# Spin-1/2 $XXZ$ Heisenberg cupolae: magnetization process and related
enhanced magnetocaloric effect

K. Karľová

(Received June 23, 2020, in final form July 30, 2020)

###### Abstract

The magnetization curves and magnetocaloric effect during the adiabatic
demagnetization of antiferromagnetic spin-1/2 $XXZ$ Heisenberg clusters with
the shape of Johnson’s solids (triangular cupola, square cupola and pentagonal
cupola) are investigated using the exact numerical diagonalization for
different values of the exchange anisotropy ranging in between the Ising and
fully isotropic limit. It is demonstrated that spin-1/2 $XXZ$ Heisenberg
cupolae display, in comparison with their Ising counterparts, at least one
more magnetization plateau. The novel magnetization plateaux extends over a
wider range of the magnetic fields with increasing of a quantum part of the
$XXZ$ exchange coupling at the expense of the original plateaux present in the
limiting Ising case. It is shown that the spin-1/2 $XXZ$ Heisenberg triangular
and pentagonal cupolae exhibit an enhanced magnetocaloric effect in the
vicinity of zero magnetic field, which makes these magnetic systems promising
refrigerants for cooling down to ultra-low zero temperatures.

Key words: $XXZ$ Heisenberg clusters, magnetization plateaux, magnetization
plateaux, magnetocaloric effect

###### Abstract

Çà äîïîìîãîþ ìåòîäó òîíî¿ äàãîíàëçàö¿ äîñëäæåíî êðèâ íàìàãíåíîñò òà
ìàãíåòîêàëîðèíèé åôåêò ïä àñ àäàáàòèíî¿ äåìàãíåòèçàö¿ àíòèôåðîìàãíòíèõ
êëàñòåðâ ó ôîðì òâåðäèõ òë Äæîíñîíà (òðèêóòíîãî, êâàäðàòíîãî òà ïÿòèêóòíîãî
êóïîëâ) äëÿ ðçíèõ çíàåíü îáìííî¿ àíçîòðîï¿ âä çíîâî¿ äî ïîâíñòþ çîòðîïíî¿
ãðàíèö. Ïðîäåìîíñòðîâàíî, ùî ñïí-1/2 XXZ Ãàéçåíáåðîâèé êóïîë âèÿâëÿ ùîíàéìåíøå
îäíå äîäàòêîâå ïëàòî íàìàãíåííÿ â ïîðâíÿíí ç âäïîâäíîþ éîìó çíîâîþ ìîäåëëþ.
Íîâå ïëàòî íàìàãíåííÿ ïîøèðþòüñÿ íà äàëüøó îáëàñòü ìàãíòíèõ ïîëâ ç çðîñòàííÿì
êâàíòîâî¿ àñòèíè XXZ îáìííî¿ âçàìîä¿ çà ðàõóíîê ïåðâñíèõ ïëàòî, ÿê ïðèñóòí ó
ãðàíèö ìîäåë çíà. Ïîêàçàíî, ùî ñïí-1/2 XXZ ìîäåëü Ãàéçåíáåðà íà òðèêóòíîìó òà
ï’ÿòèêóòíîìó êóïîëàõ âèÿâëÿ ïîñèëåíèé ìàãíåòîêàëîðèíèé åôåêò ïîáëèçó íóëüîâîãî
ìàãíòíîãî ïîëÿ, ùî ðîáèòü òàê ìàãíòí ñèñòåìè ïåðñïåêòèâíèìè îõîëîäæóâààìè äî
óëüòðàíèçüêèõ òåìïåðàòóð.

Ключов слова: XXZ Ãàéçåíáåðîâ êëàñòåðè, ïëàòî íàìàãíåííÿ, ìàãíåòîêàëîðèíèé
åôåêò

##  1 Introduction

Magnetic systems composed of a finite number of interacting spins have
attracted considerable attention due to the possibility of exploring many
fundamental physical phenomena [1, 2]. Antiferromagnetic small spin clusters
can exhibit plethora features in the zero-temperature magnetization curve, for
instance magnetization steps and jumps [3, 4], magnetization plateaux [5, 6]
or quasi-plateau [7, 8]. Intermediate magnetization plateaux were predicted
for the Ising spin clusters with the shape of Platonic solids [9], ‘star of
David’ [10] or tetrahedra-based Ising clusters [11], while the magnetization
curve of Heisenberg spin clusters was studied with the shape of triangle [12],
tetrahedron [13], truncated tetrahedron [14, 15], cuboctahedron [16, 17],
icosidodecahedron [16] and many more [18, 19, 20, 21, 22, 23]. It is
noteworthy that besides the theoretical predictions, the magnetization
plateaux were indeed observed in experimental representatives of molecular
magnets, for example for linear trimer compound A3Cu(PO4)4 [24], cubane-based
compounds [25], or homodinuclear nickel complexes [26, 27].

In addition, antiferromagnetic spin clusters open the way towards several
applications. Among them there could be mentioned the magnetic cooling based
on the magnetocaloric effect [28, 29, 30, 31], which is especially pronounced
in frustrated quantum spin systems [32]. The magnetocaloric effect has
attracted much attention in the study of low-dimensional magnetic systems both
from theoretical [34, 35, 36, 37, 33] and experimental [38, 39] points of view
due to the fact that an enhanced magnetocaloric effect represents the basis
for economically cheaper and environmentally friendly refrigeration technology
compared to standard refrigeration technology based on vapor expansion [40].
An enhanced magnetocaloric effect was found in the vicinity of magnetization
jumps, around which the rapid change of temperature was observed upon the
variation of the magnetic field during the process of adiabatic
demagnetization. Moreover, it was shown in our recent study [41] that the
absence of zero magnetization plateau causes an enhanced magnetocaloric effect
when shutting down the external magnetic field during the process of adiabatic
demagnetization. It was found that the Ising octahedron, dodecahedron and
cuboctahedron exhibit this feature and they are promising candidates for
reaching ultra-low temperatures [17, 41]. Unfortunately, by considering the
($xy$) part of the exchange interaction, the temperature achieves only finite
values by switching-off the magnetic field during the adiabatic
demagnetization due to the presence of zero magnetization plateau in the
magnetization process of the spin-1/2 $XXZ$ Heisenberg octahedron,
dodecahedron and cuboctahedron. The Heisenberg spin clusters with the absence
of zero magnetization plateau as promising candidates for future applications
in magnetic refrigeration has been much less studied. Therefore, it appears
worthwhile to investigate the low-temperature magnetization process and
isentropes in the field-temperature plane of the spin-1/2 $XXZ$ Heisenberg
cupolae, for two of which we are able to predict the absence of zero
magnetization plateau in the zero-temperature magnetization curve. It should
be pointed out that magnetization occurs in the magnetization curve of
spin-1/2 clusters at the following fractional values

$\displaystyle\frac{m}{m_{\text{s}}}=\frac{\frac{N}{2}-n}{\frac{N}{2}}\,,$
(1.1)

where $N$ is the total number of spins of a given cluster, $N/2$ denotes the
maximum value for $z$-component of the total spin $S_{\text{T}}^{z}$, and
$n=0,1,\ldots,\lfloor N/2\rfloor$ determines deviation of the $z$-component of
the total spin $S_{\text{T}}^{z}$ from its maximum value. It is clear from
equation (1.1) that the existence of the zero magnetization plateau is
possible only if $N=2n$, i.e., if $N$ is an even number.

