kong, and J.-X. Cheng, “Spectroscopic stimulated raman scattering imaging of highly dynamic specimens through matrix completion,” Light: Science & Applications, vol. 7, no. 5, pp. 17179–17179, 2018.
  * [33] M. Kruskopf, D. M. Pakdehi, K. Pierz, S. Wundrack, R. Stosch, T. Dziomba, M. Götz, J. Baringhaus, J. Aprojanz, C. Tegenkamp, et al., “Comeback of epitaxial graphene for electronics: large-area growth of bilayer-free graphene on SiC,” 2D Materials, vol. 3, no. 4, p. 041002, 2016.
  * [34] D. Momeni Pakdehi, K. Pierz, S. Wundrack, J. Aprojanz, T. T. N. Nguyen, T. Dziomba, F. Hohls, A. Bakin, R. Stosch, C. Tegenkamp, et al., “Homogeneous large-area quasi-free-standing monolayer and bilayer graphene on SiC,” ACS Applied Nano Materials, vol. 2, no. 2, pp. 844–852, 2019\.
  * [35] D. Momeni Pakdehi, J. Aprojanz, A. Sinterhauf, K. Pierz, M. Kruskopf, P. Willke, J. Baringhaus, J. Stöckman, G. Traeger, F. Hohls, et al., “Minimum resistance anisotropy of epitaxial graphene on SiC,” ACS Applied Materials & Interfaces, vol. 10, no. 6, pp. 6039–6045, 2018.

## Appendix

### Details about the sample

Epitaxial graphene was grown on the Si-face of SiC samples (5 × 10) mm2 cut
from a semi-insulating 6H-SiC wafer with a nominal miscut angle of about 0.06∘
toward the (1100) plane. The graphene samples were prepared using the polymer-
assisted sublimation growth (PASG) technique, which involves polymer
adsorbates formed on the SiC surface by liquid phase deposition from a
solution of a photoresist (AZ5214E) in isopropanol followed by sonication and
short rinsing with isopropanol. The graphene layer growth was processed at
1750 ∘C (argon atmosphere $\approx$1 bar, 6 min, zero argon flow) with pre-
vacuum-annealing at 900 ∘C [33, 34]. The applied PASG method allows for the
growth of large-area homogeneous monolayer graphene, with almost an isotropic
resistance characteristics [35]. The fabrication of a large-area two-
dimensional gallium layer at the interface of SiC and graphene has been
realized by using the liquid metal intercalation technique (LiMIT) [22]. This
method achieves the lateral intercalation and diffusion of Ga atoms at room
temperature resulting in the conversion of epitaxial graphene to quasi-
freestanding bilayer graphene (QFBLG). The average thickness of the confined
gallium layer after the intercalation process is approx. 1 nm at the interface
between SiC and graphene. In addition, Van der Pauw measurements revealed low
carrier mobility as well as strong electron doping of n $\approx$ 4.5$\times
10\textsuperscript{12}$cm-2 in the SiC- confined Ga – QFBLG sample
(SiC/2DGa/QFBLG) at room temperature [22].

### Details about the employed original Lissajous sub-sampling scheme

In Section 2D a general description of the original Lissajous sub-sampling
scheme is given. In the present case for a data cube of dimension $31\times
31\times 1024$ the following parameters have been used for the employed
original Lissajous scheme: $A_{0}=15.5$, $B_{0}=15.5$, $C_{0}=512$,
$A=97.3894$, $B=97.3894$, $C=3216.9908$, $\omega_{1}=0.0119$,
$\omega_{2}=0.0084$, $\omega_{3}=0.0002$, $\varphi_{1}=0.75$,
$\varphi_{2}=0.75$, and $\varphi_{3}=0.75$.