# Magnetic Interaction in Doped $2D$ Perovskite Cuprates with Nanoscale
Inhomogeneity: Lattice Nonlocal Effects vs Superexchange

Vladimir A. Gavrichkov<EMAIL_ADDRESS>Kirensky Institute of Physics,
Akademgorodok 50, bld.38, Krasnoyarsk, 660036 Russia  Semyon I. Polukeev
Kirensky Institute of Physics, Akademgorodok 50, bld.38, Krasnoyarsk, 660036
Russia

###### Abstract

We have studied the superexchange interaction $J_{ij}$ in doped $2D$ cuprates.
The AFM interaction strongly depends on the state of the lattice of $CuO_{2}$
layer surrounded by two $LaO$ rock salt layers. In a static $U$ and $D$ stripe
nanostructure, the homogeneous AFM interaction is impossible due to the $U/D$
periodic stripe sequence and $T_{N}=0$. In a dynamic stripe nanostructure, the
ideal $CuO_{2}$ layer with nonlocal effects and the homogeneous AFM
interaction are restored. However the interaction $J_{ij}$ decreases by the
exponential factor due to partial dynamical quenching. The meaning of the
transition from the dynamic to the static cases lies in spontaneous symmetry
$\theta$-breaking with respect to a rotation of the tilted all $CuO_{6}$
octahedra by an orientation angle
$\delta\theta=n\cdot\left({45^{\circ}}\right)$ in the $U$ and $D$ stripe
nanostructure of $CuO_{2}$ layer.

superexchange interaction, nanoscale inhomogeneity, doped $2D$ perovskite
cuprates

###### pacs:

75.30.Et 75.30.Wx 75.47.Lx 74.62.Fj 74.72.Cj

##  I Introduction

The unique functionality of several materials with perovskite structure, like
cuprates, Bussmann-Holder  _et al._ (2017); Egami (2017); Benedek and Muller
(1993); Bianconi  _et al._ (2001) can be tuned by atomic substitutions,
tolerance factor, misfit strain and pressure which control structural tilts
and nanoscale phase separation.

The scanning tunneling microscopy and spectroscopy (STM and STS) Zhao  _et
al._ (2019); Maggio-Aprile  _et al._ (2010); Fischer  _et al._ (2007); da
Silva Neto  _et al._ (2014); Ma (2009) and advanced experimental X-ray methods
Lanzara  _et al._ (1997); Campi  _et al._ (2015); Comin and Damascelli (2016);
Mitrano  _et al._ (2019); Campi and Bianconi (2021) in a wide range of
temperature and doping unambiguously indicate there is a clear connection
between the multiscale stripe texture and the quantum coherence of
quasiparticles in $2D$ perovskite high-Tc superconductors. The stripe charge
nanostructure is also accompanied by spin inhomogeneity Zaanen and Gunnarsson
(1989); Tranquada  _et al._ (1995); Fujita  _et al._ (2004); Hucker  _et al._
(2011); Fujita  _et al._ (2012); Wen  _et al._ (2019); Miao  _et al._ (2021);
Ma  _et al._ (2021), and the high-temperature superconductivity (HTSC) is
believed to originate from magnetic spin excitations that bind Cooper pairs
Anderson (1987); Scalapino (1995); Dagotto (1994); Izyumov (1999). A magnetic
interaction in parent $AFM$ $2D$ cuprates is well known superexchange
interaction. Anderson (1959) The $AFM$ interaction between the nearest spins
of $Cu^{2+}$ ions in the $CuO_{2}$ layer is strong $\left({\sim
0.146eV}\right)$ Coldea  _et al._ (2001) and is much stronger than the
interplanar exchange, that mainly responsible for the long range magnetic
ordering observed in the nondoped cuprates. For $La_{2}CuO_{4}$ (LCO), the
Neel temperature is $T_{N}\approx 300K$. In the traditional picture of BCS
superconductivity, magnetism destroys Cooper pairs. However, superconductivity
in these HTSCs develops from a ”bad metal”, whose resistivity is higher than
that of BCS superconductors, and the AFM magnetism itself is related to the
initial ”bad metal” state, where superconductivity occurs when long-range
magnetic ordering is suppressed at a relatively weak hole doping $\left({x\sim
0.01}\right)$. Indeed, in the resonating-valence bond (RVB) approach and the
$t-J$ model, Anderson (1987); Bascaran  _et al._ (1987); Gros  _et al._
(1987); Kotliar and Liu (1988); Suzumura  _et al._ (1988) the $HTSC$ emerges
due to the condensation of hole pairs, induced by the exchange interaction
$J_{ij}$. Pairing of holes in this mechanism is caused by interband
quasiparticles transfer in the $CuO_{2}$ layer with a short-range $AFM$
ordering. Retardation effects for this mechanism are insignificant. Another
view on HTSC state in the cuprates (see for example Prelovsek and Ramsak
(2005); Plakida  _et al._ (2003)) has been that $AFM$ spin fluctuations,
becoming particularly longer ranged and soft at low hole doping. In fact,
there is the resonant magnetic mode Rossat-Mignod  _et al._ (1991); Fong  _et
al._ (2000); Dai  _et al._ (2001) within the $HTSC$ phase, where the spin
fluctuations scenario represent the lowest bosonic mode relevant for the
d-wave $HTSC$ pairing in an energy range $\sim J_{ij}$.
If the interaction $J_{ij}$ is a reliable candidate for glue for holes, then
how will it change in the inhomogeneity nanostructure of doped 2D cuprates?
Here we have studied a specific stripe mechanism that establishes the
relationship between the observed nanoscale inhomogeneity and the
superexchange $J_{ij}$. Furthermore, this is a good reason to test your
feelings in accordance with the opinion: ”…there remains the nagging feeling
that even if the basic pairing mechanism ultimately arises from the short-
range antiferromagnetic correlations in the layer, some important ingredient
may be missing from the theory we have described.” (see the review Scalapino
(1995)).

##  II  Color approach to nanoscale inhomogeneity and $D$ and $U$ stripe
structure

The idea of the color approach Gavrichkov  _et al._ (2019) is stripe
identification due to the tilting effects (tilting $CuO_{6}$ octahedron as a
whole). Bianconi  _et al._ (1996) The fact is that different stripes differ
not only in the hole concentration and spin, but in the type of nuclear
configuration (see Fig.1)

Figure 1:  View of the tilted $CuO_{6}$ octahedra of the D (a) and U (b)
stripes with different tilting effects in $LSCO$. Insets show the tilting
effects in accordance with Glazer’s notations. Glazer (2011)