In the present paper we focus our attention on the magnetization process and
magnetocaloric effect of the spin-1/2 $XXZ$ Heisenberg triangular and
pentagonal cupola with an odd number of spins $N=9$ and $N=15$. In addition,
we investigate the spin-1/2 $XXZ$ Heisenberg square cupola with the total
number of spins $N=12$, for which several experimental realizations were found
in the magnetic compounds Sr(TiO)Cu4(PO4)4 [42, 43], Pb(TiO)Cu4(PO4)4 [44],
Ba(TiO)Cu4(PO4)4 [45, 46, 47, 48, 49].

The organization of this paper is as follows. The spin-1/2 $XXZ$ Heisenberg
cupolae are introduced in section 2, where the method of calculation used is
also presented. The most interesting results for the ground-state phase
diagram, magnetization process and isentropes in the field-temperature plane
are discussed in section 3. Finally, several concluding remarks are mentioned
in section 4.

##  2 Spin-1/2 $XXZ$ Heisenberg cupolae

Figure 1: (Colour online) A steric arrangement of the spin-1/2 Heisenberg
cupolae: (a) triangular cupola; (b) square cupola; (c) pentagonal cupola. The
number $N$ determines the total number of spins of a given $XXZ$ Heisenberg
cupola.

Let us examine the spin-1/2 $XXZ$ Heisenberg clusters with the shape of
Johnson’s cupolae (see figure 1), which are defined through the following
Hamiltonian

$\displaystyle\hat{\cal H}=J\sum_{\langle
i,j\rangle}\left[\Delta\left(\hat{S}_{i}^{x}\hat{S}_{j}^{x}+\hat{S}_{i}^{y}\hat{S}_{j}^{y}\right)+\hat{S}_{i}^{z}\hat{S}_{j}^{z}\right]-h\sum_{i=1}^{N}\hat{S}_{i}^{z},$
(2.1)

where $\hat{S}_{i}^{\alpha}$ denotes the spatial projections ($\alpha=x,y,z$)
of the spin-1/2 operator placed at $i$-th site of Johnson cupolae, the first
summation takes into account the exchange interaction of antiferromagnetically
coupled nearest-neighbour spins ($J>0$) and $\Delta\in\langle 0;1\rangle$
represents the anisotropy parameter in the $XXZ$ exchange interaction, i.e.,
$\Delta=0$ corresponds to the Ising model and $\Delta=1$ corresponds to the
isotropic Heisenberg model. The second term in the Hamiltonian (2.1) expresses
the Zeeman energy of magnetic moments in nonzero magnetic field $h>0$, $N$
denotes the total number of spins. Exact diagonalization data for the
isothermal magnetization curves and entropy of the spin-1/2 $XXZ$ Heisenberg
cupolae were obtained by the use of the subroutine fulldiag from the
Algorithms and Libraries for Physics Simulations (ALPS) project [50]. This
technique allows us to gain magnetization curves and magnetocaloric properties
of the spin-1/2 $XXZ$ Heisenberg cupolae.

##  3 Results and discussion

Figure 2: (Colour online) The ground-state phase diagram of the spin-$1/2$
$XXZ$ Heisenberg triangular cupola in the $\Delta{-}h/J$ plane. Each delimited
area represents lowest-energy eigenstate with the magnetization normalized
with respect to its saturation values as quoted by numbers given in circles.

Let us start our discussion with the ground-state phase diagram of the
spin-1/2 $XXZ$ Heisenberg triangular cupola, which is depicted in figure 2 in
the $\Delta{-}h/J$ plane. There are five distinct phases in the ground-state
phase diagram of the $XXZ$ Heisenberg triangular cupola for any $\Delta\neq
0$, which are delimited by displayed level-crossing fields and which differ by
the value of the total magnetization normalized with respect to its saturation
value. Contrary to this, five-ninth plateau is missing in the ground-state
phase diagram of the spin-1/2 Ising triangular cupola ($\Delta=0$).

Figure 3: The magnetization curves of a spin-1/2 $XXZ$ Heisenberg triangular
cupola for several values of the anisotropy parameter $\Delta$ and two
different temperatures: (a) $k_{\textrm{B}}T/J=0.001$; (b)
$k_{\textrm{B}}T/J=0.1$.

To better understand how the low-energy eigenstates influence the
magnetization process at finite temperatures, we depict in figure 3 the field
dependence of the isothermal magnetization for several values of the
anisotropy parameter $\Delta$ at two different temperatures
$k_{\textrm{B}}T/J=0.001$ and $0.1$. It can be noticed in figure 3 (a) that
the low-temperature magnetization curve at $k_{\textrm{B}}T/J=0.001$ strongly
resembles zero-temperature variations of the magnetizations, showing almost
abrupt magnetization jumps at the level-crossing fields. As one can see in
figure 3 (a), the low-temperature magnetization curve of spin-1/2 Ising
triangular cupola indeed exhibits, beside saturation value, three intermediate
plateaux at one-ninth, one-third and seven-ninth of the saturation
magnetization. In addition, one more five-ninth plateau can be observed in a
low-temperature magnetization curve of the spin-1/2 $XXZ$ Heisenberg
triangular cupola for any $\Delta\neq 0$, which extends over a wider range of
magnetic fields with increasing of the anisotropy parameter $\Delta$. The
magnetization curves of the spin-1/2 $XXZ$ Heisenberg cupola at the moderate
temperature $k_{\textrm{B}}T/J=0.1$ are depicted in figure 3 (b). As expected,
magnetization curves and jumps are gradually smoother upon increasing the
temperature. It can be seen in figure 3 (b) that all magnetization plateaux
except the widest ones are reflected by inflection points only.

Figure 4: (Colour online) The isentropes of a spin-1/2 $XXZ$ Heisenberg
triangular cupola in the $h/J{-}k_{\textrm{B}}T/J$ plane for three different
values of the anisotropy parameter: (a) $\Delta=0$; (b) $\Delta=0.5$; (c)
$\Delta=1$. Bottom horizontal lines correspond to the lowest temperature
$k_{\textrm{B}}T/J=0.01$ used for numerical calculations.

