# Constraints on $\Xi^{-}$ nuclear interactions from capture events in
emulsion

E. Friedman  A. Gal  Racah Institute of Physics, The Hebrew University,
Jerusalem 91904, Israel

###### Abstract

Five $\Xi^{-}p\to\Lambda\Lambda$ two-body capture events in 12C and 14N
emulsion nuclei, in which a pair of single-$\Lambda$ hypernuclei is formed and
identified by their weak decay, have been observed in $(K^{-},K^{+})$ emulsion
exposures at KEK and J-PARC. Applying a $\Xi^{-}$-nucleus optical potential
methodology to study atomic and nuclear transitions, we confirm that these
capture events occur from Coulomb assisted $1p_{\Xi^{-}}$ nuclear states.
Long-range $\Xi N$ shell-model correlations are found essential to achieve
consistency between the 12C and 14N events. The resulting $\Xi$-nuclear
interaction is strongly attractive, with $\Xi$ potential depth in nuclear
matter $V_{\Xi}\gtrsim 20$ MeV. Implications to multi-strangeness features of
dense matter are outlined.

###### keywords:

hyperon strong interactions; $\Xi^{-}$ atoms and hypernuclei.

††journal: Physics Letters B

##  1 Introduction and background

Recent two-particle correlation studies of $p\Lambda$, $\Lambda\Lambda$ and
$\Xi^{-}p$ pairs measured by ALICE [1, 2, 3, 4] in $pp$ and $p$-Pb ultra-
relativistic collisions at TeV energies have triggered renewed interest in
Strangeness ${\cal S}\neq 0$ baryon-baryon interactions and consequences
thereof to strange hadronic matter. In particular, the $\Xi^{-}p$ interaction
was shown to be attractive [3], in good agreement with the recent HAL-QCD
lattice calculations reaching $m_{\pi}=146$ MeV [5]. Understanding the
strength of the ${\cal S}=-2$ $\Xi N$ interaction, particularly when embedded
in nuclear media, is vital for resolving the Hyperon Puzzle which addresses
the fate of hyperons in dense neutron-star matter [6].

Little is known from experiment on the nuclear interaction of $\Xi$ hyperons
[7, 8]. A standard reaction production is the nuclear $(K^{-},K^{+})$
reaction, driven by $K^{-}p\to K^{+}\Xi^{-}$ strangeness exchange on protons.
Owing to its large momentum transfer, the produced $\Xi^{-}$ hyperons populate
dominantly the quasi-free continuum region, with less than 1% expected to
populate $\Xi^{-}$-nuclear bound states that decay subsequently by the
$\Xi^{-}p\to\Lambda\Lambda$ strong-interaction capture reaction. Analysis of
old emulsion events attributed to the formation of $\Xi$ hypernuclei suggested
attractive Woods-Saxon $\Xi$ nuclear potential depth $V_{\Xi}$=21-24 MeV [9].
While this range of values is considered sufficient for $\Xi$ hyperons to play
an active role in strange hadronic matter [10] and in neutron stars [11],
somewhat smaller values follow from studies of dedicated $(K^{-},K^{+})$
counter experiments: $V_{\Xi}\lesssim 20$ MeV in KEK PS-E224 [12],
$V_{\Xi}\sim 14$ MeV in BNL AGS-E885 [13], both on 12C, and $V_{\Xi}=17\pm 6$
MeV on 9Be [14] from BNL AGS-E906 [15]. New results from the J-PARC E05 and
E70 $(K^{-},K^{+})$ experiments on 12C are forthcoming [16]. However, no
$\Xi^{-}$ or $\Lambda\Lambda$ hypernuclear bound states have ever been
observed unamiguously in these experiments. Powerful future experiments by the
PANDA Collaboration at FAIR [17], using ${\bar{p}}p\to\Xi^{-}{\bar{\Xi}}^{+}$
or ${\bar{p}}n\to\Xi^{-}{\bar{\Xi}}^{0}$ production modes, and also at BESIII
[18] focusing on the $J/\psi\to\Xi^{-}{\bar{\Xi}}^{+}$ $O(10^{-3})$ decay
branch, are likely to change this state of the art.

The situation is different in exposures of light-emulsion CNO nuclei to the
$(K^{-},K^{+})$ reaction, in which a tiny fraction of the produced high-energy
$\Xi^{-}$ hyperons slow down in the emulsion, undergoing an Auger process to
form high-$n$ atomic levels, and cascade down radiatively. Cascade essentially
terminates, when strong-interaction capture takes over, in a 3D atomic state
bound by 126, 175, 231 keV in C, N, O, respectively. The 3D strong-interaction
shift is less than 1 keV [19]. Capture events are recorded by observing
$\Lambda$ hyperon or hypernuclear decay products. Interestingly, whereas the
few observed double-$\Lambda$ hypernucleus production events are consistent
with $\Xi^{-}$ capture from atomic 3D states [8], formation of pairs of
single-$\Lambda$ hypernuclei requires capture from a lower $\Xi^{-}$ orbit.
Expecting the final two $\Lambda$ hyperons in $\Xi^{-}p\to\Lambda\Lambda$
capture to be formed in a $S=0,\,1s_{\Lambda}^{2}$ configuration, the initial
$\Xi^{-}$ hyperon and the proton on which it is captured must satisfy
$l_{\Xi^{-}}=l_{p}$ [20] which for $p$-shell nuclear targets favors the choice
$l_{\Xi^{-}}=1$. Indeed, all two-body $\Xi^{-}$ capture events,
$\Xi^{-}+{{}^{A}Z}\to$
${}^{A^{\prime}}_{~{}\Lambda}$Z’+${}^{A^{\prime\prime}}_{~{}\Lambda}$Z”, to
twin single-$\Lambda$ hypernuclei reported from KEK and J-PARC light-emulsion
$K^{-}$ exposures [21, 22, 23], as listed here in Table 1, are consistent with
$\Xi^{-}$ capture from Coulomb-assisted $1p_{\Xi^{-}}$ nuclear states bound by
$\sim$1 MeV.

Table 1: Reported two-body $\Xi^{-}$ capture events $\Xi^{-}+{{}^{A}Z}\to$ ${}^{A^{\prime}}_{~{}\Lambda}$Z’+${}^{A^{\prime\prime}}_{~{}\Lambda}$Z” in light-emulsion nuclei to a pair of single-$\Lambda$ hypernuclei, some in ground states, some in specific excited states marked by asterisk. Only the first and last events are uniquely assigned. Fitted $\Xi^{-}$ binding energies $B_{\Xi^{-}}$ are listed. Experiment | Event | ${}^{A}Z$ |  ${}^{A^{\prime}}_{\Lambda}$Z’+${}^{A^{\prime\prime}}_{\Lambda}$Z” |  $B_{\Xi^{-}}$ (MeV)
---|---|---|---|---
KEK E176 [21] | 10-09-06 |  12C |  ${}^{4}_{\Lambda}$H+${}^{9}_{\Lambda}$Be | 0.82$\pm$0.17
KEK E176 [21] | 13-11-14 |  12C |  ${}^{4}_{\Lambda}$H+${}^{9}_{\Lambda}$Be∗ | 0.82$\pm$0.14
KEK E176 [21] | 14-03-35 |  14N |  ${}^{3}_{\Lambda}$H+${}^{12}_{~{}\Lambda}$B | 1.18$\pm$0.22
KEK E373 [22] | KISO |  14N |  ${}^{5}_{\Lambda}$He+${}^{10}_{~{}\Lambda}$Be∗ | 1.03$\pm$0.18
J-PARC E07 [23] | IBUKI |  14N |  ${}^{5}_{\Lambda}$He+${}^{10}_{~{}\Lambda}$Be | 1.27$\pm$0.21

