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# Semi-Dirac and Weyl Fermions in Transition Metal Oxides

Narayan Mohanta  Materials Science and Technology Division, Oak Ridge National
Laboratory, Oak Ridge, TN 37831, USA  Jong Mok Ok  Materials Science and
Technology Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA
Jie Zhang  Materials Science and Technology Division, Oak Ridge National
Laboratory, Oak Ridge, TN 37831, USA  Hu Miao  Materials Science and
Technology Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA
Elbio Dagotto  Materials Science and Technology Division, Oak Ridge National
Laboratory, Oak Ridge, TN 37831, USA  Department of Physics and Astronomy, The
University of Tennessee, Knoxville, TN 37996, USA  Ho Nyung Lee  Materials
Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, TN
37831, USA  Satoshi Okamoto  Materials Science and Technology Division, Oak
Ridge National Laboratory, Oak Ridge, TN 37831, USA

###### Abstract

We show that a class of compounds with I4/mcm crystalline symmetry hosts
three-dimensional semi-Dirac fermions. Unlike the known two-dimensional semi-
Dirac points, the degeneracy of these three-dimensional semi-Dirac points is
not lifted by spin-orbit coupling due to the protection by a nonsymmorphic
symmetry — screw rotation in the a-b plane and a translation along the c axis.
This crystalline symmetry is found in tetragonal perovskite oxides, realizable
in thin films by epitaxial strain that results in a0a0c--type octahedral
rotation. Interestingly, with broken time-reversal symmetry, two pairs of Weyl
points emerge from the semi-Dirac points within the Brillouin zone, and an
additional lattice distortion leads to enhanced intrinsic anomalous Hall
effect. We discuss possible fingerprints of this symmetry-protected band
topology in electronic transport experiments.

Protected band degeneracy continues being a fundamental topic of interest to
understand the physics of various topological semimetals, e.g. Dirac, Weyl,
and nodal loop semimetals Herring (1937); Burkov  _et al._ (2011); Burkov
(2016); Bzdušek ; Armitage  _et al._ (2018); Burkov (2018). The guiding
principle to classify these topological semimetals has been the presence of
one or more symmetries that allow degenerate eigenstates Chiu  _et al._
(2016); Yang  _et al._ (2018). Spin-orbit coupling (SOC) lifts the band
degeneracies everywhere except at momenta where symmetry enforces that the
matrix elements of the SOC operator vanish Allen and Pickett (2018). Since SOC
is present in all materials, the implication is that unless there is a special
symmetry that protects the band degeneracy, the topological nodal semimetals
exist in the theoretically-idealized case when SOC is zero.

Semi-Dirac fermions in two-dimensional solids connect the physics of monolayer
graphene that exhibits linear band dispersion near the Dirac points and
bilayer graphene that has quadratic band dispersion near the band-touching
points Castro Neto  _et al._ (2009); McCann and Koshino (2013). The semi-Dirac
points, in which both linear and quadratic band dispersions occur along
different momentum directions, exist in the absence of SOC, in various systems
including hexagonal lattices under a magnetic field Dietl  _et al._ (2008),
VO2-TiO2 heterostructures Pardo and Pickett (2009); Banerjee  _et al._ (2009),
BEDT-TTF2I3 salt under pressure Katayama  _et al._ (2006), silicene oxide
Zhong  _et al._ (2017), photonic crystals Wu (2014), and Hofstadter spectrum
Delplace and Montambaux (2010). The quadratic dispersion along one of the
momentum directions leads to an enhanced density of states at low energies as
compared to linear Dirac dispersion, making instabilities relatively easier to
occur due to the enlarged phase space for quantum fluctuations Uryszek  _et
al._ (2019). A semi-Dirac semimetal becomes a Chern insulator, revealing
quantized anomalous Hall conductivity with broken time-reversal symmetry
(TRS), in the presence of SOC that opens an energy gap at the semi-Dirac point
Huang  _et al._ (2015). A similar Chern insulating state is also induced in a
semi-Dirac semimetal with two gapless Dirac nodes by impinging a circularly
polarized light Saha (2016). In multilayer (TiO2)m$/$(VO2)n heterostructures,
the semi-Dirac point is not destroyed by SOC in a special situation when the
spins are aligned along the rutile $c$ axis Pardo and Pickett (2010).

However, the questions that remain open are whether there exists a natural
material in which the semi-Dirac point is gapless generically when SOC is
present and whether these semi-Dirac points exist in a higher dimension. Here,
we show that a class of materials hosts three-dimensional (3D) gapless semi-
Dirac fermions in the presence of SOC. Tetragonal perovskite oxides with
I4/mcm crystalline symmetry (space group 140) exhibit such 3D semi-Dirac
points at a high-symmetry point, and its nonsymmorphic symmetry protects the
semi-Dirac points against the gap-opening by SOC. Nonsymmorphic symmetry-
protected two-dimensional Dirac semimetallic phase was proposed in a
theoretical model Young and Kane (2015), and subsequent studies using ab
initio calculations in layered compounds Li  _et al._ (2018); Shao  _et al._
(2018). Cubic perovskite oxide thin films can be transformed into the
tetragonal phase to realize the I4/mcm space group under epitaxial strain that
produces a0a0c--type (in the Glazer notation) octahedral rotation Rondinelli
and Spaldin (2010); Ok  _et al._ . The out-of-phase octahedral rotation brings
in the nonsymmorphic symmetry: screw rotation $2\pi/\theta$ ($\theta$ is the
rotation angle), translation along [001] $z\rightarrow z+1/2$, as shown in
Fig. 1(a). A consequence of the fractional translational symmetry is that the
unit cell of the original symmorphic crystal is doubled. In this case, the
folded back bands stick together necessarily at the Brillouin-zone boundary if
the nonsymmorphic symmetry is present, even with SOC.

Figure 1: (a) Illustration of the octahedral rotation-driven tetragonal phase
of perovskite oxides. a0a0c--type rotation in a cubic crystal produces I4/mcm
space group that exhibits the nonsymmorphic symmetry (screw rotation in the
$a$-$b$ plane and translation along the $c$ axis). (b) Dispersion of two
electronic bands in the $\rm P$-$\rm Q$-$\rm N$ plane. (c)-(e) show the
dispersion curves along different momentum paths. The inset in the middle
shows the high-symmetry points in the Brillouin zone.

As a particular physical realization of the I4/mcm space group, we primarily
focus on tetragonal SrNbO3 that manifestly displays the defining
characteristics of this nonsymmorphic symmetry, and a 3D semi-Dirac point
appears close to the Fermi energy at a high-symmetry point P at the Brillouin-
zone boundary. Three different kinds of dispersion emerge from this semi-Dirac
point—linear along the P-X direction, quadratic along the P-Q direction, and
linear in the close vicinity of the P point along along the P-N direction, as
shown in Fig. 1(b)-(e). With a TRS-breaking element such as an external
magnetic field, we show that two pairs of Weyl points appear within the
Brillouin zone. The opposite zone boundaries share one pair of Weyl points.
The presence of Weyl points in a semi-Dirac semimetal leads to unique
properties such as highly-anisotropic plasmon frequency, unusual carrier
density-dependence of the Hall coefficient, and the divergence of the ratio of
orbital to spin susceptibilities at low carrier doping Banerjee and Pickett
(2012). The Weyl points in tetragonal SrNbO3 open up an energy gap in the
presence of an additional broken inversion symmetry, obtained by a small
displacement at a Nb site. We show that this gapped Weyl points, at the $\rm
P$ point, exhibits an enhanced intrinsic anomalous Hall effect. With these
intriguing properties, perovskite oxides of the I4/mcm space group provide a
unique example of a class of compounds that hosts crystalline symmetry-
protected 3D semi-Dirac fermions. Our Berry curvature analysis sheds light on
the unusual features in transport experiments, such as the Shubnikov de Haas
oscillations and anomalous Hall effect measurements. Furthermore, the ability
to control the Berry phase by octahedral rotation shows a pathway towards
developing devices for oxide-based electronics.

