prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: What is the effect of the proximity of superconductivity on the Kondo effect?

Read the following text and answer briefly.

Context 1: \section{Introduction}
\label{sec:Intro}

The exchange interactions control the magnetic order and properties of a vast number of materials
\cite{White2006Dec}
and lead to many fascinating phenomena, such as various types of the Kondo effect 
\cite{Kondo,NozieresBlandin,Pustilnik_Glazman} 
Double quantum dots (DQDs), and in general multi-impurity systems, constitute
a convenient and controllable playground,
where nearly as much different exchange mechanisms compete with each other to
shape the ground state of the system 
\emph{Local exchange} between the spin of a quantum dot (QD)
and the spin of conduction band electrons gives rise to the
Kondo effect \cite{Kondo,Hewson_book}  
\emph{Direct exchange} arriving with an additional side-coupled QD may destroy it or lead to the 
two-stage Kondo screening \cite{Pustilnik_Glazman,Cornaglia,Granger,ZitkoBonca,ZitkoPRB2010,Ferreira} 
In a geometry where the two QDs contact the same lead, conduction band electrons 
mediate the \emph{RKKY exchange} \cite{RK,K,Y}  The RKKY interaction competes
with the Kondo effect and leads to the quantum phase transition of a still debated nature
\cite{Doniach,Jones,Affleck,Bork,Neel,KondoRKKYexp,Hans,Hans2,Fabian} 
Moreover, in DQDs coupled in series also \emph{superexchange} can alter the Kondo physics significantly
\cite{Zitko_2QDEx,Sela} 

Recently, hybrid quantum devices, in which the interplay between various magnetic correlations
with superconductivity (SC) plays an important role, have become an important direction of research
\cite{hybridQDs,SCspintronics}  In particular, chains of magnetic atoms on SC surface have proven 
to contain self-organized Majorana quasi-particles and exotic spin textures
\cite{Braunecker,Klinovaja,Vazifeh,Yazdani},
while hybrid DQD structures have been used to split the Cooper pairs coherently into two entangled 
electrons propagating to separated normal leads \cite{CPS1,CPS2,CPS4,CPS5,CPS9} 
The latter is possible due to non-local (\emph{crossed}) Andreev reflections (CARs),
in which each electron of a Cooper pair tunnels into different QD, and
subsequently to attached lead  Such processes give rise to an exchange mechanism \cite{Yao},
that we henceforth refer to as \emph{the CAR exchange}, which can greatly modify
the low-temperature transport behavior of correlated hybrid nanostructures 

The CAR exchange may be seen as RKKY-like interaction between
two nearby impurities on SC surface \cite{Yao} 
The effect can be understood as a consequence
of spin-dependent hybridization of the Yu-Shiba-Rusinov (YSR)
states \cite{Yu,Shiba,Rusinov} in SC contact,
caused both by the overlap of their wave functions
and their coupling to Cooper-pair condensate 
This process is the most effective when the YSR states 
are close to the middle of the SC gap, {\it e g } in the YSR-screened phase \cite{YSRscreening} 
The mechanism presented here is essentially the same,
yet in the considered regime can be understood
perturbatively without referring to YSR states,
as a consequence of the non-local pairing induced by SC electrode  
In particular, the presence of YSR bound states close to the Fermi level 
is not necessary for significant consequences for the Kondo physics, 
as long as some inter-dot pairing is present  


The proximity of SC induces pairing in QDs \cite{RozhkovArovas,Buitelaar} 
and tends to suppress the Kondo effect if the superconducting energy gap $2\Delta$ 
becomes larger than the relevant Kondo temperature $T_K$ 
\cite{Buitelaar2002Dec,adatomsSC,Kondo_vs_SC1,Kondo_vs_SC2,Zitko_Kondo-Andreev,Zitko_S-QD-N,IW_Sau,YSRscreening} 
Moreover, the strength of SC pairing can greatly affect the Kondo physics in the sub-gap transport regime:
For QDs attached to SC and normal contacts, it can enhance the Kondo effect
\cite{DomanskiIW,KWIW,part1}, while
for DQD-based Cooper pair splitters, it tends to suppress both the $\mathrm{SU}(2)$ and $\mathrm{SU}(4)$ Kondo effects \cite{IW_Kacper} 
Our main result is that the non-local pairing induced by superconducting 
proximity effect, which gives rise to CAR exchange, can be the sole cause of the Kondo screening 
Moreover, relatively small values of coupling to SC, $\GS{}\ll U$, are sufficient for the effect to occur 
This is in contrast to the DQD system considered in Ref ~\cite{part1},
where only one of the quantum dots is proximized, such that 
CAR exchange cannot arise,
and the Kondo physics becomes qualitatively
affected only for $\GS{}\sim U/2$ %


\begin{figure}[bt]
\centering
\includegraphics[width=1\linewidth]{Fig1 png}
\caption{
		 (a) Schematic of the considered system  Left/right (L/R) lead
		 is coupled to the first quantum dot (QD1), while superconductor
		 is attached to both QD1 and QD2 
		 (b)-(d) illustrate an example of direct spin exchange:
		 spin-up electron from the initial state (b) hops to the other QD (c) and spin-down electron 
		 hops back (d)  Note, that the final state is in fact the same singlet state, 
		 only with opposite sign 
		 (e)-(g) show an example of process contributing to crossed Andreev reflection (CAR) exchange 
		 A Cooper pair from SC approaches DQD (e) and two singlets of the same charge 
		 are formed (f), before the Cooper pair is re-emitted (g) 
		 (h)-(j) present an example of RKKY process: an electron scattered off
		 one QD (h) mediates the spin exchange towards the other (i), before it is finally scattered
		 off there, too (j) 
		 }
\label{fig:system}
\end{figure}


In this paper we discuss the CAR-induced Kondo screening in a setup comprising T-shaped DQD
with normal and superconducting contacts, see \fig{system}(a) 
We note that despite quite generic character of CAR exchange,
and its presence in systems containing at least two localized electrons
coupled close to each other to the same SC bath,
to best of our knowledge CAR-induced screening
has hardly been identified in previous studies
\cite{CPS1,CPS2,CPS4,CPS5,CPS9,IW_Kacper,IW_Sau,Zitko_Josephson,Zitko_S2QD,Martinek2017} 
In the system proposed here [\fig{system}(a)], its presence is evident 
Moreover, CAR exchange magnitude can be directly related to the relevant energy scales, such as the Kondo 
temperature, which provides a fingerprint for quantitative experimental verification of our predictions  

The paper is organized as follows  In \Sec{model} we describe the considered system 
and present the model we use to study it  In \Sec{scales} the relevant energy scales are estimated
to make the discussion of main results concerning CAR-induced Kondo effect in \Sec{main} more clear  
Finally, the influence of effects neglected in \Sec{main} are presented in the following sections,
including CAR exchange interplay with RKKY interaction (\Sec{RKKY}), particle-hole asymmetry (\Sec{asym}),
couplings asymmetry (\Sec{x}) and reduced efficiency of CAR coupling (\Sec{coef})  In summary,
the effects discussed in \Sec{main} remain qualitatively valid in all these cases 
The paper is concluded in \Sec{conclusions} 