In Table 1 only the first and last listed events are uniquely assigned in
terms of initial emulsion nucleus ${}^{A}Z$ and final single-$\Lambda$
hypernuclei ${}^{A^{\prime}}_{\Lambda}$Z’+${}^{A^{\prime\prime}}_{\Lambda}$Z”
ground states, providing thereby a unique value of $B_{\Xi^{-}}$ per each
event. The events fitted by assuming specific excited states
${}^{A^{\prime\prime}}_{\Lambda}$Z”∗ are equally well fitted each by g.s.
assignments ${}^{A^{\prime\prime}}_{\Lambda}$Z”, with values of $B_{\Xi^{-}}$
as high as $\sim$4 MeV, and the middle event is equally well fitted as a
capture event in 16O, to ${}^{3}_{\Lambda}$H+${}^{14}_{~{}\Lambda}$C with
$B_{\Xi^{-}}=0.46\pm 0.39$ MeV or to
${}^{4}_{\Lambda}$H+${}^{13}_{~{}\Lambda}$C with $B_{\Xi^{-}}=0.40\pm 0.27$
MeV, both consistent with $\Xi^{-}$ capture from an atomic 3D state. We note
that the listed $\Xi^{-}$ binding energy $B_{\Xi^{-}}$ values are all around 1
MeV, significantly higher than the purely-Coulomb atomic 2P binding energy
values which are approximately 0.3, 0.4, 0.5 MeV in C,N,O atoms, respectively.
These $\sim$1 MeV $B_{\Xi^{-}}$ values correspond to $1p_{\Xi^{-}}$ nuclear
states that evolve from 2P atomic states upon adding a strong-interaction
$\Xi$ nuclear potential.111Nuclear single-particle (s.p.) states are denoted
by lower-case letters: $1s,1p,1d,...$ for the lowest $l$ values, in
distinction from atomic s.p. states denoted by capitals: 1S,2P,3D,… for the
lowest L values. This interpretation is the only one common to all five
events.

Not listed in the table are multi-body capture events that require for their
interpretation undetected capture products, usually neutrons, on top of a pair
of single-$\Lambda$ hypernuclei. Most of these new J-PARC E07 events [24]
imply $\Xi^{-}$ capture from $1s_{\Xi^{-}}$ nuclear states, with capture rates
${\cal O}(10^{-2})$ of capture rates from the $1p_{\Xi^{-}}$ nuclear states
considered here [20, 25].

In the present work we consider $\Xi^{-}$ atomic and nuclear transitions in
light emulsion atoms, first with a $t\rho$ optical potential, to substantiate
that $\Xi^{-}p\to\Lambda\Lambda$ capture indeed occurs from a Coulomb-assisted
nuclear $1p_{\Xi^{-}}$ state in light emulsion nuclei. The strength of this
$\Xi$-nuclear potential is determined by requiring that it reproduces a
$1p_{\Xi^{-}}$ state in 12C(0${}^{+}_{\rm g.s.}$) bound by 0.82$\pm$0.15 MeV
from Table 1. Disregarding temporarily the $s_{\Xi^{-}}=\frac{1}{2}$ Pauli-
spin degree of freedom, we proceed to discuss the implications of identifying
the value $B_{\Xi^{-}}\approx 1.15\pm 0.20$ MeV for 14N from Table 1 with the
binding energy of ${\cal L}^{\pi}=(0^{-},1^{-},2^{-})$ triplet of
$1p_{\Xi^{-}}$ nuclear states built on $J^{\pi}(^{14}$Ng.s.)=1+, thereby
linking the capture process to properties of the $\Xi$-nucleus residual
interaction. This provides the only self consistent deduction of the
$\Xi$-nuclear interaction strength from analysis of the five $\Xi^{-}$ capture
events in light nuclear emulsion, fitted to two-body formation of specific
$\Lambda$ hypernuclei, as listed in Table 1. The resulting $t\rho$ $\Xi$
potential depth at nuclear-matter density $\rho_{0}$=0.17 fm-3 is
$V_{\Xi}\approx 24$ MeV. We then improve upon the $t\rho$ leading term of the
optical potential by introducing the next, Pauli correlation term in the
optical potential density expansion [26]. This leads to $\approx$10% reduction
of $V_{\Xi}$, down to $V_{\Xi}\approx 22$ MeV, keeping it within the optical
potential approach well above 20 MeV. Our results suggest that $1s_{\Xi^{-}}$
nuclear bound states exist all the way down to 4He, with potentially far-
reaching implications for the role of $\Xi$ hyperons in multi-strange dense
matter.

A value $V_{\Xi}\gtrsim 20$ MeV implies a substantially stronger in-medium
$\Xi N$ attraction than reported by some recent model evaluations (HAL-QCD
[27], EFT@NLO [28, 29], RMF [30]) all of which satisfy $V_{\Xi}\lesssim 10$
MeV. A notable exception is provided by versions ESC16*(A,B) of the latest
Nijmegen extended-soft-core $\Xi N$ interaction model [31], in which values of
$V_{\Xi}$ higher than 20 MeV are derived. However, these large values are
reduced substantially by $\Xi NN$ three-body contributions within the same
ESC16* model.

##  2 Methodology

The starting point in optical-potential analyses of hadronic atoms [32] is the
in-medium hadron self-energy $\Pi(E,\vec{p},\rho)$ that enters the in-medium
hadron (here $\Xi$ hyperon) dispersion relation

$E^{2}-{\vec{p}}^{~{}2}-m_{\Xi}^{2}-\Pi(E,\vec{p},\rho)=0,$ (1)

where ${\vec{p}}$ and $E$ are the $\Xi$ momentum and energy, respectively, in
nuclear matter of density $\rho$. The resulting $\Xi$-nuclear optical
potential $V_{\rm opt}$, defined by $\Pi(E,\vec{p},\rho)=2EV_{\rm opt}$,
enters the near-threshold $\Xi^{-}$ wave equation

$\left[\nabla^{2}-2{\mu}(B+V_{\rm opt}+V_{c})+(V_{c}+B)^{2}\right]\psi=0,$ (2)

where $\hbar=c=1$. Here $\mu$ is the $\Xi^{-}$-nucleus reduced mass, $B$ is
the complex binding energy, $V_{c}$ is the finite-size Coulomb potential of
the $\Xi^{-}$ hyperon with the nucleus, including vacuum-polarization terms,
all added according to the minimal substitution principle $E\to E-V_{c}$.
Strong-interaction optical-potential $V_{\rm opt}$ terms other than $2\mu
V_{\rm opt}$ are negligible and omitted here. The use of a Klein-Gordon wave
equation (2) for the $\Xi^{-}$ fermion rather than Dirac equation provides an
excellent approximation when $Z\alpha\ll 1$ and fine-structure effects are
averaged on, as for the X-ray transitions considered here. $\Xi^{-}$ nuclear
spin-orbit effects are briefly mentioned below.

For $V_{\rm opt}$ in Eq. (2) we first use a standard $t\rho$ form [32]

$V_{\rm
opt}(r)=-\frac{2\pi}{\mu}(1+\frac{A-1}{A}\frac{\mu}{m_{N}})[b_{0}\rho(r)+b_{1}\rho_{\rm
exc}(r)],$ (3)

where the complex strength parameters $b_{0}$ and $b_{1}$ are effective,
generally density dependent $\Xi N$ isoscalar and isovector c.m. scattering
amplitudes respectively. The density $\rho=\rho_{n}+\rho_{p}$ is a nuclear
density distribution normalized to the number of nucleons $A$ and $\rho_{\rm
exc}=\rho_{n}-\rho_{p}$ is a neutron-excess density with
$\rho_{n}=(N/Z)\rho_{p}$, implying that $\rho_{\rm exc}=0$ for the $N=Z$
emulsion nuclei 12C and 14N analyzed in the next section. Here we used mostly
nuclear density distributions of harmonic-oscillator (HO) type [33] where the
r.m.s. radius of $\rho_{p}$ was set equal to that of the known nuclear charge
density [34]. Folding reasonably chosen $\Xi N$ interaction ranges other than
corresponding to the proton charge radius, or using Modified Harmonic
Oscillator (MHO) densities, or replacing HO densities by realistic three-
parameter Fermi (3pF) density distributions [35, 36] made little difference:
all the calculated binding energies changed by a small fraction, about 0.03
MeV of the uncertainty imposed by the $\pm$0.15 MeV experimental uncertainty
of the 0.82 MeV $1p_{\Xi}$ binding energy in 12C listed in Table 1. We note
that the central density $\rho(0)$ in all density versions used here is within
acceptable values for nuclear matter, i.e., between roughly 0.15 and 0.20
fm-3.