Electronic structure: The band dispersion of tetragonal SrNbO3 in the presence
of SOC, obtained from our ab initio calculations SM  along various high-
symmetry momenta in the Brillouin zone is shown in Figs. 2(a). The three
$t_{2g}$ orbitals of Nb $4d$ state predominantly populate the Fermi level, as
in the case of cubic SrNbO3 Bigi  _et al._ (2020); Park  _et al._ (2020). Due
to the zone folding at the Brillouin zone boundary, the bands along the $\rm
P$-$\rm N$ momentum direction are four-fold degenerate in the absence of SOC,
creating line nodes. The degeneracy of these nodal lines is lifted everywhere,
except at the high symmetry points, when SOC is turned on, as described in
2(b). The TRS and the nonsymmorphic symmetry protect the four-fold degeneracy
at the $\rm P$ point, giving rise to the 3D semi-Dirac point at an energy
approximately 0.0662 eV above the Fermi level. Three other Dirac points appear
at the $\rm N$ point at energies 0.5 eV above the Fermi level.

Figure 2: (a) Band dispersion of tetragonal SrNbO3 in the presence of spin-
orbit coupling, through various high-symmetry momenta in the Brillouin zone.
(b) Expanded view of the dispersion along the P-N direction, without and with
spin-orbit coupling. (c)-(e) The three Fermi surfaces of the spin-orbit
coupled $t_{2g}$ bands at the Fermi level. The semi-Dirac point (sDP) is
denoted by the dashed circle at the P point in (a) and by the yellow dot in
(c).

Being away from the Fermi level, the Dirac points at the $\rm N$ point may not
be accessible to electronic transport; here we, therefore, focus on the semi-
Dirac point at the $\rm P$ point. Yet, replacing Nb with other transition-
metal element such as Mo might tune the Fermi level close to the Dirac points
at the N point. The Fermi surfaces of the three spin-orbit coupled $t_{2g}$
orbitals are depicted in Fig. 2(c)-(e). Due to the linear dispersion near the
semi-Dirac point, a small doping by external gating or chemical doping can
tune the Fermi level to the semi-Dirac point, creating the possibility to
obtain a quantized Berry phase and a very high-mobility conduction Wang  _et
al._ (2011); Liang  _et al._ (2015); Veit  _et al._ (2018). Also, the presence
of the semi-Dirac points in the asymmetric, convex Fermi surface is expected
to produce a large nonsaturating magnetoresistance He  _et al._ (2019);
Vashist  _et al._ (2019); Zhang  _et al._ (2019).

Figure 3: Berry curvature $\Omega(\mathbf{k})$ for Nb spin polarization (a)
$M\\!=\\!0$, and (b) $M\\!=\\!0.2$, plotted in a momentum plane containing the
$\rm P$ point at the center, and the $\rm N$ points at the side centers. We
use a dense 100$\times$100 momentum grid. (c), (d) The Wannier charge centers
$\bar{x}_{n}$ along momentum $k_{y}$ with $k_{x}\\!=\\!0$ for (a) $M\\!=\\!0$,
and (b) $M\\!=\\!0.2$. The blue line represents the center of the maximum gap
between the Wannier charge centers. In (d), the gap center crosses the Wannier
charge centers an odd number of times, resulting in a topological index
$Z_{2}\\!=\\!1$.

Topological invariant: We investigate the topological properties of the
compound under consideration, tetragonal SrNbO3, by using two protocols—first,
by looking at the Berry curvature, and second, by computing the topological
invariant of the Bloch bands. The Berry curvature is given by Wang  _et al._
(2006)

$\displaystyle\Omega(\mathbf{k})\\!=\\!-\sum_{m,n}^{\varepsilon_{{m}\mathbf{k}}\neq\varepsilon_{{n}\mathbf{k}}}2~{}{\rm
Im}\frac{\langle\psi_{n\mathbf{k}}|v_{x}|\psi_{m\mathbf{k}}\rangle\langle\psi_{m\mathbf{k}}|v_{y}|\psi_{n\mathbf{k}}\rangle}{(\varepsilon_{{m}\mathbf{k}}-\varepsilon_{{n}\mathbf{k}})^{2}},$
(1)

where $|\psi_{n\mathbf{k}}\rangle$ is the spinor Bloch wave function of the
$n^{\rm th}$ band with eigenenergy $\varepsilon_{{n}\mathbf{k}}$, and
$\mathbf{v}\\!=\\!(v_{x},v_{y})$ is the velocity operator. We use Wannier
interpolation based on maximally-localized Wannier functions Marzari and
Vanderbilt (1997); Souza  _et al._ (2001). The Berry curvature in the $\rm
P\\!-\\!N\\!-\\!Q$ plane, the same plane considered in Fig. 1(b), is shown in
Fig. 3(a). The semi-Dirac point, being gapless, reveals zero Berry curvature
at the center $\rm P$ point. We introduce TRS breaking and opening of an
energy gap at the semi-Dirac point by constraining to a spin polarization at
the Nb sites. Fig. 3(b) shows a large Berry curvature contributions at and
near the $\rm P$ point with $M\\!=\\!0.2$, which represents 20$\%$ spin
polarization along the $+z$ direction. $M$ is defined as $M=\sum_{a\in{\rm
Nb}~{}t_{2g}{\rm~{}orbitals}}\langle n_{a,\uparrow}\rangle-\langle
n_{a,\downarrow}\rangle$, where $n_{a,\uparrow}$ is the occupation of
electrons at orbital $a$ and spin $\uparrow$. The enhanced Berry curvature
induced by a nonzero $M$ strongly suggests the nontrivial band topology. To
check this possibility, we compute the topological invariant via the Wannier
charge centers $\bar{x}_{n}$, corresponding to the Wannier functions that are
maximally localized in one dimension, defined as Soluyanov and Vanderbilt
(2011)

$\displaystyle\bar{x}_{n}\\!=\\!\frac{i}{2\pi}\int_{-\pi}^{\pi}d\mathbf{k}\langle
u_{{n}\mathbf{k}}|\partial_{\mathbf{k}}|u_{{n}\mathbf{k}}\rangle,$ (2)

Figure 4: (a) Illustration of the effect of broken symmetries on the semi-
Dirac points. TRS breaking splits the semi-Dirac point into a pair of Weyl
points, and additional lattice distortion opens up an energy gap. (b) Band
dispersion along the momentum path X-P-Q in the vicinity of the P point, in
the presence of the broken time-reversal symmetry that lifts the four-fold
degeneracy of the semi-Dirac point at the P point, and creates a pair of Weyl
points near the P point. (c) The Weyl point acquires an energy gap in the
presence of an inversion symmetry-breaking lattice distortion. (d) The
anomalous Hall conductivity, with varying energy, reveals a plateau within the
energy gap at the Weyl point. Inset in (d) shows the two pairs of the Weyl
points within the Brillouin zone, appearing near the P point.

where
$|u_{{n}\mathbf{k}}\rangle\\!=\\!e^{-i\mathbf{k}\cdot\mathbf{r}}|\psi_{n\mathbf{k}}\rangle$
represents the periodic parts of the Block states. In Fig. 3(c) and 3(d), we
show the evolution of the Wannier charge centers with the momentum $k_{y}$
for, respectively, $M\\!=\\!0$, and $M\\!=\\!0.2$. With $M\\!=\\!0.2$, the
center of the largest gap between the Wannier charge centers (blue line)
crosses the Wannier charge centers odd number of times in traversing a path
from $k_{y}\\!=\\!0$ to $k_{y}\\!=\\!\pi/a$, yielding a topological invariant
$Z_{2}\\!=\\!1$. A non-zero $Z_{2}$ index in a Chern insulator with broken TRS
usually predicts a quantum anomalous Hall effect. We, however, find that the
Hall conductivity is not quantized, and the semi-Dirac points in tetragonal
SrNbO3 may, therefore, be classified as type I Huang  _et al._ (2015), similar
to that proposed in VO2-TiO2 heterostructures Banerjee  _et al._ (2009). This
is also verified by the absence of a band crossing of the surface states SM .

