# Reaching Quantum Critical Point by Adding Non-magnetic Disorder
in Single Crystals of (CaxSr1-x)3Rh4Sn13 Superconductor

Elizabeth H. Krenkel  Ames National Laboratory, Ames, Iowa 50011, U.S.A.
Department of Physics & Astronomy, Iowa State University, Ames, Iowa 50011,
U.S.A.  Makariy A. Tanatar  Ames National Laboratory, Ames, Iowa 50011, U.S.A.
Department of Physics & Astronomy, Iowa State University, Ames, Iowa 50011,
U.S.A.  Romain Grasset  Laboratoire des Solides Irradiés, CEA/DRF/lRAMIS,
École Polytechnique, CNRS, Institut Polytechnique de Paris, F-91128 Palaiseau,
France  Marcin Kończykowski  Laboratoire des Solides Irradiés, CEA/DRF/lRAMIS,
École Polytechnique, CNRS, Institut Polytechnique de Paris, F-91128 Palaiseau,
France  Shuzhang Chen  Condensed Matter Physics and Materials Science
Department, Brookhaven National Laboratory, Upton, New York 11973, U.S.A.
Department of Physics and Astronomy, Stony Brook University, Stony Brook, New
York 11794-3800, U.S.A.  Cedomir Petrovic  Condensed Matter Physics and
Materials Science Department, Brookhaven National Laboratory, Upton, New York
11973, U.S.A.  Department of Physics and Astronomy, Stony Brook University,
Stony Brook, New York 11794-3800, U.S.A.  Shanghai Key Laboratory of Material
Frontiers Research in Extreme Environments (MFree), Shanghai Advanced Research
in Physical Sciences (SHARPS), Pudong, Shanghai 201203, China  Department of
Nuclear and Plasma Physics, Vinca Institute of Nuclear Sciences, University of
Belgrade, Belgrade 11001, Serbia  Alex Levchenko  Department of Physics,
University of Wisconsin–Madison, Madison, Wisconsin 53706, U.S.A.  Ruslan
Prozorov  Ames National Laboratory, Ames, Iowa 50011, U.S.A.  Department of
Physics & Astronomy, Iowa State University, Ames, Iowa 50011, U.S.A.

(21 June 2024)

###### Abstract

The quasi-skutterudites (CaxSr1-x)3(Rh, Ir)4Sn13 show a rare nonmagnetic
quantum critical point associated with the second-order charge-density-wave
(CDW) and structural distortion transition extended under the superconducting
“dome”. So far, the non-thermal tuning parameters for accessing the QCP
included changing stoichiometry, pressure, and a magnetic field. Here we add
another parameter – a nonmagnetic point-like disorder induced by 2.5 MeV
electron irradiation. The non-Fermi liquid regime was inferred from the
analysis of the temperature-dependent resistivity, $\rho\left(T\right)$, in
single crystals of (CaxSr1-x)3Rh4Sn13. Starting at compositions below the
known QCP concentration of $x_{c}=0.9$, added disorder resulted in a
progressively larger linear term and a reduced quadratic term in
$\rho\left(T\right)$. This behavior is supported by theoretical analysis based
on the idea of superconducting fluctuations encompassing the crossover from
quantum to thermal regimes. Our results strongly support the concept that the
nonmagnetic disorder can drive the system toward the quantum critical regime.

## Introduction

Among the quasi-skutterudite Remeika 3-4-13 compounds, Ca3Rh4Sn13 has one of
the highest superconducting transition temperatures, at $T_{c}=8.3\>\text{K}$
[1]. It does not show a charge-density wave (CDW) or the associated structural
distortion. In contrast, its sibling, Sr3Rh4Sn13, develops CDW below
$T_{\text{CDW}}=136\>\text{K}$ and superconductivity below
$T_{c}=4.6\>\text{K}$. Naturally, alloy compositions, (CaxSr1-x)3Rh4Sn13, have
been investigated for a possible quantum phase transition (QPT), which was
found at ambient pressure at the critical concentration, $x_{c}=0.9$, and also
by varying isostatic pressure, $P$ [2, 3, 4]. X-ray measurements confirmed
that the $T_{\text{CDW}}\left(x,P\right)$ exists below the superconducting
dome [5, 6]. A variety of properties exhibit singular behavior at this
structural/CDW quantum critical point (QCP). In addition to “classical” non-
Fermi behavior from the temperature dependence of the resistivity above
$T_{c}$, the specific heat jump at the transition is larger than a weak-
coupling value. This is in line with the enhanced superfluid density and
pairing strength at the QCP determined by muon spin rotation spectroscopy [7].
The critical current also shows a sharp peak at the QCP composition at
temperatures which are deep inside the superconducting dome, further
confirming its existence [8]. In another ’sibling’ compound, the refractory
metal site is instead permuted, bounded by Ca3Rh4Sn13 (no CDW and
$T_{c}=8.3\>\text{K}$) and Ca3Ir4Sn13 ($T_{\text{CDW}}=39\>\text{K}$ and
$T_{c}=6.9\>\text{K}$). Recent transport and magnetization measurements
indicate that in Ca3(RhxIr1-x)4Sn13 the CDW is suppressed somewhere between
$x=0.53$ and $0.58$, and the temperature-dependent resistivity deviates from
the Fermi liquid behavior, perhaps implying a QCP in this region [9].

Figure 1: Temperature-dependent resistivity, $\rho\left(T\right)$, of pristine
samples of (CaxSr1-x)3Rh4Sn13. Inset shows compositional evolution of
resistivity at two characteristic temperatures,
$\rho\left(T=300\;\text{K}\right)$ (left axis), and at the onset of the
superconducting transition, $\rho\left(T_{c}\right)$ (right axis).

To date, three nonthermal parameters have been used to tune various systems to
a quantum critical point: composition, pressure, and magnetic field. However,
it has been suggested that disorder might also serve as such a control
parameter, particularly in cases where the long-range ordered-to-disordered
quantum phase transition is not perfectly of the second order [10]. In these
cases, disorder may push the transition toward a pure second-order transition
and tune the system toward the QCP [10]. To our knowledge, there have been no
reports of a controlled disorder-tuned QCP in any material or for any type of
long-range order. We note that while chemical substitution influences the
chemical potential level via charge doping [11] and the electronic band
structure via chemical pressure [12], it also introduces additional
scattering, which was taken into account in the theoretical analysis of QCP
[11]. However, it is impossible to separate these effects and isolate only the
effects of scattering.

In this paper, we investigate the effects of disorder on coexisting CDW and
superconductivity and demonstrate that the QCP can indeed be reached, by
introducing a non-magnetic point-like disorder through 2.5 MeV electron
irradiation in (CaxSr1-x)3Rh4Sn13 quasi-skutterudite compounds starting with
compositions well below the “compositional” QCP located at $x_{c}=0.9$.

## Results

Figure 1 shows the temperature-dependent resistivity, $\rho\left(T\right)$, of
several compositions studied of pristine samples of (CaxSr1-x)3Rh4Sn13,
indicated in the legend. The inset shows the compositional evolution of
$\rho\left(T\right)$ at two characteristic temperatures,
$\rho\left(T=300\;\text{K}\right)$ (left $y-$axis), and at the onset of the
superconducting transition, $\rho\left(T_{c}\right)$ (right $y-$axis). To
determine the absolute value of the resistivity, we use the geometric factor
generated by matching the slope of each curve at high temperature to the known
slope of the resistivity of Ca3Rh4Sn13 [13]. As expected, the stoichiometric
compositions, $x=0$ and $x=1$, show the lowest values because in mixed
compositions, the resistivity increases due to substitutional disorder. This
makes the initial starting scattering rate of the pristine state $x-$dependent
and not well defined. Therefore, we should rely on the changes in the
properties introduced by repeated irradiation runs.

Figure 2: Left panel: Measurements of the temperature-dependent resistivity,
$\rho(T)$ with repeated electon irratiation, for the (Ca0.75Sr0.25)3Rh4Sn13
composition. The legend shows the accumulated irradiation doses. (Right
panel): Method of determination of the charge density wave regions of the
material, via subtraction of a linear fit of the resistivity above the
transition temperature. Also clearly displays the crossover from predominately
quadratic behavior (blue pristine curve) to predominantly linear
resistivity(black curve).

Figure 2 shows the evolution of temperature-dependent resistivity,
$\rho\left(T\right)$, after irradiation in $x=0.5$ and $x=0.75$, close to the
QCP composition, $x_{c}=0.9$. The sample with $x=0.75$ is the main focus of
this study. The suppression of CDW by disorder is evident. Not only does the
transition temperature shift, but the transition itself becomes broader and
less obvious. Similar behavior is observed in previously studied
stoichiometric Sr3Rh4Sn13 and Sr3Ir4Sn13 [13]. In alloyed compositions, the
CDW features are significantly less sharp compared to stoichiometric ones [9].

Figure 3: (Color online) The evolution of the characteristic temperatures
(CaxSr1-x)3Rh4Sn13 with point-like disorder introduced by electron
irradiation. (Top panels): the charge density wave transition,
$T_{\text{CDW}}\left(x\right)$. The uncertainty in determining
$T_{\text{CDW}}$, shown as shaded areas, is discussed in the text. (Bottom
panels): superconducting transition temperature, $T_{c}\left(x\right)$. Due to
a small variation, the latter is shown as the deviation from the initial
$T_{c0}$ of the pristine samples. Note an initial increase of $T_{c}$ after
irradiation in stoichiometric Sr3Rh4Sn13.

The T${}_{\text{CDW}}$ was determined by subtracting a linear fit of the data
above the transition temperature and then taking the point where the data
began to deviate from that line, as the upper bound, and the peak of the
difference between the two as the lower bound. This procedure is shown in
figure 2, and the results are shown in Fig.3 for samples of the compositions
indicated in each panel. The top set of panels shows the evolution of the CDW
transition, with a shaded area marking the temperature range between onset and
offset, as discussed above.