The form of $V_{\rm opt}$ given by Eq. (3) corresponds to a central-field
approximation of the $\Xi$-nuclear interaction. Spin and isospin degrees of
freedom induced by the most general two-body $s$-wave $\Xi N$ interaction
$V_{\Xi N}$,

$V_{\Xi
N}=V_{0}+V_{\sigma}\sigma_{\Xi}\cdot\sigma_{N}+V_{\tau}\tau_{\Xi}\cdot\tau_{N}+V_{\sigma\tau}\sigma_{\Xi}\cdot\sigma_{N}\,\tau_{\Xi}\cdot\tau_{N}$
(4)

with $V$s functions of $r_{\Xi N}$, are suppressed in this approach. Choosing
12Cg.s. with isospin $I=0$ and spin-parity $J^{\pi}=0^{+}$ for a nuclear
medium offers the advantage that only $V_{0}$ is operative in leading order
since the nuclear expectation value of each of the other three terms in Eq.
(4) vanishes. But for 14Ng.s., with $I(J^{\pi})$=0(1+), $V_{\sigma}$ is
operative as well, adding unavoidable model dependence of order ${\cal
O}(1/A)$ to $\Xi^{-}$-nuclear potential depth values derived from capture
events assigned to this emulsion nucleus. For this reason we start the present
analysis with the two $\Xi^{-}$-12C emulsion events of Table 1.

Figure 1: Energy levels (in keV) of the lowest $\Xi^{-}$ atomic states for L=0
(1S,2S) and L=1 (2P,3P) in 12C as a function of the strength parameter
Re$\,b_{0}$ (in fm) of the $\Xi^{-}$ optical potential (3). Spin-orbit
splittings of L=1 states are suppressed in this figure. The dashed and dotted
horizontal lines mark the value $B_{\Xi^{-}}=0.82\pm 0.15$ MeV from Table 1.

##  3 $\Xi^{-}$ capture in 12C

Figure 1 shows a portion of the combined atomic plus nuclear spectrum of
$\Xi^{-}$ in 12C, $B_{\Xi^{-}}\leq 2$ MeV, as a function of Re$\,b_{0}$, Eq.
(3), for a fixed Im$\,b_{0}=0.01$ fm corresponding to a nuclear-matter
$\Xi^{-}$ capture width $\Gamma_{\Xi^{-}}\approx 1.5$ MeV, compatible with the
HAL-QCD weak $\Xi^{-}p\to\Lambda\Lambda$ transition potential [5]. Plotted are
the energies of the two lowest states of each orbital angular momentum
$l_{\Xi^{-}}=0,1$, starting at Re$\,b_{0}$=0 with almost purely atomic states
1S,2P,2S,3P from bottom up. Of these states the 1S state with Bohr radius
about 3.8 fm is indistinguishable from a nuclear $1s$ state, and indeed it
dives down in energy as soon as Re$\,b_{0}$ is made nonzero. It takes
considerable strength, Re$\,b_{0}\gtrsim 0.25$ fm, before the next atomic
state, 2P with Bohr radius about 15 fm, overlaps appreciably with the 12Cg.s.
nuclear core, diving down in energy to become a nuclear $1p$ state. The higher
two states that start as atomic 2S,3P remain largely atomic as Re$\,b_{0}$ is
varied in Fig. 1. Their slowly decreasing energies indicate a rearrangement of
the atomic spectrum [37]: 2S$\to$1S and 3P$\to$2P. Judging by the marked band
of values $B_{\Xi^{-}}$=0.82$\pm$0.15 MeV for the two KEK E176 events listed
in Table 1, the figure suggests that they are compatible with a $1p_{\Xi^{-}}$
nuclear state corresponding to a $\Xi$-nuclear potential strength of
Re$\,b_{0}=0.32\pm 0.01$ fm. The sensitivity to variations of Im$\,b_{0}$ is
minimal: choosing Im$\,b_{0}=0.04$ fm [19] instead of 0.01 fm increases
Re$\,b_{0}$ by 0.01 fm to $0.33\pm 0.01$ fm.

Radiative rates for E1 transitions from the $\Xi^{-}$ atomic 3D state to the
$\Xi^{-}$ atomic 3P state, and to the $1p_{\Xi}$ nuclear state that started as
atomic 2P state are found comparable to each other, accounting together for
7.6% of the total 3D width $\Gamma_{\rm 3D}=3.93$ eV as obtained using the
optical potential (3). However, the subsequent $\Xi^{-}p\to\Lambda\Lambda$
capture will proceed preferentially from the $1p_{\Xi}$ nuclear state that
offers good overlap between the $1p_{\Xi^{-}}$ and $1p_{p}$ valence-proton
orbits. Since the final $1s_{\Lambda}^{2}$ configuration has $J_{f}=0$, and
the $p$-shell protons in 12C are mostly in $j=\frac{3}{2}$ orbits, the
requirement $J_{i}=J_{f}=0$ imposes $j_{\Xi^{-}}=\frac{3}{2}$ on the spin-
orbit doublet members of the $1p_{\Xi^{-}}$ state. The shift of this
$1p_{\Xi^{-}}(\frac{3}{2})$ sub level from the $1p_{\Xi^{-}}$ $(2j+1)$-average
is estimated, based on the 152 keV $1p_{\Lambda}$ spin-orbit splitting
observed in ${}^{13}_{~{}\Lambda}$C [38] to be less than 100 keV upward [39]
and, hence, within the 0.15 MeV listed uncertainty introduced here for the
position of the $(2j+1)$-averaged $1p_{\Xi^{-}}$ state.

##  4 Spectroscopy of 14N${}_{\rm g.s.}+1p_{\Xi^{-}}$ states

Having derived the strength parameter Re$\,b_{0}=0.32\pm 0.01$ fm of $V_{\rm
opt}$ by fitting it to $B_{\Xi^{-}}^{1p}(^{12}$C)=0.82$\pm$0.15 MeV, we apply
this optical potential to 14N where it yields
$B_{\Xi^{-}}^{1p}(^{14}$N)=2.08$\pm$0.28 MeV, considerably higher than the
value $B_{\Xi^{-}}$=1.15$\pm$0.20 MeV obtained from the three events assigned
in Table 1 to $\Xi^{-}$ capture in 14N. However, this calculated
$B_{\Xi^{-}}^{1p}(^{14}$N) corresponds to a $(2{\cal L}+1)$-average of binding
energies for a triplet of states ${\cal L}^{\pi}=(0^{-},1^{-},2^{-})$ obtained
by coupling a $1p_{\Xi^{-}}$ state to $J^{\pi}(^{14}$N${}_{\rm g.s.})$=1+, as
shown in Fig. 2. We now discuss the splitting of these triplet states. Effects
of $\Xi^{-}$ Pauli spin, $s_{\Xi^{-}}=\frac{1}{2}$, are introduced at a later
stage.

Figure 2: Energies (in MeV) of ${\cal L}^{\pi}=(0^{-},1^{-},2^{-})$ triplet of
14N${}_{\rm g.s.}+1p_{\Xi^{-}}$ states, split by a $Q_{N}\cdot Q_{\Xi}$
residual interaction (5). The $(2{\cal L}+1)$-averaged energy $-2.08\pm 0.28$
MeV was calculated using the same optical potential parameter $b_{0}$ that
yields a 12C${}_{\rm g.s.}+1p_{\Xi^{-}}$ state at $-0.82\pm 0.15$ MeV,
corresponding to the $\Xi^{-}$ capture events in 12C listed in Table 1.