\section{Model}
\label{sec:model}

The schematic of the considered system is depicted in \fig{system}(a) 
It contains two QDs attached to a common SC lead 
Only one of them (QD1) is directly attached to the left (L) and right (R) normal leads,
while the other dot (QD2) remains coupled only through QD1 
The SC is modeled by the BCS Hamiltonian, 
$H_{\mathrm{S}}=\sum_{\mathbf{k}\sigma}\xi_{\mathbf{k}}a_{\mathbf{k}\sigma}^{\dag}a_{\mathbf{k}\sigma}-\Delta\sum_{\mathbf{k}}(a^\dag_{\mathbf{k}\uparrow}a_{-\mathbf{k}\downarrow}^{\dag}+a_{-\mathbf{k}\downarrow}a_{\mathbf{k}\uparrow})$,
with energy dispersion $\xi_{\mathbf{k}}$, energy gap $2\Delta>0$ and $a_{\mathbf{k}\sigma}$ annihilation operator 
of electron possessing spin $\sigma$ and momentum $\mathbf{k}$  The coupling between
SC and QDs is described by the hopping Hamiltonian 
$H_{\mathrm{TS}}=\sum_{i\mathbf{k}\sigma}v_{\mathrm{S}i}(d^\dagger_{i\sigma}a^{}_{\mathbf{k}\sigma}+h c )$,
with $d^\dagger_{i\sigma}$ creating a spin-$\sigma$ electron at QD$i$  The matrix element 
$v_{\mathrm{S}i}$ and the normalized density of states of SC in normal state, $\rho_{\rm S}$, 
contribute to the coupling of QD$i$ to SC electrode as $\GS{i} = \pi \rho_{\rm S} |v_{{\rm S}i}|^2$  
We focus on the sub-gap regime, therefore, we integrate out SC degrees of freedom lying outside the energy gap \cite{RozhkovArovas} 
This gives rise to the following effective Hamiltonian,
$H_{\mathrm{eff}}=H_{\mathrm{SDQD}}+H_{\rm L}+H_{\rm R}+H_{\rm T}$, 
where 
\begin{eqnarray}
H_{\rm SDQD} 	& = & 
				\sum_{i\sigma} \varepsilon_{i} n_{i\sigma} 
				+\sum_{i} U n_{i\uparrow} n_{i\downarrow} 
				+U' (n_1-1)(n_2-1) 
				\nonumber\\
				&+&\sum_\sigma t(d^\dagger_{1\sigma}d^{}_{2\sigma} + h c ) 
				+J \vec{S}_1\vec{S}_2
				\nonumber\\
				&+&\sum_{i} \!\!\left[ \Gamma_{{\rm S}i} (d^\dagger_{i\uparrow} d^\dagger_{i\downarrow} \!+\! h c )
				+\Gamma_{\rm SX} (d^\dagger_{i\uparrow} d^\dagger_{\bar{i}\downarrow} \!+\! h c ) \right]
	\label{H_DQD} 
\end{eqnarray}
is the Hamiltonian of the SC-proximized DQD
\cite{IW_Kacper,Walldorf2018Feb}, with QD$i$ energy level $\varepsilon_i$,
inter-site (intra-site) Coulomb interactions $U'$ ($U$),
inter-dot hopping $t$, and CAR coupling $\GS{\rm X}$ 
$n_{i\sigma}=d^\dagger_{i\sigma}d^{}_{i\sigma}$ denotes the electron number operator 
at QD$i$, $n_i=n_\uparrow+n_\downarrow$, and $\bar{i}\equiv 3-i$  
Our model is strictly valid in the regime where $\Delta$ is the largest 
energy scale  Nevertheless, all discussed phenomena are
present in a full model for energies smaller than SC gap 
Moreover, by eliminating other consequences of the presence of SC lead,
our model pinpoints the fact that the non-local pairing is 
sufficient for the occurrence of the CAR exchange 
The presence of out-gap states shall result mainly in additional broadening of DQD energy levels,
changing the relevant Kondo temperatures 
We note that the procedure of integrating out out-gap states neglects the 
RKKY interaction mediated by SC lead and other possible indirect exchange mechanisms%
 \footnote{
 Note, that by RKKY interaction we mean only such an effective exchange, 
 which arises due to multiple scattering of a single electron or hole, see \fig{system}(h)-(j) 
 Other mechanisms leading to the total indirect exchange are considered separately 
 In particular, in the large gap limit, exchange described in Ref ~\cite{Yao} is in fact reduced to
 the CAR exchange, and additional antiferromagnetic contribution would arise for finite gap 
 }  
To compensate for this,
we explicitly include the Heisenberg term $ J \vec{S}_1\vec{S}_2$ in
$H_{\rm SDQD}$, with $\vec{S}_i$ denoting the spin operator of QD$i$
and a Heisenberg coupling $J$ substituting the genuine RKKY exchange 

The normal leads are treated as reservoirs of noninteracting electrons,
$H_{r}=\sum_{\mathbf{k}\sigma}\varepsilon_{r\mathbf{k}}c^\dagger_{r\mathbf{k}\sigma}c^{}_{r\mathbf{k}\sigma}$,
where $c^{}_{r\mathbf{k}\sigma}$ annihilates an electron of spin 
$\sigma$ and momentum $\mathbf{k}$ in lead $r$ ($r={\rm L,R}$) with the corresponding energy $\varepsilon_{r\mathbf{k}\sigma}$ 
The tunneling Hamiltonian reads,
$H_{\rm T} = \sum_{r\mathbf{k}\sigma} v_{r} (d^\dagger_{1\sigma}c^{}_{r\mathbf{k}\sigma} + h c )$,
giving rise to coupling between lead $r$ and QD$i$ of strength $\Gamma_r = \pi \rho_r |v_r|^2$,
with $\rho_r$ the normalized density of states of lead $r$ and $v_r$ the 
local hopping matrix element, assumed momentum-independent 
We consider a wide-band limit, assuming constant $\Gamma_r=\Gamma/2$
within the cutoff $\pm D = \pm 2U$ around the Fermi level  

For thorough analysis of the CAR exchange mechanism and its consequences
for transport, we determine the linear conductance between the two normal leads from
\begin{equation}
G = \frac{2e^2}{h} \pi \Gamma \int \left[ -\frac{\partial f_T}{\partial\omega} \right] \mathcal{A}(\omega) {\rm d} \omega ,
\label{G}
\end{equation}
where $f_T$ is the Fermi function at temperature $T$,
while $\mathcal{A}(\omega)$ denotes the normalized local spectral density 
of QD1 \cite{fn1} 
Henceforth, unless we state otherwise, we assume a maximal CAR coupling, 
$\GS{\rm X} = \sqrt{\GS{1}\GS{2}}$ \cite{IW_Kacper,Walldorf2018Feb},
$\GS{1}=\GS{2}=\GS{}$ and consider DQD tuned to the particle-hole symmetry point, 
$\varepsilon_1=\varepsilon_2=-U/2$  However, these assumptions are not crucial for the results presented
here, as discussed in Secs ~\ref{sec:asym}-\ref{sec:coef} 

\section{Estimation of relevant energy scales}
\label{sec:scales}

Since we analyze a relatively complex system, let us build up the understanding of its behavior starting
from the case of a QD between two normal-metallic leads, which can be obtained in our 
model by setting $t=\GS{}=J=U'=0$  Then, the conductance as a function of temperature, $G(T)$, grows
below the Kondo temperature $T_K$ and reaches maximum for $T\to 0$, $G(T\!=\!0)=G_{\rm max}$ 
At particle-hole symmetry point, the unitary transmission is achieved, $G_{\rm max}= G_0 = 2e^2/h$;
see short-dashed line in \fig{G-T}(a) 
An experimentally relevant definition of $T_K$ is that at $T=T_K$ 
$G(T)=G_{\rm max}/2$  $T_K$ is exponentially small in 
the local exchange $J_0 = 8\Gamma / (\pi \rho U)$, and is approximated by
$T_K \approx D \exp[-1/(\rho J_0)]$ \cite{Hewson_book} 

The presence of a second side-coupled QD, $t,U'>0$, significantly enriches the physics of the system 
by introducing direct exchange between QDs, see \fig{system}(b-d) 
In general, effective inter-dot exchange can be defined as energy difference between 
the triplet and singlet states of isolated DQD, 
$J^{\mathrm{eff}} = E_{S=1} - E_{\rm GS}$  Unless $U$ becomes very large, superexchange can be neglected
\cite{Zitko_2QDEx} and $J^{\mathrm{eff}}$ is determined by \emph{direct exchange}, $J^{\mathrm{eff}}\approx 4t^2/(U-U')>0$ 
When the hopping $t$ is tuned small \cite{CPS1}, one can expect $J^{\mathrm{eff}}\lesssim T_K$, which 
implies the two-stage Kondo screening \cite{Pustilnik_Glazman,Cornaglia} 
Then, for $T \ll T_K$, the local spectral density of QD1 serves as a band of width $\sim T_K$ for QD2 
The spin of an electron occupying QD2 
experiences the Kondo screening below the associated Kondo temperature
\begin{equation}
T^* = a T_K \exp(- b T_K / J_{\rm eff})
\label{Tstar}
\end{equation}
with $a$ and $b$ constants of order of unity \cite{Pustilnik_Glazman,Cornaglia} 
This is reflected in conductance, which drops to $0$ with lowering $T$, maintaining characteristic 
Fermi-liquid 
$G\sim T^2$ dependence \cite{Cornaglia}; see the curves indicated with squares 
in \fig{G-T}(a)  Similarly to $T_K$, experimentally relevant definition of $T^*$ is that 
$G(T\!=\!T^*) = G_{\rm max}/2$  Even at the particle-hole 
symmetry point $G_{\rm max} < G_0$, because the single-QD strong-coupling fixed point 
is unstable in the presence of QD2 and $G(T)$ does not achieve $G_0$ exactly,
before it starts to decrease 


The proximity of SC gives rise to two further exchange mechanisms that
determine the system's behavior  First of all, the (conventional)
\emph{RKKY interaction} appears, $J \sim \GS{}^2$ \cite{RK,K,Y}  
Moreover, the \emph{CAR exchange} emerges as a consequence of finite $\GS{}$ \cite{Yao}  
It can be understood on the basis 
of perturbation theory as follows  DQD in the inter-dot singlet state may absorb
and re-emit a Cooper pair approaching from SC; see \fig{system}(e)-(g)  As a second-order
process, it reduces the energy of the singlet, which is the ground state of isolated DQD 
A similar process is not possible in the triplet state due to spin conservation 
Therefore, the singlet-triplet energy splitting $J^{\mathrm{eff}}$ is increased (or generated for $t=J=0$)  
More precisely, the leading ($2$nd-order in $t$ and $\GS{}$) terms
in the total exchange are 
\begin{equation}
J^{\mathrm{eff}} 	\approx 	J + \frac{4t^2}{U-U'+\frac{3}{4}J} + \frac{4\GS{}^2}{U+U'+\frac{3}{4}J} 
\label{Jeff}
\end{equation}
Using this estimation, one can predict $T^*$ for finite $\GS{}$, $t$ and $J$ with \eq{Tstar} 
Apparently, from three contributions corresponding to:
(i) RKKY interaction, (ii) direct exchange and (iii) CAR exchange, only the first may bear a negative (ferromagnetic) sign 
The two other contributions always have an anti-ferromagnetic nature 
More accurate expression for $J^{\mathrm{eff}}$ is derived in Appendix~\ref{sec:downfolding}
[see \eq{A_J}] by the Hamiltonian down-folding procedure  The relevant terms differ 
by factors important only for large $\GS{}/U$  
Finally, it seems worth stressing that normal leads are not necessary for CAR exchange to occur 
At least one of them is inevitable for the Kondo screening though, and two symmetrically coupled 
normal leads allow for measurement of the normal conductance 