The bottom set of panels shows the evolution of the change in superconducting
transition temperature, $\Delta T_{c}=T_{c}-T_{c0}$, where $T_{c0}$ is the
transition temperature before irradiation. Interestingly, the stoichiometric
Sr3Rh4Sn13 shows initial increase of $T_{c}$ demonstrating the intrinsic
interplay between the CDW and superconductivity. In the simplest scenario, the
CDW gapping of the Fermi surface is weakened by the disorder, thereby
enhancing superconductivity. Other mechanisms such as softening of the phonon
modes are also possible, as observed in NbSe2 [14]. For the other
compositions, the superconducting transition is suppressed, which is
consistent with our previous conclusion of the unconventional pairing in
stannide compounds [13]. However, the rate of change of $T_{c}$ in alloys is
difficult to compare directly with the change of $T_{c}$ in the stoichiometric
material. As mentioned above, there is an elevated substitutional disorder in
the pristine state of the intermediate compositions, see Fig.1, and the
corresponding discussion. Since in the absence of nodes,
$T_{c}\left(\Gamma\right)$ is a saturating function of the dimensionless
scattering rate, $\Gamma$, the rate of change,
$dT_{c}\left(\Gamma\right)/d\Gamma$, is also a decreasing function of
$\Gamma$.

We now analyze the temperature-dependent resistivity above the transition
temperature, $T_{c}$. As can be directly seen in Fig.2, $\rho\left(T\right)$,
is nonlinear in the pristine state and is practically linear at higher levels
of disorder. We therefore performed a polynomial fitting of the resistivity
curves, $\rho\left(T\right)=A_{0}+A_{1}T+A_{2}T^{2}$, for all compositions
from above the transition temperature to 20 K. Our goal is to determine the
evolution of the coefficients $A_{0,1,2}$ with disorder. The result is
summarized in Fig.4 for five different compositions. Note that each set of
horizontal panels has a common vertical and horizontal scale to facilitate
visual comparison.

The residual resistivity, $A_{0}$, increases for all samples as expected,
indicating that the disorder was indeed added in each case. The most striking
result is presented in the middle set of panels, where the linear coefficient
indicative of non-Fermi liquid behavior is plotted. Although no linear term is
detected for $x=0$, it becomes progressively more prominent with increasing
disorder for $x=0.5$. In $x=0.75$ initial increase is followed by a decrease.
In $x=0.8$, we only see a decrease. Our interpretation is that this is because
we overshot the QCP in these two compositions. Note that in these cases, we
are adding disorder to already significant substitutional disorder level.

Finally, the quadratic term, $A_{2}$, behaves in a way that is reciprocal to
$A_{1}$. It drops significantly for intermediate compositions and remains
practically unchanged for larger values.

Overall, Fig.4 shows a proliferation of the linear temperature behavior of
$\rho\left(T\right)$ with added non-magnetic disorder in the compositions not
too far from QCP on the CDW ordered side of the compositional phase diagram.
We interpret this behavior as non-magnetic disorder scattering acting as a
nonthermal variable driving the system towards the quantum critical point, and
through it for larger scattering rates.

Figure 4: (Color online) Summary of the doping/disorder evolution of the
parameters of the polynomial fit $\rho=A_{0}+A_{1}T+A_{2}T^{2}$ of resistivity
in the temperature range from 20 K to $T_{c}$. Panels from left to right
$x=$0, $x=$0.5, $x=$0.75 and $x=$0.8, and $x=$1.

## Discussion

We now provide general theoretical arguments and estimates for the evolution
of $T$-linear and $T$-quadratic terms in $\rho(T)$ above $T_{c}$. Our analysis
is limited to the parameter range close to the onset of superconductivity near
$T_{c}(x)$, and, as such, it cannot quantitatively explain the data across the
entire range of observed behavior at $T\gg T_{c}$. Nevertheless, it offers
useful insights into the relevant physics, defining the possible mechanisms
behind the observed features in the temperature and doping dependence of the
resistivity, in particular near the critical point.

The idea we explore is based on the picture of superconducting fluctuations
encompassing the crossover from quantum to thermal regimes, which are defined
by proximity to the quantum critical point in the $T-x$ plane. Since our
analysis pertains to temperatures $T>T_{c}(x)$, a microscopic model that
captures the delicate interplay of superconducting (SC) and charge-density
wave (CDW) orders beneath the dome will not be necessary. However, such a
model in principle can be constructed based on earlier works, see e.g. Ref.
[15], with the necessary generalization to include disorder.

Near the critical temperature, the dependence of resistivity $\rho(T)$ is
strongly affected by superconducting fluctuations [16]. The corresponding
transport theory is well established for the classical regime of fluctuations,
which can be formally defined by a condition $(T-T_{c0})\ll T_{c0}$, where
$T_{c0}$ is the critical temperature of a pristine system. In situations when
it is suppressed, $T_{c}(x)<T_{c0}$, one has to consider generalizations as
appropriate for the quantum regime of fluctuations $T<T_{c0}$. The existing
theoretical approaches to this complicated problem of fluctuation-induced
transport in disordered bulk superconductors are typically based on the pair-
breaking scenario [17, 18, 19, 20]. In its simplest form $T_{c}$ suppression
is described by the Abrikosov-Gor’kov equation [21]

$\ln\frac{T}{T_{c0}}=\psi\left(\frac{1}{2}\right)-\psi\left(\frac{1}{2}+\frac{\Gamma}{2\pi
T}\right),$ (1)

where $\psi(z)$ is the digamma function. At a given pair-breaking strength
$\Gamma$, superconductivity is destroyed at $T=T_{c}(\Gamma)$, and conversely,
at a given temperature $T$, at $\Gamma=\Gamma_{c}(T)$. In the BCS model,
$\Gamma$ is related to the spin-flip scattering. In our estimates we will keep
$\Gamma$ as the phenomenological parameter. Microscopically, it has to be
determined from the proper model of SC-CDW interplay in the presence of
disorder scattering, therefore, the actual form of Eq.(1) will be different in
that case. In models where $\Gamma$ characterizes the rate of electron
collisions with impurities, the estimation of Born scattering on the impurity
potential $U$ gives for the scattering rate
$\Gamma\sim\nu|U|^{2}n_{\text{imp}}$, where $\nu$ is the single particle
density of states and $n_{\text{imp}}$ is the impurity concentration. When
irradiation acts as a source of point-like disorder, one may roughly assume
$n_{\text{imp}}\propto x$. We notice from the data of Fig.3 that $T_{c}(x)$
suppression in our samples is the most significant in the range of
concentrations $x$ across the QCP. We therefore expect fluctuation to be
especially pronounced in that part of the phase diagram.

In applying the pair-breaking scenario to determine the fluctuation correction
to the normal state conductivity, we follow the most recent diagrammatic
calculation of Ref. [22], which can be straightforwardly generalized to the
three-dimensional case. Focusing first on the temperature interval $T<T_{c0}$,
it can be shown that the main contribution is determined by the Aslamazov-
Larkin (AL) diagram, with the corresponding analytical expression for the
conductivity correction given by

$\delta\sigma=\left(\frac{e\nu D}{4\pi T}\right)^{2}\int\frac{q^{2}(\Re
B^{\prime}_{q,\omega})^{2}(\Im
L^{R}_{q,\omega})^{2}}{T\sinh^{2}\frac{\omega}{2T}}d\Upsilon_{q,\omega}$ (2)

where $D$ is the diffusion coefficient and integration spans the phase space
$d\Upsilon_{q,\omega}=d\omega d^{3}q/(2\pi)^{4}$. In these notations, the
vertex function, triangular block of the AL diagram, is given by

$B_{q,\omega}=\psi\left(\frac{1}{2}+\frac{\Gamma_{q}-i\omega}{4\pi T}\right),$
(3)

where $\Gamma_{q}=\Gamma+Dq^{2}$. The pair-propagator that describes
proliferation of superconducting fluctuations, namely its retarded component,
is given by [16]

$L^{R}_{q,\omega}=-\frac{\nu^{-1}}{\ln\frac{T}{T_{c0}}-\psi\left(\frac{1}{2}\right)+\psi\left(\frac{1}{2}+\frac{\Gamma_{q}-i\omega}{4\pi
T}\right)}.$ (4)

The pole of $L_{q,\omega}$ at $\\{Dq^{2},\omega\\}\to 0$ traces the boundary
between the superconducting and normal phases. We analyze Eq.(2) analytically
in two limits: (i) quantum fluctuations $T<\Xi<T_{c0}$, and (ii) thermal
fluctuations $\Xi<T<T_{c0}$, where energy scale $\Xi=|\Gamma-\Gamma_{c}|$
measures the proximity to quantum critical regime.

For case (i) one can neglect the frequency dependence of the vertex function,
which implies $\Re B^{\prime}_{q,\omega}\approx 4\pi T/\Gamma_{q}$. In the
low-temperature limit, the propagator of fluctuations reduces to

$L^{R}_{q,\omega}\approx\nu^{-1}\ln^{-1}\left(\frac{\Gamma_{c}}{\Gamma_{q}-i\omega}\right),$
(5)

and it is sufficient to retain only the pole of the logarithm, since the
branch-cut singularity yields a smaller contribution to the integral in
Eq.(2). This implies $\Im
L^{R}_{q,\omega}\approx-\nu\Gamma_{c}\omega/(Dq^{2}+\Xi)^{2}$. These
approximations for $B_{q,\omega}$ and $L^{R}_{q,\omega}$ taken together give a
simplified form of Eq.(2)

$\delta\sigma=\frac{(eD)^{2}}{T}\int\frac{q^{2}\omega^{2}\Gamma^{2}_{c}d\Upsilon_{q,\omega}}{\sinh^{2}\frac{\omega}{2T}(Dq^{2}+\Gamma)^{2}(Dq^{2}+\Xi)^{4}}.$
(6)

Calculating the remaining integrals by noting that the most important domain
of fluctuation momenta are $Dq^{2}\sim\Xi\ll\Gamma$, we obtain for $T<\Xi$

$\delta\sigma\simeq
e^{2}\sqrt{\frac{\Gamma_{c}}{D}}\left(\frac{\Gamma_{c}}{\Xi}\right)^{\frac{3}{2}}\left(\frac{T}{\Gamma_{c}}\right)^{2}=\delta
A_{2}T^{2}.$ (7)

Here we omitted the numerical prefactor for brevity and retained only the main
parametric dependence.