The construction of the 14N${}_{\rm g.s.}+1p_{\Xi^{-}}$ spectrum in Fig. 2
follows a similar 12C(2+;4.44 MeV)$\,+1p_{\Lambda}$ spectrum construction in
${}^{13}_{~{}\Lambda}$C [40]. The energy splittings marked in the figure are
obtained from a two-body spin-independent interaction $V_{0}(r_{\Xi N})$, Eq.
(4), between a $p$-shell $\Xi$ hyperon and $p$-shell nucleons, expressed in
terms of its shell-model (SM) quadrupole-quadrupole residual interaction
${\cal V}_{\Xi N}$,

${\cal V}_{\Xi N}=F^{(2)}_{\Xi N}Q_{N}\cdot
Q_{\Xi},\,\,\,\,\,\,Q_{B}=\sqrt{\frac{4\pi}{5}}Y_{2}({\hat{r}}_{B}),$ (5)

where $F^{(2)}$ is the corresponding Slater integral [41]. A representative
value of $F^{(2)}_{\Xi N}=-3$ MeV is used here, smaller than the value
$F^{(2)}_{\Lambda N}=-3.7$ MeV established empirically for $p$-shell $\Lambda$
hypernuclei [42], in accordance with a $\Xi N$ strong interaction somewhat
weaker than the $\Lambda N$ strong interaction (see next section). A single
${}^{3}D_{1}$ 14Ng.s. SM component providing a good approximation to the full
SM intermediate-coupling g.s. wavefunction [43] was assumed in the present
evaluation.

Fig. 2 exhibits a triplet of 14N${}_{\rm g.s.}+1p_{\Xi^{-}}$ levels, spread
over more than 1 MeV. The least bound triplet state, with ${\cal
L}^{\pi}=0^{-}$, is shifted upward by 0.84 MeV from the $(2{\cal L}+1)$
averaged position at $-2.08\pm 0.28$ MeV to $E(0^{-})=-1.24\pm 0.28$ MeV. This
is consistent with the averaged position $\bar{E}=-1.15\pm 0.20$ MeV of the
three $\Xi^{-}\,{}^{14}$Ng.s. capture events listed in Table 1. We are not
aware of any good reason why capture has not been seen from the other two
states with ${\cal L}^{\pi}=1^{-},2^{-}$. This may change when more events are
collected at the next stage of the ongoing J-PARC E07 emulsion experiment.

Table 2: Quadrupole-quadrupole contributions to the energies $E({\cal L}^{\pi})$ of the 14N${}_{\rm g.s.}+1p_{\Xi^{-}}$ triplet of states shown in Fig. 2 with respect to $E(^{14}$N${}_{\rm g.s.})$, using $F^{(2)}_{\Xi N}=-3$ MeV, and spin contributions to the splittings $\Delta E({\cal L})=E(J={\cal L}+\frac{1}{2})-E(J={\cal L}-\frac{1}{2})$ of the ${\cal L}\neq 0$ states. $A_{ls}$ and $A_{ss}$ are spin-orbit ($l_{\Xi}=1,s_{\Xi}=\frac{1}{2}$) and spin-spin ($s_{N}=s_{\Xi}=\frac{1}{2}$) energy splittings, respectively, see text. Interaction | $E(0^{-})$ | $E(1^{-})$ | $E(2^{-})$ | $\Delta E(1^{-})$ | $\Delta E(2^{-})$
---|---|---|---|---|---
$Q_{N}\cdot Q_{\Xi}$ | $-\frac{7}{25}\,F^{(2)}_{\Xi N}$ | $\frac{7}{50}\,F^{(2)}_{\Xi N}$ | $-\frac{7}{250}\,F^{(2)}_{\Xi N}$ | – | –
$l_{\Xi}\cdot s_{\Xi}$ | – | – | – | $\frac{1}{2}\,A_{ls}$ | $\frac{5}{6}\,A_{ls}$
$s_{N}\cdot s_{\Xi}$ | – | – | – | $\frac{3}{8}\,A_{ss}$ | $\frac{5}{8}\,A_{ss}$

Introducing Pauli spin, $s_{\Xi^{-}}=\frac{1}{2}$, the total angular momentum
of the uppermost level marked $0^{-}$ in Fig. 2 becomes
$J^{\pi}={\frac{1}{2}}^{-}$, but its position is unaffected by spin-orbit and
spin-spin interactions. Each of the other two levels in Fig. 2 splits into a
doublet $J={\cal L}\pm\frac{1}{2}$ whose $(2J+1)$-average remains in the
unsplit position. Estimated splittings are listed in Table 2 in terms of two
constituent spin matrix elements: $A_{ls}\lesssim 300$ keV [39] for the
$l_{\Xi}\cdot s_{\Xi}$ spin-orbit splitting
$E_{1p_{\Xi}}({\frac{3}{2}}^{-})-E_{1p_{\Xi}}({\frac{1}{2}}^{-})$, and
$A_{ss}\approx 400\pm 80$ keV for the $s_{N}\cdot s_{\Xi}$ spin-spin splitting
$E(S_{\Xi N}=0)-E(S_{\Xi N}=1)$ for $p$-shell nucleon and $\Xi$ hyperon. For
estimating $A_{ss}$ we used the HAL-QCD [5] volume integral of $V_{\sigma}$,
Eq. (4), relative to that of $V_{0}$, thereby generating about 20% systematic
uncertainty. Incorporating these spin splittings into the ${\cal
L}^{\pi}=(0^{-},1^{-},2^{-})$ triplet in Fig. 2 keeps the ${\cal
L}^{\pi}=0^{-}$ state of interest, which has become
$J^{\pi}={\frac{1}{2}}^{-}$, well separated by at least 0.5 MeV from the rest
of the split states.

##  5 Density dependence, $\Xi$ nuclear potential depth and $1s_{\Xi^{-}}$
states

So far we have discussed a density independent $t$-matrix element $b_{0}$ in
$V_{\rm opt}$, Eq. (3), to fit the $\Xi^{-}$ capture events in 12C from Table
1. The resulting value Re$\,b_{0}=0.32\pm 0.01$ fm implies, in the limit
$A\to\infty$ and $\rho(r)\to\rho_{0}$=0.17 fm-3, a value $V_{\Xi}=24.3\pm 0.8$
MeV in nuclear matter, in accordance with the extraction of $V_{\Xi}$ from old
emulsion events [9] but exceeding considerably other values reviewed in the
Introduction. To explore how robust this conclusion is, we introduce the next
to leading-order density dependence of $V_{\rm opt}$, replacing Re$\,b_{k}$
($k$=0,1) in Eq. (3) by