It is also noteworthy that inter-dot Coulomb interactions
decrease the energy of intermediate states contributing to direct exchange 
[\fig{system}(c)], while increasing the energy of intermediate
states causing the CAR exchange [\fig{system}(f)] 
This results in different dependence of corresponding terms in \eq{Jeff} on $U'$ 
As can be seen in \figs{G-T}(b) and \ref{fig:G-T}(c), it has a significant effect 
on the actual values of $T^*$ 

\begin{figure}
\includegraphics[width=1\linewidth]{Fig2 pdf}
\caption{(a) Linear conductance $G$ as function of $T$ calculated for 
		 $\varepsilon_1=\varepsilon_2=-U/2$, $\Gamma=U/5$, $U'=U/10$ and different situations, 
		 as indicated  The quantity $\xi\equiv\sqrt{\GS{}^2+t^2}$ is fixed 
		 for different curves drawn with the same dashing style 
		 Note the logarithmic scale on both axes 
		 %
		 (b) Points show $T^*/T_K$ calculated by NRG from curves in subfigure (a)  
		 Lines present the fit to \eq{Tstar} with $J^{\mathrm{eff}}$ obtained from \eq{Jeff} 
		 %
		 (c) The same as (b), only for $U'=0$ 
		 %
		 (d) and (e) show the residual conductance $G_{\mathrm{min}} \equiv G(T \!=\! 0)$ as a function of
		 $\GS{}$ for $t=0$ (denoted "CAR") and $t=\GS{}$ (denoted "Both")  
		 Dotted line is a guide for eyes  $U'=U/10$ in (b) and (d) and $U'=0$ in (c) and (e) 
		}
\label{fig:G-T}
\end{figure}

\section{CAR exchange and Kondo effect}
\label{sec:main}

To verify \eqs{Tstar}-(\ref{Jeff}) we calculate $G$ using
accurate full density matrix numerical renormalization group (NRG) technique \cite{WilsonNRG,Weichselbaum,FlexibleDMNRG,fn2} 
We compare $U'=0$ case with experimentally relevant value $U'=U/10$ \cite{Keller2013Dec} 
While for two close adatoms on SC surface RKKY interactions may lead to prominent consequences
\cite{Klinovaja}, the conventional ({\it i e } non-CAR) contribution should 
vanish rapidly when the inter-impurity distance $r$ exceeds a few lattice constants \cite{RKKYrange,SC_RKKY}  
Meanwhile, the CAR exchange may remain significant for $r$ of the order
of coherence length of the SC contact \cite{Yao}  Therefore, we first neglect the conventional RKKY coupling and analyze its consequences in Sec ~\ref{sec:RKKY} 

The main results are presented in \fig{G-T}(a), showing the temperature dependence of $G$
for different circumstances  
For reference, results for $\GS{}=0$ are shown, exhibiting 
the two-stage Kondo effect caused by \emph{direct} exchange mechanism 
As can be seen in \figs{G-T}(b) and \ref{fig:G-T}(c), an excellent agreement of $T^*$ found from NRG calculations and \eq{Tstar} 
is obtained with $a=0 42$ and $b=1 51$, the same for both $U'=0$ and $U'=U/10$  Note, 
however, that $J^{\mathrm{eff}}$ is different in these cases, cf  \eq{Jeff},
and $U'$ leads to increase of $T^*$ 

Furthermore, for $t=0$ and $\GS{}>0$ the two-stage Kondo effect caused solely by the \emph{CAR
exchange} is present; see \fig{G-T}(a) 
Experimentally, this situation
corresponds to a distance between the two QDs smaller than the superconducting coherence length,
but large enough for the exponentially suppressed direct hopping to be negligible 
While intuitively one could expect pairing to compete with any kind of magnetic ordering,
the Kondo screening induced by CAR exchange is a beautiful example of a superconductivity
in fact leading to magnetic order, namely the formation of the Kondo singlet 
This CAR-exchange-mediated Kondo screening is our main finding 
For such screening, \eq{Tstar} is still fulfilled with very similar 
parameters, $a=0 37$ ($a=0 35$) and $b=1 51$ ($b=1 50$) for $U'=0$ ($U'=U/10$),
correspondingly; see \figs{G-T}(b-c) 
Moreover, as follows from \eq{Jeff}, $U'$ reduces CAR exchange, and therefore diminishes $T^*$ 
For the same values of $J^{\mathrm{eff}}$, the dependence of $G(T)$ for $t=0$ and $\GS{}>0$ is hardly different 
from the one for $\GS{}=0$ and $t>0$ for $T\geq T^*$ (results not shown) 
However, $G(T)$ saturates at residual value $G_{\mathrm{min}}$ as $T\to 0$ only for finite
$\GS{}$, which at particle-hole symmetry makes $G_{\mathrm{min}}$
the hallmark of SC proximity and the corresponding CAR exchange processes 
From numerical results, one can estimate it as
\begin{equation}
G_{\mathrm{min}} = \frac{e^2}{h} \cdot c \, \frac{\GS{}^2}{U^2} 
	\qquad {\scriptstyle (\GS{1}=\GS{2}=\GS{})} ,
\label{Gmin}
\end{equation}
with $c\approx 2 25$, barely depending on $U'$ and getting smaller for $t>0$  
This is illustrated in \figs{G-T}(d-e), where the dotted line corresponds to \eq{Gmin} with $c=2 25$  

Lastly, in \fig{G-T}(a) we also present the curves obtained for $t=\GS{}$ chosen such, 
that the quantity $\xi=\sqrt{t^2+\GS{}^2}$ remains the same 
in all the cases 
This is to illustrate what happens when \emph{both} (direct and CAR) exchange interactions are
present  \fig{G-T}(c) clearly shows that $T^*$ remains practically unaltered for $U'=0$ 
The comparison with \fig{G-T}(b) proves that in this case it practically does not depend 
on $U'$  The enhancement of direct exchange is compensated by the decrease of the CAR one  
On the contrary, $G_{\mathrm{min}}$ decreases for larger $t$ below the estimation given by Eq ~(\ref{Gmin}), 
as can be seen in \figs{G-T}(d-e)  

While analyzing the results concerning $G_{\mathrm{min}}(\GS{})$ plotted in \figs{G-T}(d-e) 
one needs to keep in mind that $G_{\mathrm{min}}$ is obtained at deeply cryogenic conditions  To illustrate
this better, $G(\GS{})$ obtained for $t=0$ and $T=10^{-6}U$ is plotted with solid line 
in \fig{3}  Clearly, for weak $\GS{}$ the system exhibits rather conventional (single-stage)
Kondo effect with $G=G_{\mathrm{max}}\approx 2e^2/h$, while QD2 is effectively decoupled ($G_{\mathrm{max}}<2e^2/h$
in the proximity of SC lead \cite{KWIW})  Only for larger values of $\GS{}$
the CAR exchange is strong enough, such that $T^*>T$ and the dependence $G(\GS{})$ continuously 
approaches the $T=0$ limit estimated by \eq{Gmin} and presented in \figs{G-T}(d-e) 

\section{CAR-RKKY competition}
\label{sec:RKKY}

\begin{figure}
\includegraphics[width=0 98\linewidth]{Fig3 pdf}
\caption{Linear conductance $G$ vs  $\GS{}$ calculated
		 for $t=0$, $\Gamma=U/5$, $U'=U/10$, finite $T=10^{-6}U$
		 and different values of RKKY coupling $J$, as indicated  
		 Inset shows QD1 spectral function $\mathcal{A}(\omega)$ as a function of energy $\omega$
		 for points on $J=-0 1U$ curve, indicated with corresponding symbols 
		}
\label{fig:3}
\end{figure}

Let us now discuss the effects introduced by the conventional RKKY interaction 
We choose $t=0$ for the sake of simplicity and
analyze a wide range of $\GS{}$, starting from the case of anti-ferromagnetic 
RKKY interaction ($J>0$)  Large $J>0$ leads to the formation of a molecular singlet in the 
nanostructure  This suppresses the conductance, unless $\GS{}$ becomes of the order of $U/2$, 
when the excited states of DQD are all close to the ground state  This is illustrated 
by double-dotted line in \fig{3} 
Smaller value of $J>0$ causes less dramatic consequences, namely it just increases $J^{\mathrm{eff}}$ according
to \eq{Jeff}, leading to enhancement of $T^*$, cf  \eq{Tstar}  This is presented with
dot-dashed line in \fig{3} 

