# Generating Stable Spin Squeezing by Squeezed-Reservoir Engineering

Si-Yuan Bai  Lanzhou Center for Theoretical Physics, Key Laboratory of
Theoretical Physics of Gansu Province, Lanzhou University, Lanzhou 730000,
China  Jun-Hong An<EMAIL_ADDRESS>Lanzhou Center for Theoretical Physics,
Key Laboratory of Theoretical Physics of Gansu Province, Lanzhou University,
Lanzhou 730000, China

###### Abstract

As a genuine many-body entanglement, spin squeezing (SS) can be used to
realize the highly precise measurement beyond the limit constrained by
classical physics. Its generation has attracted much attention recently. It
was reported that $N$ two-level systems (TLSs) located near a one-dimensional
waveguide can generate a SS by using the mediation effect of the waveguide.
However, a coherent driving on each TLS is used to stabilize the SS, which
raises a high requirement for experiments. We here propose a scheme to
generate stable SS resorting to neither the spin-spin coupling nor the
coherent driving on the TLSs. Incorporating the mediation role of the common
waveguide and the technique of squeezed-reservoir engineering, our scheme
exhibits the advantages over previous ones in the scaling relation of the SS
parameter with the number of the TLSs. The long-range correlation feature of
the generated SS along the waveguide in our scheme may endow it with certain
superiority in quantum sensing, e.g., improving the sensing efficiency of
spatially unidentified weak magnetic fields.

##  I Introduction

A quantum-science revolution is in the making. It is expected to bring a lot
of profound impacts to technological innovations. A distinguished example is
quantum metrology or sensing Degen  _et al._ (2017); Pezzè  _et al._ (2018).
It pursues the development of measurement protocols with higher precision of
physical quantities than the limit constrained by classical physics, i.e., the
shot-noise limit, by using quantum resources. As a kind of many-body
entanglement Wineland  _et al._ (1992); Kitagawa and Ueda (1993); Lücke  _et
al._ (2014); Ma  _et al._ (2011), spin squeezing (SS) is one such resources.
It has exhibited a wonderful power in beating the shot-noise limit Bohnet  _et
al._ (2014, 2016); Hosten  _et al._ (2016a, b); Luo  _et al._ (2017); Zou  _et
al._ (2018); Bao  _et al._ (2020), with promising applications in quantum
gyroscope Jo  _et al._ (2007); Berg  _et al._ (2015), atomic clocks Leroux
_et al._ (2010); Kessler  _et al._ (2014); Kómár  _et al._ (2014); Kruse  _et
al._ (2016); Pedrozo-Peñafiel  _et al._ (2020), magnetometers Sewell  _et al._
(2012); Muessel  _et al._ (2014), and gravimetry Szigeti  _et al._ (2020). Its
efficient generation is a prerequisite for further applications. The widely
used method exploits the coherent spin-spin coupling in the one- and two-axis
twisting models Bohnet  _et al._ (2014, 2016); Hosten  _et al._ (2016a, b);
Kitagawa and Ueda (1993); Liu  _et al._ (2011). However, it is dynamically
transient and experiences a degradation under the realistic decoherence
Sørensen and Mølmer (2002); Wang  _et al._ (2010); Xue (2013). Other methods
via atom-photon couplings Zhang  _et al._ (2017); Masson and Parkins (2019);
Qin  _et al._ (2020); Groszkowski  _et al._ (2020) and quantum nondemolition
measurements Kuzmich  _et al._ (2000); Appel  _et al._ (2009); Kritsotakis
_et al._ (2021) have also been proposed.

Waveguide quantum electrodynamics (QED) refers to a scenario where arrays of
quantum emitters are coupled to a waveguide Lund-Hansen  _et al._ (2008); van
Loo  _et al._ (2013); Petersen  _et al._ (2014); Mitsch  _et al._ (2014);
Sipahigil  _et al._ (2016); Kannan  _et al._ (2020a, b, a); Kim  _et al._
(2021). It allows for long-range interactions among the quantum emitters
mediated by photons in the waveguide that is particularly interesting for
quantum network applications Lemonde  _et al._ (2018). Several schemes on
dissipative preparation of the SS Krauter  _et al._ (2011); Shahmoon and
Kurizki (2013); Gonzalez-Tudela  _et al._ (2011); González-Tudela and Porras
(2013); González-Tudela  _et al._ (2015); Dalla Torre  _et al._ (2013); Song
_et al._ (2017); Qiao  _et al._ (2020) have been proposed based on the idea of
reservoir engineering Kienzler  _et al._ (2015); Metelmann and Clerk (2015);
Haakh and Scheel (2015); Kepesidis  _et al._ (2016); Yanay and Clerk (2018) to
the waveguide modes. The SS generated in such a method exists in steady states
and does not depend on initial states, which endows it with the features of a
long lifetime and robustness. However, a coherent driving laser on each
quantum emitter is needed to stabilize the SS in these schemes. It
dramatically increases the experimental difficulties when a huge number of
quantum emitters are involved.

In this Letter, we propose a scheme to deterministically generate a stable SS
of $N$ distant quantum emitters formed by two-level systems (TLSs) in a
waveguide QED system without resorting to a coherent driving on each TLS. The
main idea is based on the combined action of the technique of squeezed-
reservoir engineering, which is widely used in quantum state preparation
Roßnagel  _et al._ (2014); Yang  _et al._ (2015); Zeytinoğlu  _et al._ (2017);
Kono  _et al._ (2017); Qin  _et al._ (2018); Zippilli and Vitali (2021), and
the mediation role of the waveguide. The waveguide enables us to manipulate
the phase difference between the reservoir-induced long-range coherent and
incoherent couplings of TLSs such that an effective collective spin mode of
TLSs is induced by precisely controlling the positions of TLSs. Then, acting
as a mold, the squeezed reservoir imprints its squeezing feature to the steady
state of the collective spin. It is found that the Wineland SS parameter
$\xi_{R}^{2}$, as a characterization of the improvement of sensitivity in
Ramsey spectroscopy Wineland  _et al._ (1992), for our generated SS scales
with the TLS number $N$ as $1.64N^{-0.54}$, which beats the ones in the one-
and two-axis twisting Sørensen and Mølmer (2002); Wang  _et al._ (2010); Xue
(2013) and Heisenberg Lee  _et al._ (2013) models with the realistic
dissipation considered. It implies the superiority of the SS of our scheme in
quantum sensing. The spatial separation of the TLSs along the waveguide in our
scheme also is helpful in improving the sensing efficiency of a weak field via
effectively increasing the contact area.

Figure 1: (a) Schematics of $N$ TLSs along $z$ axis in a one-dimensional
waveguide, which is driven by a broadband squeezed field. Evolution of the
inverse of the SS parameter of the TLSs in different $r$ when the initial
state is $|j,-j\rangle$ with $j=N/2$ (b) and in different initial states when
$r=0.5$ (c). The red dashed line in (b) denotes $\xi_{R}^{2}=1$. Other
parameters are $N=10$, $\omega_{0}=\Delta=1.0A$, and $\alpha=0.5$.

##  II System and dynamics

We consider an array of $N$ TLSs coupled to a common electromagnetic field in
a waveguide [see Fig. 1(a)]. The TLSs can be superconductor qubits Liu and
Houck (2017), nitrogen vacancy centers Song  _et al._ (2017), or natural atoms
Krinner  _et al._ (2018). The waveguide may be a coupled cavity array Chen
_et al._ (2016), a metal-dielectric surface plasmon Fang and Sun (2015); Yang
_et al._ (2019), or a photonic crystal Dutta  _et al._ (2016). Its Hamiltonian
is $\hat{H}=\hat{H}_{\text{S}}+\hat{H}_{\text{R}}+\hat{H}_{\text{I}}$ with
($\hbar=1$)

$\displaystyle\hat{H}_{\text{S}}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{N}\omega_{0}\hat{\sigma}_{i}^{\dagger}\hat{\sigma}_{i},~{}\hat{H}_{\text{R}}=\sum_{k}\omega_{k}\hat{a}_{k}^{\dagger}\hat{a}_{k},$
(1) $\displaystyle\hat{H}_{\text{I}}$ $\displaystyle=$
$\displaystyle\sum_{k,i}(g_{ki}\hat{a}_{k}+g_{ki}^{*}\hat{a}_{k}^{\dagger})(\hat{\sigma}_{i}^{\dagger}+\hat{\sigma}_{i}),$
(2)

where $\hat{\sigma}_{i}$ is the transition operator from the excited state
$|e\rangle$ to the ground state $|g\rangle$ of the $i$th TLS at
$\mathbf{r}_{i}$, with transition frequency $\omega_{0}$, and $\hat{a}_{k}$ is
the annihilation operator of the $k$th field mode with frequency $\omega_{k}$.
The coupling strength is
$g_{ki}=\sqrt{\omega_{k}/(2\epsilon_{0})}\mathbf{d}_{i}\cdot\mathbf{u}_{k}({\bf
r}_{i})$, where $\epsilon_{0}$ is the vacuum permittivity, ${\bf d}_{i}$ is
the dipole moment of the $i$th TLS, and ${\bf u}_{k}({\bf r}_{i})$ is the
spatial function of the $k$th mode.