In a similar manner, for case (ii) we can still neglect $\omega$ dependence of
$B_{q,\omega}$, we must retain the full $\omega$ dependence of
$L^{R}_{q,\omega}$, and we can also expand $\sinh(\omega/2T)\approx\omega/2T$,
since in this limit $\\{Dq^{2},\omega\\}\sim\Xi<T$. This leads us to the
approximate expression instead of Eq.(2)

$\delta\sigma=(eD)^{2}\int\frac{4Tq^{2}\Gamma^{2}_{c}d\Upsilon_{q,\omega}}{(Dq^{2}+\Gamma)^{2}((Dq^{2}+\Xi)^{2}+\omega^{2})^{2}}.$
(8)

After integration, one finds as a result for $\Xi<T<T_{c0}$

$\delta\sigma\simeq
e^{2}\sqrt{\frac{\Gamma_{c}}{D}}\sqrt{\frac{\Gamma_{c}}{\Xi}}\frac{T}{\Gamma_{c}}=\delta
A_{1}T.$ (9)

It is worth noting that this expression does not diverge at the critical
point, since at low temperature $\Gamma_{c}\approx\Gamma_{c0}-\alpha
T^{2}/\Gamma_{c0}$, where $\alpha\sim 1$.

These results should be contrasted to the behavior in the classical
fluctuations regime $T\gtrsim T_{c0}$. In that case, one finds [16]

$\delta\sigma=\left(\frac{\pi\nu eD}{8T}\right)^{2}\int\frac{q^{2}(\Im
L^{R}_{q,\omega})^{2}}{3T\sinh^{2}\frac{\omega}{2T}}d\Upsilon_{q,\omega}$ (10)

where

$L^{R}_{q,\omega}\approx-\nu^{-1}\left[\frac{T-T_{c0}}{T}+\frac{\pi}{8T}(Dq^{2}-i\omega)\right]^{-1}.$
(11)

After integration, the correction can be reduced to the form

$\delta\sigma\simeq\frac{e^{2}}{\sqrt{D}}\frac{T}{\sqrt{T-T_{c0}}},\quad
T-T_{c0}\ll T_{c0}$ (12)

As usual, the apparent divergence has to be regularized by the applicability
of the perturbation theory, namely $\delta\sigma<\sigma$, where $\sigma$ is
the bare normal-state conductivity. This limits applicability of the above
asymptotic behavior is to be outside of the critical region of strong
fluctuations, $(T-T_{c0})/T_{c0}>Gi_{3}$. The Ginzburg number in 3D case can
be estimated as $Gi_{3}\sim(\nu T_{c0}\xi^{3})^{-2}\ll 1$, where $\xi$ is the
superconducting coherence length in the diffusive limit.

Finally, at the much higher temperatures, $T\gg T_{c0}$, one may deduce from
the available results [23, 24]

$\delta\sigma\simeq-e^{2}\sqrt{\frac{T}{D}}\frac{1}{\ln(T/T_{c0})},\quad T\gg
T_{c0}.$ (13)

In this limit, however, fluctuations are dominated by the density of states
and so-called regular part of the Maki-Thompson diagram. This is reflected in
the opposite sign of the correction term, as compared to Eqs. (7), (9), and
(12), thus increasing resistivity.

This analysis suggests that superconducting fluctuations contribute to the
$T-$linear and $T-$quadratic dependence of the resistivity observed in the
experiment. Within the limitations of the perturbation theory, presented
above, the corresponding corrections to resistivity can be presented as
$\delta\rho=-\rho^{2}\delta\sigma$, where $\rho$ should be understood as a
bare disorder-limited conductivity $\rho=A_{0}$. In this approximation, the
coefficients $\delta A_{1,2}(\Gamma)$ from Eqs. (7) and (9) can be attributed
to the renormalization of the fitting parameters $A_{1,2}$. The above model
considerations indicate that once the system is tuned toward the quantum
critical regime in concentration $x$, the regime of $T$-linear dependence
expands, since $\Xi$ becomes smaller and the range of validity of Eq.(9)
becomes wider. This is qualitatively consistent with the observed trend.
Furthermore, the sign of the correction term from Eq.(9) is also consistent
with the observed reduction in the $T$-linear slope of the resistivity
governed by the coefficient $A_{1}$. The fact that the correction term changes
sign underscores the point that the coefficients $A_{1,2}$ exhibit
nonmonotonic behavior, seeing in Fig.4 .

## Conclusions

In conclusion, it is shown that a controlled non-magnetic point-like disorder
can tune the CDW/SC quasi-skutterudites (CaxSr1-x)3(Rh, Ir)4Sn13 to and
through the structural/CDW quantum critical point. Starting at compositions
below the known structural QCP at $x_{c}=0.9$, disorder induced by 2.5 MeV
electron irradiation resulted in a progressively larger linear term and a
reduced quadratic term in resistivity, $\rho\left(T\right)$, just above
$T_{c}$. These novel experiments and qualitative theoretical analysis strongly
support the general theoretical prediction that the non-magnetic disorder can
drive the system toward the quantum critical regime.

## Materials and Methods

Samples: Single crystals (CaxSr1-x)3Rh4Sn13 were grown from tin flux [7]. A
mixture of 3 parts (Ca + Sr of the desired ratio), 4 parts Rh and 93 parts Sn,
was placed in an alumina crucible and sealed under vacuum in a quartz ampule
along with a bit of quartz wool for filtration. The ampule was heated up to
1100 oC, kept for six hours, fast cooled to 800 oC and then cooled slowly to
490 oC where the excess Sn was decanted in a centrifuge outside the furnace.
The process is described elsewhere [25]. The obtained samples were
characterized by powder X-ray diffraction and energy-dispersive X-ray
compositional analysis. This well-studied system shows good sample homogeneity
and reproducibility of resistivity and EDX compositions between samples.

Transport measurements: Electrical resistivity was measured in a standard
four-probe configuration in bar-shaped single crystals of typical length of
1-3 mm, width of 0.3-0.7 mm and thickness between 50 and 300 $\mu$m. To
prepare the samples, the crystals were etched with HCl, cut to dimensions
using a wire saw, and polished. The electrical contacts were formed by
soldering 50 $\mu$m silver wires with tin-silver solder [13, 9] with typical
contact resistances below 100 $\mu\Omega\cdot$cm.

Electron irradiation: The 2.5 MeV electron irradiation was performed at the
“SIRIUS” accelerator facility at Laboratoire des Solides Irradiés, École
Polytechnique, Palaiseau, France. The acquired irradiation dose is measured by
a calibrated Faraday trap behind the sample and is conveniently expressed in
$\textrm{C}/\textrm{cm}^{2}$, where 1 $\textrm{C}/\textrm{cm}^{2}=6.24\times
10^{18}\>\textrm{e}^{-}/\textrm{cm}^{2}$. Collisions of electrons with ions
produce Frenkel pairs (vacancy + interstitial) because the relativistic energy
transfer upon collisions matches well with the knock out energy threshold,
typically between 20 and 80 eV. At these energies, the electron scattering
cross-section is on the order of 100 barns. Other particles used for
irradiation have either much lower cross-section (neutrons and gamma-rays), or
may produce clusters (protons), and other correlated defects, including
columnar tracks (heavy ions).

Another important parameter for irradiation is the particles penetration depth
into the material. While heavier particles require very thin samples because
of beam attenuation, electrons produce relatively uniform disorder at mm-range
depths for typical samples. Importantly, for the doses of irradiation studied,
the produced defects are in the dilute limit, typically with about one defect
per a thousands of host lattice ions. Furthermore, according to extensive
Hall-effect measurements [26, 27], the electron irradiation does not “charge-
dope” the samples and does not cause a shift of the chemical potential.

The irradiation is conducted at low temperatures with the sample immersed in
liquid hydrogen around 22 K. This is done to remove heat generated upon
collisions and to prevent immediate recombination and clustering of the
defects. Upon warming to room temperature, some pairs recombine, and some
defects migrate to various sinks such as lattice defects, dislocations, and
surfaces leaving a metastable population of point-like defects behind,
typically about 70% of the initially produced population. The final amount of
the induced disorder is directly estimated from the changes in residual
resistivity. Importantly, to avoid variation between different samples, in our
experiments, the same sample was repeatedly irradiated and then measured to
observe the progression of the results. Several compositions around the QCP
concentration were studied. For a more detailed discussion of the use of
controlled electron irradiation to study materials, the reader is referred to
specialized books [28, 29].

## Acknowledgments

This work was supported by the National Science Foundation under Grant No.
DMR-2219901. M.A.T. and K.R.J. were supported by the U.S. Department of Energy
(DOE), Office of Science, Basic Energy Sciences, Materials Science and
Engineering Division. Ames National Laboratory is operated for the U.S. DOE by
Iowa State University under Contract No. DE-AC02-07CH11358. A.L. acknowledges
the financial support by the National Science Foundation Grant No. DMR-
2203411 and a Research Fellowship funded by the Alexander von Humboldt
Foundation. Work in France was supported by “Investissements d’Avenir” LabEx
PALM (Grant No. ANR10-LABX-0039-PALM). The authors acknowledge support from
the EMIR&A French network (FR CNRS 3618) on the platform SIRIUS, proposals No.
20-5925 and 23-4663. Work at BNL (materials synthesis) was supported by the
U.S. Department of Energy, Basic Energy Sciences, Division of Materials
Science and Engineering, under Contract No. DE-SC0012704. C.P. acknowledges
support from the Shanghai Key Laboratory of Material Frontiers Research in
Extreme Environments, China (No. 22dz2260800) and Shanghai Science and
Technology Committee, China (No. 22JC1410300).