Next, let us also examine the isentropes of the spin-1/2 $XXZ$ Heisenberg
triangular cupola, which can be especially fascinating in proximity of the
magnetization jumps. A few isentropes are displayed in figure 4 (a) for the
spin-1/2 Ising triangular cupola in the $h/J{-}k_{\textrm{B}}T/J$ plane. The
plotted isentropes determine a magnetocaloric effect of the spin-1/2 $XXZ$
Heisenberg triangular cupola, in particular, the adiabatic temperature change
with the change of external magnetic field. The temperature of the Ising
triangular cupola sharply varies in the vicinity of all critical fields, in
which magnetization jumps occur. Owing to this fact, the Ising triangular
cupola exhibits a direct magnetocaloric effect just above the level-crossing
fields, while an inverse one is detected just below the level-crossing fields.
It should be noted that a giant magnetocaloric effect in the proximity of zero
magnetic field is potentially applicable for achieving ultra-low temperatures
quite similarly to the Ising octahedron, dodecahedron and cuboctahedron [41,
17]. Furthermore, the existence of an enhanced magnetocaloric effect, from the
technological point of view, is most valuable near the zero magnetic field,
which is most frequently used during the process of adiabatic demagnetization
as a final value of the external magnetic field.

An adiabatic demagnetization of the $XXZ$ Heisenberg triangular cupola with
relatively strong anisotropy parameter $\Delta=0.5$ is displayed in figure 4
(b), which shows a contour plot of entropy as a function of temperature and
magnetic field. A giant magnetocaloric effect can be found in the proximity of
all magnetization jumps of the spin-1/2 $XXZ$ Heisenberg triangular cupola
[see figures 4 (a) and 4 (b)]. Moreover, the absence of zero-magnetization
plateau for all values of anisotropy parameter $\Delta$, brings about the
possibility of effective cooling during the adiabatic demagnetization. This is
in contrast to the spin-1/2 $XXZ$ Heisenberg octahedron, dodecahedron and
cuboctahedron [41, 17], which exhibit a giant magnetocaloric effect when
switching-off the external magnetic field only for the Ising limiting case
$\Delta=0$. A few isentropes of the spin-1/2 isotropic Heisenberg triangular
cupola ($\Delta=1$) are plotted in figure 4 (c). As one can see in figure 4
(c), the density plot of the entropy of isotropic Heisenberg triangular cupola
is in qualitative concordance with the $XXZ$ triangular cupola at the
relatively strong anisotropy parameter $\Delta=0.5$.

Figure 5: (Colour online) The ground-state phase diagram of the spin-$1/2$
$XXZ$ Heisenberg square cupola in the $\Delta{-}h/J$ plane. Each delimited
area represents lowest-energy eigenstate with the magnetization normalized
with respect to its saturation values as quoted by numbers given in circles.

The ground-state phase diagram of the spin-1/2 $XXZ$ Heisenberg square cupola
is shown in figure 5 in the $\Delta{-}h/J$ plane. The Ising square cupola
($\Delta=0$) exhibits three intermediate plateaux at zero, one-third and two-
thirds of the saturation magnetization in zero-temperature magnetization
process. When introducing $xy$-part of the $XXZ$ exchange interaction
$\Delta>0$, two additional magnetization plateaux evolve in a wider interval
of magnetic fields. In addition, zero-temperature magnetization process of the
$XXZ$ Heisenberg square cupola exhibits one extra plateau if the anisotropy
parameter reaches the value $\Delta\approx 0.395$.

Figure 6: Magnetization curves of a spin-1/2 $XXZ$ Heisenberg square cupola
for several values of the anisotropy parameter $\Delta$ and two different
temperatures: (a) $k_{\textrm{B}}T/J=0.001$; (b) $k_{\textrm{B}}T/J=0.1$.

Figure 7: (Colour online) Isentropes of a spin-1/2 $XXZ$ Heisenberg square
cupola in the $h/J{-}k_{\textrm{B}}T/J$ plane for three different values of
the anisotropy parameter: (a) $\Delta=0$; (b) $\Delta=0.5$; (c) $\Delta=1$.
Bottom horizontal lines correspond to the lowest temperature
$k_{\textrm{B}}T/J=0.01$ used for numerical calculations.

To confirm this statement, we have depicted in figure 6 the isothermal
magnetization curves of a spin-1/2 $XXZ$ Heisenberg square cupola for several
values of the anisotropy parameter $\Delta$ and two different temperatures. As
one can see in figure 6 (a), the low-temperature magnetization curve indeed
exhibits three magnetization plateaux if $\Delta=0$ and totally five (six)
intermediate plateaux if $0<\Delta<0.395$ ($\Delta>0.395$). The influence of
higher temperature on the magnetization process of the spin-1/2 $XXZ$
Heisenberg square cupola is shown in figure 6 (b). At a moderate temperature
$k_{\textrm{B}}T/J=0.1$, only those intermediate plateaux are still visible
which were found in low-temperature magnetization curve in a relatively wide
interval of magnetic fields. Furthermore, it is clear from figure 7 that the
magnetocaloric response of the spin-1/2 $XXZ$ Heisenberg square cupola depends
on the choice of the exchange anisotropy $\Delta$. If $\Delta=0$ is set, the
spin-1/2 Ising square cupola exhibits a steep change of temperature around
three different magnetic fields. Contrary to this, the spin-1/2 $XXZ$
Heisenberg square cupola shows an enhanced magnetocaloric effect in the
proximity of five (six) level-crossing fields for $\Delta=0.5$ ($\Delta=1$).

Figure 8: (Colour online) The ground-state phase diagram of the spin-$1/2$
$XXZ$ Heisenberg pentagonal cupola in the $\Delta{-}h/J$ plane. Each delimited
area represents lowest-energy eigenstate with the magnetization normalized
with respect to its saturation values as quoted by numbers given in circles.

Figure 9: The magnetization curves of a spin-1/2 $XXZ$ Heisenberg pentagonal
cupola for several values of the anisotropy parameter $\Delta$ and two
different temperatures: (a) $k_{\textrm{B}}T/J=0.001$; (b)
$k_{\textrm{B}}T/J=0.1$.

Last but not least, the ground-state phase diagram of the spin-1/2 $XXZ$
Heisenberg pentagonal cupola is depicted in figure 8 in the $\Delta{-}h/J$
plane. It can be seen in figure 8 that the zero-temperature magnetization
curve of the spin-1/2 Ising pentagonal cupola ($\Delta=0$) should exhibit
three magnetization jumps at level-crossing fields $h/J=0.5$, 1.5 and $2.0$,
which determine rise and fall of one-fifteenth, one-third and eleven-
fifteenths magnetization plateau. On the other hand, the spin-1/2 $XXZ$
Heisenberg pentagonal cupola with $\Delta\neq 0$ should additionally display
four further lowest-energy eigenstates, which correspond to the one-fifth,
seven-fifteenths, three-fifths and thirteen-fifteenths magnetization plateaux
in the zero-temperature magnetization curve.

The above mentioned findings can be corroborated by the isothermal
magnetization curves, which are depicted in figure 9 (a) for a spin-1/2 $XXZ$
Heisenberg pentagonal cupola at low temperature $k_{\textrm{B}}T/J=0.001$. As
one can see in figure 9 (a), the low-temperature magnetization process of the
spin-1/2 Ising pentagonal cupola exhibits three intermediate magnetization
plateaux if $\Delta=0$, while the spin-1/2 Heisenberg pentagonal cupola shows
seven intermediate magnetization plateaux whenever $\Delta\neq 0$. It can be
found from figure 9 (b) that the moderate temperature $k_{\textrm{B}}T/J=0.1$
is sufficient to suppress all intermediate plateaux except the widest one-
fifth and one-third plateau at relatively weak and relatively strong values of
the exchange anisotropy $\Delta$, respectively.