We use the technique of squeezed-reservoir engineering Yang  _et al._ (2015);
Zeytinoğlu  _et al._ (2017); Kono  _et al._ (2017); Qin  _et al._ (2018);
Zippilli and Vitali (2021) to generate the SS of TLSs. It is realized by
feeding the waveguide a broadband squeezed field, which can be implemented via
an optical parametric down conversion Vahlbruch  _et al._ (2016); Serikawa
_et al._ (2016) or Josephson parametric amplification Bourassa  _et al._
(2012); Macklin  _et al._ (2015). The waveguide under the driving of the
squeezed field acts as a squeezed vacuum reservoir. The initial state is
$\rho_{\text{T}}(0)=\rho(0)\otimes\prod_{k}\hat{S}_{k}|0_{k}\rangle\langle
0_{k}|\hat{S}_{k}^{\dagger}$, where
$\hat{S}_{k}=\exp[r_{k}(e^{-i\alpha_{k}}\hat{a}_{k}^{2}-e^{i\alpha_{k}}\hat{a}_{k}^{{\dagger}2})/2]$
with $r_{k}$ and $\alpha_{k}$ being the squeezing strength and angle, and
$|0_{k}\rangle$ is the $k$th-mode vacuum state Roßnagel  _et al._ (2014). Both
$r_{k}$ and $\alpha_{k}$ relate to the amplitude of the pump field, the
second-order nonlinearity, and the length of nonlinear material in parametric
amplification. The master equation of the TLSs under the Born-Markovian and
secular approximations is Breuer and Petruccione (2007); Mäkelä and Möttönen
(2013); Sup

$\displaystyle\dot{\tilde{\rho}}(t)$ $\displaystyle=$ $\displaystyle-{i\mkern
3.0mu}[\Delta_{\mathcal{N}}\sum_{i}\hat{\sigma}_{i}^{\dagger}\hat{\sigma}_{i}+\hat{H}_{\text{DD}},\tilde{\rho}(t)]$
(3)
$\displaystyle+\sum_{i,j}\big{\\{}\gamma_{ij}^{-}/2\big{[}\mathcal{N}\mathcal{\check{\mathcal{D}}}_{\hat{\sigma}_{i}^{\dagger},\hat{\sigma}_{j}}+(\mathcal{N}+1)\mathcal{\check{\mathcal{D}}}_{\hat{\sigma}_{i},\hat{\sigma}_{j}^{\dagger}}\big{]}$
$\displaystyle-\gamma_{ij}^{+}/2\big{[}\mathcal{M}\mathcal{\check{\mathcal{D}}}_{\hat{\sigma}_{i}^{\dagger},\hat{\sigma}_{j}^{\dagger}}+\mathcal{M}^{*}\mathcal{\check{\mathcal{D}}}_{\hat{\sigma}_{i},\hat{\sigma}_{j}}\big{]}\big{\\}}\tilde{\rho}(t),$

where
$\tilde{\rho}(t)=e^{i\hat{H}_{\text{S}}t}\rho(t)e^{-i\hat{H}_{\text{S}}t}$ is
the density matrix of TLSs in the interaction picture,
$\mathcal{N}\equiv\sinh^{2}r$ with $r\equiv r(\omega_{0})$,
$\mathcal{M}\equiv\sqrt{\mathcal{N}^{2}+\mathcal{N}}e^{i\alpha}$ with
$\alpha\equiv\alpha(\omega_{0})$, and
$\mathcal{\check{\mathcal{D}}}_{\hat{A},\hat{B}}\tilde{\rho}\equiv
2\hat{A}\tilde{\rho}\hat{B}-\tilde{\rho}\hat{B}\hat{A}-\hat{B}\hat{A}\tilde{\rho}$.
The first term of Eq. (3) is the reservoir-induced coherent dynamics, where
$H_{\text{DD}}=-\sum_{i\neq
j}\Delta_{ij}(\hat{\sigma}_{i}^{\dagger}\hat{\sigma}_{j}+\hat{\sigma}_{i}\hat{\sigma}_{j}^{\dagger})/2$
is the dipole-dipole interaction of the TLSs. The interaction strengths are
$\Delta_{ij}=\Delta_{ij}^{-}+\Delta_{ij}^{+}$ with
$\Delta_{ij}^{\pm}=\mathcal{P}\int_{0}^{\infty}d\omega{G^{-}_{ij}(\omega)/(\omega\pm\omega_{0})}$,
where $\mathcal{P}$ denotes the Cauchy principal value and
$G^{-}_{ij}(\omega)\equiv\sum_{k}g_{ki}g_{kj}^{*}\delta(\omega-\omega_{k})$
are the correlated spectral densities. The frequency shift equals to
$\Delta_{\mathcal{N}}\equiv(2\mathcal{N}+1)\Delta$ with
$\Delta=(\Delta_{ii}^{+}-\Delta_{ii}^{-})$, which is independent of the
positions of TLSs. The second and third lines of Eq. (3) are the incoherent
dissipation and squeezing with rates $\gamma_{ij}^{\pm}=2\pi
G_{ij}^{\pm}(\omega_{0})$, where
$G_{ij}^{+}(\omega)\equiv\sum_{k}g_{ki}g_{kj}\delta(\omega-\omega_{k})$. It is
interesting to see that the squeezed vacuum reservoir, as a common medium of
the TLSs, can not only induce individual dissipation, squeezing, and frequency
shift to each TLSs, but also induce correlated dissipation, squeezing, and
coherent dipole-dipole interactions to the TLSs by exchanging the photons in
the waveguide. It gives us sufficient room to generate a long-range
correlation of TLSs via engineering the reservoir in the waveguide.

Figure 2: Evolution of the Husimi’s $Q$ function in the $\mathbf{n}_{\perp}$
plane for the initial state $|j,-j\rangle$ when $t=0$ (a), $0.01A^{-1}$ (b),
$0.1A^{-1}$ (c), and $t=4.0A^{-1}$ (d). We use $N=30$, $r=0.8$, and others are
the same as Fig. 1.