## References

  * Remeika  _et al._ [1980] J. P. Remeika, G. P. Espinosa, A. S. Cooper, H. Barz, J. M. Rowell, D. B. McWhan, J. M. Vandenberg, D. E. Moncton, Z. Fisk, L. D. Woolf, H. C. Hamaker, M. B. Maple, G. Shirane, and W. Thomlinson, A new family of ternary intermetallic superconducting/magnetic stannides, Solid State Comm. 34, 923 (1980).
  * Klintberg  _et al._ [2012] L. E. Klintberg, S. K. Goh, P. L. Alireza, P. J. Saines, D. A. Tompsett, P. W. Logg, J. Yang, B. Chen, K. Yoshimura, and F. M. Grosche, Pressure and composition induced structural quantum phase transition in the cubic superconductor (Sr,Ca)3Ir4Sn13, Phys. Rev. Lett. 109, 237008 (2012).
  * Goh  _et al._ [2015] S. K. Goh, D. A. Tompsett, P. J. Saines, H. C. Chang, T. Matsumoto, M. Imai, K. Yoshimura, and F. M. Grosche, Ambient pressure structural quantum critical point in the phase diagram of (CaxSr1-x)3Rh4Sn13, Phys. Rev. Lett. 114, 097002 (2015).
  * Terasaki  _et al._ [2021] Y. Terasaki, R. Yamaguchi, Y. Ishii, Y. Tada, A. Yamamoto, and S. Mori, Superconductivity enhanced by abundant low-energy phonons in (Sr1-xCax)3Rh4Sn13, Journal of the Physical Society of Japan 90, 113704 (2021).
  * Veiga  _et al._ [2020] L. S. I. Veiga, J. R. L. Mardegan, M. v. Zimmermann, D. T. Maimone, F. B. Carneiro, M. B. Fontes, J. Strempfer, E. Granado, P. G. Pagliuso, and E. M. Bittar, Possible quantum fluctuations in the vicinity of the quantum critical point of (Sr,Ca)3Ir4Sn13 revealed by high-energy x-ray diffraction, Phys. Rev. B 101, 104511 (2020).
  * Carneiro  _et al._ [2020] F. B. Carneiro, L. S. I. Veiga, J. R. L. Mardegan, R. Khan, C. Macchiutti, A. López, and E. M. Bittar, Unveiling charge density wave quantum phase transitions by x-ray diffraction, Phys. Rev. B 101, 195135 (2020).
  * Krieger  _et al._ [2023] J. A. Krieger, Z. Guguchia, R. Khasanov, P. K. Biswas, L. Li, K. Wang, C. Petrovic, and E. Morenzoni, Enhancement of superconductivity at a quantum critical point in (CaxSr1-x)3Rh4Sn13, Journal of Physics: Conference Series 2462, 012060 (2023).
  * Liu  _et al._ [2022] X. Liu, W. Zhang, K. T. Lai, K. Moriyama, J. L. Tallon, K. Yoshimura, and S. K. Goh, Peak in the critical current density in (CaxSr1-x)3Rh4Sn13 tuned towards the structural quantum critical point, Phys. Rev. B 105, 214524 (2022).
  * Krenkel  _et al._ [2024] E. H. Krenkel, M. A. Tanatar, S. Ghimire, K. R. Joshi, S. Chen, C. Petrovic, and R. Prozorov, Robust superconductivity and the suppression of charge-density wave in the quasi-skutterudites $\text{Ca}_{3}(\text{Ir}_{1-x}\text{Rh}_{x})_{4}\text{Sn}_{13}$ single crystals at ambient pressure, J. Phys. Condens. Matter 1, 1 (2024).
  * Belitz and Vojta [2005] D. Belitz and T. Vojta, How generic scale invariance influences quantum and classical phase transitions, Rev. Mod. Phys. 77, 579 (2005).
  * Joshi  _et al._ [2020] K. R. Joshi, N. M. Nusran, M. A. Tanatar, K. Cho, S. L. Bud’ko, P. C. Canfield, R. M. Fernandes, A. Levchenko, and R. Prozorov, Quantum phase transition inside the superconducting dome of Ba(Fe1-xCox)2As2 from diamond-based optical magnetometry, New J. Phys. 22, 053037 (2020).
  * Hashimoto  _et al._ [2012] K. Hashimoto, K. Cho, T. Shibauchi, S. Kasahara, Y. Mizukami, R. Katsumata, Y. Tsuruhara, T. Terashima, H. Ikeda, M. A. Tanatar, H. Kitano, N. Salovich, R. W. Giannetta, P. Walmsley, A. Carrington, R. Prozorov, and Y. Matsuda, A Sharp Peak of the Zero-Temperature Penetration Depth at Optimal Composition in BaFe2(As1-xPx)2, Science 336, 1554 (2012).
  * Krenkel  _et al._ [2022] E. H. Krenkel, M. A. Tanatar, M. Kończykowski, R. Grasset, E. I. Timmons, S. Ghimire, K. R. Joshi, Y. Lee, L. Ke, S. Chen, C. Petrovic, P. P. Orth, M. S. Scheurer, and R. Prozorov, Possible unconventional pairing in (Ca,Sr)3(Rh,Ir)4Sn13 superconductors revealed by controlling disorder, Phys. Rev. B 105, 094521 (2022).
  * Cho  _et al._ [2018] K. Cho, M. Kończykowski, S. Teknowijoyo, M. A. Tanatar, J. Guss, P. B. Gartin, J. M. Wilde, A. Kreyssig, R. J. McQueeney, A. I. Goldman, V. Mishra, P. J. Hirschfeld, and R. Prozorov, Using controlled disorder to probe the interplay between charge order and superconductivity in NbSe2, Nat. Comm. 9, 2796 (2018).
  * Machida [1984] K. Machida, Charge density wave and superconductivity in anisotropic materials, J. Phys. Soc. Japan 53, 712 (1984).
  * Larkin and Varlamov [2005] A. Larkin and A. Varlamov,  _Theory of Fluctuations in Superconductors_ (Oxford University Press, 2005).
  * Ramazashvili and Coleman [1997] R. Ramazashvili and P. Coleman, Superconducting Quantum Critical Point, Phys. Rev. Lett. 79, 3752 (1997).
  * Mineev and Sigrist [2001] V. P. Mineev and M. Sigrist, Critical fluctuation effects near the normal-metal–superconductor phase transition at low temperatures, Phys. Rev. B 63, 172504 (2001).
  * Adachi and Ikeda [2001] H. Adachi and R. Ikeda, Fluctuation conductivity in unconventional superconductors near critical disorder, Journal of the Physical Society of Japan 70, 2848 (2001).
  * Lopatin  _et al._ [2005] A. V. Lopatin, N. Shah, and V. M. Vinokur, Fluctuation conductivity of thin films and nanowires near a parallel-field-tuned superconducting quantum phase transition, Phys. Rev. Lett. 94, 037003 (2005).
  * Abrikosov and Gor’kov [1961] A. A. Abrikosov and L. P. Gor’kov, Contribution to the theory of superconducting alloys with paramagnetic impurities, Sov. Phys. JETP 12, 1243 (1961).
  * Dzero  _et al._ [2023] M. Dzero, M. Khodas, and A. Levchenko, Transport anomalies in multiband superconductors near the quantum critical point, Phys. Rev. B 108, 184513 (2023).
  * Aslamazov and Varlamov [1980] L. G. Aslamazov and A. A. Varlamov, Fluctuation conductivity in intercalated superconductors, Journal of Low Temperature Physics 38, 223 (1980).
  * Al’tshuler  _et al._ [1983] B. L. Al’tshuler, A. A. Varlamov, and M. Y. Reizer, Interelectron effects and the conductivity of disordered twodimensional electron systems, Sov. Phys. JETP 57, 1329 (1983).
  * Wang and Petrovic [2012] K. Wang and C. Petrovic, Ca3Ir4Sn13: A weakly correlated nodeless superconductor, Phys. Rev. B 86, 024522 (2012).
  * Prozorov  _et al._ [2019] R. Prozorov, M. Kończykowski, M. A. Tanatar, H.-H. Wen, R. M. Fernandes, and P. C. Canfield, Interplay between superconductivity and itinerant magnetism in underdoped Ba1-xKxFe2As2 (x=0.2) probed by the response to controlled point-like disorder, npj Quantum Mater. 4, 34 (2019).
  * Prozorov  _et al._ [2014] R. Prozorov, M. Kończykowski, M. A. Tanatar, A. Thaler, S. L. Bud’ko, P. C. Canfield, V. Mishra, and P. J. Hirschfeld, Effect of Electron Irradiation on Superconductivity in Single Crystals of Ba(Fe1-xRux)2As2 ($x=0.24$), Phys. Rev. X 4, 041032 (2014).
  * Damask and Dienes [1963] A. C. Damask and G. J. Dienes,  _Point Defects in Metals_ (Gordon & Breach Science Publishers Ltd, 1963).
  * Thompson [1969] M. W. Thompson,  _Defects and Radiation Damage in Metals_ , revised september 27, 1974 ed., Cambridge Monographs on Physics (Cambridge University Press, 1969).


# Ignition from a Fire Perimeter in a WRF Wildland Fire Model111WRF Summer
Workshop 2011. This research was supported by the National Science Foundation
under grant AGS-0835579, and by U.S. National Institute of Standards and
Technology Fire Research Grants Program grant 60NANB7D6144.

Volodymyr Y. Kondratenko222Department of Mathematical and Statistical
Sciences, University of Colorado Denver, Denver, CO, Jonathan D.
Beezley222Department of Mathematical and Statistical Sciences, University of
Colorado Denver, Denver, CO, Adam K. Kochanski333Department of Meteorology,
University of Utah, Salt Lake City, UT, and Jan Mandel222Department of
Mathematical and Statistical Sciences, University of Colorado Denver, Denver,
CO

###### Abstract

The current WRF-Fire model starts the fire from a given ignition point at a
given time. We want to start the model from a given fire perimeter at a given
time instead. However, the fuel balance and the state of the atmosphere depend
on the history of the fire. The purpose of this work is to create an
approximate artificial history of the fire based on the given fire perimeter
and time and an approximate ignition point and time. Replaying the fire
history then establishes a reasonable fuel balance and outputs heat fluxes
into the atmospheric model, which allow the atmospheric circulation to
develop. Then the coupled atmosphere-fire model takes over. In this
preliminary investigation, the ignition times in the fire area are calculated
based on the distance from the ignition point to the perimeter, assuming that
the perimeter is convex or star-shaped. Simulation results for an ideal
example show that the fire can continue in a natural way from the perimeter.
Possible extensions include algorithms for more general perimeters and running
the fire model backwards in time from the perimeter to create a more realistic
history. The model used extends WRF-Fire and it is available from openwfm.org.