We have classified the observed stripe configurations into a ninth-order
symmetric Abelian group $G(\alpha)$ consisting of two types of stripes
$U(\theta_{U})$ and $D(\theta_{D})$ rotated at right angle relative to each
other. Gavrichkov  _et al._ (2019) The sought stripe group $G(\alpha)$
represents all possible reaction products between different initial nuclear
configurations in the form of group multiplication, where an ideal nontilted
prototype of a perovskite $CuO_{6}$ octahedron used as the group unit. In all
”structural reactions” of nuclear configurations of stripes, the tilting angle
$\alpha$ of the perovskite octahedron with respect to the rock salt $LaO$
layers is saved, but not their initial symmetry, as is the case, for example,
in chemical reactions, in accordance with the Woodworth-Hoffman rules.
Woodward and Hoffmann (1971) The number of nuclear configurations of the
observed stripes in $LSCO$, is reduced to one of the two $U(\theta_{U})$
(where $\theta_{U}=45^{\circ},135^{\circ},225^{\circ},315^{\circ}$ ) or
$D(\theta_{D})$ (where $\theta_{D}=0,90^{\circ},180^{\circ},270^{\circ}$)
stripes, and their spatial distribution in $CuO_{2}$ layer is represented by
possible plane graphs with chromatic number $\chi\leq 4$ in the four-color
theorem. In four colors, $U(45^{\circ}),U(225^{\circ})$ \- red (R),
$U(135^{\circ}),U(315^{\circ})$ \- blue (B) and also
$D(0^{\circ}),D(180^{\circ})$ \- green (G); $D(90^{\circ}),D(270^{\circ})$ \-
yellow (Y), we can always color an arbitrary plane map, and the four colors R,
G, B, Y are just four subgroups in $G(\alpha)$. The group can also be
represented as a direct product of subgroups of any different colors, e.g.
$G(\alpha)=R\times B=G\times Y$ and so on. $D(\theta_{D})$ stripes occur as
overlapping different $U(\theta_{U})$ and $U(\theta_{U}\pm 90^{\circ})$
stripes in an $R/G/B$ sequence, for example, when disordering $U(\theta_{U})$
stripes with doping, and vice versa, $U(\theta_{U})$ stripes can occur as the
overlapping $D(\theta_{D})$ and $D(\theta_{D}\pm 90^{\circ})$ stripes in a
$G/R/Y$ sequence. These two stripe sequences can be combined into one, for
example into $Y/R/G/B$, from which, by rotating all tilted $CuO_{6}$ octahedra
around from the $c$ axis by the angle
$\delta\theta=n\cdot\left({45^{\circ}}\right)$, we can obtain three more
$B/Y/R/G$, $G/B/Y/R$ and $R/G/B/Y$. The rotation also leads to an shift over
$G(\alpha)$ group and transverse shift of linear stripe and diagonal shift of
checkerboard structures in Fig.2a,b. This means that the ground state of the
$CuO_{2}$ layer in Figs.2,3 is fourfold degenerate with respect to the
rotation. Indeed, we can also form a novel $JT$ cell (see Fig.3), during the
translation of which the number of hole carriers and spins in the $JT$ cell
are saved. The state $\Psi_{\theta_{n}}$ of the $JT$ cell is fourfold
degenerate by the initial phase
$\theta_{n}=0,45^{\circ},90^{\circ},135^{\circ}$, which splits at tunneling
$CuO_{6}$ octahedra over states of the $D\left({\theta_{D}}\right)$ and
$U\left({\theta_{U}}\right)$ stripes. In particular, if the $D$ and $U$ stripe
widths are equal, any $CuO_{6}$ octahedron can be located simultaneously both
in $D$, and $U$ stripes without a certain the angle $\theta$, and ideal
$CuO_{2}$ layer structure being restored. At the spontaneous $\theta$-symmetry
breaking, a static picture that is formed with the help of $JT$ cell is shown
in Fig.3, where the latter contains more than a dozen $CuO_{6}$ octahedra,i.e.
the four octahedra on the single stripe.

Figure 2:  Regular checkerboard (a) and linear (b) stripe structures
corresponding to the chromatic number $\chi=2$. The arrows show the shift of
the stripe structure when all $CuO_{6}$ octahedra are rotated by the angle
$\delta\theta=45^{\circ}$. The stripe structure are also shown in Fig.3 in
more detail.

Let us consider the $JT$ cell to understand a nature of inhomogeneous
distribution of hole and spin density in the $CuO_{2}$ layer. The cell has a
Jahn-Teller ($JT$) nature, shown in Fig.4, where the (pseudo)$JT$ effect is
non-local, and is associated with the presence of positive charged $LaO$ rock
salt layers. The tilting angle $\alpha_{U}$ at the orientation angle
$\theta_{U}$ corresponds to relaxation $a_{1g}$ modes, and the tilting angle
$\alpha_{D}$ at the orientation angle $\theta_{D}$ corresponds to $JT$ active
$b_{1g}$ modes.

Figure 3:  Structural motive for the fourfold degenerate state
$\Psi_{\theta_{n}}$ of the novel JT cell. On the right is the initial phase
$\theta_{n}$, $\Psi_{\theta_{n}}$ \- color code and coloring of the graph with
$\chi=2$.

The absence of the superconductivity in the AFM $CuO$ layers on graphene Kano
_et al._ (2017); Kvashnin  _et al._ (2019) points to a key role of rock salt
$LaO$ layers in $2D$ doped cuprates. Bianconi  _et al._ (2001) The direct
similarity of the structures of the nondoped $LCO$ and the non-$JT$
$La_{2}NiO_{4}$ materials, Rodriguez-Carvajal  _et al._ (1991) where ions
$Ni^{2+}$ is in the high spin state $S=1$, also points to missing $JT$ effect
in the $N_{0}(d^{9})$ sector of the configuration space of $CuO_{2}$ layer.
However, in the $N_{-}(d^{8})$ sector of doped $LSCO$ the hole carriers are at
the Zhang-Rice singlet state ${}^{1}A_{1g}$, Zhang and Rice (1988) and it’s a
$JT$ state. Bersuker  _et al._ (1992) Moreover, $JT$ effect depends on the
doping concentration in a threshold way. Indeed the hole carriers in
$D(\theta_{D})$ stripes can induce a non-local $JT$ (pseudo-)effect with a
hamiltonian (see Fig.4a,b)

$\hat{H}_{JT}\left({Q_{\alpha},Q_{\theta}}\right)=\varepsilon_{{}^{1}B_{1g}}\sum\limits_{\sigma}{b_{i\sigma}^{+}b_{i\sigma}+}\varepsilon_{{}^{1}A_{1g}}\sum\limits_{\sigma}{a_{i\sigma}^{+}a_{i\sigma}+}V_{b_{1g}}Q_{\alpha}\sum\limits_{\sigma}{\left({a_{i\sigma}^{+}b_{i\sigma}+b_{i\sigma}^{+}a_{i\sigma}}\right)}+U\left({Q_{\alpha},Q_{\theta}}\right)$
(1)

Figure 4:  Graphic scheme: (a) $b_{1g}$ tilting modes active in the non-local
$CuO_{6}$ octahedron $JT$ effect, as well as (b) $a_{1g}$ relaxation modes in
$D$ and $U$ stripes. Arrows in (a) and (b) show the relationship between
tilting and conventional local modes. (c) Non-local $JT$ effect in $CuO_{2}$
layer surrounded by the symmetrical $LaO$ rock salt layers.

in the hole sector $N_{-}(d^{8})$ of the configuration space of the $CuO_{2}$
layer (see Fig.4). Here is
$a_{i\sigma}^{+}=\eta\left(\sigma\right)X_{i}^{{}^{1}B_{1g},\bar{\sigma}_{b}}$
and
$b_{i\sigma}^{+}=\eta\left(\sigma\right)X_{i}^{{}^{1}A_{1g},\bar{\sigma}_{b}}$
are two hole parts of the quasiparticle operator

$\displaystyle
d_{i\lambda\sigma}^{+}==X_{i}^{\sigma_{b},A_{1g}}+\sum\limits_{h={}^{1}A_{g},{}^{1}B_{g}}{\left\langle
h\right|}d_{i\lambda\sigma}^{+}\left|{\sigma_{b}}\right\rangle
X_{i}^{h,\bar{\sigma}_{b}}=$ (2)
$\displaystyle=X_{i}^{\sigma_{b},A_{1g}}+\left\\{{\begin{array}[]{*{20}c}{a_{i\sigma}^{+},\lambda=3z^{2}-r^{2}}\\\
{b_{i\sigma}^{+},\lambda=x^{2}-y^{2}}\\\ \end{array}}\right.$ (5)

generating the two-hole ${}^{1}A_{1g}$ and ${}^{1}B_{1g}$ states in the
$N_{-}(d^{8})$ sector. If
$\left|h\right\rangle=\left|{{}^{1}A_{1g}}\right\rangle$, then the operator
$d_{i\lambda\sigma}^{+}$ takes the well-known form for the Hubbard model.
Hubbard (1964) It is assumed the $U(Q_{\alpha},Q_{\theta})$ potential is
created by the $LaO$ rock salt layers, and the equilibrium ion positions in
$CuO_{2}$ layer can be obtained from the equation
$\frac{{\partial\hat{H}_{JT}\left({Q_{\alpha},Q_{\theta}}\right)}}{{\partial
Q_{\alpha}\partial Q_{\theta}}}=0$. Here we just use the set $\theta_{D}$ and
$\alpha_{D}\approx 14^{\circ}\div 18^{\circ}$, which are observed at $T=100K$
for D stripes. Bianconi  _et al._ (1996) The hole concentration dependence of
the magnitude $\alpha_{D}(x)$ in the $JT$ effect can be obtained using $u/v$
transformation in the hamiltonian (1) with coefficients