${\rm Re}\,b_{k}(\rho)=\frac{{\rm Re}\,b_{k}}{1+\frac{3k_{F}}{2\pi}{\rm
Re}\,b_{0}^{\rm
lab}},\,\,\,\,\,\,\,\,\,\,k_{F}=(3{\pi}^{2}\rho/2)^{\frac{1}{3}},$ (6)

where $k_{F}$ is the Fermi momentum corresponding to nuclear density $\rho$
and $b_{0}^{\rm lab}=(1+\frac{m_{\Xi^{-}}}{m_{N}})b_{0}$ is the lab
transformed form of the c.m. scattering amplitude $b_{0}$. Eq. (6) accounts
for Pauli exclusion correlations in $\Xi N$ in-medium multiple scatterings
[26, 44]. Variants of the form (6) have been used in kaonic atoms [45] and
mesic nuclei [46, 47, 48] calculations. Shorter-range correlations,
disregarded here, were shown in Ref. [46] to contribute less than $\sim$30% of
the long-range Pauli correlation term. Applying Eq. (6) in the present
context, $B_{\Xi^{-}}^{1p}(^{12}$C)=0.82 MeV is refitted by Re$\,b_{0}$=0.527
fm, lowering the former value $B_{\Xi^{-}}^{1p}(^{14}$N)=2.08 MeV to 1.95 MeV
without any substantive change in the conclusions drawn above regarding the
five two-body capture events deciphered here. The nuclear-matter $\Xi$-nuclear
potential depth $V_{\Xi}$ decreases from 24.3$\pm$0.8 to 21.9$\pm$0.7 MeV, a
decrease of merely 10%, with additional systematic uncertainty of less than 1
MeV. This value of $V_{\Xi}$ is sufficient to bind $1s_{\Xi^{-}}$ states in
$p$-shell nuclei, with systematic uncertainty of less than 0.5 MeV, as
demonstrated in Table 3 which shows a steady decrease of $B_{\Xi^{-}}^{1s}$
and $\Gamma_{\Xi^{-}}^{1s}$ down to 4He. The increased
$\Gamma_{\Xi^{-}}^{1s}(^{4}$He) reflects a denser 4He medium. However,
expecting corrections of order ${\cal O}(1/A)$ to the optical potential
methodology, our 4He result should be taken with a grain of salt. It is worth
noting that all listed $1s_{\Xi^{-}}$ g.s. levels remain bound also when the
attractive finite-size Coulomb interaction $V_{c}$ is switched off. None of
such $1s_{\Xi^{-}}$ states have been observed conclusively in dedicated
experiments.

Table 3: Binding energies $B_{\Xi^{-}}^{1s}$ and widths $\Gamma_{\Xi^{-}}^{1s}$ (in MeV) in core nuclei AZ($J_{c}$), g.s. spin $J_{c}$, obtained by solving Eq. (2) with $b_{0}=0.527+{\rm i}\,0.010$ fm and $b_{1}=-0.225$ fm in $V_{\rm opt}$, Eqs. (3),(6). A finite-size Coulomb interaction $V_{c}$ is included. |  14N(1) |  12C(0) |  11B($\frac{3}{2}$) |  10B(3) |  6Li(1) |  4He(0)
---|---|---|---|---|---|---
$B_{\Xi^{-}}^{1s}$ | 11.5 | 9.8 | 8.4 | 7.6 | 2.1 | 2.0
$\Gamma_{\Xi^{-}}^{1s}$ | 1.02 | 0.93 | 0.89 | 0.77 | 0.26 | 0.45

The $T=\frac{1}{2}$ 11B nucleus, the only $T\neq 0$ nucleus listed in Table 3,
requires in addition to the isoscalar parameter $b_{0}=0.527+{\rm i}\,0.010$
fm also a knowledge of the isovector parameter $b_{1}$. Here we used the HAL-
QCD [5] volume integral of $V_{\tau}(r_{\Xi N})$ relative to that of
$V_{0}(r_{\Xi N})$, Eq. (4), to estimate $b_{1}$ relative to $b_{0}$, thereby
deriving a value $b_{1}=-0.225$ fm.222Note that Im$\,b_{1}$=0 because the
charge exchange $\Xi^{-}$+11B$\to$$\Xi^{0}$+11Be is kinematically blocked. The
resulting value $B_{\Xi^{-}}^{1s}(^{11}$B) listed in the table is lower by 530
keV than obtained disregarding $b_{1}$. Next, we introduce
$s_{\Xi^{-}}$=$\frac{1}{2}$ Pauli spin, splitting each of the listed
$1s_{\Xi^{-}}$ levels in $J_{c}\neq 0$ core nuclei into two sub levels
$J=J_{c}\pm\frac{1}{2}$. Using HAL-QCD [5] ratios of volume integrals of
$V_{\sigma}(r_{\Xi N})$ and $V_{\sigma\tau}(r_{\Xi N})$ to that of
$V_{0}(r_{\Xi N})$, Eq. (4), as done above for $V_{\tau}$, we estimate the
$1s_{\Xi^{-}}$ spin splittings to be well below 1 MeV. Other potential sources
of $\Xi^{-}$ spin splittings that are relevant in $\Lambda$ hypernuclei, such
as tensor or induced nuclear spin-orbit terms, are likely to be considerably
weaker than evaluated in Ref. [49] and are disregarded here. Of particular
interest is the (11B${}_{\rm g.s.}+1s_{\Xi^{-}}$) $J^{\pi}=1^{-}$ doublet
member expected to be formed in the 12C($K^{-},K^{+}$) production reaction
when the outgoing $K^{+}$ meson is detected in the forward direction. Our
estimates place it about 0.5 MeV deeper than the listed $(2J+1)$-averaged
$B_{\Xi^{-}}^{1s}(^{11}$B)=8.4 MeV, contrasting statements, e.g. [8], that
adopt $B_{\Xi^{-}}^{1s}(^{11}$B)$\sim$5 MeV from the BNL AGS-E885
12C($K^{-},K^{+}$) experiment [13]. In fact, the E885 poor resolution prevents
making any such conclusive statement.

##  6 Conclusion

We have shown that all five light nuclear emulsion events identified in KEK
and J-PARC $K^{-}$ exposure experiments as two-body $\Xi^{-}$ capture in 12C
and 14N into twin $\Lambda$ hypernuclei correspond to capture from
$1p_{\Xi^{-}}$ Coulomb-assisted bound states. This involved using just one
common strength parameter of a density dependent optical potential. Long-range
$\Xi N$ shell-model correlations were essential in making the 14N events
consistent with the 12C events. Earlier attempts to explain these data
overlooked this point, therefore reaching quite different conclusions [50, 51,
52, 53, 54]. Predicted then are $1s_{\Xi^{-}}$ bound states with
$B^{1s}_{\Xi^{-}}\sim 10$ MeV in 12C and somewhat larger in 14N, deeper by 4–5
MeV than the $1s_{\Xi^{-}}$ states claimed by a recent J-PARC E07 report of
multibody capture events [24]. The $\Xi$ nuclear-matter potential depth
derived here within an optical potential methodology, $V_{\Xi}=21.9\pm 0.7$
MeV, is considerably larger than $G$-matrix values below 10 MeV derived from
recent LQCD and EFT $\Xi N$ potentials [27, 29]. A systematic optical-
potential model uncertainty of less than 1 MeV as discussed in Sect. 5 is
short of bridging the gap noted above. Substantial $\Xi NN$ three-body
attractive contributions to the $\Xi$ nuclear potential depth would be
required to bridge this gap. Intuitively one expects repulsive $BNN$ three-
body contributions for octet baryons $B$, e.g. Ref. [31], but in chiral EFT
studies, focusing on decuplet-$B^{\ast}$ intermediate $B^{\ast}NN$ and
$BN\Delta$ configurations, this has been proven so far only for $B=\Lambda$
[55, 56].

To check the procedure practised in Sect. 5 for fitting $V_{\Xi}$ to just one
$\Xi^{-}$-12C bound state datum, we apply it to the $1s_{\Lambda}$ binding
energy in 12C, $B_{\Lambda}^{1s}=11.69\pm 0.12$ MeV [57]. The fitted strength
$b_{0}=0.866\pm 0.010$ fm amounts to a nuclear-matter $\Lambda$ potential
depth $V_{\Lambda}=31.7\pm 0.2$ MeV, in good agreement with the accepted value
$V_{\Lambda}\approx 30$ MeV [7]. One may slightly improve the derived value of
$V_{\Lambda}$ by subtracting from $B_{\Lambda}^{1s}$ a nuclear induced spin-
orbit contribution that vanishes in the limit $A\to\infty$, thereby reducing
our input $B_{\Lambda}^{1s}$ to 10.85 MeV [49]. This gives $b_{0}=0.798\pm
0.010$ fm and $V_{\Lambda}=30.5\pm 0.2$ MeV. Here too it is not possible to
separate the contribution of $\Lambda NN$ three-body potential terms from that
of the main $\Lambda N$ two-body potential term.