##  1 Introduction

Fire models generally start the fire from a given ignition point at a given
time, and sophisticated ignition parameterizations exist, including line
ignition and submesh ignition procedures Mandel et al. (2011). However, in
practice one is often faced with the need to start a fire model from a fire
already in progress. Often only perimeter data pertaining to some time are
available, such as from the US Forest Service at activefiremaps.fs.fed.us.
This need arizes in analyses of existing fires currently, and it will become
even more important for forecasting of the behavior of fires in progress in
future.

With models that do not include two-way interaction with the atmosphere,
continuing from an existing fire state is essentially straightforward. While
the fuel can be partially burned in some areas, fuel in locations untouched by
the fire is unchanged, and the model can simply progress to the new areas
regardless of the fire history. (A model that may include long-range effects
such as preheating in front of the fire, would be an exception.) In a coupled
atmosphere-fire model, however, the situation is very different. First, simply
igniting the whole area inside the given perimeter is not an option, because
the large instantaneous heat release will cause the model to break down. More
importantly, the state of the atmosphere evolves in interaction with the fire,
and the buyoancy caused by the heat flux causes significant changes to the
wind field, which in turn influences the future progress of the fire.

Starting the fire from a state already developed is essentially a data
assimilation problem, and it could be treated as such by a shooting method:
the fire starts from a ignition point at a time in the past, then at the given
simulation time, the state of the fire is compared with the given perimeter,
and adjustments can be made to the ignition time and location, much as in
variational data assimilation methods such as 4DVAR. We plan to study such
approaches in future as a part of our effort in the area of data assimilation
for wildland fires Beezley (2009); Beezley and Mandel (2008); Mandel et al.
(2008, 2009, 2010).

The approach adopted here is different. Given a fire perimeter, we create an
artificial history. Then the fire history is replayed, which produces heat
output into the atmosphere, and the atmospheric model spins up to a state that
is plausible for the fire at the stage given by the perimeter. The artificial
history is essentially a parameterization of the process that leads to the
development of the fire perimeter. Parameterizations of various levels of
sophistication can be considered, up to and including running a fire model
backwards in time to find an ignition point and iterating to find a matching
atmospheric state. In this initial study, we consider a very simple artificial
history model, and show that it results in acceptable fire and atmosphere
states for the perimeter. Approximate perimeter states obtained by such method
could provide also a good starting point for data assimilation in future.

##  2 The model

Fire models range from tools based on Rothermel (1972) fire spread rate
formulas, such as BehavePlus (Andrews 2007) and FARSITE (Finney 1998),
suitable for operational forecasting, to sophisticated 3-D computational fluid
dynamics and combustion simulations suitable for research and reanalysis, such
as FIRETEC (Linn et al. 2002) and WFDS (Mell et al. 2007). BehavePlus, the PC-
based successor of the calculator-based BEHAVE, determines the fire spread
rate at a single point from fuel and environmental data; FARSITE uses the fire
spread rate to provide a 2-D simulation on a PC; while FIRETEC and WFDS model
combustion in 3D, which is much more expensive. See the survey by Sullivan
(2009) for a number of other models.

The model considered here couples the mesoscale atmospheric code WRF-ARW
Skamarock et al. (2008) with a fire spread module, based on the Rothermel
model Rothermel (1972) and implemented by the level set method. In each time
step, the fire model inputs the atmospheric winds and outputs surface sensible
and latent heat fluxes into the atmosphere. Only the finest domain in WRF is
coupled with the fire model. The fire model works in conjunction with WRF land
use models, and it interpolates horizontal winds from the ideal logarighmic
wind profile to appropriate heights above the surface, for each fuel.

The model has grown out of NCAR’s CAWFE code Clark et al. (1996b, a, 2004);
Coen (2005), which couples the Clark-Hall atmospheric model with fire spread
implemented by tracers, and it got its start from a prototype code coupling
the fire model in CAWFE with WRF in LES mode Patton and Coen (2004). The
tracers, however, were replaced by a level set method, which we considered
more flexible and more suitable for data assimilation and WRF parallel
infrastructure. The coupled model is capable of running faster than real time
in LES mode, with resolution of tens of meters for the atmosphere, and meters
for the fire, with the matching time step of a fraction of a second, on the
innermost modeling domain of many kilometers in size Jordanov et al. (2011).
Fuel data and topography can be obtained from government databases in the
United States and from satellite images and GIS elsewhere Jordanov et al.
(2011). See Mandel et al. (2009, 2011) for futher details and references. The
model is currently available from the Open Wildland Fire Modeling environment
at openwfm.org, along with utilities for data preparation, visualization, and
diagnostics and a wiki with many user guides for the specific features and
utilities, discussion, and support. A code containing a subset of the features
is distributed with WRF as WRF-Fire.

##  3 Encoding and replaying the fire history

The state of the fire model consists of a level set function, $\Phi$, given by
its values on the nodes of the fire model mesh, and time of ignition
$T_{\mathrm{i}}$. The level set function is interpolated linearly. At a given
simulation time $t$, the fire area is the set of all points $\left(x,y\right)$
where $\Phi\left(t,x,y\right)\leq 0$. The level set function and the ignition
time satisfy the consistency condition

$\Phi\left(t,x,y\right)\leq 0\Longleftrightarrow
T_{\mathrm{i}}\left(x,y\right)\leq t,$ (1)

as both of these inequalities express the condition that the location
$\left(x,y\right)$ is burning at the time $t$. In every time step of the
simulation, the level set function is advanced by one step of a Runge-Kutta
scheme for the level set equation

$\frac{d\Phi}{dt}=-R\left\|\nabla\Phi\right\|,$

where $R=R\left(t,x,y\right)$ is the fire rate of spread, which depends on the
fuel, wind speed, and slope. The ignition time at nodes is then computed for
all newly ignited nodes, and it satisfies the consistency condition (1).

The fire history is encoded as an array of ignition times
$T_{\mathrm{i}}\left(x,y\right)$, prescribed at all fire mesh nodes. To replay
the fire in the period $0\leq t\leq T_{\mathrm{per}}$, the numerical scheme
for advancing $\Phi$ and $T_{\mathrm{i}}$ is suspended, and instead the level
set function is set to

$\Phi\left(t,x,y\right)=T_{\mathrm{i}}\left(x,y\right)-t.$

After the end of the replay is reached, the numerical scheme of the level set
method is started from the level set function $\Phi$ at $t=T_{\mathrm{per}}$.

For reasons of numerical accuracy and stability, the level set function needs
to have approximately uniform slope. For example, a very good level set
function, which has slope equal to one, is the signed distance from a given
closed curve $\Gamma$,

$\Phi\left(x,y\right)=\pm\operatorname*{dist}\left(\left(x,y\right),\Gamma\right),$

where the sign is taken to be negative inside the region limited by $\Gamma$,
and positive outside Osher and Fedkiw (2003). Thus, the ignition times
$T_{\mathrm{i}}$ need to be given on the whole domain and they need to be such
that $T_{\mathrm{i}}$ decreases with the distance from the given perimeter
inside the fire region, and increases outside. The ignition times
$T_{\mathrm{i}}$ outside of the given fire perimeter are perhaps best thought
of as what the ignition times might be in future as the fire keeps burning.

##  4 Creating an artificial fire history

The purpose of this algorithm is to create the artificial values of the time
of ignition on the fire model mesh, given ignition point ($x_{\mathrm{ign}}$,
$y_{\mathrm{ign}}$), ignition time $T_{\mathrm{ign}}$, fire perimeter
$\Gamma$, and the time when the fire reached the perimeter $T_{\mathrm{per}}$,
assuming that the fire perimeter is convex, or at least star-shaped with
respect to the ignition point. The fire perimeter is given as a set of points
$(x_{k},y_{k})$ in the fire model domain, $k=1,\ldots,n$ which form a closed
curve consisting of line segments
$\left[(x_{k},y_{k}),(x_{k+1},y_{k+1})\right]$ between each two successive
points. We take $(x_{1},y_{1})=(x_{n+1},y_{n+1})$ so that the starting and the
ending point are identical. The coordinates of the point of ignition and of
the points defining of the fire perimeter do not need to coincide with mesh
points of the grid.

The method consists of linear interpolation of the ignition time between
$T_{\mathrm{ign}}$ at the ignition point and $T_{\mathrm{per}}$ on the
perimeter, along straight lines connecting the ignition point with points on
the perimeter. The ignition time is also extrapolated beyond the perimeter in
the same manner to provide a suitable level set function, as discussed in the
previous section. Given a mesh point with coordinates $\left(x,y\right)$, the
algorithm to determine the ignition time $T_{\mathrm{i}}\left(x,y\right)$
consists of the following steps.

  1. 1.

Find the intersection $\left(x_{\mathrm{b}},y_{\mathrm{b}}\right)$ of the fire
perimeter and the half-line starting at the ignition point and passing through
the point $\left(x,y\right)$ (Fig. 1). For this purpose, we use the function

$F(x,y,x_{b},y_{b})=(y_{b}-y_{\mathrm{ign}})(x-x_{\mathrm{ign}})-(x_{b}-x_{\mathrm{ign}})(y-y_{\mathrm{ign}}),$

which is zero if point $\left(x_{\mathrm{b}},y_{\mathrm{b}}\right)$ lies on
the line defined by $(x,y)$ and $(x_{\mathrm{ign}},y_{\mathrm{ign}})$, and it
is positive in one half-plane and negative in the other. We then find segment
$\left[(x_{k},y_{k}),(x_{k+1},y_{k+1})\right]$ such that
$F(x,y,x_{k},y_{k})F(x,y,x_{k+1},y_{k+1})<0$, that is, the points
$(x_{k},y_{k})$ and $(x_{k+1},y_{k+1})$ lie on opposite sides of the line
passing through $(x,y)$ and $(x_{\mathrm{ign}},y_{\mathrm{ign}})$. Since the
line intersects the fire perimeter at two points, one on each side of the
ignition point, we choose correct segment as follows:

     * •

If $\left(x,y\right)$ is inside $\Gamma$, that is, closer to the ignition
point than to the intersection, then the desired segment is the one that lies
on the same side from the ignition point as the point $\left(x,y\right)$;

     * •

If $\left(x,y\right)$ is outside of $\Gamma$, then the needed segment lies on
the same side from the ignition point as $\left(x,y\right)$.