For completeness, a few isentropes of the spin-1/2 $XXZ$ Heisenberg pentagonal
cupola are shown in figure 10 for three different values of the parameter
anisotropy $\Delta$, which demonstrate a magnetocaloric response reached upon
the adiabatic change of the magnetic field. Since the zero magnetization
plateau is absent in magnetization curves of the spin-1/2 $XXZ$ Heisenberg
pentagonal cupola [see figure 9 (a)], the temperature decreases to zero value
when approaching zero magnetic field. It means that a magnetic compound with
the magnetic structure of the spin-1/2 $XXZ$ Heisenberg pentagonal cupola
should have a potential to be applied for reaching ultra-low temperatures
quite similarly to the experimental realization of the spin-1/2 $XXZ$
Heisenberg triangular cupola. Moreover, if $\Delta=0$, the spin-1/2 Ising
pentagonal cupola shows an enhanced magnetocaloric effect if the magnetic
field is close to three values, at which the magnetization jumps in zero-
temperature magnetization curve occur ($h/J=0.5,1.5$ and $2.0$). In addition,
the spin-1/2 Heisenberg pentagonal cupola exhibits a steep change of
temperature around eight values of magnetic field including zero magnetic
field if $\Delta=0.5$ and $\Delta=1$.

Figure 10: (Colour online) The isentropes of a spin-1/2 $XXZ$ Heisenberg
pentagonal cupola in the $h/J{-}k_{\textrm{B}}T/J$ plane for three different
values of the anisotropy parameter: (a) $\Delta=0$; (b) $\Delta=0.5$; (c)
$\Delta=1$. Bottom horizontal lines correspond to the lowest temperature
$k_{\textrm{B}}T/J=0.01$ used for numerical calculations.

##  4 Conclusion

The present article deals with the ground-state phase diagram, magnetization
curves and isentropes of the spin-1/2 $XXZ$ Heisenberg cupolae (triangular,
square, pentagonal), which were studied using the exact numerical
diagonalization. It is shown that the studied spin-1/2 $XXZ$ Heisenberg
cupolae exhibit three intermediate magnetization plateaux in the limiting
Ising case. Moreover, it was demonstrated that quantum ($xy$) part of the
$XXZ$ exchange interaction induces the appearance of additional intermediate
magnetization plateaux, which are absent in the Ising limit. While most of the
novel magnetization plateaux are present at an arbitrary non-zero anisotropy
parameter $\Delta>0$, there may be an exception as evidenced by a one-half
plateau of a spin-1/2 $XXZ$ Heisenberg square cupola.

In the present work, the magnetocaloric effect of the spin-1/2 $XXZ$
Heisenberg cupolae was also studied. A giant magnetocaloric effect was
detected in proximity of all magnetization jumps. There is a difference
between an even-numbered (square) cupola and odd-numbered (triangular and
pentagonal) cupolae when considering the magnetization process. Due to the
adiabatic demagnetization, the former one finally shows a small temperature
rise when switching-off the magnetic field, while for the latter ones it is
theoretically possible to reach the absolute zero temperature due to the
absence of zero magnetization plateau. Thus, it becomes evident that the
absence of zero magnetization plateau in magnetic clusters is a necessary
condition to achieve a possible cooling down to the absolute zero temperature.
The odd-numbered (triangular and pentagonal) cupolae with half-integer spins
fit well for this description. Hence, the experimental realization of the
antiferromagnetic finite-size magnetic spin-1/2 clusters with odd total number
of spins makes them promising candidates for future application in magnetic
refrigeration.

## Acknowledgements

The author would like to thank to T. Verkholyak and O. Krupnitska for their
valuable remarks on the first version of the manuscript. This work was
financially supported by the grants of The Ministry of Education, Science,
Research and Sport of the Slovak Republic under the contract Nos. VEGA
1/0531/19 and the Slovak Research and Development Agency provided under
contract No. APVV-18-0197.

## References

  * [1] Friedman J.R., Sarachik M.P., Annu. Rev. Condens. Matter Phys., 2010, 1, 109–128,
doi:10.1146/annurev-conmatphys-070909-104053.

  * [2] Hotynska M., Single-Molecule Magnets, John Wiley and Sons, Hoboken, NJ, USA, 2018.
  * [3] Shapira Y., Bindilatti V., J. Appl. Phys., 2002, 92, 4155–4185, doi:10.1063/1.1507808.
  * [4] Schulenburg J., Honecker A., Schnack J., Richter J., Schmidt H.-J., Phys. Rev. Lett., 2002, 88, 167207, doi:10.1103/PhysRevLett.88.167207.
  * [5] Lacroix C., Mendels P., Mila F., Introduction to Frustrated Magnetism, Springer-Verlag, Berlin, Heidelberg, 2011.
  * [6] Honecker A., Schulenburg J., Richter J., J. Phys.: Condens. Matter, 2004, 16, S749–S758,
doi:10.1088/0953-8984/16/11/025.

  * [7] Bellucci S., Ohanyan V., Rojas O., EPL, 2014, 105, 47012, doi:10.1209/0295-5075/105/47012.
  * [8] Ohanyan V., Rojas O., Strečka J., Bellucci S., Phys. Rev. B, 2015, 92, 214423, doi:10.1103/PhysRevB.92.214423.
  * [9] Strečka J., Karl’ová K., Madaras T., Physica B, 2015, 466-467, 76–85, doi:10.1016/j.physb.2015.03.031.
  * [10] Žukovič M., Semjan M., J. Magn. Magn. Mater., 2018, 451, 311–318, doi:10.1016/j.jmmm.2017.11.076.
  * [11] Mohylna M., Žukovič M., Phys. Lett. A, 2019, 383, 2525–2534, doi:10.1016/j.physleta.2019.05.015.
  * [12] Haraldsen J.T., Barnes T., Musfeldt J.L., Phys. Rev. B, 2005, 71, 064403, doi:10.1103/PhysRevB.71.064403.
  * [13] Bose I., Tribedi A., Phys. Rev. A, 2005, 72, 022314, doi:10.1103/PhysRevA.72.022314.
  * [14] Schnalle R., Schnack J., Phys. Rev. B, 2009, 79, 104419, doi:10.1103/PhysRevB.79.104419.
  * [15] Coffey D., Trugman S.A., Phys. Rev. B, 1992, 46, 12717–12722, doi:10.1103/PhysRevB.46.12717.
  * [16] Rousochatzakis I., Läuchli A.M., Mila F., Phys. Rev. B, 2008, 77, 094420, doi:10.1103/PhysRevB.77.094420.
  * [17] Karl’ová K., Strečka J., J. Low Temp. Phys., 2017, 187, 727–733, doi:10.1007/s10909-016-1676-8.
  * [18] Arian Zad H., Sabeti M., Zoshki A., Ananikian N., J. Phys.: Condens. Matter, 2019, 31, 425801, doi:10.1088/1361-648X/ab2854.
  * [19] Konstantinidis N.P., Phys. Rev. B, 2005, 72, 064453, doi:10.1103/PhysRevB.72.064453.
  * [20] Konstantinidis N.P., Phys. Rev. B, 2009, 80, 134427, doi:10.1103/PhysRevB.80.134427.
  * [21] Strečka J., Čisárová J., Acta Phys. Pol. A, 2014, 126, 26, doi:10.12693/APhysPolA.126.26.
  * [22] Szałowski K., Kowalewska P., Materials, 2020, 13, 485, doi:10.3390/ma13020485.
  * [23] Arian Zad H., Kenna R., Ananikian N., Physica A, 2020, 538, 122841, doi:10.1016/j.physa.2019.122841.
  * [24] Belik A., Matsuo A., Azuma M., Kindo K., Takano M., J. Solid State Chem., 2005, 178, 709–714, doi:10.1016/j.jssc.2004.12.020.
  * [25] Aronica C., Chastanet G., Pilet G., Le Guennic B., Robert V., Wernsdorfer W., Luneau D., Inorg. Chem., 2007, 46, 6108–6119, doi:10.1021/ic700645q.
  * [26] Strečka J., Jaščur M., Hagiwara M., Narumi Y., Kuchár J., Kimura S., Kindo K., J. Phys. Chem. Solids, 2005, 66, 1828–1837, doi:10.1016/j.jpcs.2005.09.089.
  * [27] Strečka J., Hagiwara M., Baláž P., Jaščur M., Narumi Y., Kimura S., Kuchár J., Kindo K., Physica B, 2008, 403, 3146–3153, doi:10.1016/j.physb.2008.03.025.
  * [28] Evangelisti M., Luis F., de Jongh L.J., Affronte M., J. Mater. Chem., 2006, 16, 2534–2549,
doi:10.1039/B603738K.