The surface plasmonic as a waveguide experiences severe loss due to the metal
absorption Fang and Sun (2015). The photonic crystal as a waveguide reflects
light only in a certain narrow frequency range Dutta  _et al._ (2016). We
consider that the waveguide is formed by a rectangular hollow metal due to its
high $Q$ factor Pucci  _et al._ (2013) and wide permissable bandwidth. Its
allowable electromagnetic modes are the transverse modes TEmn and TMmn above
the cutoff frequency $\omega_{mn}=c[(m\pi/a)^{2}+(n\pi/b)^{2}]^{1/2}$, where
$a$, $b$ are the transverse lengths of the waveguide, and $m$, $n$ are
nonnegative integers. Their dispersion relations are
$\omega^{mn}_{k}=[(ck)^{2}+\omega_{mn}^{2}]^{1/2}$ with $k$ being the
longitudinal wave number and $c$ being the speed of light Kong (1986).
Assuming that the TLSs are polarized in the $z$ direction
($\mathbf{d}_{i}=d\mathbf{e}_{z}$), we have Shahmoon and Kurizki (2013)

$G^{\pm}_{ij}(\omega)=\sum_{mn}\frac{\Gamma_{mn}}{2\pi}\frac{\cos[k(z_{i}\pm
z_{j})]}{[(\omega/\omega_{mn})^{2}-1]^{1/2}}\Theta(\omega-\omega_{mn})$ (4)

with $\Theta(x)$ as the Heaviside step function,
$\Gamma_{mn}={4\omega_{mn}\tilde{u}_{mn,i}\tilde{u}_{mn,j}\over\epsilon_{0}cab}$,
and $\tilde{u}_{mn,i}=d\sin(\frac{m\pi}{a}x_{i})\sin(\frac{n\pi}{b}y_{i})$.
Considering the dominated mode with $m=n=1$ and $\omega_{0}>\omega_{11}$, we
can derive Sup

$\displaystyle\Delta_{ij}$ $\displaystyle=$
$\displaystyle-[\Gamma_{11}\zeta\omega_{11}/(2c)]\sin({|z_{i}-z_{j}|/\zeta}),$
(5) $\displaystyle\gamma^{\pm}_{ij}$ $\displaystyle=$
$\displaystyle(\Gamma_{11}\zeta\omega_{11}/c)\cos({|z_{i}\pm z_{j}|/\zeta}),$
(6)

where $\zeta=c(\omega_{0}^{2}-\omega_{11}^{2})^{-1/2}$. Equations (5) and (6)
originate from the interference of the $N$ individual interaction channels of
the TLSs with the common reservoir Chen  _et al._ (2016). Remarkably, the
waveguide as the reservoir medium enables us to modulate the phase difference
of Eqs. (5) and (6) via tailoring the position $z_{i}$. It allows the switch-
on-off for either of the two couplings and offers an opportunity for
engineering the multipartite quantum correlation of the TLSs. By positioning
the TLSs such that $|z_{i}\pm z_{j}|=2n_{\pm}\pi\zeta$
($n_{\pm}\in\mathbb{Z}$), we have $\Delta_{ij}=0$,
$\gamma^{\pm}_{ij}=\Gamma_{11}\zeta\omega_{11}/c\equiv A$, and Eq. (3) reduced
to Sup

$\displaystyle\dot{\tilde{\rho}}(t)=$ $\displaystyle-{i\mkern
3.0mu}\Delta_{\mathcal{N}}[\hat{S}^{z},\tilde{\rho}(t)]+\frac{A}{2}\Big{[}\mathcal{N}\mathcal{\check{\mathcal{D}}}_{\hat{S}^{+},\hat{S}^{-}}+(\mathcal{N}+1)\mathcal{\check{\mathcal{D}}}_{\hat{S}^{-},\hat{S}^{+}}$
$\displaystyle-\mathcal{M}\mathcal{\check{\mathcal{D}}}_{\hat{S}^{+},\hat{S}^{+}}-\mathcal{M}^{*}\mathcal{\check{\mathcal{D}}}_{\hat{S}^{-},\hat{S}^{-}}\Big{]}\tilde{\rho}(t),$
(7)

where
$\hat{S}^{z}\equiv\sum_{i}(\hat{\sigma}_{i}^{\dagger}\hat{\sigma}_{i}-1/2)$
and $\hat{S}^{-}=(\hat{S}^{+})^{\dagger}=\sum_{i}\hat{\sigma}_{i}$. Thus, a
collective spin mode of the TLSs is induced to interplay with the common
squeezed reservoir by the constructive interference among the interaction
channels.

##  III Stable spin squeezing

An SS is featured with a reduced quantum fluctuation in certain spin
component. Defining a mean direction
$\mathbf{n}=\text{Tr}(\hat{\mathbf{S}}\rho)/|\text{Tr}(\hat{\mathbf{S}}\rho)|\equiv\left(\sin\theta_{0}\cos\varphi_{0},\sin\theta_{0}\sin\varphi_{0},\cos\theta_{0}\right)$,
a spin state $\rho$ is squeezed if its minimal variance in the
$\mathbf{n}_{\perp}$ plane spanned by
$\mathbf{n}_{1}=\left(\cos\theta_{0}\cos\varphi_{0},\cos\theta_{0}\sin\varphi_{0},-\sin\theta_{0}\right)$
and $\mathbf{n}_{2}=\left(-\sin\varphi_{0},\cos\varphi_{0},0\right)$ is
smaller than that of the spin coherent state, i.e. $N/4$ Kitagawa and Ueda
(1993). It is quantified by the SS parameter Wineland  _et al._ (1992)
$\xi_{R}^{2}={N[\operatorname{Var}(\hat{\mathbf{S}}^{\bot})]_{\operatorname{min}}}\big{/}{|\text{Tr}(\hat{\mathbf{S}}\rho)|^{2}}$,
where $\hat{\mathbf{S}}^{\bot}$ is the spin in the $\mathbf{n}_{\perp}$ plane,
$\operatorname{Var}(\hat{O})=\langle\hat{O}^{2}\rangle-\langle\hat{O}\rangle^{2}$,
and the superscript $\operatorname{min}$ means the minimum in all directions.
Exhibiting multipartite entanglement, the state is squeezed if $\xi_{R}^{2}<1$
Tóth  _et al._ (2009). Another way to visually depict the SS is the Husimi’s
$Q$ function
$Q(\theta,\varphi)=(2j+1)/(4\pi)\left\langle\theta,\varphi\right|\rho\left|\theta,\varphi\right\rangle$
Leonhardt and Paul (1993), where
$\left|\theta,\varphi\right\rangle=(1+|\eta|^{2})^{-j}e^{-\eta^{*}\hat{S}^{-}}|j,j\rangle$
with $|j,m\rangle$ ($m=-j,\cdots,j$) being the eigenstates of
$\\{\hat{\mathbf{S}}^{2},\hat{S}^{z}\\}$ and
$\eta=-\tan(\theta/2)e^{-i\varphi}$. The $Q$ function maps $\rho$ to a
quasiclassical probability distribution in the phase space defined by $\theta$
and $\varphi$.

Via numerically solving Eq. (II), we show in Fig. 1(b) the evolution of
$1/\xi_{R}^{2}$ for the initial state $|j,-j\rangle$ with $j=N/2$ in different
squeezing strengths $r$. It is found that a stable SS can be formed in the
regime of a moderate $r$. Thanks to the constructive role played by the
squeezed reservoir in the waveguide, the system spontaneously evolves to a
spin squeezed state uniquely dependent on $r$. This is in sharp contrast to
the previous results González-Tudela and Porras (2013); González-Tudela  _et
al._ (2015); Dalla Torre  _et al._ (2013); Song  _et al._ (2017); Qiao  _et
al._ (2020), where a coherent driving field is applied on each TLS to
stabilize the SS. The evolution of $1/\xi_{R}^{2}$ with chosen $r$ in
different initial states $|j,m\rangle$ verifies the uniqueness of the steady
state [see Fig. 1(c)]. We plot in Fig. 2 the evolution of the projections of
the $Q$ function in the $\mathbf{n}_{\perp}$ plane for the initial state
$|j,-j\rangle$. Being isotropically distributed, it shows no SS initially.
With increasing time, it shrinks in one direction at the expense of expanding
in its orthogonal direction. Such an SS is kept to its steady state [see Fig.
2(d)]. It confirms that the TLSs as a collective spin are squeezed by the
squeezed reservoir.

To figure out the relation between the SS and $r$, we calculate the steady
state Agarwal and Puri (1990); Sup

$\tilde{\rho}(\infty)=\sum_{m,n=-j}^{j}p_{m}p_{n}^{*}\langle\phi_{m}|\phi_{n}\rangle|\psi_{m}\rangle\langle\psi_{n}|.$
(8)

Satisfying $\hat{R}_{z}|\psi_{m}\rangle=m|\psi_{m}\rangle$ and
$\hat{R}_{z}^{\dagger}|\phi_{m}\rangle=m|\phi_{m}\rangle$ with
$\hat{R}_{z}=i(4|\mathcal{M}|)^{-1/2}(\hat{S}^{+}e^{i\frac{\alpha}{2}}\sinh
r-\hat{S}^{-}e^{-i\frac{\alpha}{2}}\cosh r)$, $|\psi_{m}\rangle$ and
$|\phi_{m}\rangle$ forms a complete set of biorthogonal basis. The recurrence
relation of coefficient $p_{m}$ is $p_{m+1}={Am-i\Delta_{\mathcal{N}}\over
A(m+1)+i\Delta_{\mathcal{N}}}p_{m}$, which can be fixed by
$\operatorname{Tr}[\tilde{\rho}(\infty)]=1$.