$\displaystyle
u^{2}\left({Q_{\alpha}}\right)=\frac{1}{2}\left({1-\frac{{\varepsilon_{{}^{1}A_{1g}}-\varepsilon_{{}^{1}B_{1g}}}}{{D\left({Q_{\alpha}}\right)}}}\right),$
(6) $\displaystyle
v^{2}\left({Q_{\alpha}}\right)=\frac{1}{2}\left({1+\frac{{\varepsilon_{{}^{1}A_{1g}}-\varepsilon_{{}^{1}B_{1g}}}}{{D\left({Q_{\alpha}}\right)}}}\right),$

where
$D\left({Q_{\alpha}}\right)=\sqrt{\left({\varepsilon_{{}^{1}A_{1g}}-\varepsilon_{{}^{1}B_{1g}}}\right)^{2}+4\left({IQ_{\alpha}}\right)^{2}}$
and
$a_{i\sigma}^{+}=v\left({Q_{\alpha}}\right)\tilde{a}_{i\sigma}^{+}+u\left({Q_{\alpha}}\right)\tilde{b}_{i\sigma}^{+}$,
$b_{i\sigma}^{+}=u\left({Q_{\alpha}}\right)\tilde{a}_{i\sigma}^{+}-v\left({Q_{\alpha}}\right)\tilde{b}_{i\sigma}^{+}$.
Then hamiltonian (1) may be written

$\hat{H}_{JT}\left({Q_{\alpha},Q_{\theta_{D}}}\right)=\varepsilon_{+}\left({Q_{\alpha}}\right)\sum\limits_{\sigma}{\tilde{b}_{i\sigma}^{+}\tilde{b}_{i\sigma}}+\varepsilon_{{}_{-}}\left({Q_{\alpha}}\right)\sum\limits_{\sigma}{\tilde{a}_{i\sigma}^{+}\tilde{a}_{i\sigma}},$
(7)

where
$\varepsilon_{\pm}\left({Q_{\alpha}}\right)=\frac{1}{2}\left[{\varepsilon_{{}^{1}A_{1g}}+\varepsilon_{{}^{1}B_{1g}}\pm
D\left({Q_{\alpha}}\right)}\right]$ are the energies of hole quasiparticles in
two-hole singlets $\left|{{}^{1}\tilde{A}_{1g}}\right\rangle$ and
$\left|{{}^{1}\tilde{B}_{1g}}\right\rangle$ taking into account the $JT$
interaction $V_{b_{1g}}$.

The equilibrium tilting angle $\alpha_{D}$ is obtained from the condition
$\frac{{\partial\hat{H}_{JT}\left({Q_{\alpha},Q_{\theta_{D}}}\right)}}{{\partial\alpha}}=0$
for the minimum adiabatic potential
$\varepsilon_{\pm}\left({Q_{\alpha}}\right)$. One at
$U\left({Q_{\alpha},Q_{\theta_{D}}}\right)\approx{{\left({k_{\alpha}\alpha^{2}}\right)}\mathord{\left/{\vphantom{{\left({k_{\alpha}\alpha^{2}}\right)}2}}\right.\kern-1.2pt}2}$
take the form
$\alpha_{D}=\pm\sqrt{\left({\frac{{R_{D}V_{b_{1g}}}}{{k_{\alpha_{D}}}}}\right)^{2}-\left({\frac{\Delta}{{V_{b_{1g}}}}}\right)^{2}}$
where
$R_{D}=\sum\limits_{\sigma}{\left\langle{\tilde{b}_{\sigma}^{+}\tilde{b}_{\sigma}-\tilde{a}_{\sigma}^{+}\tilde{a}_{\sigma}}\right\rangle}$.
Thus, in doped LSCO cuprates there is a hole doping level
$x_{c}=\left\langle{X_{i}^{{}^{1}\tilde{A}_{1g}{}^{1}\tilde{A}_{1g}}-X_{i}^{{}^{1}\tilde{B}_{1g}{}^{1}\tilde{B}_{1g}}}\right\rangle=\frac{{k_{\alpha}\Delta}}{{2V_{b_{1g}}^{2}}}$
, above which in $D$ stripes, a $JT$ pseudo effect with a $\alpha_{D}\neq 0$
can be observed. At
$\Delta=\varepsilon_{{}^{1}B_{1g}}-\varepsilon_{{}^{1}A_{1g}}=0$ Gavrichkov
and Gavrichkov (1990) we obtain the quadratic concentration dependence of the
$JT$ contribution:

$\hat{H}_{JT}\left({Q_{\alpha_{D}},Q_{\theta_{D}}}\right)=-E_{JT}R\sum\limits_{\sigma}{\left({\tilde{b}_{i\sigma}^{+}\tilde{b}_{i\sigma}-\tilde{a}_{i\sigma}^{+}\tilde{a}_{i\sigma}}\right)},$
(8)

where
$E_{JT}={{V_{b_{1g}}^{2}}\mathord{\left/{\vphantom{{V_{b_{1g}}^{2}}{2k_{\alpha}}}}\right.\kern-1.2pt}{2k_{\alpha}}}$.
Any homogeneous hole density at $x<x_{c}$ will be unstable to creating the
local $D$ areas with a higher hole density $x_{D}>x_{c}$. Without a dependence
of elasticity value $k_{\alpha}\left(x\right)$, a minimum of the adiabatic
potential corresponds to the hole concentration $x=1$.

##  III  Electron-hole pairs in superexchange interaction

3.1 Let’s consider the superexchange interaction $J_{ij}$ in a static stripe
structure in Fig.3. In the case of spontaneous $\theta$-symmetry breaking (the
JT cell state $\Psi_{\theta_{n}}$ is nondegenerate), the doped LSCO cuprate
has a static nanostructure, where the exchange interaction are missing in the
$D$ stripes, and in the $U$ stripes the superexchange interaction is equal to
the one in nondoped $LCO$ material:

Figure 5:  Graphic representation of electron-hole pairs for interacting $i$
-th and $j$ \- th $Cu^{2+}$ ions. Each pair corresponds to the double exchange
loop (solid or dotted lines)

$\hat{H}_{S}=-\sum\limits_{ij}{J_{ij}^{tot}\hat{S}_{i}\hat{S}_{j}}$ (9)

where the interaction
$J_{ij}^{tot}=\sum\limits_{h,e}{J_{ij}\left({h,e}\right)}$ can be calculated
in the adiabatic approximation. Superexchange interaction (9) arises as a
result of the superposition of contributions from all possible virtual
electron-hole pairs on interacting $i$ -th and $j$-th ions (see Fig.5). The
sign of the $J_{ij}\left({h,e}\right)$ contribution to the $J_{ij}^{tot}$
interaction is determined by a simple rule. If the spins of an electron and a
hole in a virtual pair are equal , $\hat{S}_{h}=\hat{S}_{e}$ this is the AFM
contribution. If $\hat{S}_{h}=\hat{S}_{e}\pm 1$ it’s FM contribution. Other
terms are missing, and all electron-hole wave functions $\left|h\right\rangle$
and $\left|e\right\rangle$ in the hopping integral
$t_{ij}^{hn,en}=\sum\limits_{\lambda\lambda^{\prime}}{t_{ij}\left({\lambda\lambda^{\prime}}\right)\sum\limits_{\sigma}{\gamma_{\lambda\sigma}^{*}\left({hn}\right)\gamma_{\lambda^{\prime}\sigma}\left({en}\right)}}$
Gavrichkov  _et al._ (2020, 2017); Mikhaylovskiy  _et al._ (2020) in the
superexchange

$\displaystyle
J_{ij}^{tot}=\sum\limits_{h,e}{J_{ij}\left({hn,en}\right)},J_{ij}\left({hn,en}\right)=\frac{{2\left({t_{ij}^{hn,en}}\right)^{2}}}{{\Delta\left({hn,en}\right)}},$
(10)
$\displaystyle\Delta\left({hn,en}\right)=\varepsilon_{h}+\varepsilon_{e}-2\varepsilon_{n},$

are calculated in the adiabatic approximation. The pairs of indices $hn$ and
$en$ run over all possible quasiparticle excitations $\left({n,e}\right)$ and
$\left({n,h}\right)$ between many-electron states $\left|n\right\rangle$ and
$\left|{e\left(h\right)}\right\rangle$ with energies $\varepsilon_{n}$ and
$\varepsilon_{n}$ in the sectors $N_{0}$ and $N_{\pm}$ of configuration space
in Fig.6a,b.