The situation changes qualitatively for ferromagnetic RKKY coupling, $J<0$ 
Then, RKKY exchange and CAR exchange have opposite signs and compete with each other 
Depending on their magnitudes and temperature, one
of the following scenarios may happen 

For $J^{\mathrm{eff}} > 0$, {\it i e } large enough $\GS{}$, and $T<T^*$, the system is in the 
singlet state due to the two-stage Kondo screening of DQD spins  $G(T\!=\!0)$ is reduced 
to $G_{\mathrm{min}}$, which tends to increase for large negative $J$; see dashed lines in \fig{3}  
In the inset to \fig{3}, the spectral density of QD1 representative for this regime is plotted 
as curve indicated by triangle  It corresponds to a point on the $J=-0 1U$ curve in the main 
plot, also indicated by triangle  The dip in $\mathcal{A}(\omega)$ has width of order of $T^*$ 

For finite $T$, there is always a range of sufficiently small $|J^{\mathrm{eff}}|$, where QD2 becomes effectively
decoupled, and, provided $T<T_K$, $G$ reaches $G_{\mathrm{max}}$ due to conventional Kondo effect 
at QD1  This is the case for sufficiently small $\GS{}$ for $J=0$ or $J=-0 01U$, and in the narrow
range of $\GS{}$ around the point indicated by a circle in \fig{3} for $J=-0 1U$ (for $J=0 01U$, 
the considered $T$ is close to $T^*$ and $G$ does not reach $G_{\rm max}$)  The conventional Kondo effect manifests itself with 
a characteristic peak in $\mathcal{A}(\omega)$, as illustrated in the inset in \fig{3} with line denoted by circle 

Finally, large enough $J^{\mathrm{eff}} < 0$ and low $T$, give rise to an effective ferromagnetic coupling of DQDs 
spins into triplet state  Consequently, the underscreened Kondo effect occurs 
\cite{Mattis,NozieresBlandin} for weak $\GS{}$ and, {\it e g }, $J=-0 1U$; 
see the point indicated by square in \fig{3} 
This leads to $G=G_{\mathrm{max}}$ and a peak in $\mathcal{A}(\omega)$, whose shape is significantly different from the
Kondo peak, cf  the curve denoted by square in the inset in \fig{3}  



\section{Effects of detuning from the particle-hole symmetry point}
\label{sec:asym}

\begin{figure}
\includegraphics[width=0 98\linewidth]{Fig4 pdf}
\caption{
         (a) Linear conductance between the normal leads $G$ as a function of temperature $T$
         for parameters corresponding to \fig{G-T}(a) with $\xi=U/10$, and additional curves for finite 
         detuning from particle-hole symmetry point, $\delta_1=-\delta_2$, 
         and two values of $\xi=\sqrt{t^2+\GS{}^2}$, as indicated in the figure 
         (b) $G_{\mathrm{min}} \equiv G(T \!=\! 0)$ as a function of QD1 detuning $\delta_1$ for different
         exchange mechanisms, $\xi=U/10$ and $\delta_2=\pm\delta_1$ (as indicated) 
		}
\label{fig:asym}
\end{figure}

At PHS $G_{\mathrm{min}}=G(T \!=\! 0)=0$ in the absence of superconducting lead, making $G_{\mathrm{min}} > 0$ a hallmark
of SC-induced two-stage Kondo effect  However, outside of PHS point $G_{\mathrm{min}} > 0$ even in the case of 
the two-stage Kondo effect caused by the direct exchange  
Exact PHS conditions are hardly possible in real systems, and the fine-tuning of the QD energy
levels to PHS point is limited to some finite accuracy 
Therefore, there may appear a question, if the results obtained at PHS are of any importance for the
realistic setups  As we show below --- they are,
in a reasonable range of detunings $\delta_i=\varepsilon_i +U/2$ 

In \fig{asym}(a) we present the $G(T)$ dependence in and outside the PHS, corresponding to 
parameters of \fig{G-T}(a)  
Clearly, for considered small values of $\delta_1=\delta_2=\delta$, 
$G_{\mathrm{min}}<10^{-3}e^2/h$ for direct exchange only, while $G_{\mathrm{min}}$ in the presence of a superconductor is 
significantly increased and close to the PHS value  Furthermore, for $|\delta_1| \sim |\delta_2| 
\sim \delta$, the residual conductance caused by the lack of PHS, $G_{\mathrm{min}} \approx e^2/h \cdot (\delta/U)^2$,
which is a rapidly decreasing function in the vicinity of PHS point, as illustrated in \fig{asym}(b)
with lines denoted by a square  Evidently, in the regime $|\delta_i| < 0 01U$ the residual conductance
caused by SC is orders of magnitude larger, leading to the plateau in $G_{\mathrm{min}}(\delta_1)$ dependence,
visible in \fig{asym}(b) 
Taking into account that the realistic values of $U$ in the semiconductor quantum dots are rather 
large, this condition seems to be realizable by fine-tuning of QD gate voltages 

Lastly, let us point out that while in the presence of only one exchange mechanism, \emph{CAR} or
\emph{direct}, $G_{\mathrm{min}}(\delta_1)$ dependencies depicted in \fig{asym}(b) are symmetrical with respect
to sign change of $\delta_1$, for \emph{both} exchange mechanisms the dependence is non-symmetric  

\section{Effects of asymmetry of couplings to superconductor}
\label{sec:x}

\begin{figure}
\includegraphics[width=0 98\linewidth]{Fig5 pdf}
\caption{
		 (a) Linear conductance between the normal leads, $G$, as a function of temperature, $T$,
		 for parameters corresponding to \fig{G-T}(a) with $\xi=U/10$, for different values 
		 of asymmetry coefficient $x$ [see \eq{xGS}], in the presence of \emph{CAR} exchange only 
		 %
		 (b) The second-stage Kondo temperature $T^*$ normalized by $T_K$ as a function of $x$, 
		 calculated with the aid of NRG (points) and a fit to \eq{Tstar} (lines) 
		 with $J^{\mathrm{eff}}$ from \eq{Jeff} 
		 %
		 (c) The zero-temperature conductance $G_{\mathrm{min}}$ as a function of QD1 coupling to SC lead, $\GS{1}$,
		 compiled from data obtained at different circumstances (as indicated in the legend)
		 for different $x$  Dotted line corresponds to \eq{Gmin2} with $c=2 25$ 
		}
\label{fig:x}
\end{figure}

Similarly to PHS, the ideal symmetry in the coupling between respective QDs and SC lead is hardly possible
in experimental reality  As shown below, it does not introduce any qualitatively new features 
On the other hand, it decreases the second stage Kondo temperature, which is already small, therefore,
quantitative estimation of this decrease may be important for potential experimental approaches 
To analyze the effects of $\GS{1}\neq\GS{2}$, we introduce the asymmetry parameter $x$ and extend
the definition of $\GS{}$,
\beq
x = \frac{\GS{1}-\GS{2}}{\GS{1}+\GS{2}}, \quad \GS{} = \frac{\GS{1}+\GS{2}}{2} 
\label{xGS}
 \end{equation} 
Note, that even for a fixed $\GS{}$, the actual CAR coupling $\GS{\rm X}=\GS{}\sqrt{1-x^2}$ decreases
with increasing $|x|$, which is a main mechanism leading to a decrease of $T^*$ outside the $x=0$ point
visible in \figs{x}(a) and (b)  To illustrate this, the curves corresponding to \emph{both} exchange
mechanisms were calculated using $x$-dependent $t=\GS{\rm X}$ instead of $t=\xi/\sqrt{2}$  
Therefore, $\xi$ was generalized for $x\neq 0$ by setting $\xi=\sqrt{t^2(1-x^2)^{-1}+\GS{}^2}$ 
Clearly, in \fig{x}(b) the curves for different exchange mechanisms are very similar and differ mainly 
by a constant factor, resulting from different influence of $U'$; see \Sec{scales}  
The magnitude of $T^*$ changes is quite large, exceeding an order of magnitude for $x=\pm 0 5$ 
and $\xi=U/20$  Moreover, $T^* \to 0$ for $x\to\pm 1$  Consequently, for strongly asymmetric
devices one cannot hope to observe the second stage of Kondo screening 

A careful observer can note that the $T^*(x)$ dependency is not symmetrical; note for example different 
$T^*$ for $x=\pm 0 5$ in \fig{x}(a)  This is caused by the dependence of the first stage Kondo temperature
$T_K$ on $\GS{1}$ \cite{part1,DomanskiIW},
\beq
\widetilde{T}_K(\GS{1}) = T_K \cdot \exp\!\left( \frac{\pi}{2} \frac{\GS{1}^2}{\Gamma U}\right) 
 \end{equation} 
Here, $T_K$ is, as earlier, defined in the absence of SC, while $\widetilde{T}_K$ is a function 
of $\GS{1}$, such that $G(\widetilde{T}_K) = G_{\rm max}(\GS{1})/2$ in the absence of QD2  
As $\widetilde{T}_K$ grows for increasing $\GS{1}$ (or $x$), $T^*$ decreases according to \eq{Tstar}  
Its $\GS{}$ dependence can be accounted for by small changes in the coefficients $a$ and $b$ in \eq{Tstar}, 
as long as $x$ is kept constant  