We plot in Fig. 3(a) the comparison of the steady-state squeezing obtained by
the analytical solution (8) with the numerical one. It verifies the
correctness of Eq. (8) in describing the steady state of Eq. (II). One
observation from Fig. 3(a) is that, with increasing $N$, the range of $r$
supporting the stable SS becomes wider and wider. Thus, the stable SS can be
generated in a fairly wider parameter range if more TLSs are involved [see
Fig. 3(b)]. Another feature of Fig. 3(a) is that, even for sufficiently large
$N$, the SS still tends to vanish in a gradual manner with increasing $r$. It
endows our scheme with a difference from the previous ones based on the
driven-dissipative Dicke model González-Tudela and Porras (2013); Song  _et
al._ (2017), where a nonequilibrium phase transition manifested by an abrupt
disappearance of the SS is presented. It is understandable based on the fact
that our SS is generated via purely incoherent interactions of TLSs mediated
by the reservoir, while theirs is via the combined actions of the incoherent
interactions and coherent driving.

Figure 3: (a) Comparison of $1/\xi_{R}^{2}$ from the analytical solution (8)
(purple dot dashed, red dashed, and blue dotted lines) with the one by
numerically solving Eq. (II) at $t=200A^{-1}$ (gray lines) in different $N$.
Range of $r$ supporting stable SS (b), maxima of $1/\xi_{R}^{2}$ (c), and
optimal $r$ (d) as a function of $N$. The numerical fitting gives
$(\xi_{R}^{2})_{\text{Max}}=1.64N^{-0.54}$ and $r_{\text{opt}}=0.27\ln N-0.1$.
Parameters are the same as Fig. 1.

The SS parameter $\xi_{R}^{2}$ describes the improvement of the sensitivity to
measure the atomic frequency in Ramsey spectroscopy Wineland  _et al._ (1992).
Therefore, it itself is an important quantity to characterize quantum
superiority in quantum metrology Gross (2012); Tóth and Apellaniz (2014);
Giovannetti  _et al._ (2011); Lee  _et al._ (2014). The analytical solution
(8) permits us to investigate the SS in the large-$N$ limit, where the
numerical calculation is hard. The relation between the optimal SS and the
number $N$ of the TLSs is calculated via Eq. (8) [see red dots in Fig. 3(b)].
Via numerical fitting, we obtain its scaling relation as

$(\xi_{R}^{2})_{\text{min}}=1.64\times N^{-0.54}.$ (9)

According to the definition of $\xi_{R}^{2}$ Wineland  _et al._ (1992), the
metrology error using this state in Ramsey spectroscopy outperforms the shot-
noise limit achieved using the spin coherent state by a factor of
$\sqrt{1.64}N^{-0.27}$. It is better than the previous schemes. Explicitly,
the SS generated via the one-axis twisting scales as $\xi_{R}^{2}\propto
N^{-2/3}$ and via two-axis twisting as $N^{-1}$ in the ideal situation
Kitagawa and Ueda (1993). However, they tend to $N^{-1/2}$ at optimized time
and to be divergent in the long-time limit when the dissipation is considered
Sørensen and Mølmer (2002); Wang  _et al._ (2010); Xue (2013). Our SS is also
better than the ones in the ground state of Lipkin-Meshkov-Glick model scaling
as $\xi_{R}^{2}\propto N^{-1/3}$ Dusuel and Vidal (2004) and in the steady
state of dissipative Heisenberg model scaling as $\xi_{R}^{2}=1/2$ Lee  _et
al._ (2013). Figure 3(c) shows the squeezing strength to achieve the best SS
in different $N$. The numerical fitting reveals $r_{\text{opt}}=0.27\ln
N-0.11$ Sup , which gently depends on $N$. Thus, we do not bother to sharply
increase $r$ to generate the SS for a large number of TLSs. This gives a
useful guideline for experiments to optimize the working condition.

##  IV Effect of position imperfection

Consider that the position $z_{i}$ of $i$th TLS has a disorder $w\chi_{i}$,
where $\chi_{i}$ is a random number uniformly distributed in $[-1,1]$ and $w$
is disorder strength. The disorder makes the collective spin mode in Eq. (II)
not exist anymore. Solving the original master equation (3), we plot in Figs.
4(a)- 4(c) the evolution of $1/\xi_{R}^{2}$ in different $w$. It can be found
that the disorder introduces a decay factor roughly in a timescale
$\zeta/(Aw)$ to $1/\xi_{R}^{2}$ Sup . When $w$ is small, almost no observable
influence can be found in a sufficiently wide time window. With increasing
$w$, the decay becomes obvious. By setting the left-hand side of Eq. (3) to
zero, we plot in Fig. 4(d) the steady-state $1/\xi_{R}^{2}$ in different $w$.
It shows that the stable SS is preserved until the disorder strength is as
high as $0.4\pi\zeta$. This reveals the robustness of our scheme to the
position imperfection of the TLSs.

Figure 4: Evolution of $1/\xi_{R}^{2}$ in the ideal (red line) and disordered
(blue dashed line) cases with the standard deviations marked in orange when
$w=0.03\pi\zeta$ (a), $0.06\pi\zeta$ (b), and $0.3\pi\zeta$ (c) by averaging
over $100$ random configurations. (d) Steady-state $1/\xi_{R}^{2}$ in
different $w$ by averaging over $200$ random configurations. $N=5$ and others
are the same as Fig. 1.

##  V Discussion and conclusion

The current experimental advances in nanophotonics and circuit QED provide a
support to our scheme Petersen  _et al._ (2014); Mitsch  _et al._ (2014); van
Loo  _et al._ (2013); Sipahigil  _et al._ (2016); Kannan  _et al._ (2020a).
The transmission-line waveguide mediated couplings of two 18.6mm separated
TLSs have been observed van Loo  _et al._ (2013). A 15 dB squeezed light
corresponding to $r=1.7$ has been realized Vahlbruch  _et al._ (2016). A 20 dB
squeezing for microwave with bandwidth up to a few GHz has been realized by
the Josephson parametric amplifier Macklin  _et al._ (2015), which fulfills
our requirement. The precise positioning of TLSs with 20 nm accuracy was
reported Mohan  _et al._ (2010); Felici  _et al._ (2020). Our scheme is also
realizable in SiV centers as TLSs in a diamond waveguide Wang  _et al._
(2017); Kepesidis  _et al._ (2016); Lemonde  _et al._ (2018). Comparing with
the SS in an atom ensemble Bohnet  _et al._ (2014, 2016); Hosten  _et al._
(2016a, b); Luo  _et al._ (2017); Zou  _et al._ (2018); Bao  _et al._ (2020),
our SS shows a long-range correlation. It hopefully is useful in developing
quantum sensing in extremal conditions, e.g., improving the sensing efficiency
of a spatially unidentified weak magnetic field via effectively increasing the
contact area.

In summary, we have proposed a scheme to generate stable SS of $N$ distant
TLSs in a waveguide QED system by squeezed-reservoir engineering. A collective
effect of the far separated TLSs is efficiently created by the mediation role
of the common squeezed reservoir in the waveguide via well-positioned TLSs. It
makes the TLSs spontaneously evolve from any initial state to a spin squeezed
state in the long-time limit. Our analysis reveals that the generated SS
scales with the number of TLSs as $N^{-0.54}$, which outperforms the two-axis
twisting and Heisenberg models with the realistic dissipation considered.
Without resorting to the coherent driving on each TLS, our scheme reduces the
difficulty of experiment realization in previous schemes. The recent
advancement of the waveguide QED experiments indicates that our scheme is
within the present experimental state of the art.

##  VI Acknowledgments

This work is supported by the National Natural Science Foundation (Grants No.
11875150, No. 11834005, and No. 12047501).