Figure 6: Graphic scheme of energy levels of $CuO_{6}$ octahedron in different
sectors $N_{0}$ and $N_{\pm}$ configuration space in $U$ and $D$ stripe of the
$CuO_{2}$ layer. (a) The dashed line shows the level of the hypothetical
nontilted octahedron $\left|{{}^{1}b_{1g}^{\left(0\right)}}\right\rangle$, as
well as a symmetrical $a_{1g}$ relaxation processes. (b) The wavy line shows
the JT interaction between $\left|{{}^{1}\tilde{A}_{1g}}\right\rangle$ and
$\left|{{}^{1}\tilde{B}_{1g}}\right\rangle$ states in $N_{-}$ sector. Loops
show the virtual electron-hole pairs on interacting $i$ -th and $j$ \- th
$Cu^{2+}$ ions.

These quasiparticle excitations are described by nondiagonal elements
$t_{ij}^{nh,ne}$. In Hubbard’s model there is only one such element
corresponding to the excitations between lower and upper Hubbard bands. In
cuprates, the main contribution from the electron-hole pair with
$\left|n\right\rangle={}^{2}b_{1g}$, $\left|h\right\rangle={}^{1}A_{1g}$ and
$\left|e\right\rangle={}^{1}A$ has an AFM character $J_{ij}^{tot}\approx
J_{ij}^{\left(U\right)}\left({{}^{1}A{}^{2}b_{1g},{}^{1}A_{1g}{}^{2}b_{1g}}\right)$
(see Fig.6a,b), Gavrichkov  _et al._ (2016) where

$J_{ij}^{\left(U\right)}\left({{}^{1}A{}^{2}b_{1g},{}^{1}A_{1g}{}^{2}b_{1g}}\right)=\frac{{2t_{ij}^{2}}}{{\Delta\left({{}^{1}A{}^{2}b_{1g},{}^{1}A_{1g}{}^{2}b_{1g}}\right)}}\approx
0.15eV$ (11)

where
$\Delta\left({{}^{1}A{}^{2}b_{1g},{}^{1}A_{1g}{}^{2}b_{1g}}\right)=\varepsilon_{{}^{1}A_{1g}}-2\left({\varepsilon_{{}^{2}b_{1g}}-V_{a_{1g}}\alpha_{U}}\right)=U_{H}$
and
$\varepsilon_{{}^{1}A_{1g}}=2\left({\varepsilon_{{}^{2}b_{1g}}-V_{a_{1g}}\alpha_{U}}\right)+U_{H}$
. However, the transverse percolation of the interaction exchange interaction
$J_{i_{R}j_{B}}^{tot}$, where $i_{R}$ and $j_{B}$ octahedra $CuO_{6}$ are into
different red(R) and blue(B) stripes at the static regular Y/R/G/B… stripe
structure, is unlikely.

3.2 If there is no spontaneous $\theta$-symmetry breaking and the JT cell
state is four-fould degenerate, then the exchange interaction in the JT cell
presented above is incomplete. In the novel configutation space, the states of
$CuO_{6}$ octahedra with the tilting $\alpha_{D}$ angle and orientation
$\theta_{D}$ in $U$ stripes, and with the tilting $\alpha_{U}$ angle and
orientation $\theta_{U}$ in $D$ stripes are impossible. It’s just that the
$i$-th $CuO_{6}$ octahedron (the black point in Fig.7b) can simultaneously be
at the $U$ and $D$ stripe states of any color of four R,B and G,Y possible.
The novel quasiparticle excitations do not destroy the configuration space, if
these quasiparticles are accompanied by changes
$\delta\alpha=\alpha_{D}-\alpha_{U}$ and $\delta\theta=\theta_{D}-\theta_{U}$
shown in Fig.7)b

Figure 7: (a) Graphical representation of anisotropic non-local effects in the
$CuO_{6}$ octahedron located in the JT cell with fourfold degeneracy. (b)
Graphical diagram of electron-hole pairs contributing to the exchange
interaction when both $i$ -th and $j$ \- th magnetic ions are in a JT cell
with fourfold degeneracy. Here, the $U$ and $D$ stripe widths are shown the
same to demonstrate the appearance of an ideal $CuO_{2}$ layer with mobile
hole carriers only at the equal stripe widths

To obtain the contributions into $J_{ij}\left({hn,en}\right)$ from the
electron-hole pairs with changing angles $\delta\alpha$ and $\delta\theta=\pm
45^{\circ}$ in the JT cell we need $i$-th ($CuO_{6}$ octahedron
$\left|{h_{\theta_{D}\alpha_{D}}}\right\rangle$ and
$\left|{n_{\theta_{U}\alpha_{U}}}\right\rangle$ states, where the indices $U$
and $D$ denote the stripe affiliation for the $i$-th octahedron

$\left|{h_{\theta_{D}\alpha_{D}}}\right\rangle=\left|{\tilde{h}}\right\rangle\left|{\chi_{\theta_{D}}\chi_{\alpha_{D}}}\right\rangle;\left|{n_{\theta_{U}\alpha_{U}}}\right\rangle=\left|n\right\rangle\left|{\chi_{\theta_{U}}\chi_{\alpha_{U}}}\right\rangle$
(12)

where
$\left|{\tilde{h}}\right\rangle=\left|{{}^{1}\tilde{A}_{1g}}\right\rangle,\left|{{}^{1}\tilde{B}_{1g}}\right\rangle$
and $\left|n\right\rangle=\left|{{}^{2}b_{1g}}\right\rangle$. Each of R,G,B,Y
colors in Fig.7b corresponds to a harmonic oscillator wave function
$\left|{\chi_{\theta}}\right\rangle$ for a displaced 2D oscillator. These
states differ by
$\left|{\chi_{\alpha_{D}}\left({Q_{\alpha}+\alpha_{D}}\right)}\right\rangle$,
$\left|{\chi_{\theta_{D}}\left({Q_{\theta_{D}}}\right)}\right\rangle$ and
$\left|{\chi_{\alpha_{U}}\left({Q_{\alpha}+\alpha_{U}}\right)}\right\rangle$,
$\left|{\chi_{\theta_{U}}\left({Q_{\theta_{U}}}\right)}\right\rangle$, and the
matrix elements $\gamma_{\lambda\sigma}\left(r\right)$ in Eq.(III) at the
$i$-th $CuO_{6}$ octahedron cell will be calculated on the functions
$\left|{h_{\theta_{D}\alpha_{D}}}\right\rangle$ and
$\left|{n_{\theta_{U}\alpha_{U}}}\right\rangle$ located at the four equivalent
nuclear configurations of the JT cell corresponding to the phase $\theta_{n}$
(see black dot in Fig.7b).