A strong $\Xi$-nuclear interaction, such as derived here, may have far-
reaching implications to ($N,\Lambda,\Xi$) strange hadronic matter [10] and
particularly to dense neutron star matter [11]. In the latter case a strong
$\Xi$-nuclear interaction might cause a faster depletion of $\Lambda$ hyperons
by $\Lambda\Lambda\to\Xi^{-}p$, a process inverse to the $\Xi^{-}$ capture
reaction considered in the present work. More work is necessary in this
direction.

## Acknowledgments

Recent related correspondence with Johan Haidenbauer, Jiří Mareš, John
Millener, Tomofumi Nagae, Josef Pochodzalla and Tom Rijken is gratefully
acknowledged. The present work is part of a project funded by the European
Union’s Horizon 2020 research & innovation programme, grant agreement 824093.

## References

  * [1] S. Acharya, et al. (ALICE Collaboration), Phys. Rev. C 99 (2019) 024001.
  * [2] S. Acharya, et al. (ALICE Collaboration), Phys. Lett. B 797 (2019) 134822.
  * [3] S. Acharya, et al. (ALICE Collaboration), Phys. Rev. Lett. 123 (2019) 112002.
  * [4] S. Acharya, et al. (ALICE Collaboration), Nature 588 (2020) 232\.
  * [5] K. Sasaki, et al. (HAL QCD Collaboration), Nucl. Phys. A 998 (2020) 121737.
  * [6] L. Tolos, L. Fabbietti, Prog. Part. Nucl. Phys. 112 (2020) 103770\.
  * [7] A. Gal, E.V. Hungerford, D.J. Millener, Rev. Mod. Phys. 88 (2016) 035004.
  * [8] E. Hiyama, K. Nakazawa, Annu. Rev. Nucl. Part. Sci. 68 (2018) 131\.
  * [9] C.B. Dover, A. Gal, Ann. Phys. 146 (1983) 309.
  * [10] J. Schaffner-Bielich, A. Gal, Phys. Rev. C 62 (2000) 034311, and references cited therein to earlier work.
  * [11] S. Weissenborn, D. Chatterjee, J. Schaffner-Bielich, Nucl. Phys. A 881 (2012) 62.
  * [12] T. Fukuda, et al. (E224 Collaboration), Phys. Rev. C 58 (1998) 1306\.
  * [13] P. Khaustov, et al. (The AGS E885 Collaboration), Phys. Rev. C 61 (2000) 054603.
  * [14] T. Harada, Y. Hirabayashi, Phys. Rev. C 103 (2021) 024605.
  * [15] J.K. Ahn, et al., Phys. Rev. Lett. 87 (2001) 132504.
  * [16] T. Nagae, et al., AIP Conf. Proc. 2130 (2019) 020015.
  * [17] J. Pochodzalla, et al. (for the PANDA Collaboration), JPS Conf. Proc. 17 (2017) 091002.
  * [18] C.Z. Yuan, M. Karliner, Phys. Rev. Lett. 127 (2021) 012003.
  * [19] C.J. Batty, E. Friedman, A. Gal, Phys. Rev. C 59 (1999) 295.
  * [20] D. Zhu, C.B. Dover, A. Gal, M. May, Phys. Rev. Lett. 67 (1991) 2268.
  * [21] S. Aoki, et al. (KEK E176 Collaboration), Nucl. Phys. A 828 (2009) 191, and earlier E176 publications cited therein.
  * [22] H. Nakazawa, et al. (KEK PS-E373), Prog. Theor. Exp. Phys. 2015 (2015) 033D02.
  * [23] S.H. Hayakawa, et al. (J-PARC E07 Collaboration), Phys. Rev. Lett. 126 (2021) 062501.
  * [24] M. Yoshimoto, et al. (J-PARC E07), Prog. Theor. Exp. Phys. 2021 (2021) 073D02.
  * [25] T. Koike, JPS Conf. Proc. 17 (2017) 033011.
  * [26] C.B. Dover, J. Hüfner, R.H. Lemmer, Ann. Phys. 66 (1971) 248\.
  * [27] T. Inoue (for HAL QCD Collaboration), AIP Conf. Proc. 2130 (2019) 020002.
  * [28] J. Haidenbauer, U.-G. Meißner, Eur. Phys. J. A 55 (2019) 23\.
  * [29] M. Kohno, Phys. Rev. C 100 (2019) 024313.
  * [30] T. Gaitanos, A. Chorozidou, Nucl. Phys. A 1008 (2021) 122153\.
  * [31] M.M. Nagels, Th.A. Rijken, Y. Yamamoto, Phys. Rev. C 102 (2020) 054003; see in particular Tables XXI, XXIV for models ESC16*(A,B) respectively.
  * [32] E. Friedman, A. Gal, Phys. Rep. 452 (2007) 89.
  * [33] L.R.B. Elton, Nuclear Sizes, (Oxford Univ. Press, Oxford, 1961).
  * [34] I. Angeli, K.P. Marinova, At. Data Nucl. Data Tables 99 (2013) 69\.
  * [35] C.W. de Jager, H. de Vries, C. de Vries, At. Data Nucl. Data Tables 14 (1974) 479.
  * [36] H. de Vries, C.W. de Jager, C. de Vries, At. Data Nucl. Data Tables 36 (1987) 495.
  * [37] A. Gal, E. Friedman, C.J. Batty, Nucl. Phys. A 606 (1996) 283\.
  * [38] S. Ajimura, et al., Phys. Rev. Lett. 86 (2001) 4255.
  * [39] J. Mareš, B.K. Jennings, Phys. Rev. C 49 (1994) 2472.
  * [40] E.H. Auerbach, A.J. Baltz, C.B. Dover, A. Gal, S.H. Kahana, L. Ludeking, D.J. Millener, Phys. Rev. Lett. 47 (1981) 1110, Ann. Phys. 148 (1983) 381 where values of $-F^{(2)}_{\Lambda N}=3.0-3.4$ MeV were proposed.
  * [41] A. de-Shalit, I. Talmi, Nuclear Shell Theory (AP, New York, 1963).
  * [42] R.H. Dalitz, A, Gal, Ann. Phys. 131 (1981) 314.
  * [43] D.J. Millener, in Topics in Strangeness Nuclear Physics, Edited by P. Bydžovský, A. Gal, and J. Mareš, Lecture Notes in Physics 724 (Springer, Heidelberg, 2007) 31-79.
  * [44] T. Waas, M. Rho, W. Weise, Nucl. Phys. A 617 (1997) 449.
  * [45] E. Friedman, A. Gal, Nucl. Phys. A 899 (2013) 60, Nucl. Phys. A 959 (2017) 66.
  * [46] W. Weise, R. Härtle, Nucl. Phys. A 804 (2008) 173.
  * [47] E. Friedman, A. Gal, J. Mareš, Phys. Lett. B 725 (2013) 334\.
  * [48] A. Cieplý, E. Friedman, A. Gal, J. Mareš, Nucl. Phys. A 925 (2014) 126.
  * [49] D.J. Millener, Nucl. Phys. A 881 (2012) 298.
  * [50] M. Yamaguchi, K. Tominaga, Y. Yamamoto, T. Ueda, Prog. Theor. Phys. 105 (2001) 627.
  * [51] E. Hiyama, Y. Yamamoto, H. Sagawa, Phys. Scr. 91 (2016) 093001\.
  * [52] T.T. Sun, E. Hiyama, H. Sagawa, H.-J. Schulze, J. Meng, Phys. Rev. C 94 (2016) 064319.
  * [53] J. Hu, H. Shen, Phys. Rev. C 96 (2017) 054304.
  * [54] Y. Jin, X.-R. Zhou. Y.-Y. Cheng, H.-J. Schulze, Eur. Phys. J. A 56 (2020) 135.
  * [55] S. Petschauer, J. Haidenbauer, N. Kaiser, U.-G. Meißner, W. Weise, Nucl. Phys. A 957 (2017) 347.
  * [56] D. Gerstung, N. Kaiser, W. Weise, Eur. Phys. J. A 56 (2020) 175\.
  * [57] D.H. Davis, Nucl. Phys. A 754 (2005) 3c.