  * [29] Sessoli R., Angew. Chem. Int. Ed., 2012, 51, 43–45, doi:10.1002/anie.201104448.
  * [30] Evangelisti M., Molecular Magnets, Springer-Verlag, Berlin, Heidelberg, 2014.
  * [31] Zheng Y.Z., Zhou G.J., Zheng Z., Winpenny R.E.P., Chem. Soc. Rev., 2014, 43, 1462–1475,
doi:10.1039/C3CS60337G.

  * [32] Zhitomirsky M.E., Phys. Rev. B, 2003, 67, 104421, doi:10.1103/PhysRevB.67.104421.
  * [33] Schnack J., Heesing C., Eur. Phys. J. B, 2013, 86, 46, doi:10.1140/epjb/e2012-30546-7.
  * [34] Topilko M., Krokhmalskii T., Derzhko O., Ohanyan V., Eur. Phys. J. B, 2012, 85, 278,
doi:10.1140/epjb/e2012-30359-8.

  * [35] Derzhko O., Richter J., Eur. Phys. J. B, 2006, 52, 23–36, doi:10.1140/epjb/e2006-00273-y.
  * [36] Richter J., Derzhko O., Krokhmalskii T., Phys. Rev. B, 2006, 74, 144430, doi:10.1103/PhysRevB.74.144430.
  * [37] Derzhko O., Richter J., Phys. Rev. B, 2004, 70, 104415, doi:10.1103/PhysRevB.70.104415.
  * [38] Hooper T.N., Inglis R., Lorusso G., Ujma J., Barran P.E., Uhrin D., Schnack J., Piligkos S., Evangelisti M., Brechin E.K., Inorg. Chem., 2016, 55, 10535–10546, doi:10.1021/acs.inorgchem.6b01730.
  * [39] Hooper T.N., Schnack J., Piligkos S., Evangelisti M., Brechin E.K., Angew. Chem. Int. Ed., 2012, 51, 4633–4636, doi:10.1002/anie.201200072.
  * [40] Tishin A.M., Spichkin Y.I., The Magnetocaloric Effect and its Applications, IOP Publishing, Bristol, 2003.
  * [41] Karľová K., Strečka J., Richter J., J. Phys.: Condens. Matter, 2017, 29, 125802, doi:10.1088/1361-648X/aa53ab.
  * [42] Kato Y., Kimura K., Miyake A., Tokunaga M., Matsuo A., Kindo K., Akaki M., Hagiwara M., Kimura S., Kimura T., Motome Y., Phys. Rev. B, 2019, 99, 024415, doi:10.1103/PhysRevB.99.024415.
  * [43] Islam S.S., Ranjith K.M., Baenitz M., Skourski Y., Tsirlin A.A., Nath R., Phys. Rev. B, 2018, 97, 174432, doi:10.1103/PhysRevB.97.174432.
  * [44] Kimura K., Kato Y., Yamauchi K., Miyake A., Tokunaga M., Matsuo A., Kindo K., Akaki M., Hagiwara M., Kimura S., Toyoda M., Motome Y., Kimura T., Phys. Rev. Mater., 2018, 2, 104415,
doi:10.1103/PhysRevMaterials.2.104415.

  * [45] Babkevich P., Testa L., Kimura K., Kimura T., Tucker G.S., Roessli B., Rønnow H.M., Phys. Rev. B, 2017, 96, 214436, doi:10.1103/PhysRevB.96.214436.
  * [46] Kato Y., Kimura K., Miyake A., Tokunaga M., Matsuo A., Kindo K., Akaki M., Hagiwara M., Sera M., Kimura T., Motome Y., Phys. Rev. Lett., 2017, 118, 107601, doi:10.1103/PhysRevLett.118.107601.
  * [47] Kimura K., Sera M., Nakano T., Nozue Y., Kimura T., Physica B, 2018, 536, 93–95,
doi:10.1016/j.physb.2017.10.101.

  * [48] Rästa R., Heinmaa I., Kimura K., Kimura T., Stern R., Phys. Rev. B, 2020, 101, 054417,
doi:10.1103/PhysRevB.101.054417.

  * [49] Kimura K., Babkevich P., Sera M., Toyoda M., Yamauchi K., Tucker G.S., Martius J., Fennell T., Manuel P., Khalyavin D.D., Johnson R.D., Nakano T., Nozue Y., Rønnow H.M., Kimura T., Nat. Commun., 2016, 7, 13039, doi:10.1038/ncomms13039.
  * [50] Bauer B., Carr L.D., Evertz H.G., Feiguin A., Freire J., Fuchs S., Gamper L., Gukelberger J., Gull E., Guertler S., et al., J. Stat. Mech.: Theory Exp., 2011, 2011, P05001, doi:10.1088/1742-5468/2011/05/P05001.

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# Evaluation and Optimization of Rendering Techniques for Autonomous Driving
Simulation

Chengyi WANG  Chunji XU  Peilun WU Chengyi WANG, Chunji XU, Peilun WU are with
the Southern University of Science and Technology, Shenzhen, 518055, China

###### Abstract

In order to meet the demand for higher scene rendering quality from some
autonomous driving teams (such as those focused on CV), we have decided to use
an offline simulation industrial rendering framework instead of real-time
rendering in our autonomous driving simulator. Our plan is to generate lower-
quality scenes using a game engine, extract them, and then use an IQA
algorithm to validate the improvement in scene quality achieved through
offline rendering. The improved scenes will then be used for training.