Anomalous Hall effect: In the absence of TRS, e.g. with an external magnetic
field that is applied perpendicular to the $a$-$b$ plane, the semi-Dirac point
in the I4/mcm space group compounds splits into two Weyl nodes, as illustrated
in Fig. 4(a). In the considered tetragonal SrNbO3, we find that a finite
magnetization at the Nb sites generates a Weyl point close to the P point, as
shown in Fig. 4(b). The pair of Weyl points appears along the P-X-P momentum
direction, totaling two pairs in the entire Brillouin zone, as shown in the
inset of Fig. 4(d). By lowering the lattice symmetry further, an energy gap is
induced at the Weyl points. We consider a lattice distortion that lowers the
inversion symmetry but preserves the nonsymmorphic symmetry. This is achieved
by shifting the center Nb site vertically by a small amount. Such a distortion
may be realized naturally by epitaxial strain induced by lattice mismatch with
the substrate at a polar interface, or by applying an electric field
perpendicular to the interface. Fig. 4(c) shows a distortion-induced energy
gap at a Weyl point. In a truly semimetallic system, such an energy gap gives
rise to quantum anomalous Hall effect in the vicinity of the Dirac or semi-
Dirac point. In tetragonal SrNbO3, a net quantization is not achieved due to
the presence of other bands at the Fermi level and instead an enhanced
intrinsic anomalous Hall effect is expected to appear within the energy gap,
as shown in Fig. 4(d). We find that the anomalous Hall conductivity is
dependent on the amount of distortion in the tetragonal lattice, and the
octahedral rotation, suggesting another route to obtain tunable anomalous Hall
effect in oxide interfaces.

Tunable Berry phase: Understanding the Berry phase-driven phenomena in
perovskite oxides is fundamentally important, not only to realize emergent
electromagnetism in a feasible way, but also to identify the origin of non-
trivial Hall signatures. In oxide compounds where magnetism coexists with a
non-trivial band topology, such as SrRuO3, the intrinsic anomalous Hall effect
appears without any external magnetic field Groenendijk  _et al._ (2020). In
the non-magnetic tetragonal perovskites, like the one considered here i.e.
SrNbO3, doping magnetic impurities or applying an external magnetic field
leads to a finite Berry phase in the Bloch bands. We find that the Berry
curvature $\Omega$ can be tuned externally by changing the octahedral rotation
angle $\theta$ SM . Particularly, the Berry curvature and the resulting
anomalous Hall conductivity vary non-monotonically with increasing $\theta$.
Previous experiments have demonstrated that external strain modulation can
control octahedral rotations in oxide thin films Herklotz  _et al._ (2016);
Choquette  _et al._ (2016). Our presented results, therefore, show an
effective way to control the Berry phase and consequently the topological
properties of perovskite transition-metal oxides by epitaxial strain
engineering.

Effective low-energy model: To study the low-energy behavior of the 3D semi-
Dirac dispersion, we derive an effective tight-binding model for the SOC-
coupled $t_{2g}$ electrons in tetragonal perovskite oxides SM . From this
tight-binding Hamiltonian, we obtain the following low-energy Hamiltonian that
describes the 3D semi-Dirac dispersion around the P point

$\displaystyle H_{\rm low\\!-\\!E}=$
$\displaystyle\\!-A\bigl{(}k_{x}^{2}+k_{y}^{2}\bigr{)}\tau_{0}\sigma_{0}\\!-\\!B(-k_{x}\tau_{3}\sigma_{1}\\!-\\!k_{y}\tau_{3}\sigma_{2})$
$\displaystyle-\\!\Bigl{\\{}C\bigl{(}k_{x}^{2}-k_{y}^{2}\bigr{)}-Dk_{z}\Bigr{\\}}\tau_{1}\sigma_{0},$
(3)

where $\sigma_{i}$ and $\tau_{i}$ are the Pauli matrices in the spin space and
the sublattice space; $A$, $B$, $C$, and $D$ are parameters. The partial
density of states surrounding the semi-Dirac point, exhibits a depletion at
the energy at which the semi-Dirac point appears SM , rather than a minimum as
in the isotropic Dirac or 2D semi-Dirac cases Chen  _et al._ (2021). A
depletion of the density of states near the Fermi energy usually indicates a
strong effect of electronic correlation or disorder Noh  _et al._ (2000). The
mutual interplay of these additional effects in the present semi-Dirac
compound may, therefore, produce unconventional properties.

Discussion: We further analyzed the electronic properties of four other
compounds from the I4/mcm space group, viz. CaNbO3, SrRuO3, SrMoO3, and SrTiO3
and confirmed the presence of the 3D semi-Dirac fermions at the P point SM .
This establishes that the discussed symmetry-protected band topology is
generic to tetragonal perovskite oxides that belong to this space group, the
only difference being the location in energy of the semi-Dirac points. For
CaNbO3 and SrRuO3, the semi-Dirac point at the P point is closer to the Fermi
level than in SrNbO3. On the other hand, SrTiO3, having a large energy gap at
the Fermi level, the semi-Dirac points are situated far away from the Fermi
level. Using SrTiO3 as a substrate is, therefore, beneficial for providing
epitaxial strain. For SrMoO3 and SrRuO3, three Dirac points at the N point are
also located close to the Fermi level which indicates that these two compounds
can be interesting also to explore the influence of the non-trivial band
topology in transport experiments.

In contrast to the isotropic Dirac semimetals, the semi-Dirac semimetals can
be realized at a quantum critical point between a 3D Dirac semimetal and a
topologically trivial insulator Yang and Nagaosa (2014); Parameswaran  _et
al._ (2013). In response to a circularly polarized light, the semi-Dirac
semimetals with a gapless Dirac node do not open an energy gap, unlike the
isotropic Dirac semimetals Narayan (2015). Tetragonal perovskite oxides,
therefore, offer a feasible material platform to study new phenomena that
cannot be found in known Dirac semimetals.

The interplay of Coulomb interaction and disorder in a two-dimensional semi-
Dirac semimetal gives rise to a variety of quantum phase transitions and non-
Fermi liquid behaviors Zhao  _et al._ (2016). Disorder, particularly, has a
profound effect as it can drive a Lifshitz transition from an insulator to a
semimetal, and a topological transition to a Chern insulating state
Sriluckshmy  _et al._ (2018). The interplay of topology, disorder and Coulomb
interaction in the nonsymmorphic symmetry-protected semi-Dirac fermions, found
in the tetragonal perovskite oxides, can lead to even richer physical
properties due to their three-dimensional nature.

To conclude, we have shown that non-magnetic tetragonal perovskite oxides with
I4/mcm symmetry, e.g. SrNbO3, CaNbO3, and SrMoO3, host 3D semi-Dirac fermions
that are protected by a nonsymmorphic symmetry. This crystalline symmetry can
be realized by a substrate strain when the cubic perovskites undergo an
octahedral rotation of type a0a0c-. Due to the symmetry protection, the semi-
Dirac degeneracy is maintained in the presence of SOC, making this class of
compounds a unique, natural example. Breaking the TRS leads to two pairs of
Weyl points from the semi-Dirac points. In the presence of an additional
inversion symmetry-breaking lattice distortion, the Weyl points acquire an
energy gap, and an enhanced intrinsic anomalous Hall effect is realized. The
presence of the Weyl points is expected to produce negative magnetoresistance
induced by chiral anomaly Son and Spivak (2013). The Berry phase arising due
to the semi-Dirac point with a broken symmetry is tunable by controlling the
octahedral rotation angle that can be changed by the sample thickness, or
advanced interface engineering. Our findings serve as an important stepping
stone towards understanding various kinds of quasiparticles that can be found
in solids Bradlyn  _et al._ (2016); Wang  _et al._ (2016); Wieder  _et al._
(2018).

Acknowledgements: This work was supported by the U.S. Department of Energy,
Office of Science, Basic Energy Sciences, Materials Sciences and Engineering
Division.