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# IMC 2024 Methods & Solutions Review

1st Shyam Gupta  Technicshe Universitat Dortmund
Master Student Department of Statistics
Dortmund,Germany
<EMAIL_ADDRESS>2nd Dhanisha Sharma  B.Sc Physics (Honors)
DAVV (2024)
Indore, India
<EMAIL_ADDRESS>3rd Songling Huang  College of Big Data
Yunnan Agricultural University
Yunnan, China
<EMAIL_ADDRESS>
###### Abstract

For the past three years, Kaggle has been hosting the Image Matching
Challenge, which focuses on solving a 3D image reconstruction problem using a
collection of 2D images. Each year, this competition fosters the development
of innovative and effective methodologies by its participants. In this paper,
we introduce an advanced ensemble technique that we developed, achieving a
score of 0.153449 on the private leaderboard and securing the 160th position
out of over 1,000 participants. Additionally, we conduct a comprehensive
review of existing methods and techniques employed by top-performing teams in
the competition. Our solution, alongside the insights gathered from other
leading approaches, contributes to the ongoing advancement in the field of 3D
image reconstruction. This research provides valuable knowledge for future
participants and researchers aiming to excel in similar image matching and
reconstruction challenges.

###### Index Terms:

3d scene reconstruction, ALIKED, descriptors, SIFT, lightglue, keypoints,
COLMAP, image pairs, SFM, attention, descriptors.

##  I Introduction

The process of reconstructing 3D models from diverse image collections, known
as Structure from Motion (SfM)[17], is critical in Computer Vision but remains
challenging, especially with images captured under varied conditions like
different viewpoints, lighting, and occlusions. This competition [8] addresses
these complexities across six distinct categories:

  1. 1.

Phototourism and historical preservation: Includes diverse viewpoints, sensor
variations, and challenges posed by ancient historical sites.

  2. 2.

Night vs. day and temporal changes: Combines images from different times,
lighting conditions, and weather, testing algorithms against temporal
variations.

  3. 3.

Aerial and mixed aerial-ground: Involves images from drones with arbitrary
orientations, alongside ground-level shots.

  4. 4.

Repeated structures: Focuses on disambiguating perspectives of symmetrical
objects.

  5. 5.

Natural environments: Challenges include irregular structures like trees and
foliage.

  6. 6.

Transparencies and reflections: Deals with objects like glassware that lack
texture and create reflections, presenting unique computational hurdles.

The competition aims to advance understanding in Computer Vision by bridging
traditional image-matching techniques with modern machine learning approaches.
By tackling these varied categories, participants contributed to evolving
solutions for robust 3D reconstruction from real-world image datasets.

##  II Existing Methods & Techniques

Every solution has a unique way of solving the problem. However, there is a
standard flow of data most of the solutions followed.

Figure 1: General Flow of how data was processed by most of the competitors

Following this pipeline, will not result in top ranks itself. Hence, below we
mention methods used by top solutions to score higher. In further sections, we
discuss how kagglers made a mix’n’match of these techniques to get the best
performance.

## MatchFormer [1]

MatchFormer(2022) was a novel approach to matching multiple views of a scene,
crucial for tasks like Structure-from-Motion (SfM), Simultaneous Localization
and Mapping (SLAM), relative pose estimation, and visual localization.
Traditional methods using detectors and hand-crafted local features are
computationally heavy. Recent deep learning methods use Convolutional Neural
Networks (CNNs) for feature extraction but are often inefficient due to
overburdened decoders.

MatchFormer proposed a new pipeline called extract-and-match, which uses a
pure transformer model to perform feature extraction and matching
simultaneously. This approach is more intuitive and efficient compared to
previous methods. MatchFormer introduces a hierarchical transformer with a
matching-aware encoder that uses interleaved self- and cross-attention
mechanisms. This design improves computational efficiency and robustness,
especially in low-texture scenes.

## DINOv2[7]

Figure 2: using segmentation models significantly improved keypoint detection
on transparent Figure 3: as clearly visible, background keypoints detected in
transparent scene class. To solve this DINOv2 provides seperation between
foreground & background which helps detect foreground keypoints for
transparent scene

DINOv2 (Distillation of Self-supervised Vision Transformers) enhances
segmentation, keypoint detection, and extraction, making it valuable for image
matching and 3D reconstruction. DINOv2 leverages self-supervised learning to
train vision transformers without labeled data, enabling the model to learn
robust and detailed image representations.

For segmentation, DINOv2 uses its learned feature maps to identify and
delineate different regions within an image accurately. This segmentation
capability is crucial in breaking down complex scenes into manageable parts,
aiding in precise object recognition and separation, which is foundational for
subsequent processing steps.

In keypoint detection and extraction, DINOv2’s ability to generate high-
quality feature descriptors ensures that keypoints are distinctive and
repeatable. These descriptors are pivotal in matching corresponding points
across different images, a core requirement for image matching. The robustness
of these keypoints helps in achieving higher accuracy in image alignment and
feature matching, which directly impacts the quality of 3D reconstruction.

By integrating DINOv2, image matching algorithms benefit from enhanced feature
extraction, leading to more reliable keypoint matches. This improved matching
process is essential for constructing accurate 3D models, as it ensures that
the spatial relationships between points are preserved across multiple views,
resulting in more detailed and accurate 3D reconstructions.

Figure 4: detection and segmentation of foreground can be seen.

## ALIKED[2]

Efficiently and robustly extracting image keypoints and descriptors is
essential for various visual measurement applications, including simultaneous
localization and mapping (SLAM)[6], computational photography, and visual
place recognition. Traditional methods relied on hand-crafted algorithms,
which were not very efficient or robust. Modern approaches use deep neural
networks (DNNs) for better performance.

Keypoints and Descriptors

  1. 1.

Keypoints: Distinctive points in an image that are used for tasks like image
matching and 3D reconstruction.

  2. 2.

Descriptors: Descriptions of the keypoints that allow different keypoints to
be compared and matched across images.

Early DNN methods extracted descriptors at predefined keypoints, but newer
methods use a single network to extract both keypoints and descriptors
simultaneously. These newer methods generate a score map and a descriptor map
from which keypoints and descriptors are extracted.

Challenges with Existing Methods is Existing methods use fixed-size
convolutions that lack geometric invariance, which is crucial for accurate
image matching. This problem is partially solved by estimating the scale and
orientation of descriptors. However, these methods can only handle affine
transformations, not more complex geometric transformations. Deformable
Convolution Networks (DCNs) can model any geometric transformation by
adjusting each pixel’s position in the convolution, enhancing descriptor
representation. However, DCNs are computationally expensive.

Sparse Deformable Descriptor Head (SDDH) are used to improve efficiency, the
paper introduces the Sparse Deformable Descriptor Head (SDDH):

  1. 1.

SDDH: Extracts deformable descriptors only at detected keypoints instead of
the entire image, significantly reducing computational costs. It uses
adjustable positions (offsets) for better flexibility and efficiency in
modeling descriptors.

  2. 2.

ALIKED: A network designed for visual measurement tasks that includes a
solution to adapt the neural reprojection error (NRE) loss for sparse
descriptors. This adaptation minimizes computational overhead and saves memory
during training.

## Dense Matchers and Sparse Keypoint Matchers

Dense Matchers aim to find correspondences for every pixel or a dense grid of
pixels in an image. This comprehensive approach is used in tasks where fine-
grained details are important, such as optical flow estimation, depth
estimation, and image stitching.

  * •

Full Coverage: Consider the entire image, ensuring correspondences for almost
every pixel.

  * •

High Computational Cost: Require significant computational resources and
memory.

  * •

Applications: Motion tracking, 3D reconstruction, dense image alignment.

Sparse Keypoint Matchers, in contrast, detect and match distinct and
repeatable keypoints or features in images. These keypoints are typically
corners, edges, or blobs identifiable across different views.

  * •

Selective Coverage: Only a subset of points (keypoints) in the image is
considered.

  * •

Lower Computational Cost: Faster and less computationally demanding than dense
matchers.

  * •

Applications: Object recognition, image retrieval, feature-based 3D
reconstruction.

Comparison:

  * •

Coverage: Dense matchers cover the entire image, while sparse keypoint
matchers focus on specific, informative points. We leverage Sparse Matchers
for this competition since for SFM we focus on specific object and reject
noise.