To close the discussion of $T^*(x)$ dependence let us point out, that in \eq{A_J} 
there appears a correction to \eq{Jeff} for $x\neq 0$  However, it is very small due to additional
factor $\GS{}^2/U^2$ in the leading order  Its influence on curves plotted in \fig{x}(b) is hardly visible 

In turn, let us examine the $x$ dependence of the $T=0$ conductance $G_{\mathrm{min}}$  As can be seen 
in \fig{x}(a), it monotonically increases with $x$, as it crosses $x=0$ point  In fact, \eq{Gmin}
can be generalized to
\beq
G_{\mathrm{min}} = \frac{e^2}{h} \cdot c \, \frac{\GS{1}^2}{U^2} ,
\label{Gmin2}
 \end{equation} 
with $c\approx 2 25$ (indicated by a dotted line in \fig{x}(c))  Note that $G_{\mathrm{min}}$ is proportional to 
$\GS{1}^2=(x+1)^2 \GS{}^2$, instead of simply $\GS{}$, cf  \eq{Gmin}  The values of $G_{\mathrm{min}}$ obtained
from all analyzed $G(T)$ dependencies for different $x$ have been compiled in \fig{x}(c) 
It is evident, that \eq{Gmin2} is approximately fulfilled for all the considered cases 

Finally, it seems noteworthy that the normal-lead coupling asymmetry, 
$\Gamma_{\rm L}\neq \Gamma_{\rm R}$, is irrelevant for the results except for a constant factor
diminishing the conductance $G$ \cite{KWIWJB-asym} 



\section{The role of CAR efficiency}
\label{sec:coef}

\begin{figure}[tb]
\includegraphics[width=0 98\linewidth]{Fig6 pdf}
\caption{Linear conductance between the normal leads
		 $G$ as a function of coupling to SC lead, $\GS{}$, for indicated values of RKKY exchange $J$
		 and the efficiency of CAR processes reduced by factor (a) $\mathcal{C}=0 9$ and (b) $\mathcal{C}=0 5$ 
		 Other parameters as in \fig{3} 
		 Insets: QD1 local spectral density $\mathcal{A}(\omega)$ as a function of energy $\omega$
		 for points on $J=-0 1U$ curve, indicated with corresponding symbols 
		} 
\label{fig:C}
\end{figure}

Up to this point we assumed $\GS{\rm X} = \sqrt{\GS{1}\GS{2}}$, which is valid when the two 
quantum dots are much closer to each other than the coherence length in the superconductor 
This does not have to be the case in real setups, yet relaxing this assumption does not 
introduce qualitative changes  Nevertheless, the model cannot be extended to inter-dot 
distances much larger than the coherence length, where $\GS{\rm X}\to 0$ 

To quantitatively analyze the consequences of less effective Andreev coupling we define the 
CAR efficiency as $\mathcal{C} \equiv \GS{\rm X} / \sqrt{\GS{1}\GS{2}}$ and analyze $\mathcal{C} < 1$
in the wide range of $\GS{1}=\GS{2}=\GS{}$ and other parameters corresponding to \fig{3}  
The results are presented in \fig{C} 

Clearly, decreasing $\mathcal{C}$ from $\mathcal{C}=1$ causes diminishing of $\GS{\rm X}$, and consequently of CAR 
exchange  For a change as small as $\mathcal{C}=0 9$, the consequences reduce to some shift of the 
conventional Kondo regime, compare \fig{C}(a) with \fig{3}  Stronger suppression of CAR may, 
however, increase the SC coupling necessary to observe the second stage of Kondo screening caused
by CAR outside the experimentally achievable range, see \fig{C}(b)  Moreover, the reduced $T^*$
leads to narrowing of the related local spectral density dip, while the
increased critical $\GS{}$ necessary for the observation of the second stage of screening leads to the
shallowing of the dip  This is visible especially in the inset in \fig{C}(b) 


\section{Conclusions}
\label{sec:conclusions}

The CAR exchange mechanism is present in any system comprising at least
two QDs or magnetic impurities coupled to the same superconducting contact
in a way allowing for crossed Andreev reflections 
In the considered setup, comprised of two quantum dots in a T-shaped geometry 
with respect to normal leads and proximized by superconductor,
it leads to the two-stage Kondo
screening even in the absence of other exchange mechanisms 
This CAR induced exchange screening is characterized by a residual 
low-temperature conductance at particle-hole symmetric case 
We have also shown that the competition between CAR exchange and RKKY
interaction may result in completely different Kondo screening scenarios 

The presented results bring further insight into the low-temperature
behavior of hybrid coupled quantum dot systems, which hopefully could be verified
with the present-day experimental techniques 
Moreover, non-local pairing is present also in bulk systems such as non-$s$-wave superconductors 
The question if an analogue of discussed CAR exchange may play a role there
seems intriguing in the context of tendencies of many strongly correlated materials
to possess superconducting and anti-ferromagnetic phases 


\begin{acknowledgments}
This work was supported by the National Science Centre in Poland through project no 
2015/19/N/ST3/01030 
We thank J  Barna\'{s} and T  Maier for valuable discussions 
\end{acknowledgments}





Context 2: Paper Info

Title: CONTOUR COMPLETION USING DEEP STRUCTURAL PRIORS
Publish Date: 9 Feb 2023
Author List: Ali Shiraee, Morteza Rezanejad, Mohammad Khodadad, Dirk Walther, Hamidreza Mahyar

Figure

Figure 1: Just by looking at subfigure (a), we, as humans, can easily perceive a shape like the one in subfigure (b) This is an extraordinary capability of our human brain and in this paper, we tried to see whether convolutional neural networks can show such capabilities 
Figure 2: The trajectory from random noise X N to the incomplete image X I in image space The network will pass a completed version of the image, X C , throughout this trajectory 
Figure 4: This figure shows our iterative process to complete the fragmented contours of an image given as input to our pipeline 
Figure 5: This example shows how different scores change throughout a single run All three scores change in the range of [0, 100] Our goal is to maximize reconstruction_score and minimize the overfit_score, but we should consider that the minimization lower bound is data dependent and is not zero 
Figure 6: Evolutionary process of the deep structure prior The right column shows the incomplete shapes given to the model and the rest of the columns show how the model is overfitting gradually to produce the incomplete shapes In each column, we are showing an intermediate iteration of this process The loss-term setup enables our pipeline to let the completed image appears during this iterative process 
Average MSE and IoU values between the incomplete (Raw) images, the output of DIP and DSP methods, and ground truth for each image are provided in this table 
For this experiment, we ran the model over a subset of complex dataset with 500 incomplete images at various levels of alpha for 250 iterations After the image completion is done, we compared the evaluation metrics between the completed image and the ground truth to examine the performance of the model for different values of alpha 
In this table, we show the effect of the receptive filter size on our algorithm's capability to fill in bigger gap sizes The numbers in this table are showing the percentage of the time that DIP was successful to complete shapes with each gap size and corresponding receptive field size As predicted, the bigger the filter size, the more successful the algorithm is in filling in the gaps 

abstract

Humans can easily perceive illusory contours and complete missing forms in fragmented shapes  This work investigates whether such capability can arise in convolutional neural networks (CNNs) using deep structural priors computed directly from images  In this work, we present a framework that completes disconnected contours and connects fragmented lines and curves 
In our framework, we propose a model that does not even need to know which regions of the contour are eliminated  We introduce an iterative process that completes an incomplete image and we propose novel measures that guide this to find regions it needs to complete  Our model trains on a single image and fills in the contours with no additional training data 
Our work builds a robust framework to achieve contour completion using deep structural priors and extensively investigate how such a model could be implemented 