##  I Introduction

Computer graphics rendering technology is essential for autonomous driving
simulation. Specifically, when engineers attempt to run a simulation, they
must first create a 3D model that specifies how geometries should be placed in
the virtual world [1, 2].

The image rendering technology used by modern autonomous driving simulators
has been relatively complete from the algorithm level. However, due to the
limitation of calculation time, the effect of real-time rendering is often
unsatisfactory. In order to improve the quality of model rendering used in our
autonomous driving laboratory, it is necessary and feasible to use offline
rendering instead of real-time rendering to breakthough this time limit. We
expect that offline rendering will perform better than real-time rendering.
And currently, most rendering tools are real-time rendering. To perform
offline rendering, we need to render ourselves, and after offline rendering,
we also need to evaluate the rendering effect to see if offline rendering is
really better than real-time rendering.

We will first render a scene in real-time, and then render a similar scene
offline. Then we will evaluate the rendering effects of the two scenes. We
will evaluate the rendering effect in two ways. The first method is to use
object detection algorithms to detect rendered targets and see if it can
improve the detection effect. Another method is to use the IQA algorithm to
evaluate rendered images to see if the quality of offline rendered images is
higher than that of real-time rendered images.

The result is in two parts. In detection algorithms, offline rendering is
slightly better than online ones, and in image quality assessments the
advantage is much greater.

##  II Background Research

###  II-A Building Real-time Simulation Environment

Real-time refers to the ability of a system to process and respond to data in
a timely manner, typically within a few milliseconds or less. In the context
of a real-time simulator for autonomous driving, this means that the simulator
is able to simulate the driving environment and the behavior of the autonomous
vehicle in real-time, responding to inputs and changes in the environment as
quickly as possible [3].

To highlight the advantage of our offline rendering method in producing high-
quality scenes, we will first build a real-time autonomous driving simulation
environment, extract appropriate scenes from it, render them offline, and then
compare the quality of the renders.

Engineering a simulation environment requires the same features and toolsets
used in creating other types of rich interactive content: lighting and
physics, particle and weather systems, animation, machine learning [4, 5].

A number of real-time simulators for autonomous driving have been created,
including CARLA, Airsim, Udacity Simulator, Gazebo, LGSVL Simulator, Carsim
and so on. Among them, CARLA and Airsim are based on a famous game engine
Unreal Engine.

###  II-B Precomputation-Based Rendering methods

In 2001, Basri and Jacobs showed that reflection from a curved surface, lit by
an environment map, could be seen as a spherical convolution of the incident
illumination and the reflective properties or BRDF of the surface. [6][7]

Sloan et al. introduced the term precomputed radiance transfer and led to
greatly increased interest in precomputation-based relighting methods. They
use full environment maps for lighting instead of discrete light sources. The
first innovation was to build on the spherical harmonic methods, but
precompute the shadowing and interreflection components, encapsulated in a
transfer function on the object surface. [8][7]

###  II-C IQA for Rendering

A method known as CVRKD-IQA proposed by Yin et al. [9] is especially suitable
for our underground parking lot rendering research. Knowledge distillation is
used for feature extraction from pictures and training the NAR-student agent.
Under the verification of various data sets and algorithms, the results of the
IQA algorithm have sufficient credibility.

Bosse et al. [10] implemented an FR-IQA and NR-IQA method by Deep Neural
Networks named WaDIQaM. The pre-study and training of different feature fusion
strategies make the DNN system effective on FR and NR IQA scenes.

Hossein and Peyman [11] introduced a neural image assessment algorithm trained
on both aesthetic and pixel-level quality datasets. This CNN-based NR-IQA
method is called NIMA. NIMA is a pre-trained IQA algorithm and is popular for
effectively predicting the distribution of quality ratings, rather than just
the mean scores.

###  II-D Evaluate image quality

After using the previously mentioned methods for rendering, we will use other
methods and metrics to verify the image quality.

####  II-D1 PSNR (Peak Signal-to-Noise Ratio) and MSE (Mean Square Error)

Peak Signal to Noise Ratio (PSNR) is a commonly used indicator for evaluating
image quality. It is determined by calculating the ratio between the maximum
possible pixel value in the image and the mean square error caused by noise in
the image. Usually, the higher the PSNR, the better the image quality.
However, it should be noted that PSNR does not always accurately reflect the
human eye’s perception of image quality, as the human eye has different
sensitivities to different types and intensities of distortion [12].
Therefore, when evaluating image quality, other factors need to be considered,
such as structural similarity index (SSIM) and subjective visual quality
assessment [13].

Mean error refers to the difference between each data point in a set of data
and the average value of that set of data. It can be calculated by adding up
the difference between each data point and the average value and dividing it
by the number of data points. Mean error is usually used to evaluate the
accuracy and accuracy of a set of measurement results, with smaller mean
errors indicating that the measurement results are closer to the true values
[13].

The MSE formula is as follows:

$MSE=\sum_{i=1}^{M}\sum_{j=1}^{N}|R(i,j)-F(i,j)|^{2}$ (1)

Where R is the reference image, F is the image to be evaluated. And the PSNR
formula is as follows:

$PSNR=20log_{10}MAX_{p}-10log_{10}MSE$ (2)

Where $MAX_{p}$ is the maximum possible pixel value in an image (usually 255).
This algorithm has the advantage of convenient computation, but its evaluation
results are significantly different from the intuitive perception of the human
eye, as the human visual system is more sensitive to brightness information
than chromaticity information.

####  II-D2 SSIM(Structural Similarity Index)

SSIM is an abbreviation for Structural Similarity Index, which is an objective
evaluation indicator used to measure image quality. It can compare the
structural similarity and differences in brightness, contrast, structure, and
other aspects between two images to evaluate their similarity. The higher the
SSIM index, the more similar the two images are. The SSIM index is widely used
in the field of image processing, such as image compression, denoising, and
enhancement.[14]

The SSIM formula is as follows:

$SSIM=[l(x,y)c(x,y)s(x,y)]$ (3)

where x and y represent two images, while l(x,y), c(x,y), and s(x,y) represent
brightness similarity, contrast similarity, and structural similarity,
respectively. Specifically, their calculation formula is as follows:

$l(x,y)=\frac{2\mu_{x}\mu_{y}+C_{1}}{\mu_{x}^{2}+\mu_{y}^{2}+C_{1}}$ (4)

$c(x,y)=\frac{2\sigma_{x}y+C_{2}}{\sigma_{x}^{2}+\sigma_{y}^{2}+C_{2}}$ (5)

$s(x,y)=\frac{\sigma_{xy}+C_{3}}{\sigma_{x}\sigma_{y}+C_{3}}$ (6)

Where $\mu_{x}$, $\mu_{y}$ represents the average value of image x and image
y, respectively, $\sigma_{x}$, $\sigma_{y}$ represents their standard
deviations, respectively, $\sigma_{xy}$ represents their covariance. $C_{1}$,
$C_{2}$, and $C_{3}$ are constants used to avoid instability when the
denominator is 0 or close to 0.