# A Concise Tiling Strategy for Preserving
Spatial Context in Earth Observation Imagery

Ellianna Abrahams, Tasha Snow, Matthew R.
Department of Statistics
University of California, Berkeley
Berkeley, CA 94720, USA
{ellianna<EMAIL_ADDRESS>
& Siegfried, & Fernando Pérez
Department of Geophysics
Colorado School of Mines
Golden, CO 80401, USA
<EMAIL_ADDRESS>
###### Abstract

We propose a new tiling strategy, Flip-n-Slide, which has been developed for
specific use with large Earth observation satellite images when the location
of objects-of-interest (OoI) is unknown and spatial context can be necessary
for class disambiguation. Flip-n-Slide is a concise and minimalistic approach
that allows OoI to be represented at multiple tile positions and orientations.
This strategy introduces multiple views of spatio-contextual information,
without introducing redundancies into the training set. By maintaining
distinct transformation permutations for each tile overlap, we enhance the
generalizability of the training set without misrepresenting the true data
distribution. Our experiments validate the effectiveness of Flip-n-Slide in
the task of semantic segmentation, a necessary data product in geophysical
studies. We find that Flip-n-Slide outperforms the previous state-of-the-art
augmentation routines for tiled data in all evaluation metrics. For
underrepresented classes, Flip-n-Slide increases precision by as much as
15.8%.

##  1 Introduction

The volume of geospatial satellite imagery has grown rapidly in the past
decade. Semantic segmentation presents a promising opportunity for rapidly
parsing meaningful scientific understanding from these images. Despite the
remarkable accomplishments of deep neural networks for such segmentation tasks
(Ronneberger et al., 2015; Chen et al., 2019; Tan et al., 2020; Amara et al.,
2022), these methods can underperform on data that have noisy or
underrepresented labels (Shin et al., 2011; Guo et al., 2019) or when one set
of data representations is used for a wider set of downstream tasks (Yang et
al., 2018). These are common challenges in Earth observation imagery. To
overcome these issues, data augmentation is a widely adopted technique for
generalizing a model fit to make better predictions by expanding the size and
distribution of training data through a set of transformations (Van Dyk &
Meng, 2001; Hestness et al., 2017). In recent years, much focus has been given
to upstream augmentation methods that address overfitting through data mixing
(Zhang et al., 2017; Yun et al., 2019; Hong et al., 2021; Dabouei et al.,
2021) or proxy-free augmentations (Cubuk et al., 2019; 2020; Reed et al.,
2021; Li & Li, 2023)—strategic approaches that expand the training data, but
also execute unrealistic data transformations. Furthermore, limited attention
has been given to investigating upstream augmentation techniques in the realm
of learning on tiled imagery, an approach often employed to parse large images
into smaller tiles to overcome the intractable size of the overall image for
the GPU memory (Pinckaers & Litjens, 2018; Huang et al., 2019).

Figure 1: Flip-n-Slide’s tile overlap strategy creates eight overlapping tiles
for any image region more than a 75% tile threshold away from the overall
image edge. Three tiling strategies, shown in false color to illustrate
overlap, are visualized here. a) Tiles do not overlap. b) The conventional
tile overlap strategy, shown at the recommended 50% overlap. c) Flip-n-Slide
includes more tile overlaps, capturing more OoI tile position views for the
training set.

Tiling techniques in scientific applications require intentional augmentation
choices, as certain transformations are unphysical and therefore not useful to
learn. Spatial context is needed to disambiguate between classes with similar
channel outputs and surface textures (Pereira & dos Santos, 2021). This is
particularly true in scientific use-cases where tasks like classification are
employed to answer wider questions about rare or relatively unknown phenomena.
Although most machine learning methods demonstrate robust performance with
well-represented phenomena, this performance degrades when training data fails
to accurately capture poorly represented features or the structure of the
problem. Human experts are better able to identify rare and unknown phenomena
by relying on domain context, such as the semantic or temporal proximity of
other classes (Wang & Zhu, 2023); it stands to reason that machines could
similarly use context for similar purposes.

To maintain access to context after tiling, the current state-of-the art
employs tiling as a sliding window data augmentation, overlapping tiles by a
significant percentage to extend dataset size and expose the model to multiple
contextual views (Ronneberger et al., 2015; Ünel et al., 2019; Zeng & Zheng,
2019; Akyon et al., 2022) at training time, at test time, or during both.
However, when applied upstream, this approach leads to redundancies within the
training set, as many pixel windows are repeated more than once; for objects-
of-interest (OoI) that are the size of the window slide, spatial context is
limited beyond the singular sliding parameter (Zeng & Zheng, 2019; Reina et
al., 2020). This leads to the question: How can we best tile and augment large
images with limited physically realistic transformations without losing
important spatio-contextual information in the tiling process? Further: Can we
achieve this without creating the redundancies that occur in simply
overlapping tiles?

To answer this question, we propose a concise augmentation strategy, Flip-n-
Slide, built for use with large Earth observation images where: 1) tiling is
necessary; 2) data transformations must be limited to rotations and
reflections to be realistic; and 3) there is no prior knowledge of the pixel
locations for which spatial context will be necessary. We argue that
physically realistic transformations of the data can be implemented alongside
the tiling overlap process, thereby removing redundancies when training
convolutional neural networks (CNNs), in which orientation matters for
learning (Ghosh et al., 2018; Szeliski, 2022). Our strategy allows us to
create a larger set of samples without the superfluity of simply overlapping
the tiles, enhancing downstream generalization. To achieve our goal, we slide
through multiple overlap thresholds of the tiling window, exposing the model
to more contextual views of each location (Figure 1), and distinctly permute
any overlapping windows to eliminate redundancy with other tiles that share
pixels. We run segmentation experiments using our routine on the Land Cover of
Canada (LCC) dataset (Latifovic, 2020) to classify Landsat 8 satellite imagery
(Earth Resources Observation And Science Center(2013), EROS; Roy et al.,
2014). Our findings demonstrate that our strategy improves the segmentation
performance of tiled data, especially for underrepresented classes, when
compared to the conventional method (Figure 2c).

##  2 Tiling and Augmentation Method

We present the Flip-n-Slide algorithm, a data preprocessing strategy that
tiles a large input image using a sliding window, providing eight sliding
overlaps for every tile. Redundancies on overlap are eliminated using
distinct, physically realistic transformations to permute each overlapping
tile.

###  2.1 Mathematical Formalism for the Flip-n-Slide Algorithm

Flip-n-Slide follows a concise mathematical formulation to concurrently tile
and transform a large image. In Algorithm 1 we outline the formal process for
implementing Flip-n-Slide on an input image, $I$. The algorithm is implemented
in two stages. First a sliding window captures overlapping tiles, $t\in T$, at
specified boundary ranges, $i\times j$, in the 2D image plane of $I$. The
boundary ranges are given in $S$. This operation is specified in Line 8. The
tile boundaries are defined such that any given $x\times y$ pixel location
sufficiently away from the edge of the overall image $I$ overlaps with eight
tiles. The length of $i$ is equal to the length of $j$, creating a square.
Each of these overlapping tiles is permuted correspondingly from a specified
set of eight distinct transformations, $F$. This operation is specified in
Line 9. As augmentation is implemented concurrently with tiling, Flip-n-Slide
must be implemented before the data streaming stage. This results in the
constraint of CPU storage capacity required to accommodate the complete
extended dataset. The overlap and transformation choices are explained in
greater detail in Section 2.2. We release this algorithm as a versioned,
documented, open-source Python package named flipnslide, available on PyPI and
GitHub.