  * •

Computational Efficiency: Sparse keypoint matchers are more efficient
computationally. Which proves to be more advantage, since we have 9 hours of
runtime limit on Kaggle.

  * •

Applications: Dense matchers are suitable for detailed correspondences, while
sparse matchers are ideal for tasks relying on robust and distinctive
features.

Below we discuss some keypoint matching algorithms, specially LightGlue[4] &
OmniGlue[16] since it proved to be most robust and efficient algorithm.

## LightGlue[4]

LightGlue is a deep network designed to efficiently and accurately match
sparse points between two images. It improves upon SuperGlue by addressing
computational limitations while retaining high performance. LightGlue is
adaptive, making it faster for easy-to-match image pairs and robust for
challenging ones. It is more efficient, easier to train, and suitable for low-
latency applications like SLAM[6] and large-scale mapping.

Figure 5: LigthGlue is faster at matching easy image pairs (top) than
difficult ones (bottom) because it can stop at earlier layers when its
predictions are confident.

## Complex Dense Keypoint Methods

###  II-A LoFTR (Local Feature TRansformer)

LoFTR provides dense correspondences between images without needing
descriptors. It uses a transformer-based architecture to establish
correspondences directly from image patches.

  * •

Transformer Layers: Utilizes multi-head self-attention to relate features
across the entire image.

  * •

High Computational Cost: Despite its accuracy, the dense matching process is
computationally intensive.

###  II-B SuperGlue[15]

SuperGlue matches keypoints by considering the entire context of both images
simultaneously using a graph neural network with attention mechanisms.

  * •

Graph Neural Network: Models relationships between keypoints across images.

  * •

Transformer-Based: Uses self and cross-attention to enhance matching
robustness.

  * •

Training and Computation: Requires significant computational resources for
training and inference.

### Why LightGlue is Preferred

###  II-C Efficiency

  * •

Adaptive Matching: LightGlue adjusts its computational effort based on the
difficulty of the image pair, making it faster for easy matches.

  * •

Early Discarding: Discards non-matchable points early, reducing unnecessary
computations.

While LOFTR (Learning to Optimize Frameworks)[14] and Superglue have made
significant strides in multimodal research, they do possess certain drawbacks
when compared to LightGlue.

In summary, while LOFTR and Superglue have advanced multimodal research, their
drawbacks in terms of complexity, computational requirements, and
generalization challenges highlight the potential advantages of LightGlue’s
approach in certain applications.

## COLMAP [17][19]

COLMAP (Construction and Localization MAPping) is a versatile and widely-used
photogrammetry software designed for 3D reconstruction and structure-from-
motion (SfM) tasks. It provides a suite of tools for processing images to
create 3D models by detecting, describing, and matching keypoints across
images, and then using these matches to estimate the 3D structure and camera
positions.

Figure 6: colmaps reconstruction capabilities- central rome constructed with
21k images.

Key features of COLMAP include:

  * •

Feature Extraction and Matching: Uses algorithms like SIFT for detecting and
matching keypoints across multiple images.

  * •

Structure-from-Motion (SfM): Estimates camera poses and reconstructs sparse 3D
points.

  * •

Multi-View Stereo (MVS): Generates dense 3D models by computing depth maps and
fusing them into a consistent 3D reconstruction.

  * •

Scalability: Efficiently handles large datasets with thousands of images.

  * •

User Interface: Provides a graphical interface for easy interaction and
visualization, along with command-line tools for automation.

COLMAP is favored for its robustness, accuracy, and ease of use, making it
suitable for applications in archaeology, architecture, cultural heritage
preservation, and more.

## OMNIGLUE[16]

OmniGlue addresses the generalization limitations of current learnable image
matchers, which typically excel in specific domains with abundant training
data but falter in diverse, unseen domains. Traditional methods like SIFT,
despite being hand-crafted, often outperform these advanced models in
unfamiliar contexts due to their domain-agnostic nature.

OmniGlue introduces two key innovations to enhance generalizability:

  1. 1.

Foundation Model Guidance: Utilizes the broad visual knowledge of large pre-
trained models like DINOv2[7] to guide the matching process, enhancing
performance in domains not covered during training.

  2. 2.

Keypoint-Position Guided Attention: Disentangles positional encoding from
matching descriptors, avoiding over-reliance on geometric priors from training
data, thereby improving cross-domain performance.

Experimental results demonstrate OmniGlue’s superior generalization across
various domains, including synthetic and real images, scene-level to object-
centric, and aerial datasets. Key contributions include:

  * •

Enhanced pose estimation accuracy by leveraging foundation model guidance.

  * •

Improved cross-domain transferability through innovative positional encoding
strategies.

  * •

Significant performance gains across diverse datasets, showcasing OmniGlue’s
robust generalization capabilities.

  * •

Ease of adaptation to new domains with minimal fine-tuning data.

### Comparison with SuperGlue, LightGlue, and LOFTR

SuperGlue[15] is a prominent learnable image matcher that uses attention
mechanisms to perform intra- and inter-image keypoint feature propagation,
typically leveraging SuperPoint for keypoint detection. While it demonstrates
high performance in specific domains, its generalization to unseen domains is
limited due to entanglement of local descriptors with positional information,
leading to over-specialization.

LightGlue[4] emphasizes lightweight and efficient multimodal fusion, making it
suitable for resource-constrained environments or real-time applications. By
focusing on simplicity and efficiency, it addresses computational and data
requirement issues but may not achieve the same level of performance on
diverse datasets as more complex models like OmniGlue.

LOFTR[14] (Learning-based Optical Flow with Transformers) employs a coarse-to-
fine correspondence prediction paradigm, excelling in dense image matching.
However, like other dense matchers, LOFTR struggles with generalization across
diverse domains due to its heavy reliance on domain-specific data and
computational intensity.

OmniGlue[16], compared to SuperGlue, LightGlue, and LOFTR, stands out in its
generalization capability. By leveraging foundation model guidance and novel
keypoint-position attention mechanisms, OmniGlue significantly improves
performance in unseen domains while maintaining high accuracy in the training
domain. This makes it a more versatile and robust solution for a wide range of
image matching tasks, addressing the limitations observed in its predecessors.

##  III Abbreviations and Acronyms

  1. 1.

Correspondences: Points in one image that match points in another image,
allowing the images to be aligned.

  2. 2.

Sparse Interest Points: Keypoints in an image that are distinctive and used
for matching across images.

  3. 3.

High-Dimensional Representations: Numerical descriptions of keypoints that
capture their local visual appearance.

  4. 4.

Robustness: The ability to handle variations in viewpoint, lighting, and other
changes.

  5. 5.

Uniqueness: The ability to discriminate between different points to avoid
false matches.

  6. 6.

Transformer Model: A type of neural network architecture that uses self-
attention mechanisms to process input data.

  7. 7.

Pareto-Optimal: A state where no criterion (like efficiency or accuracy) can
be improved without worsening another.

  8. 8.

Simultaneous Localization and Mapping (SLAM): A technique used in robotics and
computer vision to create a map of an environment while simultaneously keeping
track of the device’s location within that environment.

  9. 9.

Self-attention: A mechanism in neural networks where each element of a
sequence pays attention to other elements to understand its context better.

  10. 10.

Cross-attention: A mechanism where elements of one sequence pay attention to
elements of another sequence, useful in tasks like machine translation and
feature matching.

  11. 11.

Positional Patch Embedding (PosPE): A method to incorporate positional
information into patches of an image to improve feature detection.

  12. 12.

Geometric Invariance: The ability of a method to handle various
transformations (like rotation, scaling) in the input data.

  13. 13.

Deformable Convolution Network (DCN): A type of neural network that can adjust
the position of each pixel in the convolution, allowing it to model more
complex transformations.

  14. 14.

Neural Reprojection Error (NRE) Loss: A loss function used to measure the
difference between predicted and actual keypoint locations in image matching
tasks.

  15. 15.

Affine Transformations: Transformations that include scaling, rotation, and
translation.

  16. 16.

Specularities: Bright spots of light that appear on shiny surfaces when they
reflect light sources. These can create difficulties in image matching and 3D
reconstruction.

##  IV Our Solution

Our solution aims to implement a complex pipeline of image feature extraction,
matching, and 3D reconstruction, integrating a variety of advanced image
processing and 3D reconstruction tools. Our solution progressed in 3 steps as
follows:

  1. 1.