Although the results of the SSIM algorithm are more in line with the human
eye’s perception, it requires that the two images be of the same size and
undergo graying during evaluation. In addition, due to the highly nonlinear
nature of the human visual system, there are differences between the
evaluation results and the actual visual perception.

####  II-D3 NIQE (Natural Image Quality Evaluator)

The natural image quality evaluator evaluates the quality of an image by
extracting quality perception features from each image block and fitting them
into a multivariate Gaussian model. Finally, the distance between the original
image and the multivariate Gaussian model of the test image is used as the
quality of the test image. This framework has been proposed with some methods
that mainly focus on natural scene statistics (NSS) feature extraction. In the
Natural Image Quality Evaluator (NIQE), Mittal et al. used mean subtraction
and contrast normalization (MSCN) fitting parameters as quality perception
features. These features are also fundamental features of subsequent work, as
they are very effective.[15]

The NIQE formula is as follows:

$Q(T)=\frac{1}{N}\sum(\frac{||x-\mu||^{2}}{\sigma^{2}})$ (7)

Where N is the total number of blocks in the test image, $\mu$ and
$\sigma^{2}$ are the mean and variance of the multivariate Gaussian model.
This formula calculates the average Euclidean distance between the test image
and the multivariate Gaussian model, which is used to evaluate the quality of
the test image.

The NIQE algorithm can perform well in super-resolution reconstruction tasks
where other algorithms cannot perform well, but it also requires more complex
calculations and longer computational time.

###  II-E Object Detection Algorithms

Object detection algorithms are computer vision techniques that enable
machines to identify and locate objects within an image or video [16, 17].
These algorithms use various techniques, including machine learning, deep
learning, and computer vision, to detect objects within an image or video and
draw bounding boxes around them [18].

There are generally two main types of object detection algorithms: two-stage
detectors and one-stage detectors [19]. Two-stage detectors follow a two-step
process. The first stage generates a set of potential object proposals, which
are regions in the image that may contain objects. These proposals are then
refined and classified in the second stage. One popular example of a two-stage
detector is the Faster R-CNN (Region-based Convolutional Neural Network)
algorithm, which uses a region proposal network (RPN) to generate proposals
and then classifies and refines them using subsequent stages. One-stage
detectors perform object detection in a single pass without the explicit
proposal generation step. These algorithms directly predict the bounding boxes
and class labels for objects in an image [20, 21]. Single Shot MultiBox
Detector (SSD) and You Only Look Once (YOLO) are examples of popular one-stage
detectors. SSD divides the image into a grid and predicts multiple bounding
boxes and class probabilities at each grid location, while YOLO divides the
image into a grid and predicts bounding boxes and class probabilities
directly.

Compared to two-stage detectors, one-stage detectors(especially YOLO) are more
efficient and provide better performance in detecting small objects. These
advantages make them more suitable for application in the field of autonomous
driving, and that is the reason why we choose YOLOv3, YOLOv8, and Detectron2
to evaluate our results.

##  III Research methods

###  III-A Build scene from ground

For our experimental purposes, we need to use a rendering model based entirely
on offline industrial renders. In order to ensure the same work as the
recognition algorithm, we select components with the same functions as Carla
Town to build the scene.

For reasons of realism, aesthetics or consistency with the scene, some
parameters can be adjusted before using these models. We combine the models to
get the required scene, and then select a suitable perspective in the scene to
get a photo for the algorithm to identify in both online and offline engines.

###  III-B Rendering engine and optimize

To optimize the rendering result, we use an offline Cycles engine to render
images of possible autonomous driving scenes.

Cycles is a ray tracing engine in which many post-processing functions are
built in. The main feature of ray tracing engines is that the ray tracing
engine determines the position of the object first, while the rasterization
engines determine the position of the sampling point first. Above that, Cycles
implemented global illumination calculation, which resulted in more realistic
rendered images [22]. Cycles can run on CPUs, while GPU is used to accelerate
the rendering process.

Figure 1: Images of similar scenes in both online(above) and offline(below)
engines.

###  III-C Evaluation

We will use several methods of image quality evaluation mentioned earlier to
compare the quality of images generated by offline rendering with those
generated by online rendering. We will analyze in which aspects the quality of
offline-rendered images is better than that of online-rendered images, and
also note in which aspects the quality of images has not significantly
improved. We hope to improve our offline rendering through this analysis
process.

To quantitatively and visually evaluate the effectiveness of our optimization
method [23], we have decided to design the following experiment:

We will perform real-time rendering and offline rendering on a set of
identical 3D models. And then we capture scene images with the camera
positioned at the same location after real-time rendering and offline
rendering. We will apply the object detection algorithm to detect objects
(such as pedestrians and vehicles) relevant to driving in both data sets.
Finally, we will compare the recognition performance (confidence scores) of
these objects in the two sets of images.

Specifically, we set two locations for the main camera to provide the close-up
perspective and long-distance perspective of the objects.

Figure 2: close-up perspective Figure 3: long-distance perspective and
offline(below) engines.

The confidence score on the objects using Yolov8n and Detectron2 are shown in
Table I and Table II respectively.

YOLO | Close-up perspective | Long-distance perspective
---|---|---
real-time rendering |  Person:71% Car:82% |  Person: $<$ 25% Car: $<$ 25%
Offline-rendering |  Person: 88.5% Car:72% |  Person:42% Car:34%
TABLE I: YOLO results. Detectron2 | Close-up perspective | Long-distance perspective
---|---|---
real-time rendering |  Person:100% Car:99% |  Person:68% Car:90%
Offline-rendering |  Person: 100% Car:98% |  Person:99% Car:83%
TABLE II: Detectron results.

While performing better under the object detector does not necessarily
guarantee the effectiveness of our optimization method at all times, it
remains an important metric indicating that our optimization results exhibit
superior computer vision capabilities for autonomous driving compared to the
original data.

##  IV Results

###  IV-A IQA algorithm

The calculation results of peak signal-to-noise ratio are shown in Table III.

| set 1 | set 2
---|---|---
PSNR | 14.015 | 15.425
TABLE III: The results in PSNR.

In the results of real-time and offline rendering in the near and far groups,
the PSNR value of the near group is 14.015, and the PSNR value of the far
group is 15.425. This result indicates that the images of the two rendering
methods can be identified as similar in terms of visual recognition.

The calculation results of Structural Similarity are shown in Table IV.

| set 1 | set 2
---|---|---
SSIM | 0.5126 | 0.5375
TABLE IV: The results in SSIM.

In the real-time and offline rendering results of the near and far groups, the
SSIM values of the near group are 0.5126, and the SSIM values of the far group
are 0.5375. This result indicates that the images of the two rendering methods
can be considered similar in terms of structural similarity, including
brightness, contrast, and structure.

The calculation results of Natural Image Quality Evaluator are shown in Table
V.