In other words, the hole creation in the $i$-th $CuO_{6}$ octahedron of the
$U\left({\theta_{U}}\right)$ stripe is possible, but only if one also belongs
to the $D\left({\theta_{D}}\right)$ stripe, where
$\delta\theta=\theta_{D}-\theta_{U}=\pm 45^{\circ}$. Thus, the matrix elements
$\gamma_{\lambda\sigma}\left({hn}\right)$ contain the vibronic reduction
factors or partial dynamical quenching: Ham (1965)

$\displaystyle
J_{ij}\left({{}^{1}\tilde{A}_{1g},{}^{1}A_{1}}\right)=\frac{{2\left({t_{ij}^{{}^{1}\tilde{A}_{1g}{}^{2}b_{1g},{}^{1}A_{1}{}^{2}b_{1g}}}\right)^{2}}}{{\Delta\left({{}^{1}\tilde{A}_{1g}{}^{2}b_{1g},{}^{1}A_{1}{}^{2}b_{1g}}\right)}}\approx$
(13) $\displaystyle\approx\frac{{2u^{2}\left({\alpha_{D}}\right)\cdot
t_{ij}^{2}\left({x^{2},x^{2}}\right)}}{{\Delta\left({{}^{1}\tilde{A}_{1g}{}^{2}b_{1g},{}^{1}A_{1}{}^{2}b_{1g}}\right)}}\cdot\exp\left\\{{-\frac{{\delta\theta^{2}+\delta\alpha^{2}}}{4}}\right\\}$

where

$\begin{array}[]{l}t_{ij}^{{}^{1}\tilde{A}_{1g}{}^{2}b_{1g},{}^{1}A_{1}{}^{2}b_{1g}}=t_{ij}\left({x^{2},x^{2}}\right)\sum\limits_{\sigma}{\gamma_{x^{2}\sigma}^{*}\left({{}^{1}\tilde{A}_{1g},{}^{2}b_{1g}}\right)\gamma_{x^{2}\sigma}\left({{}^{1}A_{1},{}^{2}b_{1g}}\right)}+t_{ij}\left({z^{2},z^{2}}\right)\sum\limits_{\sigma}{\gamma_{z^{2}\sigma}^{*}\left({{}^{1}\tilde{B}_{1g},{}^{2}b_{1g}}\right)\gamma_{z^{2}\sigma}\left({{}^{1}\tilde{B}_{1},{}^{2}b_{1g}}\right)}\approx\\\
\approx t_{ij}\left({x^{2},x^{2}}\right)\cdot
u\left({\alpha_{D}}\right)\cdot\left\langle{\chi_{i\theta_{D}}}\right|\left.{\chi_{i\theta_{U}}}\right\rangle\left\langle{\chi_{i\alpha_{D}}}\right|\left.{\chi_{i\alpha_{U}}}\right\rangle,\\\
\end{array}$ (14)

where
$\Delta\left({{}^{1}\tilde{A}_{1g}{}^{2}b_{1g},{}^{1}A_{1}{}^{2}b_{1g}}\right)=2\left[{\varepsilon_{{}_{-}}\left({\alpha_{D}}\right)-\varepsilon_{{}^{2}b_{1g}}}\right]+U_{H}$
, and $t_{ij}\left({x^{2},x^{2}}\right)\gg t_{ij}\left({z^{2},z^{2}}\right)$.
The matrix elements in Eq.(III) are equal to the following expression

$\gamma_{x^{2}\sigma}\left({{}^{1}\tilde{A}_{1g},{}^{2}b_{1g}}\right)=u\left({\alpha_{D}}\right)\cdot\left\langle{\chi_{i\theta_{D}}}\right|\left.{\chi_{i\theta_{U}}}\right\rangle\cdot\left\langle{\chi_{i\alpha_{D}}}\right|\left.{\chi_{i\alpha_{U}}}\right\rangle$
(15)

where

$\begin{array}[]{l}\left\langle{\chi_{i\theta_{D}}}\right|\left.{\chi_{i\theta_{U}}}\right\rangle\cdot\left\langle{\chi_{i\alpha_{D}}}\right|\left.{\chi_{i\alpha_{U}}}\right\rangle=\\\
\left\langle{\chi_{i\alpha_{D}}\left({Q_{\alpha}+\alpha_{D}}\right)\chi_{i\theta_{D}}\left({\theta_{D}}\right)}\right|\left.{\chi_{i\alpha_{U}}\left({Q_{\alpha}+\alpha_{U}}\right)\chi_{i\theta_{U}}\left({\theta_{U}}\right)}\right\rangle\\\
\approx\exp\left\\{{-{{\left({\delta\theta^{2}+\delta\alpha^{2}}\right)}\mathord{\left/{\vphantom{{\left({\delta\theta^{2}+\delta\alpha^{2}}\right)}4}}\right.\kern-1.2pt}4}}\right\\}\\\
\end{array}$ (16)

and $\delta\alpha=\alpha_{D}-\alpha_{U}\approx 5\div 10^{\circ}$. Thus, at an
absence of spontaneous $\theta$-symmetry breaking, the homogeneous
superexchange interaction in the hole-doped $LSCO$ cuprates is equal to

$J_{ij}^{tot}\approx
J_{ij}\left({{}^{1}\tilde{A}_{1g}{}^{2}b_{1g},{}^{1}A_{1}{}^{2}b_{1g}}\right)\cdot
u^{2}\left({\alpha_{D}}\right)\cdot\exp\left\\{{-\frac{{\delta\theta^{2}+\delta\alpha^{2}}}{4}}\right\\}$
(17)

Note that, even with the at $\delta\alpha\to 0$ (weak JT effect), the result
(17) does not reduce to Eq.(11). The result is independent of the choice of a
$i$ and $j$ pair of interacting $CuO_{6}$ octahedra and the exchange
$J_{ij}^{tot}$ is homogeneous in the ideal $CuO_{2}$ layer with nonlocal
effects. Indeed the neutron experiments in $LCO$ cuprates demonstrate the
termal anisotropic motion Hafliger  _et al._ (2014) similar to the oxygen ions
motion shown in Fig.7a. Similar reduction factors in the $n$ doped $LNCO$
cuprates are impossible, since there is no JT effect in the
$N_{+}\left({d^{10}}\right)$ electron sector.

##  IV  Conclusions

The JT pseudo effect in the $N_{-}\left({d^{8}}\right)$ hole sector with a
hole concentration exceeding a threshold magnitude $x_{c}$ is accompanied by
the charge and spin inhomogeneities due to the nonzero tilting angle
$\alpha_{D}\approx 10\div 14^{\circ}$ of $CuO_{6}$ octahedra at the
orientation angle $\theta_{D}$ ($b_{1g}$-tilting mode) in
$D\left({\theta_{D}}\right)$ stripes surrounded by the two rock salt $LaO$
layers. $U\left({\theta_{U}}\right)$ stripes with $CuO_{6}$ octahedra tilted
by the angle $\alpha_{U}\approx 5^{\circ}$ and oriented at the angle
$\theta_{U}$ form dielectric regions. Note our arguments in favor of the JT
nature of the charge inhomogeneity is an alternative to that competition
between kinetic energy and Coulomb repulsion could cause holes to segregate
into strip structures. Imada (2021)