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Supplemental Materials on "Semi-Dirac and Weyl Fermions in Transition Metal
Oxides"

Narayan Mohanta1, Jong Mok Ok1, Jie Zhang1, Hu Miao1, Elbio Dagotto1,2, Ho
Nyung Lee1, Satoshi Okamoto1

1Materials Science and Technology Division, Oak Ridge National Laboratory, Oak
Ridge, TN 37831, USA
2Department of Physics and Astronomy, The University of Tennessee, Knoxville,
TN 37996, USA

I. Details of ab initio calculations
The electronic structure was computed within ab initio density functional
theory (DFT) Hohenberg and Kohn (1964) as implemented in the QUANTUM ESPRESSO
code Giannozzi  _et al._ (2009) with the plane-wave pseudopotential method. We
adopted the Perdew-Burke-Ernzerhof generalized gradient approximation
exchange-correlation functional Perdew  _et al._ (1996) and used a $16\times
16\times 16$ Monkhorst-Pack momentum-point mesh to discretize the first
Brillouin zone. A plane wave cut-off of 600 eV, which were found to be
sufficient to achieve convergence of the total energy. The energy convergence
criterion was set to 10-6 eV during the minimization process of the self-
consistent cycle. Starting with the experimental lattice constants
$a\\!=\\!5.518$ Å, $b\\!=\\!5.518$ Å, and $c\\!=\\!8.280$ Å Ok  _et al._ ,
full optimization of the crystal structure was performed using the force
convergence criterion of $10^{-3}$ eV/Å.

The Bloch wave functions, obtained from the DFT calculations, were used to
construct the maximally localized Wannier functions Marzari and Vanderbilt
(1997); Souza  _et al._ (2001); Mostofi  _et al._ (2008); Pizzi  _et al._
(2020). Starting from an initial projection of atomic d-basis functions
belonging to the $t_{2g}$ manifold and centered on metal sites, the
$t_{2g}$-like Wannier functions were obtained. Using the tight-binding
Hamiltonian, formulated using the $t_{2g}$ orbitals and obtained from
Wannier90, the topological invariant was computed within the Wannier charge
center formalism using the WannierTools code Soluyanov and Vanderbilt (2011);
Wu  _et al._ (2018).

II. Tight-binding model
To gain insight into the nature of the semi-Dirac dispersions in $t_{2g}$
perovskite transition-metal oxides with the $I4/mcm$ crystalline symmetry, we
develop an effective tight-binding Hamiltonian.

For the sake of simplicity, we define the local orbitals as
$\\{a,b,c\\}=\\{d_{xy},d_{yz},d_{zx}\\}$, and denote the hopping integral from
orbital $\beta$ to $\alpha$ along the $h\hat{x}+k\hat{y}+l\hat{z}$ direction
as $t_{hkl}^{\alpha\beta}$. This leads to the following tight-binding
Hamiltonian:

$\displaystyle H_{\rm
TB}=-\sum_{\alpha,\beta}\sum_{\vec{r},\vec{r}^{\prime}}\sum_{\sigma}t_{\vec{r}-\vec{r}^{\prime}}^{\alpha\beta}\alpha_{\vec{r}\sigma}^{\dagger}\beta_{\vec{r}^{\prime}\sigma},$
(E1)

where $\alpha_{\vec{r}\sigma}^{({\dagger})}$ is an annihilation (creation)
operator of an electron on orbital $\alpha$ at position $\vec{r}$ with spin
$\sigma$. Typical hopping integrals are presented in Fig. S1. The amplitude
and the sign of $t_{hkl}^{\alpha\beta}$ are determined by the relative
direction and the sign of two orbitals and must satisfy the underlying
$I4/mcm$ symmetry. This leads to the following relations:

$\displaystyle
t_{100}^{aa}=t_{\bar{1}00}^{aa}=t_{010}^{bb}=t_{0\bar{1}0}^{bb},$
$\displaystyle
t_{010}^{aa}=t_{0\bar{1}0}^{aa}=t_{100}^{bb}=t_{\bar{1}00}^{bb},$
$\displaystyle
t_{001}^{aa}=t_{00\bar{1}}^{aa}=t_{001}^{bb}=t_{00\bar{1}}^{bb},$
$\displaystyle
t_{100}^{a^{\prime}b}=t_{\bar{1}00}^{a^{\prime}b}=t_{010}^{a^{\prime}b}=t_{0\bar{1}0}^{a^{\prime}b}=-t(\\{a,b\\}\leftrightarrow\\{a^{\prime},b^{\prime}\\}),$
$\displaystyle t_{001}^{ab}=t_{00\bar{1}}^{ab},$ $\displaystyle
t_{100}^{cc}=t_{\bar{1}00}^{cc}=t_{010}^{cc}=t_{0\bar{1}0}^{cc},$
$\displaystyle t_{001}^{cc}=t_{00\bar{1}}^{cc}$ (E2)

for the nearest-neighbor bonds, and

$\displaystyle
t_{110}^{aa}=t_{\bar{1}\bar{1}0}^{aa}=t_{\bar{1}10}^{bb}=t_{1\bar{1}0}^{bb}=t_{110}^{b^{\prime}b^{\prime}}=t_{\bar{1}\bar{1}0}^{b^{\prime}b^{\prime}}=t_{\bar{1}10}^{a^{\prime}a^{\prime}}=t_{1\bar{1}0}^{a^{\prime}a^{\prime}},$
$\displaystyle
t_{1\bar{1}0}^{aa}=t_{\bar{1}10}^{aa}=t_{110}^{bb}=t_{\bar{1}\bar{1}0}^{bb}=t_{1\bar{1}0}^{b^{\prime}b^{\prime}}=t_{\bar{1}10}^{b^{\prime}b^{\prime}}=t_{110}^{a^{\prime}a^{\prime}}=t_{\bar{1}\bar{1}0}^{a^{\prime}a^{\prime}},$
$\displaystyle
t_{110}^{ab}=t_{\bar{1}\bar{1}0}^{ab}=-t_{1\bar{1}0}^{ab}=-t_{\bar{1}10}^{ab},$
$\displaystyle
t_{110}^{cc}=t_{\bar{1}\bar{1}0}^{cc}=t_{\bar{1}10}^{cc}=t_{1\bar{1}0}^{cc},$
$\displaystyle
t_{011}^{aa}=t_{0\bar{1}\bar{1}}^{aa}=t_{0\bar{1}1}^{aa}=t_{01\bar{1}}^{aa}=t_{101}^{bb}=t_{\bar{1}0\bar{1}}^{bb}=t_{\bar{1}01}^{bb}=t_{10\bar{1}}^{bb},$
$\displaystyle
t_{101}^{aa}=t_{\bar{1}0\bar{1}}^{aa}=t_{\bar{1}01}^{aa}=t_{1\bar{0}1}^{aa}=t_{011}^{bb}=t_{0\bar{1}\bar{1}}^{bb}=t_{0\bar{1}1}^{bb}=t_{01\bar{1}}^{bb},$
$\displaystyle
t_{101}^{ac}=t_{\bar{1}0\bar{1}}^{ac}=-t_{\bar{1}01}^{ac}=-t_{10\bar{1}}^{ac}=t_{011}^{bc}=t_{0\bar{1}\bar{1}}^{bc}=-t_{0\bar{1}1}^{bc}=-t_{01\bar{1}}^{bc},$
$\displaystyle
t_{101}^{bc}=t_{\bar{1}0\bar{1}}^{bc}=t_{011}^{ac}=t_{0\bar{1}\bar{1}}^{ac}=-t_{10\bar{1}}^{bc}=-t_{\bar{0}1}^{bc}=-t_{01\bar{1}}^{ac}=-t_{0\bar{1}1}^{ac}=-t(\\{a,b,c\\}\leftrightarrow\\{a^{\prime},b^{\prime},c^{\prime}\\})$
(E3)

for the second-neighbor bonds, and the elements that are not listed here are
zero up to second-neighbor bonds. When the hopping integral depends explicitly
on the sublattice A and B, we use $\\{a^{\prime},b^{\prime},c^{\prime}\\}$ for
sublattice B. Further-neighbor hopping integrals could be considered, but
including these terms does not alter our discussion.