Introduction

The human visual system is used to seeing incomplete outlines  Our brains can effortlessly group visual elements and fragmented contours that seem to be connected to each other  This power enables us to make shapes, organize disconnected visual features, and even properties of 3D surfaces when projected on 2D planes 
demonstrated how early vision may quickly complete partially-occluded objects using monocular signals  This capability of perceptual grouping has been studied in vision science for decades   Although there has been some work on perceptual grouping in the past couple of years, it has been less studied in the past decade due to the enormous progress of deep neural networks and their success in dealing with the pixel-by-pixel inference of images 
Different types of lines and curves have been studied to maximize the connectivity of two broken ends in the planer contour completion problem   Different types of lines and curves have been studied to maximize the connectivity of two broken ends in the planer contour completion problem  Geometry-based constraints can be utilized to address some challenges of contour completion problems, such as smoothness and curvature consistency  
However, such approaches only work for simple, smooth contours and usually fail in more complex settings  On the other hand, we currently have deep models that could easily take an incomplete image and complete the missing regions using enough training data   The amazing capability of such models especially those that are trained on different modalities with millions or billions of training data raises the question of whether we need such a large amount of training to perceive all the visual cues that are present in an image, which underlies visual perception by humans 
In human vision, Gestalt psychology suggests that our brain is designed to perceive structures and patterns that are grouped by some known rules  In this work, we show that some perceptual structures can also be learned from the image itself directly using architectures that enable such learning  Earlier work has shown  This is an extraordinary capability of our human brain and in this paper, we tried to see whether convolutional neural networks can show such capabilities 
that some forms of perceptual grouping can be achieved using computational models, such as stochastic completion fields   This type of learning resonates with some of the Gestalt perceptual grouping principles including "proximity", "good continuation" and "similarity"  In scenarios where color and/or texture are present, the cue of "similarity" helps us group regions with consistent patterns  
When color and texture are present, they provide a collection of rich information for such cues  In the present article, we probe convolutional neural networks in a scenario where both are absent, and the neural network is dealing with just forms and shapes  Specifically, we explore whether the convolutional neural network architecture itself can give rise to some of these grouping cues when they are fed just contours and shapes alone 
For years, neural networks have been treated as black boxes that can generalize images very well to multiple classes when there are enough training exemplars  One of the reasons that neural networks are trained on many exemplars is to avoid the problem of overfitting  On the other hand, we know that CNNs that generalize well to large classes of exemplars can easily overfit when those class labels are randomly permuted  
Inspired by this observation, suggest that image priors can be learned to a large extent through a generator network architecture that is solely trained on a single image  This encouraged us to take a deeper look at what structural information can be learned from a single-shape image and whether we can reconstruct some of those perceptual grouping capabilities using a generator network 
Inspired by , in this work, we adopt a novel training regime to complete shapes and contours where we use a UNet architecture with random initial weights and try to complete the contours within a single image without any training data  In our model, we imagine that the input image (i e , the only image used to update the model's weights) is an image of fragmented contours 
In this work, instead of training the model on multiple images fetched from a big image dataset, we imagine a random fixed tensor noise image as input to this model  At each iteration, the random noise tensor is inferred through our generative network and the network produces an outcome image  We introduce a novel loss function that enables this network to complete contours 
This process repeats, and the weights of our network are updated gradually based on this loss function, which is an energy term defined based on the input image and the output of the network  The model will reconstruct the missing structures i e , group fragmented contours that perceptually seem to be connected, before it fully overfits to the incomplete input image 
Contributions of our work are summarized as follows: 1  In our pipeline, we propose a novel algorithm that enables us to complete contours that appear to be connected to each other in an illusory form  2  Our model is trained on just one single query image and does not need any training data  3  Our model does not need to know which regions of the image are masked or occluded, i e , we remove the dependency of the algorithm on the guiding mask (a guiding mask is a mask that informs the model on where the missing regions are located at) 
We also introduce two metrics to produce a stopping criterion to know when to stop training before the model fully overfits to the incomplete image, i e , we guide the model to stop when the completed image is produced 

Methods

Our eyes are trained to predict a missing region of an occluded object within a scene  We can easily perceive or make guesses about parts of objects or shapes that we do not necessarily see  Even when we are looking at an image, we might guess about the shape, property, or other attributes of an unknown object within a scene 
Such capability extends beyond just known objects or shapes  We can look at a disconnected set of contours and guess what the connected form may look like  This capability is rooted in our prior knowledge about the world  (see Figure )  In this work, we aim to achieve a similar capability using deep generative networks 
Most neural networks that we work with these days are trained with a massive amount of data and one might think that this is the only way that a neural network can obtain prior information  Authors of Deep Image Prior (DIP) suggest that the convolutional architecture can capture a fair amount of information about image distribution 
They show that the hourglass architectures like UNet can show some good performances in some inverse problems such as image denoising, super-resolution, and inpainting  In this work, we focus on completing fragmented contours end-to-end just by using a single image  To be able to address this problem, we first look at a similar problem in image editing, known as image inpainting 
Image inpainting is the task of completing an image where some regions of that image are covered or filtered by a mask  In image inpainting, the generative model receives a masked image with the mask that guides the algorithm to fill in those missing regions  Although in the problem of contour completion, we have a very similar goal, the additional challenge that we suffer from is that we do not necessarily have a mask that covers the regions of interest for us 
For example, when we look at Figure (left), we are not provided that which regions of the image are incomplete by a guiding mask  Our brain figures this out just by looking at the form and predicting those missing regions  Inspired by the image inpainting work of DIP , we propose a novel algorithm for the contour completion problem (see Figure ), where, unlike DIP, we do not have a guiding mask to know where to fill in the missing regions of our disconnected contours 
Let us assume that we are given a degraded image x I containing a fragmented contour  We propose an iterative process (see Figure ) that can connect those discontinuities and glue those fragmented pieces together as follows  We propose an hour-glass model structure (f ) that is initially set up with completely random parameters (θ 0 ) at first 
Through an iterative process, we start feeding our network with a fixed random tensor noise z signal and obtain the inferred output (f (z)) from that network  We then back-propagate the difference between the inferred output and the incomplete image to the network  We then repeat this process until the difference between the generated outcome of the network (f θ (z)) and the incomplete image (x I ) gets smaller and smaller and finally overfits the incomplete image (x I ) 
In this work, we propose a novel error metric to backpropagate in the model and update its weights  we set the metric in a way that enables us to complete the incomplete image before it overfits the incomplete image  This is where the magic of our algorithm happens  We also propose a stopping criterion, so that when the image is complete, we no longer overfit the outcome of the model and instead produce a plausible connected set of fragmented contour pieces 
As illustrated in Figure , this trajectory will pass through a complete version of the image in image space, which is close to the actual connected ground truth x gt , which we do not have access to directly 

Energy Function

We can model this iterative process mathematically by maximizing a posterior distribution  Let us assume that the optimal image x * that we want to achieve is on a path that connects a random tensor noise z to the incomplete image x I   With this assumption, we can eventually overfit any random tensor noise to the incomplete image x I , and we can formulate the posterior distribution of our wanted optimal image x * as follows:
No Prior To better recapitulate what we want to achieve using our generative model, we solve an energy minimization problem on the parameter space of the model, rather than explicitly working with probability distributions and optimizing on x (image space)  Thus, we solve an energy minimization problem that incorporates the incomplete image (x I ) and model parameters (f θ (z)):
As shown in Figure , the pipeline starts from a random initialized set of parameters θ and updates those weights until it reaches a local minimum θ *   The only information provided for the network is the incomplete image x I   When we reach the optimal θ * , the completed image is obtained as x * = f θ * (z) where z is random tensor noise 
In this work, we use a U-Net architecture with skip connections as the generator model  " As we mentioned previously, in this work we were inspired by an earlier work known as Deep Image Prior (DIP)   In this work, the authors suggested a mean-squared-error loss term that enables the network to compare the output of the generator to the incomplete input image:
where x I is the incomplete image with missing pixels in correspondence of a binary mask m ∈ {0, 1} H×W and operator is for point-wise multiplication of two image matrices  In the inpainting tasks, the existence of a mask is essential as the algorithm needs to know where to fill in the missing area, whereas, in our work, we wanted to know whether the network can perform completion on its own without the need for the mask 
In other words, is it possible for the network to predict where to fill in at the same time that it is trying to reconstruct the incomplete image through the iterative process? To answer this question, we tried to solve a much harder problem in which the mask is not provided to the model and the model is agnostic to it 
To better understand how a solution could be hypothesized for this problem, we first imagine that we want to consider all the available regions in our image that could be potential places to fill in, i e , we set the mask in the previous formula 1 to be equal to the incomplete image x I   This is problematic as the model quickly tries to fill in all white space and quickly reconstructs the incomplete image by doing so 
On the other hand, we can take the inverse problem of the current problem, where the model tries to just fill in the regions that fragmented contour lives in
Context 3: Weep Not, Child is a 1964 novel by Kenyan author Ngũgĩ wa Thiong'o  It was his first novel, published in 1964 under the name James Ngugi  It was among the African Writers Series  It was the first English language|English novel to be published by an East African  Thiong'o's works deal with the relationship between Africans and white settlers in colonial Kenya, and are heavily critical of colonial rule  Specifically, Weep Not, Child deals with the Mau Mau Uprising, and "the bewildering dispossession of an entire people from their ancestral land " Ngũgĩ wrote the novel while he was a student at Makerere University 

The book is divided into two parts and eighteen chapters  Part one deals mostly with the education of Njoroge, while part two deals with the rising Mau Mau movement 

Plot summary

Njoroge, a little boy, is urged to attend school by his mother  He is the first one of his family able to go to school  His family lives on the land of Jacobo, an African made rich by his dealings with white settlers, namely Mr  Howlands, the most powerful land owner in the area  Njoroge's brother Kamau works as an apprentice to a carpenter, while Boro, the eldest living son, is troubled by his experiences while in forced service during World War II, including witnessing the death of his elder brother  Ngotho, Njoroge's father and a respected man in the surrounding area, tends Mr  Howlands' crops, but is motivated by his passion to preserve his ancestral land, rather than for any compensation or loyalty 