The observed regular line and checkerboard stripe structures with the
chromatic number $\chi=2$ generate a novel element of symmetry in $CuO_{2}$
layer: simultaneously rotating all tilted $CuO_{6}$ octahedra by the
$\delta\theta=n\cdot\left({45^{\circ}}\right)$ angle around the c axis.Wherein
we can choose the novel JT cell, during the translation of which the initial
structure and the number of hole carriers and spins in the JT cell are saved.
The state $\Psi_{\theta_{n}}$ of the JT cell is fourfold degenerate by the
initial phase $\theta_{n}=0,45^{\circ},90^{\circ},135^{\circ}$, and all four
states can be obtained by the novel symmetry operation. In particular, any
$CuO_{6}$ octahedron can be located simultaneously, both in
$U\left({\theta_{U}}\right)$ and $D\left({\theta_{D}}\right)$ stripes without
a well specific orientation $\theta_{D}\left({\theta_{U}}\right)$ and tilting
$\alpha_{D}\left({\alpha_{U}}\right)$ angles. The exchange interaction
$J_{ij}^{tot}$ will be different depending on whether spontaneous phase
$\theta$-symmetry breaking occurs or not:
(i) The spontaneous $\theta$-symmetry breaking leads to a static spatial
distribution of the $U\left({\theta_{U}}\right)$ and
$D\left({\theta_{D}}\right)$ stripes in doped $LSCO$. In the static structure
of the $CuO_{2}$ layer, the superexchange interaction has Anderson’s form
$J_{ij}^{\left(U\right)}={{2t^{2}}\mathord{\left/{\vphantom{{2t^{2}}{U_{H}}}}\right.\kern-1.2pt}{U_{H}}}$
for interacting ions $Cu^{2+}$ with spin
${1\mathord{\left/{\vphantom{12}}\right.\kern-1.2pt}2}$ , but is limited in
space by $U\left({\theta_{U}}\right)$ stripes with zero hole concentration
$x=0$. However, a transverse percolation of the homogeneous exchange
interaction in a static regular line Y/R/G/B and checkerboard stripe
structures is unlikely.
(ii) Without spontaneous $\theta$-symmetry breaking, a novel JT cell can be
constructed in the periodic stripe nanostructure with the fourfold degenerate
$\Psi_{\theta_{n}}$ state. The ideal $CuO_{2}$ structure with nonlocal
anisotropic effects and the homogeneous superexchange are restored. The stripe
boundaries, spin and charge inhomogeneities disappear. The superexchange is
suppressed from its magnitude $J_{ij}^{\left(U\right)}$ (in the nondoped $LCO$
cuprate) by the exponential factor $J_{ij}^{tot}\approx
J_{ij}^{\left(U\right)}\cdot
u^{2}\left({\alpha_{D}}\right)\cdot\exp\left\\{{-{{\left({\delta\alpha_{D}^{2}+\delta\theta_{D}^{2}}\right)}\mathord{\left/{\vphantom{{\left({\delta\alpha_{D}^{2}+\delta\theta_{D}^{2}}\right)}4}}\right.\kern-1.2pt}4}}\right\\}$
due to dynamical quenching for the $CuO_{6}$ octahedron surrounded by two
$LaO$ rock salt layers. The conclusion does not contradict the results, where
the low-temperature optical pump, soft x-ray probe measurements Mitrano  _et
al._ (2019) detected the unexpected gapless nature of the charge order.
Indeed, a characteristic tunneling JT splitting $\leq
10sm^{-1}\left({1meV}\right)$ Bersuker (2010) has the observed energy scale of
the transverse fluctuations observed in $LBCO$. Mitrano  _et al._ (2019) As
follows from Fig.7b, the doping dependence of superexchange manifests itself
in the form of inhomogeneity only at the different stripe widths, since the
$CuO_{6}$ octahedra located in the $D(U)$ stripes begin to predominate at
increasing(decreasing) hole concentration $x$, and therefore the
$\theta$-symmetry disappears.

(iii) In n-type $LNCO$ cuprates, the JT effect in $N_{+}\left({d^{10}}\right)$
configuration sector is impossible. In contrast to p-type $LSCO$, the
superexchange interaction $J_{ij}$ in n-type $LNCO$ cuprates should not be
subject to dynamical quenching, but disappear at the high electron doping.

This work was supported by Russian Science Foundation research grant RSF
No.22-22-00298.

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# Chest X-ray lung and heart segmentation based on minimal training sets

Balázs Maga Eötvös Loránd University, Pázmány Péter sétány 1/C, Budapest,
H-1117 Hungary, email<EMAIL_ADDRESS>
###### Abstract

As the COVID-19 pandemic aggravated the excessive workload of doctors
globally, the demand for computer aided methods in medical imaging analysis
increased even further. Such tools can result in more robust diagnostic
pipelines which are less prone to human errors. In our paper, we present a
deep neural network to which we refer to as Attention BCDU-Net, and apply it
to the task of lung and heart segmentation from chest X-ray (CXR) images, a
basic but ardous step in the diagnostic pipeline, for instance for the
detection of cardiomegaly. We show that the fine-tuned model exceeds previous
state-of-the-art results, reaching $98.1\pm 0.1\%$ Dice score and $95.2\pm
0.1\%$ IoU score on the dataset of Japanese Society of Radiological Technology
(JSRT). Besides that, we demonstrate the relative simplicity of the task by
attaining surprisingly strong results with training sets of size 10 and 20: in
terms of Dice score, $97.0\pm 0.8\%$ and $97.3\pm 0.5$, respectively, while in
terms of IoU score, $92.2\pm 1.2\%$ and $93.3\pm 0.4\%$, respectively. To
achieve these scores, we capitalize on the mixup augmentation technique, which
yields a remarkable gain above $4\%$ IoU score in the size 10 setup.

##  1 Introduction

All around the world, a plethora of radiographic examinations are performed
day to day, producing images using different imaging modalities such as X-ray,
computed tomography (CT), diagnostic ultrasound and magnetic resonance imaging
(MRI). According to the publicly available, official data of the National
Health Service ([13]), in the period from February 2017 to February 2018, the
count of imaging activity was about 41 million only in England. The thorough
examination of the vast quantity of these images imposes a huge workload on
radiologists, which increases the number of the avoidable human mistakes.
Consequently, automated methods aiding the diagnostic processes are sought-
after.

The examination of medical images customarily includes various segmentation
tasks, in which detecting and pixelwise annotating different tissues and
certain anomalies are vital. Common examples include lung nodule segmentation
in the diagnosis of lung cancer, lung and heart segmentation in the diagnosis
of cardiomegaly, or plaque segmentation in the diagnosis of thrombosis. Even
in the case of 2-dimensional modalities, such segmentation tasks can be
extremely time-demanding, and the situation gets even worse in three
dimension. Taking into consideration that these tasks are easier to formalize
as a standard computer vision exercise than the identification of a particular
disease, it is not surprising that they sparked much activity in the field of
automated medical imaging analysis.

Semantic segmentation – that is assigning a pre-defined class to each pixel of
an image – requires a high level of visual understanding, in which state-of-
the-art performance is attained by methods utilizing Fully Convolutional
Networks (FCN) [4]. An additional challenge of the field is posed by the
strongly limited quantity of training data on which one train machine learning
models, as annotating medical imaging requires specialists in contrast to
“real-life” images. To overcome this difficulty, the so-called U-Net
architecture was proposed: its capability to being efficiently trained on
small datasets has been demonstrated in [5]. Over the past few years several
modifications and improvements have been proposed on the original
architecture, some of which involved different attention mechanisms designed
to help the network to detect the important parts of the images.

In the present paper we introduce a new network primarily based on the ideas
of [12] and [8], to which we refer to as Attention BCDU-Net. We optimize its
performance through hyperparameter tests on the depth of the architecture and
the loss function, and we demonstrate the superiority of the resulting model
compared to the state-of-the-art network presented in [15] in the task of lung
and heart segmentation on chest X-rays. Besides that, we will also give an
insight into two interesting phenomena arising during our research which might
be interesting for the AI medical imaging community: one of them is the very
small data requirement of this particular task, while the other one is the
peculiar evolution of the loss curves over the training.