Figure S1: Typical hopping integrals. Hopping integral from orbital $\beta$ to
orbital $\alpha$ along the $h\hat{x}+k\hat{y}+l\hat{z}$ direction
$t_{hkl}^{\alpha\beta}$ is indicated by an arrow. Here, red and blue circles
belong to sublattice A and B, respectively. Orbitals $a:d_{yz}$, $b:d_{zx}$,
and $c:d_{xy}$ are indicated by orange, green, blue robes, respectively. To
highlight the sign difference in electron orbitals, negative regions are shown
in white. Solid lines indicate how O6 octahedra rotate, creating bond angles
smaller than 180 degrees. In the second row, hopping is between second-
neighbor sites on two adjacent layers.  Figure S2: Tight-binding dispersions.
(a) Dispersions in the entire Brillouin zone with its first octant shown in
(b). Dispersions around the P point and the N point are shown in panels (c-e)
and (f-h), respectively. The midpoint between P and N is labeled as
$\overline{\rm PN}$.  Figure S3: Partial density of states originating from
one of the semi-Dirac dispersions at the P point.

In the undistorted perovskite structure, many hopping integrals disappear
reflecting the symmetry (positive and negative sign of $t_{2g}$ orbitals).
Among such hopping integrals, non-zero value of $t_{101}^{ac}$ and symmetry
related hopping as well as finite spin-orbit coupling are found to lift the
band degeneracy of nodal lines along the P-N direction, creating semi-Dirac
dispersions at the P point and Dirac dispersions at the N point.

In addition to the hopping integrals, there could exist crystal field
splitting between orbitals $a$, $b$, and $c$ as

$\displaystyle H_{\rm
CFS}=\sum_{\sigma}\bigl{\\{}\varepsilon_{ab}(a_{\sigma}^{\dagger}a_{\sigma}+b_{\sigma}^{\dagger}b_{\sigma})+\varepsilon_{c}c_{\sigma}^{\dagger}c_{\sigma}\bigr{\\}},$
(E4)

where $a$ and $b$ remain degenerate in the $I4/mcm$ symmetry.

Finally, as the most important ingredient, the spin-orbit coupling is
included. Here, we consider the atomic-like form given by

$\displaystyle H_{\rm
SOC}=\lambda\bigl{(}\vec{l}\cdot\vec{s}\bigr{)}_{t_{2g}}=\frac{\lambda}{2}\sum_{\alpha,\beta,\gamma}\sum_{\sigma,\sigma^{\prime}}i\varepsilon_{\alpha\beta\gamma}\alpha_{\sigma}^{\dagger}\sigma_{\sigma\sigma^{\prime}}^{\gamma}\beta_{\sigma^{\prime}},$
(E5)

where $(\ldots)_{t_{2g}}$ indicates the projection onto the $t_{2g}$ subspace,
$\alpha$, $\beta$, and $\gamma$ are orbital indices, $\sigma^{\gamma}$ is the
Pauli matrix, and $\varepsilon_{\alpha\beta\gamma}$ is the Levi-Civita
antisymmetric tensor. For $H_{\rm CFS}$ and $H_{SOC}$, site indices are
omitted for simplicity.

Thus, the effective Hamiltonian is given by the sum of the above three terms:

$\displaystyle H_{\rm eff}=H_{\rm TB}+H_{\rm CFS}+H_{\rm SOC}.$ (E6)

In an example we show here, the following parameter set is considered:
$t_{100}^{aa}=t_{001}^{aa}=t_{100}^{cc}=t$, $t_{010}^{aa}=t_{001}^{cc}=0.1t$,
$t_{100}^{a^{\prime}b}=0.05t$, $t_{001}^{a^{\prime}b}=0.1t$,
$t_{110}^{cc}=t_{011}^{aa}=0.4t$, $t_{1\bar{1}0}^{aa}=0.1t$,
$t_{110}^{ab}=0.1t$, $t_{101}^{ac}=0.05t$, $\varepsilon_{c}=0.2t$,
$\lambda=0.5t$, and others are set to zero. Resulting dispersion relation is
summarized in Fig. S2. One notices that this tight-binding dispersion relation
qualitatively reproduces the DFT dispersions, in particular semi-Dirac
dispersions at the P point and the linear Dirac dispersion at the N point.

To see the consequence of the semi-Dirac dispersions, we computed the partial
density of states (DOS) by focusing on one of the P points. The resulting DOS
is shown in Fig. S3. Here, the momentum $\vec{k}$ integral is carried out in a
small cube around the P point with $\Delta k=\pm\pi/8$ and the Dirac delta
function approximated as the Lorentzian with the width $\eta=0.02t$. One
notices a sharp dip in DOS, instead of a gap, reflecting the semi-Dirac
dispersions. With correlation effects, this may lead to novel many-body
effects as discussed in Ref. Wang2021 .

III. Low-energy effective model
The tight-binding model developed in the previous section is useful to derive
a low-energy effective model. For this purpose, we focus on the Kramers pairs
that are eigenstates of local Hamiltonian, $H_{\rm CFS}+H_{\rm SOC}$. Such
Kramers pairs may be expressed as

$\displaystyle|\sigma=\pm
1\rangle\propto|a,\sigma\rangle+i|b,\sigma\rangle+\delta|c,-\sigma\rangle.$
(E7)

Considering the hopping integrals described in the previous section, a low-
energy Hamiltonian may be parameterized as

$\displaystyle H_{\rm low\\!-\\!E}=-2A\cos k_{x}\cos
k_{y}\tau_{0}\sigma_{0}-B(\sin k_{x}\tau_{3}\sigma_{1}-\sin
k_{y}\tau_{3}\sigma_{2})\sin k_{z}-\bigl{\\{}2C(\cos k_{x}+\cos k_{y})+D\cos
k_{z}\bigr{\\}}\tau_{1}\sigma_{0},$ (E8)

where $\tau_{i}$ and $\sigma_{j}$ are $2\times 2$ Pauli matrices acting on the
sublattice space and the spin space specifying Kramers states, respectively,
and Kronecker product are abbreviated as
$\tau_{i}\otimes\sigma_{j}\equiv\tau_{i}\sigma_{j}$. Expanding cosine and sine
functions at the P point $\vec{k}=(\pi,0,\pi/2)$, and measuring the momentum
from the P point, one arrives at

$\displaystyle H_{\rm
low\\!-\\!E}\approx-A\bigl{(}k_{x}^{2}+k_{y}^{2}\bigr{)}\tau_{0}\sigma_{0}-B(-k_{x}\tau_{3}\sigma_{1}-k_{y}\tau_{3}\sigma_{2})-\Bigl{\\{}C\bigl{(}k_{x}^{2}-k_{y}^{2}\bigr{)}-Dk_{z}\Bigr{\\}}\tau_{1}\sigma_{0},$
(E9)

where a constant term is neglected from the diagonal component. The
eigenvalues of this Hamiltonian are given by

$\displaystyle
E_{\pm}=A\bigl{(}k_{x}^{2}+k_{y}^{2}\bigr{)}\pm\sqrt{B^{2}\bigl{(}k_{x}^{2}+k_{y}^{2}\bigr{)}+\Bigl{\\{}C\bigl{(}k_{x}^{2}-k_{y}^{2}\bigr{)}-Dk_{z}\Bigr{\\}}^{2}}$
(E10)

In Fig. S4, we plot the dispersion relations along three different directions
with $A=0.05$, $B=0.01$, $C=0.5$, and $D=1.0$. One can notice that this
effective model gives essential feature of dispersions obtained in DFT and the
tight-binding model; linear band crossing along the $k_{z}$ and the
$k_{x}\\!=\\!\pm k_{y}$ directions, and quadratic band crossing along the
$k_{x}$ and the $k_{y}$ directions. The band splitting along the
$k_{x}\\!=\\!\pm k_{y}$ directions, corresponding to the P-N-P direction, is
due to the parameter $B$, which corresponds to $t_{101}^{ac}$ and symmetry-
related hopping terms.