NIQE | set 1 | set 2
---|---|---
RealTime | 38.463 | 40.344
Offline | 25.129 | 30.467
TABLE V: The results in NIQE.

In the results of real-time and offline rendering in the near and far groups,
the NIQE value for the near group is 38.463, while the NIQE value for offline
rendering is 25.129. The NIQE value for the far group is 40.344, while the
NIQE value for offline rendering is 30.467. This result indicates that through
the evaluation method of NIQE, the quality of offline rendered images is
higher than that of real-time rendered images.

##  V Conclusion

Our research system has been successfully constructed. The research methods
should be able to effectively prove the effectiveness of improvement from
offline rendering for the optimization of automatic driving model training.
However, we still lack a more rigorous theoretical basis, general experiment
and a definite conclusion. As a result, the next task of our research group is
to further expand in these three directions.

Judging from the results, our current research cannot guarantee that our
expectations must be correct. But if we finally succeed in falsification, it
can also play a guiding role in the improvement of automatic driving model
training.

In the future, we will extend this work with the AI technologies, such as
Knowledge graphs [24, 25], and optimization algorithms [5, 26].

## References

  * [1] G. Lan, Z. Luo, and Q. Hao, “Development of a virtual reality teleconference system using distributed depth sensors,” in _2016 2nd IEEE International Conference on Computer and Communications (ICCC)_. IEEE, 2016, pp. 975–978.
  * [2] G. Lan, J. Sun, C. Li, Z. Ou, Z. Luo, J. Liang, and Q. Hao, “Development of uav based virtual reality systems,” in _2016 IEEE International Conference on Multisensor Fusion and Integration for Intelligent Systems (MFI)_. IEEE, 2016, pp. 481–486.
  * [3] H. Xu, G. Lan, S. Wu, and Q. Hao, “Online intelligent calibration of cameras and lidars for autonomous driving systems,” in _2019 IEEE Intelligent Transportation Systems Conference (ITSC)_. IEEE, 2019, pp. 3913–3920.
  * [4] Unity, “Efficient development of simulated environments for autonomous vehicle training,” 2020.
  * [5] G. Lan, M. van Hooft, M. De Carlo, J. M. Tomczak, and A. E. Eiben, “Learning locomotion skills in evolvable robots,” _Neurocomputing_ , vol. 452, pp. 294–306, 2021.
  * [6] R. Basri and D. Jacobs, “Lambertian reflectance and linear subspaces,” _IEEE Transactions on Pattern Analysis and Machine Intelligence_ , 2003.
  * [7] R. Ramamoorthi, “Precomputation-based rendering,” _Foundations and Trends in Computer Graphics and Vision_ , 2007.
  * [8] P.-P. Sloan, J. Kautz, and J. Snyder, “Precomputed radiance transfer for real-time rendering in dynamic, low-frequency lighting environments,” _ACM Transactions on Graphics_ , 2002.
  * [9] G. Y. et al., “Content-variant reference image quality assessment via knowledge distillation,” _The Thirty-Sixth AAAI Conference on Artificial Intelligence_ , no. 9, 2022.
  * [10] S. B. et al., “Deep neural networks for no-reference and full-reference image quality assessment,” _IEEE Transactions on Image Processing_ , 2018.
  * [11] H. Talebi and P. Milanfar, “Nima: Neural image assessment,” _IEEE Transactions on Image Processing_ , 2018.
  * [12] G. Lan, Y. Bu, J. Liang, and Q. Hao, “Action synchronization between human and uav robotic arms for remote operation,” in _2016 IEEE International Conference on Mechatronics and Automation_. IEEE, 2016, pp. 2477–2481.
  * [13] D. Hoult and R. Richards, “The signal-to-noise ratio of the nuclear magnetic resonance experiment,” _Journal of Magnetic Resonance (1969)_ , vol. 24, no. 1, pp. 71–85, 1976.
  * [14] D. Brunet, E. R. Vrscay, and Z. Wang, “On the mathematical properties of the structural similarity index,” _IEEE Transactions on Image Processing_ , vol. 21, no. 4, pp. 1488–1499, 2012.
  * [15] L. Wu, X. Zhang, H. Chen, D. Wang, and J. Deng, “Vp-niqe: An opinion-unaware visual perception natural image quality evaluator,” _Neurocomputing_ , vol. 463, pp. 17–28, 2021.
  * [16] G. Lan, J. Benito-Picazo, D. M. Roijers, E. Domínguez, and A. Eiben, “Real-time robot vision on low-performance computing hardware,” in _2018 15th international conference on control, automation, robotics and vision (ICARCV)_. IEEE, 2018, pp. 1959–1965.
  * [17] T. Xiang, F. Jiang, G. Lan, J. Sun, G. Liu, Q. Hao, and C. Wang, “Uav based target tracking and recognition,” in _2016 IEEE international conference on multisensor fusion and integration for intelligent systems (MFI)_. IEEE, 2016, pp. 400–405.
  * [18] G. Lan, L. De Vries, and S. Wang, “Evolving efficient deep neural networks for real-time object recognition,” in _2019 IEEE Symposium Series on Computational Intelligence (SSCI)_. IEEE, 2019, pp. 2571–2578.
  * [19] G. Lan, Y. Wu, F. Hu, and Q. Hao, “Vision-based human pose estimation via deep learning: A survey,” _IEEE Transactions on Human-Machine Systems_ , 2022\.
  * [20] G. Lan, Z. Gao, L. Tong, and T. Liu, “Class binarization to neuroevolution for multiclass classification,” _Neural Computing and Applications_ , vol. 34, no. 22, pp. 19 845–19 862, 2022.
  * [21] Z. Gao and G. Lan, “A neat-based multiclass classification method with class binarization,” in _Proceedings of the genetic and evolutionary computation conference companion_ , 2021, pp. 277–278.
  * [22] G. Lan, J. Sun, C. Li, Z. Ou, Z. Luo, J. Liang, and Q. Hao, “Development of uav based virtual reality systems,” in _2016 IEEE International Conference on Multisensor Fusion and Integration for Intelligent Systems (MFI)_. IEEE, 2016, pp. 481–486.
  * [23] G. Lan, J. M. Tomczak, D. M. Roijers, and A. Eiben, “Time efficiency in optimization with a bayesian-evolutionary algorithm,” _Swarm and Evolutionary Computation_ , vol. 69, p. 100970, 2022.
  * [24] T. Liu, G. Lan, K. A. Feenstra, Z. Huang, and J. Heringa, “Towards a knowledge graph for pre-/probiotics and microbiota–gut–brain axis diseases,” _Scientific Reports_ , vol. 12, no. 1, p. 18977, 2022.
  * [25] G. Lan, T. Liu, X. Wang, X. Pan, and Z. Huang, “A semantic web technology index,” _Scientific reports_ , vol. 12, no. 1, p. 3672, 2022.
  * [26] G. Lan, M. De Carlo, F. van Diggelen, J. M. Tomczak, D. M. Roijers, and A. E. Eiben, “Learning directed locomotion in modular robots with evolvable morphologies,” _Applied Soft Computing_ , vol. 111, p. 107688, 2021.