##  2 Deep learning approach

###  2.1 Related work

As already mentioned in Section 1, [5] introducing U-Nets is of paramount
importance in the field. Since then U-Nets have been used to cope with diverse
medical segmentation tasks, and numerous papers aimed to design U-Net variants
and mechanisms such that the resulting models tackles better the problem
considered. Some of these paid primary attention to the structure of the
encoder and the decoder – that is the downsampling and the upsampling path –
of the original architecture. For example in [18], the authors developed a
network (CoLe-CNN) with multiple decoder branches and Inception-v4 inspired
encoder to achieve state-of-the-art results in 2-dimensional lung nodule
segmentation. In [10] and [14], the authors introduced U-Net++, a network
equipped with intermediate upsampling paths and additional convolutional
layers, leading to essentially an efficient ensemble of U-Nets of varying
depths, and demonstrated its superiority compared to the standard U-Net in
many image domains. Other works put emphasis on the design of skip connections
and the way the higher resolution semantic information joins the features
coming through the upsampling branch. In [12], the authors proposed the
architecture BCDU-Net, in which instead of the simple concatenation of the
corresponding filters, the features of different levels are fused using a
bidirectional ConvLSTM layer, which introduces nonlinearity into the model at
this point and makes more precise segmentations available. In [8] it has been
shown that for medical image analysis tasks the integration of so-called
Attention Gates (AGs) improved the accuracy of the segmentation models, while
preserving computational efficiency. In [15], this network was enhanced by a
critic network in a GAN-like scheme following [9], and achieved state-of-the-
art results in the task of lung and heart segmentation. Other attention
mechanisms were introduced in [17] and in [16].

###  2.2 Our proposal

The network architecture Attention BCDU-Net we propose is a modification of
the Attention U-Net, shown at Figure 1.

Figure 1: Schematic architecture of Attention U-Net [8].

Figure 2: Schematic figure of the attention gate used in Attention U-Net [8],
the tensor addition to alter is highlighted by an arrow.

In [12], the authors demonstrated that it is beneficial to use bidirectional
ConvLSTM layers to introduce nonlinearity in the step of merging semantic
information gained through skip connections and the features arriving through
the decoder. This inspired us to modify the attention gates (see Figure 2) in
a similar manner, in which these pieces of information are merged via tensor
addition, that is a linear operation as well. This addition is replaced by a
bidirectional ConvLSTM layer, to which the output of $W_{g}$ and $W_{x}$ – the
processed features and the semantic information, respectively – is fed in this
order. We note that to our best knowledge, there is a slight ambiguity about
the structure of the resampling steps in the attention gate: while the
official implementation is in accordance with the figure, there are widely
utilized implementations in which the output of $W_{g}$ is upsampled instead
of downsampling the output of $W_{x}$ in order to fit their shape. We tested
both solutions and did not experience a measurable difference in the
performance. We also experimented with the usage of additional spatial and
channel attention layers as proposed by [17], however, we found that it does
not improve the performance of our model.

The depth of the network is to be determined by hyperparameter testing. Our
tests confirmed that four downsampling steps results in the best performance,
however, the differences are minuscule.

###  2.3 Loss function

A standard score to compare segmentations is the Intersection over Union
(IoU): given two sets of pixels $X,Y$, their IoU is

$IoU(X,Y)=\frac{|X\cap Y|}{|X\cup Y|}.$

In the field of medical imaging, Dice Score Coefficient (DSC) is probably the
most widespread and simple way to measure the overlap ratio of the masks and
the ground truth, and hence to compare and evaluate segmentations. It is a
slight modification of IoU: given two sets of pixels $X,Y$, their DSC is

$DSC(X,Y)=\frac{2|X\cap Y|}{|X|+|Y|}.$

If $Y$ is in fact the result of a test about which pixels are in $X$, we can
rewrite it with the usual notation true/false positive (TP/FP), false negative
(FN) to be

$DSC(X,Y)=\frac{2TP}{2TP+FN+FP}.$

We would like to use this concept in our setup. The class $c$ we would like to
segment corresponds to a set, but it is more appropriate to consider its
indicator function $g$, that is $g_{i,c}\in\\{0,1\\}$ equals 1 if and only if
the $i$th pixel belongs to the object. On the other hand, our prediction is a
probability for each pixel denoted by $p_{i,c}\in[0,1]$. Then the Dice Score
of the prediction in the spirit of the above description is defined to be

$DSC=\frac{\sum_{i=1}^{N}p_{i,c}g_{i,c}+\varepsilon}{\sum_{i=1}^{N}\left(p_{i,c}+g_{i,c}\right)+\varepsilon},$

where $N$ is the total number of pixels, and $\varepsilon$ is introduced for
the sake of numerical stability and to avoid divison by 0. The IoU of the
prediction can be calculated in a similar manner. The linear Dice Loss (DL) of
the multiclass prediction is then

$DL=\sum_{c}\left(1-DSC_{c}\right).$

A deficiency of Dice Loss is that it penalizes false negative and false
positive predictions equally, which results in high precision but low recall.
For example practice shows that if the region of interests (ROI) are small,
false negative pixels need to have a higher weight than false positive ones.
Mathematically this obstacle is easily overcome by introducing weights
$\alpha,\beta$ as tuneable parameters, resulting in the definition of Tversky
similarity index [1]:

$TI_{c}=\frac{\displaystyle\sum_{i=1}^{N}p_{i,c}g_{i,c}+\varepsilon}{\displaystyle\sum_{i=1}^{N}p_{i,c}g_{i,c}+\alpha\displaystyle\sum_{i=1}^{N}p_{i,\overline{c}}g_{i,c}+\beta\displaystyle\sum_{i=1}^{N}p_{i,c}g_{i,\overline{c}}+\varepsilon},$

where $p_{i,\overline{c}}=1-p_{i,c}$ and $g_{i,\overline{c}}=1-g_{i,c}$, that
is the overline simply stands for describing the complement of the class.
Tversky Loss is obtained from Tversky index as Dice Loss was obtained from
Dice Score Coefficient:

$TL=\sum_{c}\left(1-TI_{c}\right).$

Another issue with the Dice Loss is that it struggles to segment small ROIs as
they do not contribute to the loss significantly. This difficulty was
addressed in [11], where the authors introduced the quantity Focal Tversky
Loss in order to improve the performance of their lesion segmentation model:

$FTL=\sum_{c}\left(1-TI_{c}\right)^{\gamma^{-1}},$

where $\gamma\in[1,3]$. In practice, if a pixel with is misclassified with a
high Tversky index, the Focal Tversky Loss is unaffected. However, if the
Tversky index is small and the pixel is misclassified, the Focal Tversky Loss
will decrease significantly.

In our work we use multiclass DSC and IoU to evaluate segmentation
performance. As our initial tests demonstrated that training our network with
Focal Tversky loss results in better scores, we will use this loss function.
The optimal $\alpha,\beta,\gamma$ parameters should be determined by extensive
hyperparameter testing and grid search. We worked below with
$\alpha=0.6,\beta=0.4,\frac{1}{\gamma}=0.675$.

###  2.4 Dataset and preprocessing

For training- and validation data, we used the public Japanese Society of
Radiological Technology (JSRT) dataset ([3]), available at [2]. The JSRT
dataset contains a total of 247 images, all of them are in $2048\times 2048$
resolution, and have 12-bit grayscale levels. Both lung and heart segmentation
masks are available for this dataset.

In terms of preprocessing, similarly to [15], the images were resized to the
resolution $512\times 512$ first. As X-rays are grayscale images with
typically low contrast, which makes their analysis a difficult task. This
obstacle might be overcome by using some sort of histogram equalization
technique. The idea of standard histogram equalization is spreading out the
the most frequent intensity values to a higher range of the intensity domain
by modifying the intensities so that their cumulative distribution function
(CDF) on the complete modified image is as close to the CDF of the uniform
distribution as possible. Improvements might be made by using adaptive
histogram equalization, in which the above method is not utilized globally,
but separately on pieces of the image, in order to enhance local contrasts.
However, this technique might overamplify noise in near-constant regions,
hence our choice was to use Contrast Limited Adaptive Histogram Equalization
(CLAHE), which counteracts this effect by clipping the histogram at a
predefined value before calculating the CDF, and redistribute this part of the
image equally among all the histogram bins.