Figure S4: Dispersion relation from the low-energy effective model Eq. (E9)
along three inequivalent directions. Parameter values are chosen to be
$A=-0.05$, $B=0.01$, $C=0.5$, and $D=1.0$.  Figure S5: DFT dispersions of
tetragonal (a) CaNbO3, (b) SrRuO3, (c) SrMoO3, and (d) SrTiO3 in the presence
of spin-orbit coupling. The red circles denote the three-dimensional semi-
Dirac points at the P and the Dirac point at the N point. The lattice
constants are: (a) $a=b=5.560$ Å, $c=8.018$ Å, (b) $a=b=5.577$ Å, $c=8.087$ Å,
(c) $a=b=5.621$ Å, $c=8.078$ Å, (d) $a=b=5.567$ Å, $c=7.907$ Å. Figure S6:
Unit cell of tetragonal SrNbO3 (top view) with different amount of the
octahedral rotation: (a) $\theta\\!=\\!4^{\circ}$, (b)
$\theta\\!=\\!8^{\circ}$, and (c) $\theta\\!=\\!16^{\circ}$. (d)-(f). Berry
curvature at the respective values of $\theta$ with a Nb spin polarization
$M\\!=\\!0.2$. (g)-(i) Energy gap (in eV) between the two bands that connect
the semi-Dirac point for the above three values of $\theta$. Evidently, with
increasing the rotation angle $\theta$, there are drastic changes in the Berry
curvature $\Omega$, and the energy gap between the Bloch bands near the semi-
Dirac points.  Figure S7: Band dispersion of a (001) slab of tetragonal
SrNbO3, showing the surface states in blue and bulk states in grey along the
high-symmetry momenta in the surface Brillouin zone, shown in the inset.

IV. Semi-Dirac points in the tetragonal oxides
The presence of the 3D semi-Dirac fermions was investigated in a few other
compounds from the I4/mcm space group, viz, CaNbO3, SrRuO3, SrMoO3, and
SrTiO3. In Fig. S5, we show the electronic structures of these compounds that
reveal the semi-Dirac points at the P point and Dirac points at the N point,
similar to SrNbO3. The semi-Dirac points in CaNbO3 are located father away
from the Fermi level than in SrNbO3. In SrRuO3, a semi-Dirac point at the P
point is located very close to the Fermi level while the Dirac points at the N
point are below the Fermi level. Furthermore, SrRuO3 is known to have magnetic
order which will split the semi-Dirac points to Weyl points, as described in
the main text. In SrMoO3, the semi-Dirac point at the P point is located even
farther away from the Fermi level while three Dirac points at the N point are
located very close to the Fermi level. With slight electron doping, these
semi-Dirac points can appear at the Fermi level and contribute to the
transport properties. Therefore, SrMoO3 is another candidate material that can
be interesting to look at to investigate the signatures of the semi-Dirac
points. On the other hand, SrTiO3 has the semi-Dirac points $\sim$2 eV above
the Fermi level and hence they will not influence the transport properties.
SrTiO3, is therefore, useful as a substrate compound to control the epitaxial
strain in the metallic tetragonal oxides.

V. Octahedral rotation as a control parameter for the Berry curvature
The electronic properties of tetragonal SrNbO3 were studied at several values
of the rotation angle $\theta$ of NbO6 octahedra. In experiments, $\theta$ can
be tuned by controlling the epitaxial strain, via an effective interface
engineering, changing the sample thickness, or an external electric field. In
Fig. S6, the results at three values $\theta=4^{\circ}$, $\theta=8^{\circ}$
and $\theta=16^{\circ}$ are shown at a finite Nb spin polarization $M=0.2$.
The Berry curvature $\Omega$ exhibits peaks at the P point (center of the
plane in (d)-(f)) and the N points (side corners of the plane) — the high-
symmetry momenta at the Brilluoin zone boundary at which the semi-Dirac points
appear. At the P point, $\Omega$ increases with increasing $\theta$ up to
$\theta\approx 10^{\circ}$, after which it decreases. On the other hand,
$\Omega$ at the N points increases monotonically with increasing $\theta$ up
to $\theta=16^{\circ}$. The energy separation between the two relevant bands
near the semi-Dirac point, shown in Fig. S6(g)-(i) at the three values of
$\theta$, also captures the substantial changes in the band dispersion. The
change in the Berry curvature will consequently be noticeable in the intrinsic
anomalous Hall conductivity $\sigma_{xy}$. The octahedral rotation, therefore,
provides an additional control knob to tune the Berry phase properties in
oxides.

VI. Surface states in a slab (001) geometry
To study the nature of the surface states of tetragonal SrNbO3, we calculate
the band dispersion in a slab geometry. We consider a slab of thickness 40
unit cells oriented along the (001) direction, relevant to the common
experimental situation, and use the tight-binding Hamiltonian, expressed in
the basis of the Wannier orbitals. The band dispersion of the SrNbO3 slab,
along the high-symmetry momenta in the surface Brillouin zone, is shown in
Fig. S7. In this case, the surface states do not exhibit any crossing-like
feature, as expected. With a finite magnetization at the Nb site also, the
surface states do not reveal any topological band crossing, as confirmed by
the non-quantized anomalous Hall conductivity, as presented in the main text.

## References

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# Measuring Natural Scenes SFR of Automotive Fisheye Cameras

Daniel Jakab1, Eoin Martino Grua1, Brian Micheal Deegan2, Anthony Scanlan1,
Pepijn Van De Ven1, and Ciarán Eising1
1Dept. of Electronic and Computer Engineering, University of Limerick,
Castletroy, Co. Limerick V94 T9PX, Ireland
2Dept. of Electrical and Electronic Engineering, University of Galway, Galway
H91 TK33, Ireland

###### Abstract

The Modulation Transfer Function (MTF) is an important image quality metric
typically used in the automotive domain. However, despite the fact that
optical quality has an impact on the performance of computer vision in vehicle
automation, for many public datasets, this metric is unknown. Additionally,
wide field-of- view (FOV) cameras have become increasingly popular,
particularly for low-speed vehicle automation applications. To investigate
image quality in datasets, this paper proposes an adaptation of the Natural
Scenes Spatial Frequency Response (NS-SFR) algorithm to suit cameras with a
wide field-of-view.

## Introduction

Wide field-of-view (FOV) cameras are a unique optical and sensor combination
for low-speed vehicle automation [1, 2]. The interest in this sensor modality
has spawned several datasets [3, 4, 5], which encourages the development of
perception algorithms. However, the optical quality of such datasets is
unknown, despite the fact that the connection between optical quality and
vision performance has been demonstrated [6].

In [6] an obvious impact of optical lens quality is seen on computer vision
performance, where a Cooke Triplet lens degraded the performance of object
detection on a BDD100k dataset using a sample of 10,000 images. The BS
ISO12233:2017 standard [7] and Slanted Edge algorithm [8] were used to measure
optical quality across the spatially variant domain of the camera. In this
experiment, test charts were used to measure the optical quality of a camera
using three different lens blur models (Gaussian, Superposition, and isoplanar
patches). This study was limited to only one lens. This suggests that more
information on optical quality is required for publicly available datasets. As
of this moment, no datasets provide a means of measuring optical performance
using test charts, which suggests replicating optical quality measurements
from [6] is difficult and time-consuming. As a strategy to measure image
quality performance in the publicly available datasets, the Natural Scenes
Spatial Frequency Response (NS-SFR) developed by van Zwanenberg et al.
specializes in extracting slanted edges from natural scenes [9, 10, 11, 12].
We adapt NS-SFR to various datasets of spatially variant FOV.

In this paper, we introduce a modified pipeline for generalizing the NS-SFR
algorithm to both narrow and wide FOV lenses. The original NS-SFR is unable to
eliminate areas not part of the natural scene, which biases optical quality
measurements. This especially applies to automotive wide FOV cameras which
have ego-vehicle occlusion and lens aperture in the images. Thus, we propose
the following contributions in this paper:

  1. 1.

We propose regional masking around the scenes in wide FOV images to remove
ego-vehicle occlusion and mechanical vignetting,

  2. 2.

We propose a valid and invalid Region of Interest (ROI) Selection System which
chooses edges with the least amount of edge enhancement and noise of slanted
edges,

  3. 3.

We propose using an Adaptive Radial Distance Analysis for evaluating optical
measurements where radial segments adapted to the regional mask of scene,

  4. 4.