One day, black workers call for a strike to obtain higher wages  Ngotho is ambivalent about participating in the strike because he fears he will lose his job  However, he decides to go to the gathering, even though his two wives do not agree  At the demonstration, there are calls for higher wages  Suddenly, the white police inspector brings Jacobo to the gathering to pacify the native people  Jacobo tries to put an end to the strike  Ngotho attacks Jacobo, and the result is a riot where two people are killed  Jacobo survives and swears revenge  Ngotho loses his job and Njoroge’s family is forced to move  Njoroge’s brothers fund his education and seem to lose respect for their father 

Mwihaki, Jacobo's daughter and Njoroge's best friend, enters a girls' only boarding school, leaving Njoroge relatively alone  He reflects upon her leaving, and realizes that he was embarrassed by his father's actions towards Jacobo  For this reason, Njoroge is not upset by her exit and their separation  Njoroge switches to another school 

For a time, everyone's attention is focused on the upcoming trial of Jomo Kenyatta – a revered leader of the movement  Many blacks think that he is going to bring forth Kenya’s independence  But Jomo loses the trial and is imprisoned  This results in further protests and greater suppression of the black population 

Jacobo and a white landowner, Mr  Howlands, fight against the rising activities of the Mau Mau, an organization striving for Kenyan economic, political, and cultural independence  Jacobo accuses Ngotho of being the leader of the Mau Mau and tries to imprison the whole family  Meanwhile, the situation in the country is deteriorating  Six black men are taken out of their houses and executed in the woods 

One day Njoroge meets Mwihaki again, who has returned from boarding school  Although Njoroge had planned to avoid her due to the conflict between their fathers, their friendship is unaffected  Njoroge passes an important exam that allows him to advance to High School  His village is proud of him, and collects money to pay Njoroge's High School tuition 

Several months later, Jacobo is murdered in his office by a member of the Mau Mau  Mr  Howlands has Njoroge removed from school for questioning  Both father and son are brutally beaten before release and Ngotho is left barely alive



ground_truth: It tends to suppress the Kondo effect.

prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: How does the receptive field size affect the completion of shapes?

Read the following text and answer briefly.

Context 2:  Taking these two at the same time, we came up with a novel loss term for energy minimization term that helps us remove the need for the mask in the case of the contour completion problem:
In this term, we introduce a linear combination of the two loss terms, where one focuses on reconstructing the missing regions in the foreground, and one focuses on avoiding inpainting regions in the background  The logic behind this is that, if we assume the original image to be representative of the mask, then the model tries to reconstruct in all white regions (the foreground), and in the inverse problem we just want to reconstruct the regions that are already part of the ground truth 

Stopping Criteria

As shown in Figure , knowing when to stop iterating to the over-fitted model is a key to obtaining a completed shape  Therefore, we equipped our model with a criterion that uses two individual novel terms to know when to stop and output the result of the network  These two metrics expand the capability of the generator network beyond what it does currently and achieve a full end-to-end contour completion model that trains and infers on a single image of divided contour fragments 
These new terms are: reconstruction_score (ρ) and overfit_score (ω) 

Reconstruction Score

The first score that this paper suggests is the reconstruction score, i  e , we have to make sure that the model is trained enough that it can reconstruct at least the entire set of fragmented contours within the image  This is a trivial score and to compute the reconstruction_score (ρ), we apply a k-dimensional tree (KDTree) nearest-neighbor lookup to find the ratio of points in the original incomplete image (x 0 ) 
This score ranges from [0 − 100] 

Overfit Score

It is evident that the model overfits the fragmented contours  This is due to the fact that the error in our loss term is minimized as the x overfits to x I , i  e , replacing x with x I in the loss term would give us zero  As we hypothesize iterative process also produces the complete image before it overfits to the incomplete image, we can imagine that at some point the image is complete (x C ) and does not need to be fine-tuned any more to overfit to x I  
We suggest a new score called overfit_score  overfit_score determines how much of the reconstructed outcome is over the number of pixels that are already in the incomplete image (x I )  To compute the overfit_score (ω), we apply a k-dimensional tree (KDTree) nearest-neighbor lookup of points in the outcome of the input image and see what portions of those points are novel and not already in the incomplete image (x I ) 
Similar to reconstruction_score, the overfit_score also ranges from [0 − 100]  Our goal is to maximize reconstruction_score and minimize the overfit_score, but we should consider that the minimization lower bound is data dependent and is not zero 

Combined Score

To be able to find the best possible set of completed contours, we combine the two and have a loop that tries to achieve close to full reconstruction and avoids over-fitting at the same time  This is what we call an "ideal" stopping point in the contour completion problem  In each run throughout all iterations, we pick an output of the network that minimizes a dissimilarity term:
where δ represents our dissimilarity score  The reconstruction_score and overfit_score are obtainable given network output and the incomplete image  Ideally, we want to achieve an output image that has a reconstruction_score equal to 100 and an overfit_score of γ which is a hyperparameter that is dependent on the image and complexity of the shape 
Empirically, we observed that this value is highly correlated with the distance between gaps that are observed in fragmented contours, i  e , the larger the gap results in a larger γ value  We will discuss this in more detail in the next section (see Section 3)  For one sample image, we computed the two metrics reconstruction_score and overfit_score and the combined value of dissimilarity (δ) and showed how these values change (see Figure ) 
Our initial observations show that the reconstruction_score will increase to 100 quickly for the incomplete image indicating that the already existing fragments of the contours have been reconstructed in the output  However, as mentioned previously, we cannot solely rely on this score since we also want to minimize the overfitting 
Remember that our goal is to produce an output that: a) preserves the original contours in the incomplete image and b) fills in the gaps between the fragmented contours  It is evident that overfit_score decreases throughout an iterative run of our process until it reaches zero  The dissimilarity will also decrease along with the overfit to a point, then it will increase, as the model tries to reproduce the incomplete image 
This is where an ideal γ value can be picked, i e , where to stop when the reconstruction is good but we have not done a full overfit to the incomplete image  Thus, one should pick the value of γ empirically in the scenario that the ground truth is not available, whereas, assuming that the ground truth is available, we can easily compute the best γ value 
In our experiments, we tried two datasets of images with different gap sizes  We observed that the best the γ for one set of samples is ∼ 5 (the set with shorter gaps) while it is ∼ 23 for samples from the other set, i  e, the set with longer gaps (see Figure for some completed examples) 

Experiments and Results

Performing unsupervised contour completion is a difficult task to benchmark as one can never know what fragments exactly are connected to each other in a real-world scenario  This makes the problem of contour completion a hard problem to solve  In this paper, we tried to create artificial shapes that are occluded by some masks and then tried to see if our model can regenerate the missing pieces and glue those divided contours together 
To demonstrate our model's behavior, we will conduct experiments on datasets created for this task and will report on them in this section  To compare network results in different settings, we will use pixel-wise Mean Squared Error (MSE) and Intersection over Union (IoU) between the produced result of the network and unmasked ground truth data and the reconstructed image on black pixels (where contours live) 

Data

We prepared two datasets, one labeled "Simple" and one "Complex", in accordance with the number of gaps in each shape  Both datasets contain nine different categories of shapes  In order to generate the Complex dataset, we used FlatShapeNet which is a dataset for the educational game Ariga  The dataset includes the following categories: Circle, Kite, Parallelogram, Rectangle, Rhombus, Square, Trapezoid, Triangle and Overlap 
The "overlap" category contains images that are made as a mixture of two shapes that are overlapping from the previous categories  These are some standard shapes with a few gaps in simple dataset, while the complex dataset has some hand-drawn shapes with fragmented lines and more gaps that produce more variety in general 
For each instance, a ground truth image is available for comparison  Most of our experiments have been conducted using the complex dataset in order to evaluate the generalization of our approach  For the analysis of how γ values should be set for each shape, we used the simple dataset as a reference 

Evaluation

In this section, we compare our model to the original Deep Image Prior (DIP) inpainting model  DIP's inpainting module accepts a degraded image and a binary mask corresponding to that image  In order to make a fair comparison, instead of providing a binary mask, we used the incomplete images both as input and as a mask in order to see whether it can produce a result similar to ours 
For DIP, we run the iterative process for a maximum number of 2500 iterations with the U-net backbone  We used the exact same architecture and setting in our model for a fair comparison  Using our ground truth dataset images, we calculate the MSE loss between the network output and ground truth during each iteration instead of relying on our stopping mechanism described in the previous section 
We then store the output with minimal loss throughout all the iterations  Finally, we select the best output among all iterations, report the MSE and IoU with the ground truth, and save the iteration number which resulted in the lowest MSE  Table compares the results that are obtained using the DIP method, the DSP method (ours), and the difference between raw images and the ground truth 
We have presented the average MSE-loss, average IoU, and the average number of iterations for the best output for different methods  As can be seen from the table, our model improves both MSE and IoU between the incomplete image and ground truth in fewer iterations  The DIP method can neither generate a better result than the raw image nor provide stopping criteria to prevent overfitting 
We provide a more detailed analysis of this result in Figure   As results show, our algorithm not only provides a much faster convergence but also consistently provides a better-completed image (consistently less MSE loss and better IoU), whereas it is challenging for the DIP method to accomplish better results without a guiding mask 
We compare MSE loss between the degraded raw images that these algorithms started with (shown in blue) (a) Mean Squared Error Loss: we clearly see that for almost all images, DSP (green) achieves a lower MSE than the incomplete images (blue) whereas, the DIP completed images either do not improve the MSE or even worsen that for the incomplete images 
Note that, the MSE is computed to an available ground truth image hidden from our methods (the lower is better)  (b) Intersection Over Union: here, we are looking at the IoU metric that specifies the amount of intersection over the union between the obtained images and the ground truth data  Again, we see that DSP produces images that are much closer to the ground truth (in most cases) whereas the DIP can not achieve a similar result 
While we see few DIP completed images produce a better IoU than the degraded images (in terms of IoU), most of them are worse than the starting image (the higher is better)  (c) The number of iterations that are needed for each algorithm to obtain its best results  Here, we see that DSP can quickly produce the best outcome with the least MSE loss whereas the DIP algorithm's best results are when we run the iterative process for more iterations (the lower is better) 