  2. 2.

Calculate the time of ignition of the mesh point, based on the ratio of the
distances of the mesh point and the perimeter point to the ignition point,

$T_{\mathrm{i}}(x,y)=T_{\mathrm{ign}}+\frac{\left\|\left(x,y\right)-\left(x_{\mathrm{ign}},y_{\mathrm{ign}}\right)\right\|}{\left\|\left(x_{\mathrm{b}},y_{\mathrm{b}}\right)-\left(x_{\mathrm{ign}},y_{\mathrm{ign}}\right)\right\|}\left(T_{\mathrm{per}}-T_{\mathrm{ign}}\right).$

Figure 1: Construction of intersection of the fire perimeter and the half line
originating from the Ignition point and passing through a given mesh point.

##  5 Computational results

We have tested this algorithm on an ideal example to measure the difference in
the atmospheric winds between a simulation propagated naturally from a point
and another one advanced artificially. In this example, the topography was
flat except for a small hill roughly $500$ m in diameter and $100$ m high in
the center of a domain of size $2.4$ km $\times$ $2.4$ km. The atmospheric and
fire grid resolutions used were $60$ m and $6$ m respectively, with a $0.25$ s
time step. The background winds were approximately $9.5$ m/s traveling
southwest at the lowest atmospheric layer $30$ m above the surface. The first
simulation was ignited from a point in the northeast corner of the domain $2$
seconds from the start, and the fire perimeter was recorded after $40$
minutes. This perimeter and ignition location were used to generate an
artificial history for the first $40$ minutes, which was replayed in the
second simulation. Therefore, the fire perimeters in both simulations are
identical at $40$ minutes. Both simulations were then allowed to advance
another $28$ minutes, using the standard coupled model. The outputs were then
collected for analysis.

Any differences in the simulations after this time are a result of the error
of the artificial fire propagation. In Fig. 3, we show 3D renderings of the
simulation. The streamlines near the surface show the updraft created as a
result of the heat output from the fire. In Fig. 3a, the fire is affecting the
atmosphere despite being propagated artificially. A semi-transparent volume
rendering of QVAPOR was added to simulate the smoke release. In Fig. 2, the
differences in the wind between the two simulations at $68$ minutes and the
fire perimeter are shown. Fig. 2a, shows the difference of the wind from the
direct fire propagation minus the wind from the artificial propagation. Fig.
2b shows the relative error in the wind speed defined as the norm of the
difference from Fig. 2a, divided by the wind speed from the direct simulation.
This shows that the maximum error at the end of this $68$ minute simulation is
less than $2.5\%$. In this case, the Froude number is about $F_{c}=0.79$,
showing that the heating from the fire may significantly affect ambient wind,
therefore small differences caused by using the artificial history have an
effect. The effect is concentrated downwind from the fire, as it could be
expected.

(a) The difference in the winds of the direct simulation minus the artificial
propagation at $68$ minutes.

(b) The relative error in the speed of the wind at $68$ minutes.

Figure 2:

(a) The artificial fire simulation at $40$ minutes.

(b) The direct fire simulation at $68$ minutes.

Figure 3:

##  6 Conclusion

We have presented a parameterization of the fire up to a point in time when a
fire perimeter is given. The parameterization allows for the changes in the
atmospheric circulation to develop, caused by the heat flux from the fire.
This provides appropriate starting conditions for the computation to continue
with the full coupled fire-atmosphere simulation. We have shown on an ideal
example that the differences in the state of the atmosphere between a complete
fire simulation and when the parameterization is used are not significant. In
the studied case, the coupling between the fire and the atmosphere was strong.
It would be interesting to observe how the the differences change if the
problem moves from the wind-driven ($F_{c}>1$) to the plume driven regime
($F_{c}<1$) regime. This will be studied elsewhere. We plan to study also
algorithms for more general domains, not just star-shaped, and to take into
account different rates of fire propagation due to fuel nonhomogeneity.

## References

  * Andrews (2007) Andrews, P. L., 2007: BehavePlus fire modeling system: past, present, and future. Paper J2.1, 7th Symposium on Fire and Forest Meteorology, http://ams.confex.com/ams/pdfpapers/126669.pdf.
  * Beezley (2009) Beezley, J. D., 2009: High-Dimensional Data Assimilation and Morphing Ensemble Kalman Filters with Applications in Wildfire Modeling. Ph.D. thesis, University of Colorado Denver.
  * Beezley and Mandel (2008) Beezley, J. D. and J. Mandel, 2008: Morphing ensemble Kalman filters. Tellus, 60A, 131–140, doi:10.1111/j.1600-0870.2007.00275.x.
  * Clark et al. (2004) Clark, T. L., J. Coen, and D. Latham, 2004: Description of a coupled atmosphere-fire model. International Journal of Wildland Fire, 13, 49–64, doi:10.1071/WF03043.
  * Clark et al. (1996a) Clark, T. L., M. A. Jenkins, J. Coen, and D. Packham, 1996a: A coupled atmospheric-fire model: Convective feedback on fire line dynamics. J. Appl. Meteor, 35, 875–901, doi:10.1175/1520-0450(1996)035$<$0875:ACAMCF$>$2.0.CO;2.
  * Clark et al. (1996b) Clark, T. L., M. A. Jenkins, J. L. Coen, and D. R. Packham, 1996b: A coupled atmosphere-fire model: Role of the convective Froude number and dynamic fingering at the fireline. International Journal of Wildland Fire, 6, 177–190, doi:10.1071/WF9960177.
  * Coen (2005) Coen, J. L., 2005: Simulation of the Big Elk Fire using coupled atmosphere-fire modeling. International Journal of Wildland Fire, 14, 49–59, doi:10.1071/WF04047.
  * Finney (1998) Finney, M. A., 1998: FARSITE: Fire area simulator-model development and evaluation. Res. Pap. RMRS-RP-4, Ogden, UT: U.S. Department of Agriculture, Forest Service, Rocky Mountain Research Station, http://www.fs.fed.us/rm/pubs/rmrs_rp004.html.
  * Jordanov et al. (2011) Jordanov, G., J. D. Beezley, N. Dobrinkova, A. K. Kochanski, and J. Mandel, 2011: Simulation of the 2009 Harmanli fire (Bulgaria). 8th International Conference on Large-Scale Scientific Computations, Sozopol, Bulgaria, June 6-10, 2011, lecture notes in Computer Science, Springer, to appear.
  * Linn et al. (2002) Linn, R., J. Reisner, J. J. Colman, and J. Winterkamp, 2002: Studying wildfire behavior using FIRETEC. Int. J. of Wildland Fire, 11, 233–246, doi:10.1071/WF02007.
  * Mandel et al. (2009) Mandel, J., J. D. Beezley, J. L. Coen, and M. Kim, 2009: Data assimilation for wildland fires: Ensemble Kalman filters in coupled atmosphere-surface models. IEEE Control Systems Magazine, 29, 47–65, doi:10.1109/MCS.2009.932224.
  * Mandel et al. (2011) Mandel, J., J. D. Beezley, and A. K. Kochanski, 2011: Coupled atmosphere-wildland fire modeling with WRF-Fire version 3.3. Geoscientific Model Development Discussions, 4, 497–545, doi:10.5194/gmdd-4-497-2011.
  * Mandel et al. (2010) Mandel, J., J. D. Beezley, and V. Y. Kondratenko, 2010: Fast Fourier transform ensemble Kalman filter with application to a coupled atmosphere-wildland fire model. Computational Intelligence in Business and Economics, Proceedings of MS’10, A. M. Gil-Lafuente and J. M. Merigo, eds., World Scientific, 777–784.
  * Mandel et al. (2008) Mandel, J., L. S. Bennethum, J. D. Beezley, J. L. Coen, C. C. Douglas, M. Kim, and A. Vodacek, 2008: A wildland fire model with data assimilation. Mathematics and Computers in Simulation, 79, 584–606, doi:10.1016/j.matcom.2008.03.015.
  * Mell et al. (2007) Mell, W., M. A. Jenkins, J. Gould, and P. Cheney, 2007: A physics-based approach to modelling grassland fires. Intl. J. Wildland Fire, 16, 1–22, doi:10.1071/WF06002.
  * Osher and Fedkiw (2003) Osher, S. and R. Fedkiw, 2003: Level Set Methods and Dynamic Implicit Surfaces. Springer, New York.
  * Patton and Coen (2004) Patton, E. G. and J. L. Coen, 2004: WRF-Fire: A coupled atmosphere-fire module for WRF. Preprints of Joint MM5/Weather Research and Forecasting Model Users’ Workshop, Boulder, CO, June 22–25, NCAR, 221–223, http://www.mmm.ucar.edu/mm5/workshop/ws04/Session9/Patton_Edward.pdf.
  * Rothermel (1972) Rothermel, R. C., 1972: A mathematical model for predicting fire spread in wildland fires. USDA Forest Service Research Paper INT-115, http://www.treesearch.fs.fed.us/pubs/32533.
  * Skamarock et al. (2008) Skamarock, W. C., J. B. Klemp, J. Dudhia, D. O. Gill, D. M. Barker, M. G. Duda, X.-Y. Huang, W. Wang, and J. G. Powers, 2008: A description of the Advanced Research WRF version 3. NCAR Technical Note 475, http://www.mmm.ucar.edu/wrf/users/docs/arw_v3.pdf.
  * Sullivan (2009) Sullivan, A. L., 2009: A review of wildland fire spread modelling, 1990-present, 1: Physical and quasi-physical models, 2: Empirical and quasi-empirical models, 3: Mathematical analogues and simulation models. International Journal of WildLand Fire, 18, 1: 347–368, 2: 369–386, 3: 387–403, doi:10.1071/WF06143, 10.1071/WF06142, 10.1071/WF06144.