First analysis across multiple dynamic automotive scenes where each scene has
a unique camera lens,

Our experiments are focused on four datasets: three are real-life datasets
with 90°, 185°, and 190° FOV respectively, and the 4 dataset is based on
simulation with 190° FOV.

## Context of NS-SFR

For this paper, we use the BS ISO12233:2017 [7] Slanted Edge algorithm as a
basis for experiments where the adapted NS-SFR focuses on finding edges on
both narrow and wide FOV natural scenes. As far as we know, these experiments
are the first attempt at Edge Spatial Frequency Response (e-SFR) for wide FOV
cameras without using test charts. More sophisticated cameras exceeding $None$
FOV require test charts with greater distortion. There has been little
research on taking natural scenes as part of the measurement process for
automotive scenes.

## Methodology

The proposed Automotive NS-SFR pipeline is depicted in Fig.1. As the goal is
to measure the optical performance of a single camera, the dataset must
contain images from only one camera. For wide FOV cameras, results are biased
where NS-SFR selects the edges that are not part of the natural scene. To
avoid result bias, regional masking is used and is applied to the dataset only
if parts of the vehicle or camera aperture cover parts of the natural scene,
biasing the results. Next, the optimized NS-SFR [9] is applied to the dataset
for ROI selection as illustrated by the green box in Fig. 1. The ROI locations
of the slanted edge measurements were categorized into three spatial areas on
the images: (1) center, (2) middle, and (3) edge. Finally, the mean e-SFR of
each radial segment was calculated.

StartDataset Create custom ROI masks for camera perspective, removing ego-body
occlusion and vignetting using Freehand ROI Apply the custom ROI masks to
images and crop the images to the appropriate size Re-use Optimized NS-SFR
stage Categorize slanted edge locations into three custom radial
distances/segments determined by the shape of the Custom ROI mask Calculate
the mean MTF50 in each radial segment and display results Estimated ISO12233
mean e-SFR Stop$Pre-processing$$Part1$$Part2$

Figure 1: Adapted NS-SFR.

### Regional Masks

(a) KITTI

(b) LMS

(c) Woodscape

(d) SynWoodscape

Figure 2: _Adaptive Radial Distances for all four datasets: (a) KITTI (no
mask), (b) LMS (circular mask), (c) Woodscape (Front View mask) and (d)
SynWoodscape (Front View mask). Note: all masks are marked by a red outline,
inner radial distances (yellow circles) and the outer radial distance (orange
circle). For KITTI only the inner Radial Distances are shown between
(200-1200) pixels._

To generalize processing lenses for NS-SFR, the main interest is to use the
natural scene captured by each camera lens. To do this, certain optical
effects especially in wide FOV cameras need to be removed which are not part
of the natural scene. This involves removing effects such as mechanical
vignetting (part of the camera itself is visible in the corners of fisheye
images) and ego-vehicle occlusion. These can bias the measurements of NS-SFR
where the part of the car to which the camera is attached is visible in any
image. All custom masks are created using the Freehand ROI tool from
MATLAB111See Mathworks for information on Freehand ROI:
https://uk.mathworks.com/help/images/ref/drawfreehand.html allowing the user
to customize a particular ROI mask to isolate the natural scene. If a new
camera is used, a new mask is created to fit the scene(see Fig. 2).

### Adaptive Radial Distances

To further analyze the images three adaptive radial distances were created
which categorize the location of each slanted edge according to its relative
distance from the center of the image. The technique used to create each
segment is by taking the Euclidean distance from the central point to the
farthest edge of the custom ROI mask. In the case of KITTI, the corner of the
image was taken as the farthest distance from the center of the image. The
Euclidean distance calculation is represented by the following formula:

$r_{e}=\max\left(\sqrt{(x_{c}-x_{m})^{2}+(y_{c}-y_{m})^{2}}\right)\forall(x_{m},y_{m})\in[X_{m},Y_{m}]$
(1)

Where, $r_{e}$ is the Euclidean distance representing the radius of the
largest radial distance (i.e. the orange circles (with the exception of KITTI
where the orange circle is not visible due to the radius being the distance
from corner to centre of the image) in Fig. 2), the point ($x_{c}$, $y_{c}$)
represent the location of the centre of the image or mask, ($x_{m}$, $y_{m}$)
represent points along the periphery of the custom ROI mask $[X_{m},Y_{m}]$
and can be the location of the bottom right corner of the image if no mask is
present. (Note: In the case that no ROI is set, $[X_{m},Y_{m}]$ is the
periphery of the image). The radii of the radial distances/annuli are obtained
by dividing the maximum radius $r_{e}$ into $N$ proportional parts.

The ratios determined in these calculations are suitable mainly because they
give sufficient segments to divide up the images into three categories (i.e.,
center, middle, and edge) and generalize well regardless of camera type.
Ratios smaller than this are unsuitable (i.e., there tends to be bias towards
having too many slanted edges favored for one segment).

(a) Valid and Invalid ROI Selection

(b) Measurements

Figure 3: _An example of ROI Selection from one of the Front View of the
Woodscape dataset at (ST=0.04, esfW=10). Observe that the local maxima (red)
and local minima (blue) of each measurement are marked by an x symbol._

### Re-use Optimized NS-SFR

The ROI selection technique and parameter tuning of NS-SFR optimized algorithm
[9] is reused here. Canny Edge Detection with edge masking is used to find
edges from the natural scenes and isolate edges of interest [9]. Unlike
measurement test charts, NS-SFR does not have a strong sharpening effect on
step edges due to the surrounding scene content [9]. The NS-SFR parameters
that determine slanted edge selection in natural scenes are the following (the
parameters used are given):

  1. 1.

Contrast Range (0.1 - 0.9) - the contrast range between the transition from
dark to white in a region with a slanted edge:

    1. (a)

Low contrast ($<$0.1) is prone to noise error,

    2. (b)

High contrast ($>$0.9) is prone to non-linear sharpening and image processing,

  2. 2.

Edge Angle Range ($None$ \- $None$) - there is no restriction on the angle
range of slanted edges as it limits the number of edges that can be found in a
natural scene except for $None$ and $None$ which are filtered out and is not
possible due to errors [7],

  3. 3.

Step Edge Noise Floor (ST) (0.02 and 0.04 (for images with greater noise))
ensures that the gradient on either side of the edge is uniform. For example,
a value of 0.02 signifies a change of pixel value or digital number (DN) of
4.5 for an 8-bit image and 0.04 is twice this value at 9 [11]. It was observed
that sharper edges were found as the ST and esfW parameters increased,

  4. 4.

Edge Spread Function Width (esfW) (default value is $5$ pixels) remove edges
too close together. For example, edges can be 5 pixels apart in an image,

For initial experiments, a few default parameters were used (i.e., ST=0.02,
esfW=5 pixels) [11]. These were found to be especially suitable for narrow FOV
cameras, such as KITTI. The only difference to the default was removing the
Edge Angle range and increasing the default contrast range between (0.55-0.65)
to (0.1-0.9) to accommodate for a larger ROI selection. As a second
experiment, the default parameters were increased to (ST=0.04,esfW=10 pixels)
which showed a higher quality ROI selection for wide FOV cameras (particularly
for Woodscape). This was done to understand what happens if distortion is
applied to the images and whether the algorithm performs better with a
different parameter combination than the default used in the experiments in
the prior art [10]. In the end, we settle on a unique parameter combination
for each dataset where each radial segment of the spatial domain is well
represented (i.e., each segment should have more than 20 valid slanted edges
to calculate the mean).

### Valid & Invalid ROI Selection

We apply additional constraints to slanted edge selection where the following
constraints were applied:

  1. 1.

Local Maxima $<$ 1.4 SFR \- If the local maxima exceed a peak of 1.4 SFR
($\cong 25\%$ overshoot), the measurement is discarded. 25% overshoot or less
is acceptable222See Imattest website for information on MTF curves and Image
appearance: https://www.imatest.com/support/docs/23-1/mtf_appearance/ for
slanted edges,

  2. 2.