Correlation of γ with the Gap Size

To better understand the γ parameter of our combined score, we conducted the following experiment  A total of 12000 samples were obtained by merging all of our images from the two datasets, simple and complex  As we have access to the ground truth for each degraded image in the combined dataset, we can easily calculate reconstruction_score and overfit_score for each degraded-ground truth pair 
As expected, we obtain a reconstruction_score of 100 for all samples, but the overfit_score varies among them  Intuitively, we hypothesized that an optimal value of overfit score should be intertwined with the total area of gaps  To test this hypothesis, we did the following experiment  We first define a function φ(x) which takes a binary, black and white image x and returns the number of black pixels in it 
Then we define a gap term as follows: where x I is the incomplete image and x gt is the ground truth  In this case, gap indicates the total area of the gap with respect to the whole shape  We found out that this term and the best result have a correlation of 97 43%  This indicates that the value of γ is highly correlated with the gap size, that is something expected in a way 

Effect of α

We conducted additional experiments concerning how α affects the quality of reconstruction  In the previous section, we defined Equation 2 as the loss term that guides our iterative process  The term α specifies the amount of emphasis the model should place on reconstructing missing regions, rather than filling in fragmented contours 
A lower α indicates a better grouping quality, as shown in Equation   However, we will not achieve completion if we remove the first term completely from the loss by setting α = 0  Therefore, the first term should be kept, but its weight should be very low in order to achieve a good completion  On the other hand, if we set α = 1 and omit the second term, we lose the contour completion regularization term and obtain the same output as a vanilla deep image prior, which does not complete shapes 

Effect of Receptive Field Size

To better understand the effect of receptive field size on our algorithm, we test the following hypothesis: can models with bigger receptive field size complete shapes with bigger gaps? In Table , we report showing the results of this experiment  As we can see, the bigger the receptive field size, the more complete shapes we can reconstruct using DSP 

Implementation Details

As shown in Figure , we use a model with 5 layers and 128 channels for downsampling and upsampling convolutions, and 64 channels for skip convolutions  The upsampling and downsampling modules use 3 × 3 filters, while the skip module uses 1 × 1 filters  In the upsample part of the network, the nearest neighbor algorithm is used 
We used 256 × 256 images with three channels in all of our experiments  In training, we use the MSE loss between the degraded image and the output of the network, and we optimize the loss using the ADAM optimizer and a learning rate equal to 0 01   In our experiments, we also used α = 0 15 as an optimal proportion coefficient for reconstruction loss 

Conclusion

In this work, we introduced a novel framework for contour completion using deep structure priors (DSP)  This work offers a novel notion of a maskless grouping of fragmented contours  In our proposed framework, we introduced a novel loss metric that does not require a strict definition of the mask  Instead, it lets the model learn the perceivable illusory contours and connects those fragmented pieces using a generator network that is solely trained on just the single incomplete input image 
Our model does not require any pre-training which demonstrates that the convolutional architecture of the hour-glass model is able to connect disconnected contours  We present an extended set of experiments that show the capability of our algorithm  We investigate the effect of each parameter introduced in our algorithm separately and show how one could possibly achieve the best result for their problem using this model 
In future work, we plan to extend this model and try to see how it performs with real images  In particular, we want to determine whether we can inpaint real-world photographs while retaining perceptually aware scene structures  The importance of shape in perception by deep neural networks has been highlighted in many adversarial examples to appearance-based networks  
The outcome of this work has strong potential to impact the designing and implementation of models that are robust to such perturbations 



ground_truth: Bigger receptive field size leads to more successful shape completion.

prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: What happens to Ngotho after he attacks Jacobo at a workers' strike?

Read the following text and answer briefly.

Context 3:  Although there doesn't seem to be a connection between Njoroge's family and the murder, it is eventually revealed that Njoroge's brothers are behind the assassination, and that Boro is the real leader of the Mau Mau  Ngotho soon dies from his injuries and Njoroge finds out that his father was protecting his brothers  Kamau has been imprisoned for life  Only Njoroge and his two mothers remain free, and Njoroge is left as the sole provider of his two mothers  Njoroge fears that he cannot make ends meet; he gives up hope of continuing in school and loses faith in God 

Njoroge asks Mwihaki's for support, but she is angry because of her father’s death  When he finally pledges his love to her, she refuses to leave with him, realizing her obligation to Kenya and her mother  Njoroge decides to leave town and makes an attempt at suicide; however, he fails when his mothers find him before he is able to hang himself  The novel closes with Njoroge feeling hopeless, and ashamed of cowardice 

Characters in Weep Not, Child
 Njoroge: the main character of the book whose main goal throughout the book is to become as educated as possible 
 Ngotho: Njoroge's father  He works for Mr Howlands and is respected by him until he attacks Jacobo at a workers strike  He is fired and the family is forced to move to another section of the country  Over the course of the book his position as the central power of the family weakened, to the point where his self-realization that he has spent his whole life waiting for the prophecy (that proclaims the blacks will be returned their land) to come true rather than fighting for Kenyan independence, leads to his depression 
 Nyokabi and Njeri: the two wives of Ngotho  Njeri is Ngotho's first wife, and mother of Boro, Kamau, and Kori  Nyokabi is his second wife, and the mother of Njoroge and Mwangi 
 Njoroge has four brothers: Boro, Kamau, Kori and Mwangi (who is Njoroge's only full brother, who died in World War II) 
 Boro: Son of Njeri who fights for the Allies in World War II  Upon returning his anger against the colonial government is compounded by their confiscation of the his land  Boro's anger and position as eldest son leads him to question and ridicule Ngotho, which eventually defeats their father's will (upon realizing his life was wasted waiting and not acting)  It is eventually revealed that Boro is the leader of the Mau Mau (earlier alluded to as "entering politics") and murders Mr Howlands  He is caught by police immediately after and is scheduled to be executed by the book's end  It is highly likely that it is also Boro who kills Jacobo 
 Mwihaki: Njoroge's best friend (and later develops into his love interest)  Daughter of Jacobo  When it is revealed that his family killed Jacobo (most likely Boro), Mwihaki distances herself from Njoroge, asking for time to mourn her father and care for her mother 
 Jacobo: Mwihaki's father and an important landowner  Chief of the village 
 Mr  Howlands: A white settler who emigrated to colonial Kenya and now owns a farm made up of land that originally belonged to Ngotho's ancestors  Has three children: Peter who died in World War II before the book's beginning, a daughter who becomes a missionary, and Stephen who met Njoroge while the two were in high school 

Themes and motifs
Weep Not, Child integrates Gikuyu mythology and the ideology of nationalism that serves as catalyst for much of the novel's action  The novel explores the negative aspects of colonial rule over Kenya  Njoroge's aspiration to attend university is frustrated by both the violence of the Mau Mau rebels and the violent response of the colonial government  This disappointment leads to his alienation from his family and ultimately his suicide attempt 

The novel also ponders the role of saviours and salvation  The author notes in his The River Between: "Salvation shall come from the hills  From the blood that flows in me, I say from the same tree, a son shall rise  And his duty shall be to lead and save the people " Jomo Kenyatta, the first prime minister of Kenya, is immortalised in Weep Not, Child  The author says, "Jomo had been his (Ngotho's) hope  Ngotho had come to think that it was Jomo who would drive away the white man  To him, Jomo stood for custom and traditions purified by grace of learning and much travel " Njoroge comes to view Jomo as a messiah who will win the struggle against the colonial government 

See also

Things Fall Apart
Death and the King's Horseman

References

External links
Official homepage of Ngũgĩ wa Thiong'o
BBC profile of Ngũgĩ wa Thiong'o
Weep Not, Child at Google Books

British Empire in fiction
Novels set in colonial Africa
Historical novels
Kenyan English-language novels
Novels by Ngũgĩ wa Thiong'o
Novels set in Kenya
1964 novels
Heinemann (publisher) books
Postcolonial novels
African Writers Series
1964 debut novels



ground_truth: After attacking Jacobo at a workers' strike, Ngotho loses his job and Njoroge's family is forced to move.