# Encoder-Decoder-Based Intra-Frame Block Partitioning Decision

Yucheng Jiang12, Han Peng2, Yan Song2, Jie Yu1, Peng Zhang2 and Songping Mai1
1Shenzhen International Graduate School, Tsinghua University, Shenzhen, China
2Peng Cheng Laboratory, Shenzhen, China

###### Abstract

The recursive intra-frame block partitioning decision process, a crucial
component of the next-generation video coding standards, exerts significant
influence over the encoding time. In this paper, we propose an encoder-decoder
neural network (NN) to accelerate this process. Specifically, a CNN is
utilized to compress the pixel data of the largest coding unit (LCU) into a
fixed-length vector. Subsequently, a Transformer decoder is employed to
transcribe the fixed-length vector into a variable-length vector, which
represents the block partitioning outcomes of the encoding LCU. The vector
transcription process adheres to the constraints imposed by the block
partitioning algorithm. By fully parallelizing the NN prediction in the intra-
mode decision, substantial time savings can be attained during the decision
phase. The experimental results obtained from high-definition (HD) sequences
coding demonstrate that this framework achieves a remarkable 87.84% reduction
in encoding time, with a relatively small loss (8.09%) of coding performance
compared to AVS3 HPM4.0.

###### Index Terms:

intra prediction, block partition, CNN encoder, Transformer decoder

##  I Introduction

An efficient video coding standard is of great significance for the
development of high-definition video industry. To this end, the next-
generation video coding standards, Versatile Video Coding (VVC) and Audio
Video coding Standard 3 (AVS3), were respectively released in July and
November 2020. Compared to the widely used HEVC standard, both VVC and AVS3
achieves around 30% performance improvement [1], but at the cost of increasing
the encoding complexity by an order of magnitude [2].

The intra-frame prediction in video coding is usually used for key frame
compression. It utilizes spatial correlation between adjacent pixels to
predict the current pixel value, thereby reducing the number of bits required
to represent the current pixel value and thus reducing the amount of video
data. Considering that the intra-frame block partitioning decision algorithm
accounts for over 97% of the intra-frame encoding time [3], there has been a
lot of research on reducing the time required for this decision making.

Traditional acceleration algorithms for intra-frame block partition usually
use hand-crafted feature extraction for analysis, and skip some of the
partitioning modes of the current coding unit (CU) based on analysis results
[4, 5]. While compared with hand-crafted feature extraction, neural network-
based acceleration algorithms can usually achieve higher coding efficiency.
There are two primary categories of neural network acceleration algorithms.

One category combines neural networks with traditional intra-frame prediction
algorithms. These algorithms leverage the information acquired from the
preceding neural network to reduce the number of traversals required by
traditional intra-frame prediction algorithms. More specifically, a pooling-
variable CNN was proposed to predict the continuation of current CUs partition
in advance [6]. Tech _et al_ proposed in their various works to use CNNs to
predict different parameters related to the current LCU, which were utilized
to constrain the block partitioning process [7, 8]. Xu _et al_ used two
lightweight CNNs as classifiers, to distinguish whether the CU is to be
partitioned and the direction in which the partition should take place [9].
Moreover, one set of research employs CNNs to predict the partitioning
probabilities of basic edges within a LCU. The predicted probability vector is
then exploited in subsequent block partitioning decisions to expedite the
judgement process [10, 11].

The other category decouples neural networks from traditional intra-frame
prediction algorithms, and allows the neural network to directly output the
block partitioning result of the current LCU. The traditional intra-frame
prediction algorithm only needs to make mode decisions without block
partitioning decisions. For instance, Li _et al._ proposed a MSECNN, as
described in [12], for predicting partitioning results in a greedy way.
However, the proposed network architecture is excessively complex, leading to
a significant increase in computational load. When selecting the Top-1 result,
the method proposed in [11] can replace the block partitioning decision
process, but it demonstrates a significant decrease in performance.

In this paper, we propose a novel approach utilizing a compact network
structure to accurately predicte block partitioning results. By fully
parallelizing the process of predicting block partition, significant
reductions in encoding time can be achieved. The proposed network has been
trained and validated within the AVS3 coding framework, but can be easily
adapted to the VVC framework with minor modifications.

##  II Task Formulation

For video coding standards such as VVC and AVS3, once the partition of the
parent node has been determined in the CU block partitioning decision process,
the decision order for its child nodes is fixed. Therefore, the block
partitioning result of the LCU can be represented as a structured variable-
length integer vector, with each integer element in the vector ranging from 0
to 5, corresponding to the six partitioning modes of Non-Split (NS), QuadTree
(QT), Horizontal Binary-Tree (BT-H), Vertical Binary-Tree (BT-V), Horizontal
Ternary-Tree (TT-H) and Vertical Ternary-Tree (TT-V) in the VVC standard, or
NS, QT, BT-H, BT-V, Horizontal Extend Quad-Tree (EQT-H) and Vertical Extend
Quad-Tree (EQT-V) in the AVS3 standard.

As described in [11], the partitioning result of a 64x64-sized LCU can also be
represented as a fixed-length vector consisting of 480 binary elements of 1 or
0, where 1 represents that the corresponding 4-pixel length edge (basic edge)
belongs to the boundary of the partitioning block, and 0 represents the
opposite. Fig. 1 provides a visual representation of these two different forms
of representation discussed above, using two LCUs as examples. It should be
noted that, in the actual inference process, the elements in the fixed-length
vector are floating-point numbers ranging from 0 to 1, representing the
probability of the basic edge belonging to the block partitioning boundary.

Figure 1: Variable-length vector and Fixed-length vector representation of CTU
partitioning results. Figure 2: Neural network architecture. (a) CNN encoder.
(b) Transformer decoder. (c) Block Partitioning Constrain module

Our objective is to use a neural network to predict the variable-length vector
for each LCU. This vector can then be utilized by the video encoder to replace
the recursive process of block partitioning decisions. The retention of only
the mode decision algorithm leads to a substantial reduction in encoding time.
Based on previous studies [11], the pixels of a LCU can be encoded to a fixed-
length vector using a neural network similar to ResNet. With this encoding,
the subsequent prediction task can be reformulated as a sequence-to-sequence
(seq2seq) task. In this task, the input sequence comprises flattened block
partitioning information, while the output sequence consists of structured
block partitioning information. In other words, we need to extract structured
information from the fixed-length vector.

##  III Proposed Method

Fig. 2 illustrates the neural network based on an encoder-decoder architecture
proposed in this study. The CNN shown in subfigure (a) encodes the pixel data
into a floating-point vector of length 480. Then the Transformer decoder in
(b) performs the transformation from fixed-length vector to variable-length
vector. The purpose of Block Partitioning Constrain (BPC) module in (c) is to
ensure that the variable-length vector generated by the neural network
conforms to the AVS3 coding constrains and can be correctly decoded.

Compared with the fixed threshold method in [10] and the decision tree (DT)
method in [11], the network structure proposed in this study has the following
advantages,

  * •

The self-attention mechanism in the Transformer decoder facilitates the
integration of the previous block partitioning decision results, especially
the partitioning information of the parent nodes. This correlation between
different decision layers was usually overlooked in other methods.

  * •

The decoder can automatically extract information by enhancing the attention
of the current CU element in the encoder output feature. This replaces the
manual construction of the DT model input. Furthermore, this structure
achieves framework uniformity, eliminating the need to train different DT
models for different CU sizes.

  * •

The variable-length vector output from the model can be directly used for LCU
coding, completely replacing the recursive process of block partitioning
decision making.

###  III-A CNN Encoder

In the traditional AVS3 encoding algorithm, the optimal intra prediction mode
for the current CU is computed by utilizing the reconstructed pixels from the
top, left, and top-left neighboring CUs as references [13]. Therefore, we fold
the reference pixels corresponding to each 64x64-sized LCU and fill them into
the upper two rows and left two columns of a 66x66-sized block, while the
remaining part is filled with the luminance pixels of the current LCU. We also
input the Quantization Parameter (QP) into the neural network in addition to
the pixel block, as QP value significantly affects block partitioning results.

The architecture of the CNN encoder is similar to the standard ResNet18 [14],
consisting of an input convolutional layer, four residual blocks, and an
output fully connected layer. The difference lies in the size of the
convolutional kernel in the convolutional layer and the dimension of the fully
connected layer. Additionally, before entering the output fully connected
layer, the QP value is concatenated with the flattened feature vector,
enabling the probability vector of the model output to be affected by changes
in the QP value.

###  III-B Transformer Decoder

We employ the standard Transformer Decoder [15] for the seq2seq task,
benefiting from its multi-head self-attention mechanism and multi-head
encoder-decoder attention mechanism. These mechanisms enable us to effectively
extract structured information from the fixed-length vector, enhancing the
model’s ability to capture and utilize contextual dependencies.

To prevent limited expressive capacity of the model caused by a small feature
vector dimension, it is necessary to reshape the fixed-length vector. In
consideration of the correlation typically found among basic edges located in
the same row or column during block partition, and with each row or column
comprising 16 basic edges, we reshape the fixed-length vector of length 480
into a (30,16) shape, thereby increasing the feature vector dimension from 1
to 16.

As shown in Fig. 2(b), the decoder consists of 4 layers of Transformer decoder
modules, with a hidden layer dimension of 16 and 4 attention heads. Its output
is computed by (1).

${{l}_{t}}=Decoder({{p}_{0}},{{p}_{1}},...,{{p}_{479}},{{l}_{0}},{{l}_{1}},...,{{l}_{t-1}})\in{{R}^{N}}$
(1)

where ${{p}_{i}}$ represents the i-th element of a fixed-length vector, and
${{l}_{j}}$ represents the output of the decoder at the j-th step. $N$ is the
vocabulary size. For this task, since there are only 6 block partitioning
modes in AVS3 and VVC, $N$ is set to 6. Specifically, the reshaped fixed-
length vector is used as the key-value pairs and the concatenated historical
output of the decoder is used as the query. The output vector of the decoder
represents the probability of the 6 block partitioning methods corresponding
to the current CU.