The get_keypoints method uses a deep learning model (such as LoFTR) to extract
key points from the image. Then, the matches_merger method and the
keypoints_merger method are used to merge the key points from different images
into a unified dataset to ensure the uniqueness and consistency of the key
points.

  2. 2.

The wrapper_keypoints method and the reconstruct_from_db method use COLMAP to
perform 3D reconstruction from the key points and matching data in the
database to obtain the camera pose.

  3. 3.

Finally, the create_submission method generates a submission file and formats
the output results to participate in a specific challenge or competition. The
entire pipeline achieves efficient and accurate image processing through
accurate feature point extraction, reliable matching filtering, and efficient
3D reconstruction, which is suitable for application scenarios such as
autonomous driving, robot navigation, and virtual reality that require
sophisticated image processing and 3D reconstruction.

### What made the difference?

Here are a few points we missed on. In Winner’s solutions you can observe,
they detected these edge cases and solved them efficiently with novel methods,
which resulted them better scores.

  1. 1.

Not correcting image orientation place a significant role. since the
algorithms we used are not designed for affine transformations neither they
are scale & rotation invariant.

  2. 2.

We did not solve for transparent & Low light images.

  3. 3.

We should have used an ensemble of aliked+lightglue[2][4] for key points
detection and feature extraction.

If we would have done these changes. We could have scored higher.

## Top Solutions

We have summarized most of the terms and latest research you shold have known
for a successful submission in the competition. Top Medal Baggers in Kaggle
mostly used permutation & combination of these techniques to get best scores.

##  V 1st Place Solution

The final solution combined 3D image reconstruction (I3DR) with COLMAP[19] for
non-transparent scenes and direct image pose estimation (DIP) for transparent
scenes. They used an ensemble of ALIKED extractors and LightGlue matchers,
cross-validation, multi-GPU acceleration, and a new cropping method.
Integration of OmniGlue enhanced match accuracy, and multiple reconstructions
were merged for robust results.

### Solving Transparent Surface Keypoint Matching

The solution started with performing orientation correction[5]. For detecting
keypoints they used ALIKED + LightGlue, However faced with a problem of
keypoint detection on transparent surfaces.This was a problem which was faced
by most of the kagglers. These top solutions discuss and address such problems
in detail.

The initial SfM pipeline with COLMAP did not work with transparent scenes. To
address this, they experimented with different strategies, hypothesizing that
direct pose estimation might help compute the rotation matrix. They assumed
cameras were positioned close to the object, capturing it from all sides.

### Approach #1

They placed cameras in a circle around the object and sorted images using
several methods:

  * •

Optical Flow: Calculated magnitude for each image pair and assigned a weight
equal to the standard deviation of the magnitude.

  * •

Pixel-level Difference: Simple grayscale difference with weights based on the
difference value.

  * •

SSIM Score: Calculated SSIM index for each pair, assigning a weight of 1 -
SSIM.

  * •

ALIKED+LG Matching: Number of matches for each pair with weights as 1 /
num_matches.

They built a distance matrix from these weights and solved the ordering
problem using the Travelling Salesman Problem (TSP).

### Approach #2

Estimated image order by matching images at high resolution (4096px). The
number of matches was higher for consecutive images, using a kNN-like method
for estimation.

After this performing reconstruction using COLMAP to get rotation and
translation matrix made them rank on TOP of the table, helping them score
0.28, resulting in a gold medal.

##  VI 2nd Place Solution

The second-place solution for IMC 2024 devised separate strategies for
conventional and transparent scenes due to their distinct characteristics,
which were identified through extensive trials.

## Preprocessing

1\. Rotation Detection: Utilized a rotation detection model to predict and
correct image rotations. Retained original rotations if less than 10% of
images were predicted as rotated, acknowledging the model’s potential
inaccuracies.[5]

2\. Shared Camera Intrinsics: If image dimensions were identical, set all
cameras to share the same internal parameters, occasionally improving results
by 0.01.

3\. Transparency Detection: Calculated the average difference between images
to classify scenes as transparent or not, enabling separate handling for each
type.

## Modeling Techniques

1\. Global Features: Developed a robust global feature descriptor combining
point and patch features. Extracted point features (ALIKED) and patch features
(DINO), establishing one-to-one correspondence based on spatial relationships.
Used clustering and the VLAD algorithm to generate global descriptors. This
method outperformed existing techniques (NetVLAD, AnyLoc, DINO, SALAD) on VPR-
related datasets.

2\. Local Features: Utilized three types of local features: Dedode v2 + Dual
Softmax, DISK + LightGlue, and SIFT + Nearest Neighbor. The Dedode v2 detector
produced rich and evenly distributed feature points. The G-upright descriptor
and dual softmax matcher were selected for this purpose.

3\. MST-Aided Coarse-to-Fine SfM Solution:

  1. 1.

Constructed a similarity graph with images as vertices and similarities as
edges. Computed the Minimum Spanning Tree (MST) to obtain an optimal data
association, used for the initial SfM. This stage focused on removing
incorrect associations and improving coarse-grained accuracy.

  2. 2.

Utilized full data associations and the coarse model from Stage 1 to provide
initial camera pose priors for geometric verification. This filtered out
incorrect feature matches, maintaining coarse-grained advantages while
improving fine-grained accuracy.

4\. Post-Processing:

Employed pixsfm to optimize the SfM model. Deployed an HLoc-based
relocalization module to process unregistered images, typically resulting in a
0-0.01 score improvement.

Handling Transparent Scenes:Explored various local features (ALIKED, DISK,
LoFTR, DKMV3), but none were satisfactory.

Many Kagglers dealt with the problem by separately handling transparent and
conventional scenes with tailored preprocessing and modeling techniques, this
solution achieved significant accuracy improvements in both types of scenes.
Thereby, resulting them in 2nd place gold.

##  VII 3rd Place Solution

The third-place solution for IMC 2024, VGGSfM, is a structure-from-motion
method based on Visual Geometry Grounded Deep Structure From Motion, which was
enhanced for this competition. The approach involved several key strategies:

VGGSfM Across All Frames: Applied VGGSfM to all input frames, improving mAA by
4% compared to the baseline. However, due to GPU memory limitations on Kaggle
servers, this method had to be integrated into the existing pycolmap pipeline.

Figure 7: VGGsfm in action Figure 8: Reconstruction of In-the-wild Photos with
VGGSfM, displaying estimated point clouds (in blue) and cameras (orange).

  1. 1.

Additional Tracks: Utilized VGGSfM’s track predictor to estimate 2D matches
and fed them into pycolmap. Nearest frames were identified using NetVLAD or
DINO V2. This approach improved mAA by 3% on the evaluation set and public
leaderboard score from 0.17 to 0.18.

  2. 2.

SfM Track Refinement: Enhanced tracks predicted by pycolmap with VGGSfM’s fine
track predictor. After running the baseline with ALIKED+LightGlue, 3D points
were refined and updated. A global bundle adjustment further optimized camera
and point positions, improving mean reprojection error from 0.64 to 0.55 and
leaderboard score from 0.18 to 0.20.

  3. 3.

Relocating Missing Images: Used VGGSfM to identify and relocate missing images
in the scene. This process aligned camera poses and improved the leaderboard
score from 0.20 to 0.21.

Final Solution: which led to 3rd place.

  1. 1.

Handled image rotation to maximize matches using a pre-trained model.

  2. 2.

Used matches from both ALIKED+LightGlue and SP+LightGlue.

  3. 3.

For transparent images, extracted an area of interest using DBSCAN on
keypoints and ran keypoint detection on these areas again.

##  VIII 4th Place Solution

## Handling Transparent Images

### Foreground Segmentation

  * •

Observation: Many keypoints in transparent scenes appeared in the background,
disrupting camera pose estimation.

  * •

Solution: Employed the DINOv2 Segmenter, identifying foreground objects as
class5 (“bottle” in VOC2012). This allowed high-precision segmentation by
focusing on transparent objects.

  * •

Keypoint Detection: Detected keypoints at the original image scale (1024x1024
grid units) without resizing, which was efficient given the uniform image
size. Keypoints were detected only in the foreground area using the segmented
results from DINOv2.