Local Minima $<$ 0.4 SFR \- If the local minima exceed 0.4 SFR, the
measurement is discarded. Setting this would filter out signals with excessive
noise past the Nyquist frequency (0.5cy/px) as shown in Oliver et al. [13],

Measurements that meet the above requirements are marked by a green bounding
box and those that do not are marked by a red bounding box as illustrated in
Fig. 3a.

### Radial Distance Analysis

Instead of using three evenly distributed radial distances [9], three adaptive
radial distances are proposed dependent on the regional mask. The mean NS-SFR
per radial distance is calculated and compared visually between all four
datasets. The mean NS-SFR are plotted against each other to gain insight into
how the varying spatial domain affects image quality.

(a) KITTI (ST = 0.02, esfW = 5)

(b) LMS (ST = 0.02, esfW = 5)

(c) Woodscape (ST = 0.04, esfW = 10)

(d) SynWoodscape (ST = 0.04, esfW = 10)

Figure 4: _Horizontal Slanted Edge Locations of all four datasets (refer to
Fig. 2 for Adaptive Distances): (a) KITTI, (b) LMS, (c) Woodscape and (d)
SynWoodscape. Distributions show the pattern of ROI selection for NS-SFR for
both narrow and wide FOV cameras. Here, we can see the trend of the location
of slanted edges in the scenes for the given datasets._

## Results

### Datasets

The datasets used in this study are:

  1. 1.

1065 Front View 90° FOV KITTI city images [14],

  2. 2.

1251 LMS 185° FOV real-life images from Drive A [4],

  3. 3.

1514 Front View 190° FOV Woodscape images [3],

  4. 4.

500 Front View images of SynWoodscape [5],

Each dataset has one lens calibration and a custom ROI mask applied to the
images. The first experiment is to demonstrate the operation of the NS-SFR
algorithm on a dataset with little to no distortion (i.e., the KITTI dataset
[14]). No mask is applied to these images. The second experiment is to use a
circular fisheye dataset from [4] with wide FOV. A custom circular mask is
applied to these images. For the third experiment, a typical automotive
dataset, the Woodscape dataset [2] is used in this work for wide FOV (i.e.,
190° field-of-view) cameras. Finally, SynWoodscape is used which is a
synthetic version of Woodscape created in the CARLA simulator using 4th-order
polynomial cube-map projection [5, 15].

### KITTI camera experiments

ROI selection was iterated over 1065 images of a natural scene in the KITTI
dataset and a resulting spatial distribution of the slanted edges can be seen
in Fig. 4a. There was no adaptive regional mask required for KITTI. The yellow
radial distances from Fig. 2a were calculated by taking one of the corners of
the KITTI images which is by default the farthest location away from the
centre of each image and the radiuses were determined by a ratio of $N$
proportional segments. The mean MTF of each segment is shown in Fig. 5a.

### The LMS Circular fisheye camera experiments

The experiments were repeated on 185° FOV circular fisheye images. A circular
mask was created for the dataset illustrated in Fig. 2b where the blue radial
distance is calculated based on the furthest point of the circular mask using
(1). Fig. 4b represents the resulting spatial distribution of the LMS dataset
subsample used in the experiments. In comparison to KITTI, this has a circular
aperture and a completely different ROI distribution. The default parameters
were used in the KITTI dataset. The mean MTF of each segment is shown in Fig.
5b where all MTF50 measurements are worse than in KITTI (see Fig. 5a). This is
the first dataset to exceed the $None$ FOV in these experiments.

### Woodscape experiments

The Woodscape dataset is composed of $None$ FOV cameras [2]. Out of the four
camera perspectives available, the Front View perspective was chosen for
experiments that consisted of 1514 images. To demonstrate the proposed
pipeline on wide FOV, a sample slanted edge ROI selection is shown in Fig. 3
where the green boxes represent valid slanted edge measurements with minimal
edge enhancement and noise as specified by the local maxima and minima
constraints. As can be seen, NS-SFR has identified 10 potential edges of which
only 5 were chosen as valid which is approximately 50%. Considering the amount
of noise natural scenes tend to have it can be concluded that Woodscape has
good selection of slanted edges. Slanted Edge measurements of each ROI from
Fig. 3a are shown in Fig. 3b where all 5 slanted edges are graphed. The ROI
mask of Woodscape is shown in Fig. 2c where the images had both ego-vehicle
body occlusion and mechanical vignetting. The experiments resulted in another
unique spatial distribution shown in Fig. 4c. For this experiment, both ST and
esfW were increased to 0.04 and 10 respectively. The mean MTF of each segment
is shown in Fig. 5c.

### SynWoodscape experiments

For SynWoodscape experiments, the custom ROI mask can be seen in Fig. 2d.
Unlike Woodscape SynWoodscape does not have dark corners from mechanical
vignetting and only ego-body occlusion needs to be removed from the images.
The spatial distribution of this experiment can be seen in Fig. 4d. By
comparing Fig. 4d to Fig. 4c, SynWoodscape has a much denser and wider spread
of points than Woodscape using the same parameter settings despite having only
500 images to work with. This comparison proves that in a perfect world
simulated scenes tend to have more ideal slanted edges than in real life. For
consistency, the same NS-SFR parameters as for Woodscape were used. The mean
MTF of each segment is shown in Fig. 5d.

(a) KITTI (ST = 0.02, esfW = 5)

(b) LMS (ST = 0.02, esfW = 5)

(c) Woodscape (ST = 0.04, esfW = 10)

(d) SynWoodscape (ST = 0.04, esfW = 10)

Figure 5: _NS-SFR MTF50 Results of four automotive datasets_.

### Discussion

In Fig. 4 we show the spatial distribution of slanted edges found by NS-SFR
and we can observe distinct patterns emerging where scene dependence affects
ROI selection in images with greater distortion. For example, the selection of
slanted edges for KITTI Fig. 4a is mostly uniform whereas for wide FOV, ROI
selection follows the distortion of the natural scenes. The LMS dataset has a
significant number of slanted edges identified on the buildings or in the sky
(see Fig. 4b) whereas, for both Woodscape and SynWoodscape, clusters of
slanted edges are found along the horizon where the road meets the sky and the
buildings (see Fig. 4c and 4d). However, a greater density of slanted edges
does not imply better quality selection as is evident in the LMS MTF50 results
in Fig. 5b where the quality of slanted edge selection is poor (mean MTF50 is
between 0.15-0.18 cy/px). Parameter variations visibly provide higher quality
but fewer slanted edge selections for Woodscape (as shown in Fig. 4c) whereas
the periphery provides better-slanted edge selection than the centre as shown
in Fig. 5c. In contrast to LMS, Woodscape has better MTF50 between (0.31-0.35
cy/px). The SynWoodscape dataset has lower-quality slanted edges than
Woodscape suggesting that simulation does not imply higher-quality slanted
edges. The central radial segment has the least sharp slanted edges, the
middle segment is the second sharpest and finally, the third radial segment
has the sharpest edges. Peter Burns et al [16], shows 3 order and 5 order
polynomial measurements on curved edges with similar results. All measurements
are of 5 order in these natural scenes.

## Future Work

  * •

Consider automated regional masking for each fisheye perspective as originally
proposed by Hogan et al. [17].

  * •

Differentiate SFR of different objects in natural scenes.

## Conclusion

In this paper, we have demonstrated the adaptation of NS-SFR to automotive
datasets. We have demonstrated a novel approach to isolate natural scenes by
applying regional masking (where required) and measuring image quality using
an Adaptive Radial Distance Analysis on four datasets (i.e., KITTI, LMS,
Woodscape and SynWoodscape). It was found that Woodscape provided the best
selection of slanted edges where MTF50 measurements are the highest out of the
four datasets. For horizontal edges, image quality degrades at the centre of
the image.

## Acknowledgments

This work was supported, in part, by the Science Foundation Ireland grant
13/RC/2094_P2 and co-funded under the European Regional Development Fund
through the Southern & Eastern Regional Operational Programme to Lero - the
Science Foundation Ireland Research Centre for Software (www.lero.ie). The
authors of this work would like to thank Oliver van Zwanenberg for assisting
in the adaptation of the NS-SFR framework.

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