###  III-C Block Partitioning Constrain

Multiple constraints exist for block partitioning decision of CUs in the AVS3
video coding standard. Failure to meet these constraints can result in a
undecodable bitstream. Specifically, these constraints include: the aspect
ratio of child CUs after partition cannot exceed 8, the total partitioning
depth cannot exceed 6, non-QuadTree (QT) partitioning nodes cannot have child
nodes that are QT partitioned, and so on. We design BPC module, which selects
the block partitioning result that satisfies the encoder constraints from the
probability vector output by the decoder. This is done by choosing the maximum
probability value that meets the encoder constraint, rather than simply
selecting the highest probability value. Since we adopt the teacher-forcing
training method [15], BPC module is not used during the training process. In
this way, during both the training and testing processes, the model receives
queries that adhere to encoding constraints. This ensures that the data is
drawn from the same distribution.

###  III-D Neural Network Training

We adjusted high-definition images from two publicly available datasets [16,
17] to HD resolution, and concatenated them into YUV video format. The
resulting video consisted of a total of 19200 frames. Then we encoded the
video using AVS3 reference software, and extracted the data. Each set of data
consists of stitched 66x66 luminance pixel data, QP value, fixed-length vector
labels, and variable-length vector labels. Additionally, over 85% of the
fixed-length vectors in the dataset have less than half of their elements
equal to 1. To tackle this data imbalance and improve the accuracy of complex
LCU block partition predictions, we balanced the dataset by adjusting the
number of LCUs with deep and shallow partitions during training.

In contrast to [18], where a pre-trained CNN is required to provide input data
for the Transformer decoder, our decoder can utilize fixed-length vector
labels as input. This enables us to use a two-step training approach that
includes both independent training and joint training. Independent training
serves as a form of pre-training, supplying the initialization parameters for
both the CNN encoder and the Transformer decoder. This methodology reduces the
time required for joint training and subsequently enhances the overall
predictive accuracy of the model.

##  IV Experimental Results

###  IV-A Neural Network Prediction

Fig. 3 presents the prediction results of the trained network, which indicates
that the network exhibits higher accuracy in predicting the partition of LCU
blocks with simple textures, but lower accuracy in predicting more complex
texture LCUs. Thus, improving the model’s predictive capability on complex
textures is crucial to further enhancing network performance. On average, the
validation accuracy of the network can reach over 80%, which is higher than
the TOP1 accuracy of 67.24% reported in [11] using the DT models.

Figure 3: Comparison of block partitioning labels (left) and neural network
prediction results (right) for LCU with different textured complexity levels

###  IV-B Video Coding Performance

We deployed the trained neural network and AVS3 reference software on a
heterogeneous platform to evaluate their coding performance and efficiency.
The neural network was executed on a NVIDIA RTX3090 GPU, while the AVS3
reference software was executed on an Intel Xeon Gold 6148 CPU. It should be
noted that to reduce training cost and time, our training, testing, and
performance evaluation are all focused on HD video sequences. In the future,
LCU data of videos with different resolutions will be added to the dataset to
enhance the applicability of the network.

As shown in Table I, we conducted joint testing of neural networks with
reference software versions High-Performance Model (HPM) 14.0 and HPM4.0, and
compared the test results with the coding results obtained solely using the
reference software. The BDBR stands for Bjøntegaard delta bit rate [19], while
TS represents encoding time saving, which is calculated from (2),

$TS=\frac{{{T}_{HPM}}-\min({{T}_{NN}},T_{HPM}^{{}^{\prime}})}{{{T}_{HPM}}}$
(2)

where ${{T}_{HPM}}$ represents the encoding time of the reference software,
${T_{HPM}^{{}^{\prime}}}$ represents the encoding time of the reference
software with block partitioning decisions removed, and ${{T}_{NN}}$
represents the inference time of the block partitioning results of the neural
network. Due to the parallelized design, theoretically, the encoding time can
be reduced by up to 97%, which refers to the time required for block mode
decisions. However, limited by our GPU performance, we save 93.97% and 87.84%
of the encoding time for two versions of reference software respectively.

In terms of coding performance, as shown in Table II, the BDBR metric exhibits
a decrease of 8.09% compared to HPM4.0, which outperforms the 8.97%
performance decrease reported in [20] using the same reference standard.
Compared to the methods designed for VVC, our approach reduces more
complexity, but its performance is inferior and requires further improvement.

TABLE I: Performance of the proposed method in comparison with reference softwares in AI coding configuration Sequences | HPM14.0 | HPM4.0
---|---|---
BDBR | TS | BDBR | TS
Class B | BasketballDrive | 13.73% | 94.48% | 3.44% | 88.84%
BQTerrace | 18.02% | 93.40% | 9.41% | 87.38%
Cactus | 16.22% | 93.30% | 10.01% | 85.92%
MarketPlace | 9.73% | 95.45% | 5.24% | 91.18%
RitualDance | 15.92% | 93.21% | 12.33% | 85.88%
Average | 14.73% | 93.97% | 8.09% | 87.84%
TABLE II: Performance of the proposed method in comparison with state of the art methods in AI coding configuration Standards | Versions | Methods | BDBR | TS
---|---|---|---|---
AVS3 | HPM4.0 | Ours | 8.09% | 87.84%
HPM4.0 | Han, et al. [20] | 8.97% | 97%
VVC | VTM7.0 | Li, et al. [12] | 2.78% | 65.75%
VTM10.2 | Tissier, et al. [11] | 2.31% | 74.60%

##  V Conclusion

This paper proposes an encoder-decoder structured neural network for
predicting the intra-frame block partitioning results of the LCUs. Firstly,
the CNN encoder extracts local features from the LCU pixels. Then, a
Transformer decoder is incorporated to predict the block partitioning results.
Compared to other algorithms such as DT, the proposed structure achieves
superior prediction accuracy. Furthermore, our designed BPC module enables the
output of the Transformer decoder to be directly utilized by the video coding
frameworks. Therefore, by adopting a parallelized design, this heterogeneous
framework can save a significant amount of encoding time. Due to the
similarity of the intra-frame block partitioning decision algorithm, the model
can be easily adapted to other video coding algorithms, such as VVC, through
retraining while maintaining the same network structure.

## References

  * [1] X. Zheng, et al, ”Performance Evaluation for AVS3 Video Coding Standard,” 2020 IEEE International Conference on Multimedia & Expo Workshops (ICMEW), London, UK, 2020, pp. 1-6, doi: 10.1109/ICMEW46912.2020.9106032.
  * [2] F. Bossen, X. Li, and K. Suehring, “AHG report: Test model software development (AHG3),” JVET-T0003.
  * [3] A. Tissier, A., et al, ”Complexity Reduction Opportunities in the Future VVC Intra Encoder,” 2019 IEEE 21st International Workshop on Multimedia Signal Processing (MMSP), Kuala Lumpur, Malaysia, 2019, pp. 1-6, doi: 10.1109/MMSP.2019.8901754.
  * [4] T. Fu, H. Zhang, F. Mu and H. Chen, ”Fast CU Partitioning Algorithm for H.266/VVC Intra-Frame Coding,” 2019 IEEE International Conference on Multimedia and Expo (ICME), Shanghai, China, 2019, pp. 55-60, doi: 10.1109/ICME.2019.00018.
  * [5] H. Yang, et al, ”Low-Complexity CTU Partition Structure Decision and Fast Intra Mode Decision for Versatile Video Coding,” in IEEE Transactions on Circuits and Systems for Video Technology, vol. 30, no. 6, pp. 1668-1682, June 2020, doi: 10.1109/TCSVT.2019.2904198.
  * [6] Tang, G., et al. (2019). Adaptive CU split decision with pooling-variable CNN for VVC intra encoding. 2019 IEEE Visual Communications and Image Processing (VCIP), IEEE.
  * [7] Tech, G., et al. (2021). Fast partitioning for VVC intra-picture encoding with a CNN minimizing the rate-distortion-time cost. 2021 Data Compression Conference (DCC), IEEE.
  * [8] G. Tech et al., ”CNN-based parameter selection for fast VVC intra-picture encoding,” 2021 IEEE International Conference on Image Processing (ICIP), Anchorage, AK, USA, 2021, pp. 2109-2113, doi: 10.1109/ICIP42928.2021.9506360.
  * [9] Xu, J., et al. (2022). CNN-Based Fast CU Partitioning Algorithm for VVC Intra Coding. 2022 IEEE International Conference on Image Processing (ICIP), IEEE.
  * [10] Tissier, A. , et al. ”CNN Oriented Complexity Reduction Of VVC Intra Encoder.” 2020 IEEE International Conference on Image Processing (ICIP) IEEE, 2020.
  * [11] Tissier, A., et al. (2023). ”Machine Learning based Efficient QT-MTT Partitioning Scheme for VVC Intra Encoders.” IEEE Transactions on Circuits and Systems for Video Technology.
  * [12] Li, T., et al. (2021). ”DeepQTMT: A deep learning approach for fast QTMT-based CU partition of intra-mode VVC.” IEEE transactions on image processing 30: 5377-5390.
  * [13] Cai, Z., and W. Gao . ”Efficient Fast Algorithm and Parallel Hardware Architecture for Intra Prediction of AVS3.” IEEE International Symposium on Circuits and Systems IEEE, 2021.
  * [14] He, K., et al. ”Deep Residual Learning for Image Recognition.” IEEE Conference on Computer Vision and Pattern Recognition IEEE, 2016.
  * [15] A. Vaswani, et al. “Attention is all you need,” Advances in neural information processing systems, 2017, pp. 5998-6008
  * [16] E. Makov, Dataset image 4k. 2019. Version1. Retrieved 2023 from https://www.kaggle.com/datasets/evgeniumakov/images4k
  * [17] J. Liu, D. Liu, W. Yang, S. Xia, X. Zhang and Y. Dai, ”A Comprehensive Benchmark for Single Image Compression Artifact Reduction,” in IEEE Transactions on Image Processing, vol. 29, pp. 7845-7860, 2020, doi: 10.1109/TIP.2020.3007828.
  * [18] Chen, Kun, et al. ”Audio Captioning Based on Transformer and Pre-Trained CNN.” DCASE. 2020.
  * [19] G. Bjøntegaard, “Calculation of average PSNR difference between RD curves,” VCEGM33, Oct. 2001.
  * [20] Han, Xu, et al. ”Intra block partition structure prediction via convolutional neural network.” 2021 Data Compression Conference (DCC). IEEE, 2021.