###  2.5 Data augmentation

Concerning data augmentation, we follow [7], in which the method mixup was
used to improve glioma segmentation on brain MRI’s. This slightly counter-
intuitive augmentation technique was introduced by [6]: training data samples
are obtained by taking random convex combinations of original image-mask
pairs. That is, for $(x_{1},y_{1})$ and $(x_{2},y_{2})$ image-mask pairs, we
create a random mixed up pair $x=\lambda x_{1}+(1-\lambda)x_{2}$, $y=\lambda
y_{1}+(1-\lambda)y_{2}$, where $\lambda$ is chosen from the beta distribution
$B(\delta,\delta)$ for some $\delta\in(0,\infty)$. In each epoch, the original
samples are paired randomly, hence during the course of the training, a
multitude of training samples are fed to the network. (From the mathematical
point of view, as the coefficient $\lambda$ is chosen independently in each
case from a continuous probability distribution, the network will encounter
pairwise distinct mixed up training samples with probability 1, modulo
floating point inaccuracy.) In [6], the authors argue that generating training
samples via this method encourages the network to behave linearly in-between
training examples, which reduces the amount of undesirable oscillations when
predicting outside the training examples.

The choice of $\delta$ should be determined by hyperparameter testing for any
network and task considered. In [6], $\delta\in[0.1,0.4]$ is proposed, while
in [7] $\delta=0.4$ is applied.

##  3 Experiments

###  3.1 Training schedule

In our main tests, the JSRT dataset was randomly split so that 85% of it was
used for training and the rest for validation and testing. This split was
carried out independently in each case, enhancing the robustness of our
results. Besides that, we also experimented with small dataset training, in
which rather modest sets of 10 and 20 X-rays was utilized as training set.
(The test set remained the same.) It enabled us to measure the benefits of
mixup more transparently. In each of these cases, we trained our network with
Adam optimizer: in the former case, for 50 epochs, while in the latter cases
for 1000 and 500 epochs, respectively. As these epoch numbers are
approximately inversely proportional to the size of the training sets, these
choices correspond to each other in terms of training steps.

###  3.2 Results

Table 1 summarizes the numerical results we obtained during the testing of
Attention BCDU-Net with different train sizes and choices of $\delta$, while
Figure 3-5 display visual results. Note that the highest DSC scores slightly
exceed the ones attained by the state-of-the-art, adversarially enhanced
Attention U-Net introduced in [15] ($97.6\pm 0.5\%$) and admit higher
stability. The effect of augmentation is the most striking in the case of
training on an X-ray set of size 10, when the choice $\delta=0.2$ results in a
5% increase of IoU compared to the no mixup case. In general, we found this
case particularly interesting: it was surprising that we could achieve IoU and
DSC scores of this magnitude using such a small training set. Nevertheless the
predictions have some imperfections, displayed by Figure 3: the contours of
the segmentation are less clear and both the heart and the lung segmentation
tend to contain small spots far from the ground truth. However, such
conspicuous faults are unlikely to occur in the case of the best models for 20
train X-rays (Figure 4), which is still remarkable. The sufficiency of such
small training sets is probably due to the relative simplicity of the task.
Notably, lung and heart regions admit large similarity across a set of chest
X-rays, and they are strongly correlated with simple intensity thresholds.
Consequently, even small datasets have high representing potential. We note
that as $\delta$ gets smaller, the probability density function of
$B(\delta,\delta)$ gets more strongly skewed towards the endpoints of the
interval $[0,1]$, which results in mixed up samples being closer to original
samples in general. The perceived optimality of $\delta=0.2$ in the small
dataset cases show that a considerable augmentation is beneficial and
desirable, yet it is unadvised to use too wildly modified samples.

The benevolent effect of mixup gets more obscure as we increase the size of
the training set. Notably, the results of different augmentation setups are
almost indistinguishable from each other. We interpret this phenomena as
another consequence of the similarity of masks from different samples, which
inherently drives the network towards simpler representations in the case of a
sufficiently broad training set, even without using mixup.

We also note that in the case of 10 training samples, while the IoU
differences between the no mixup and the mixup regime are striking, the gain
in DSC is less remarkable. It hints that it is unadvised to rely merely on DSC
when evaluating and comparing segmentation models.

Figure 3: Ground truth (left) compared to the prediction of the Attention
BCDU-Net (right), train size: 10.

Figure 4: Ground truth (left) compared to the prediction of the Attention
BCDU-Net (right), train size: 20.

Figure 5: Ground truth (left) compared to the prediction of the Attention
BCDU-Net (right), train size: complete.

| Train size: 10 | Train set: 20 | Train size: 209 (Complete)
---|---|---|---
| IoU | DSC | IoU | DSC | IoU | DSC
No mixup |  $87.2\pm 1.9$% |  $94.8\pm 1.1$% |  $91.9\pm 0.6$% |  $96.9\pm 0.5$% |  $94.9\pm 0.2$% |  $98.0\pm 0.1$%
$\delta=0.1$ |  $91.9\pm 1.3$% |  $96.8\pm 0.9$% |  $92.5\pm 0.5$% |  $97.1\pm 0.5$% |  $95.2\pm 0.2$% |  $98.1\pm 0.1$%
$\delta=0.2$ |  $92.2\pm 1.2$% |  $97.0\pm 0.8$% |  $93.3\pm 0.4$% |  $97.3\pm 0.5$% |  $95.0\pm 0.1$% |  $98.0\pm 0.1$%
$\delta=0.3$ |  $91.3\pm 1.2$% |  $96.5\pm 1.0$% |  $92.9\pm 0.5$% |  $97.2\pm 0.5$% |  $94.9\pm 0.1$% |  $98.0\pm 0.1$%
$\delta=0.4$ |  $91.3\pm 1.4$% |  $96.4\pm 1.0$% |  $93.0\pm 0.5$% |  $97.2\pm 0.4$% |  $94.8\pm 0.1$% |  $97.9\pm 0.1$%

Table 1: Dice scores and IoU scores of Attention BCDU-Net with different mixup
parameters

We would also like to draw attention to the peculiar loss curves we primarily
encountered during the small dataset trainings, as displayed in Figure 6.
Notably, the curve of the validation DSC flattens far below the also
flattening curve of the train DSC, strongly inciting the usage of early
stopping. (Train DSC reaches essentially 1 in fact, which is unsurprising with
such a small training set.) However, in the later stages the validation DSC
catches up, even though the train DSC does not have any room for further
improvement. We were especially puzzled by this behaviour in the 10-sized
training setup, in which both the train and validation DSC seems completely
stabilized after from epoch 50 to epoch 400, yet validation DSC skyrockets in
the later stages in a very short amount of time. The same behaviour was
experienced during each test run. We have yet to give the intuitive or
theoretical explanation for this phenomenon that how the generalizing ability
of the model can improve further when it seems to be in a perfect state from
the training perspective. We note that these observations naturally led us to
experiment with even longer trainings, but to no avail.

Figure 6: From left to right: the evolution of the train DSC (blue) and the
validation DSC (orange) with 10 training samples, 20 training samples, and the
complete training dataset, respectively. The IoU curves admit similar
patterns.

##  4 Conclusion

In the present work, we addressed the problem of automated lung and heart
segmentation on chest X-rays. We introduced a new model, Attention BCDU-Net, a
variant of Attention U-Net equipped with modified attention gates, and
surpassed previous state-of-the-art results. We also demonstrated its ability
to attain surprisingly reasonable results with strongly limited training sets.
Performance in these cases was enhanced using the mixup augmentation
technique, resulting in highly notable contribution in the IoU score.

Concerning future work, a natural extension of this work would be adding a
structure correcting adversarial network to the training scheme, similarly to
[9] and [15], and measuring its effect on the performance, especially in the
setup of limited training sets. We would also like to give some kind of
explanation to the phenomenon of peculiar loss curves.

## Acknowledgements

The project was supported by the grant EFOP-3.6.3-VEKOP-16-2017-00002.

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