While looking at accuracte foreground seperation for 4th solution, one should
also see this figure 4 for looking how the method is employed.

Figure 9: accurate keypoint detected when using DINOv2

### Feature Matching

  * •

Strategy: Limited the search for matches to corresponding grids during
keypoint detection, significantly reducing the search range and focusing on
relevant areas, improving matching efficiency and accuracy.

## Leveraging ALIKED and LightGlue

### Non-Transparent Scenes

  * •

Keypoint Detection: Generated keypoints for images rotated in 90-degree
increments using ALIKED-n16, retaining keypoints for each rotation.

  * •

Matching Stage: Utilized LightGlue to evaluate matches. For each fixed set of
keypoints, evaluated matches with rotated counterparts, adopting the
combination with the highest number of matches. This ensured robust matching
regardless of image rotation.

### Additional Techniques

### Exhaustive Matching

  * •

Instead of searching for pairs using embedding-based similarity measures like
DINOv2 or EfficientNet, exhaustive matching for all image pairs was performed,
mitigating the risk of missing matches due to low similarity scores.

### Using All Images

  * •

By incorporating images beyond those listed in the submission file (up to 100
images for validation), the solution increased the number of triangulated
points, enhancing 3D reconstruction accuracy.

## Results

  * •

Baseline: Private LB=0.149, Public LB=0.136

  * •

Add Transparent Trick: Private LB=0.184, Public LB=0.171

  * •

Add Exhaustive Matching: Private LB=0.186, Public LB=0.176

  * •

Add All Images: Private LB=0.197, Public LB=0.194

These combined approaches led to a robust solution, achieving 4th place in the
competition.

##  IX 5th Place Solution

## Build Local Evaluation Datasets

To manage the large dataset realistically, three subsets were created using
random sampling strategies. Results across these subsets showed high
correlation, validating the chosen subset for local cross-validation (CV)
reporting.

## Build a General SfM Pipeline

The pipeline was divided into three modules:

###  IX-A Proposing Pair Candidates by Global Descriptors

Utilized pretrained models like EVA-CLIP Base, ConvNeXt Base, and Dinov2 ViT
Base to extract global features. Customized similarity thresholds based on
scene diversity (e.g., Lizard versus Cylinder) using cosine similarity.

###  IX-B Matching Pairs in the Candidate List

Focused on lightweight detector-based methods for image matching due to their
efficiency. Experimented with SIFT + NN and LightGlue combinations, noting
significant performance boosts in CV but slight drawbacks in LB performance.

###  IX-C Reconstruction with Colmap

Explored various approaches including single-camera usage and manual initial
pair settings, aiming for improved reconstruction accuracy. Incremental
mapping enhanced consistency in CV results but showed minimal impact on LB.

## Customize the Pipeline for Each Specific Category

###  IX-D Transparent Objects

Implemented segmentation models like MobileSAM for mask detection and keypoint
extraction (e.g., ALIKED). Opted for the smallest mask encompassing most
keypoints to focus solely on the object, significantly boosting CV performance
with LightGlue.

###  IX-E Finding the Best Pairs

Implemented methods to find consecutive pairs efficiently. Used exhaustive
matching for all pairs and built matrices based on match counts, achieving
nearly 100% accuracy in pair selection. But for dark/dim images, I assume that
these scenes may have plenty of false positive matches because of their
natural properties, so they tried tuning parameters with much more strict
values (e.g., increasing the matching threshold to 0.5, etc.). Nonetheless, it
didn’t show any improvement. so they tried dopplegangers [3]. First, they run
SfM one time to get all the matching pairs. then used the Doppelganger model
to filter out pairs that have high probabilities as false positives
(doppelgangers). It didn’t show any improvement on the church scene on the
local CV.

## Results

Leveraging these strategies, the solution consistently improved performance
across categories, notably advancing in the LB rankings. Techniques tailored
for specific challenges like transparent objects and day-night scenes
contributed to the overall success.
ALIKED+LG+rot+transparentcustom+tuning:0.195 Which made them rank 5th.

##  X Evaluation Metric- Mean Average Accuracy (mAA)

Submissions are evaluated based on the mean Average Accuracy (mAA) of the
registered camera centers $C=-R^{T}T$.

Given the set of cameras of a scene parameterized by their rotation matrices
$R$ and translation vectors $T$, and the hidden ground truth, the evaluation
computes the best similarity transformation $T$ (scale, rotation, and
translation altogether) that registers the highest number of cameras onto the
ground truth starting from triplets of corresponding camera centers.

A camera is registered if $\|C_{g}-T(C)\|<t$, where $C_{g}$ is the ground-
truth camera center corresponding to $C$ and $t$ is a given threshold. Using a
RANSAC-like approach, all possible (N choose 3) feasible similarity
transformations $T^{\prime}$ derived by Horn’s method on triplets of
corresponding camera centers $(C,C_{g})$ are exhaustively verified. Here, $N$
is the number of cameras in the scene.

Each transformation $T^{\prime}$ is refined into $T^{\prime\prime}$ by
registering the camera centers again using Horn’s method, incorporating
previously registered cameras with the initial triplets. The best model $T$,
among all $T^{\prime\prime}$ with the highest number of registered cameras, is
returned.

##  XI Conclusion

Joining this competition for the first time came with a lot of learning and
experiences. We learnt a lot about sfm problems, how edge cases can lead to
poor reconstruction results, how using SOTA will not guarantee top score
alone, which we saw in case of 2nd place solution. Special thanks to my co-
authors for writing this competition review with me.

## References

  * [1] Qing Wang and Jiaming Zhang and Kailun Yang and Kunyu Peng and Rainer Stiefelhagen 2022. MatchFormer: Interleaving Attention in Transformers for Feature Matching. MatchFormer.
  * [2] Xiaoming Zhao and Xingming Wu and Weihai Chen and Peter C. Y. Chen and Qingsong Xu and Zhengguo Li. 2023. ALIKED: A Lighter Keypoint and Descriptor Extraction Network via Deformable Transformation. Aliked
  * [3] Ruojin Cai and Joseph Tung and Qianqian Wang and Hadar Averbuch-Elor and Bharath Hariharan and Noah Snavely. 2023. Doppelgangers: Learning to Disambiguate Images of Similar Structures. Paper
  * [4] Philipp Lindenberger and Paul-Edouard Sarlin and Marc Pollefeys. LightGlue: Local Feature Matching at Light Speed.lightglue
  * [5] Ternaus et. al. - Check Orientation rotation github
  * [6] Y. Ding, Z. Xiong, J. Xiong, Y. Cui and Z. Cao, ”OGI-SLAM2: A Hybrid Map SLAM Framework Grounded in Inertial-Based SLAM,” in IEEE Transactions on Instrumentation and Measurement ieee
  * [7] Maxime Oquab and Timothée Darcet et. al. DINOv2: Learning Robust Visual Features without Supervision DinoV22
  * [8] Fabio Bellavia, Jiri Matas, Dmytro Mishkin, Luca Morelli, Fabio Remondino, Weiwei Sun, Amy Tabb, Eduard Trulls, Kwang Moo Yi, Sohier Dane, Ashley Chow 2024. Image Matching Challenge 2024 - Hexathlon Competition
  * [9] IGOR LASHKOV, 1st Place Solution – High Image Resolution ALIKED/LightGlue + Transparent Trick. 1st Place
  * [10] neo, sunnyykk, gdchenhao, mayunchaoamap and wangshengyi96. 2nd Place Solution: MST-Aided SfM & Transparent Scene Solution.2nd Place
  * [11] Jianyuan Wang, Nikita Karaev, Christian Rupprecht, David Novotny, 3rd Place Solution: VGGSfM 3rd Place Arxiv
  * [12] Tomoya Okazaki, 4th Place Solution: ALIKED+LightGlue is all you need. 4th Place
  * [13] KHOA NGO, 5th Place Solution: Customized Scene Matching. 5th Place
  * [14] Sun, Jiaming and Shen, Zehong and Wang, Yuang and Bao, Hujun and Zhou, Xiaowei cvpr 2021. LOFTR Detector-Free Local Feature Matching with Transformers. Paper
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  * [19] colmap code  github