###  2.2 Hyperparameter Selection for Tiling and Transformation

We justify the fixed parameter choices for Flip-n-Slide to meet the goal of
preserving spatial context for OoI, while eliminating informational
redundancies that have been introduced in past tiling strategies. Our approach
has the added benefit of expanding the initial training distribution with
physically realistic transformations, adding further downstream generalization
to the model fit.

Tile Size: Similar to past methods (Ronneberger et al., 2015; Ünel et al.,
2019), Flip-n-Slide employs a fixed tile size to enable ease-of-scale when
processing petapixel datasets. The algorithmic method is independent of tile
size as needs will vary depending on the size of OoI, allowing users to pick a
size that is most appropriate for a specific use-case. However, we limit the
algorithm to square tiles to make the data more efficient for use with CNNs.

Tile Overlap : Previous methods employ a single overlap threshold, generally
recommended to be 50%. Flip-n-Slide uses a strategy with multiple overlap
thresholds, where a tile window slides along to capture a 0%, 25%, 50%, and
75% threshold on both spatial axes, leading to eight overlaps for any given
$x\times y$ area that is more than a 75% tile threshold away from the edge of
the original large image. This method provides the model with multiple tile
views of contextual information for OoI, leading to greater downstream tile-
position-invariance. Additionally, this method reduces the overhead present in
dynamic methods by removing the need for an upstream check that small objects
are not sliced during augmentation; in Flip-n-Slide sliced objects that are
smaller than the tile size will have unsliced representations in other
overlapping tiles.

Transformation Permutations : Rotations and reflections provide alternative
views of spatial relationships to the convolutional kernel in CNNs (Van Dyk &
Meng, 2001). For each of the eight tile overlaps, Flip-n-Slide employs a
distinct rotation and/or reflection transformation from the following set of
permutations
$\\{0\degree,90\degree,180\degree,270\degree,(0\degree,\rightarrow),(0\degree,\uparrow),(90\degree,\rightarrow),(90\degree,\uparrow)\\}$,
shown in Figure 4. This set avoids commutative and indistinct transformation
compositions to reduce redundancies for tiles that overlap in some way on the
same initial image position. Additionally, it is limited to physically
realistic transformations of the original image bypassing any pixel-level
effects that can be introduced when rotating square matrices to angles
indivisible by $90\degree$.

##  3 Experimental Setup

We evaluate our strategy’s performance on a classification task with a
benchmarked semantic segmentation approach and compare it with the current
recommended tiling convention as a control. Across all experiments, we argue
that our augmentation strategy exposes the model to enough semantic context to
remove the need for augmentation and label averaging at training time. To test
this hypothesis, we only tile the test image and do not add any further
augmentations at test time.

Figure 2: In Earth observation, classes can be extremely imbalanced as is
shown here for Ellesmere Island, Nunavut, CA. a) Labels from the Land Cover of
Canada dataset (Latifovic, 2020) overlain on the corresponding Landsat
satellite imagery (Earth Resources Observation And Science Center(2013),
EROS). The legend shows the relative class distribution. Image tiles showing
(b) an over-represented class (snow and ice) and (c) an under-represented
class (lichen and moss) with the binary class masks from the ground truth data
set, segmentation after training using the Flip-n-Slide tiling algorithm, and
segmentation after training using the conventional tiling algorithm (50%
overlap). Although both algorithms perform well for the over-represented class
case, Flip-n-Slide is more precise, by up to 15.8%, than the conventional
strategy (50% tile overlap).

We perform our experiment on data from Ellesmere Island in the Canadian Arctic
to ensure that our tiling strategy performs well on datasets in wilderness
regions, which are an understudied focus of inquiry in established machine
learning datasets, (Figure 2a). We use land cover classifications from the
publicly available Land Cover of Canada (LCC) dataset (Latifovic, 2020) and
the corresponding images from NASA’s Landsat 8 imagery (Earth Resources
Observation And Science Center(2013), EROS; Roy et al., 2014) since the LCC
was derived from Landsat images. The data are classified by seven labeled
classes (Snow & Ice [39.36% of the overall dataset], Barren Rock [33.57%],
Water [17.98%], Polar Moss & Lichen [4.97%], Polar Grassland [4.23%], Urban
Development [0.0003%], Wetland [0.0001%]). There is high class imbalance in
this data as is common in geophysical land cover segmentation problems, where
often the OoI are among the least represented classes.

We evaluate the performance of Flip-n-Slide on a semantic segmentation task,
generating 12,800 256$\times$256 tiles from the dataset. All training
experiments are conducted using the benchmarked UNet architecture (Ronneberger
et al., 2015), employed with the ADAM optimizer (Kingma & Ba, 2017) at a
learning rate of 0.001 and an unweighted cross-entropy loss. We train
additional models on the two control tiling strategies to test the comparative
performance, employing the same model architecture and optimization. The
models undergo 300 training epochs with a batch size of 32.

##  4 Experimental Results

We follow the standard evaluation setup for semantic segmentation tasks using
a UNet (Ronneberger et al., 2015). We train a first model on tiles generated
by employing the Flip-n-Slide strategy. As a comparative control, the second
model is trained on non-overlapping tiles, and the third is trained on tiles
generated using the conventional 50% overlap strategy (Ünel et al., 2019; Zeng
& Zheng, 2019; Reina et al., 2020; Akyon et al., 2022). We test all three
trained models on three random draws of non-overlapping tiles from the same
geographic location, and share the performance results, averaged across the
tests, in Table 1. In each experiment, Flip-n-Slide improves the model
performance across all evaluation metrics when compared to conventional
methods. These results position Flip-n-Slide as an effective augmentation
strategy for tiling large scientific images, particularly in use-cases with
high class-imbalance.

###  4.1 Performance Improvement for Underrepresented Classes

Incorporating the Flip-n-Slide strategy enhances model performance on
underrepresented classes, even without tailoring the model architecture or
loss function to addressing class imbalance (Table 1). Excluding the two
classes in this investigation that exhibit extreme underrepresentation
($<$0.0004% of the overall data) and are therefore subject to model
noise111Flip-n-Slide also outperforms other tiling methods on the two
extremely underrepresented classes, but the performance improvement is within
error, so we exclude those results here., the next two smallest classes each
make up less than 5% of the overall dataset. Flip-n-Slide improves prediction
precision by 13%, on average, in underrepresented classes, and outperforms
other approaches in every metric tested. These results highlight the improved
performance for underrepresented classes, with no changes to architecture,
loss function, optimization, or any other model parameters.

Predicted masks for a withheld test set are shown for two classes in Figure 2.
Although all tiling methods succeed at predicting the boundaries for a well-
represented class (Snow & Ice, 39.36% of the data), only the model trained on
Flip-n-Slide tiling reasonably estimates the underrepresented class (Lichen &
Moss, 4.97%). Flip-n-Slide achieves this without altering the underlying class
distribution while providing the model with more contextual views of all
classes. In CNNs, classes can be distinguished by their channel spectrum or
surface texture, but spatial context is also important for separating between
classes (Wang & Zhu, 2023). This is especially true in scientific use-cases
where the morphometry of the scene is dictated by physical processes and can
aid in class separation for rare classes. Our results confirm that the Flip-n-
Slide method sufficiently generalizes a model fit to include a fuller
understanding of the training distribution, especially for underrepresented
classes.

Performance Comparison Between Strategies |  Performance On Underrepresented Class
---|---
| Evaluated on All Classes | Moss & Lichen (4.97% of Data)
Method | IoU | Precision | F1 Score | mAP | IoU | Precision | F1 Score
No Tile Overlap | 70.7% | 71.7% | 82.8% | 71.3 | 42.7% | 44.4% | 59.9%
50% Overlap | 85.4% | 87.1% % | 92.1% | 85.5% | 58.3% | 62.6% | 73.6%
Flip-n-Slide Ours | 87.6% | 90.1% | 93.4% | 87.8% | 69.7% | 78.4% | 82.2%
Table 1: Performance results for three tiling strategies across all class
predictions. Flip-n-Slide outperforms other strategies in all metrics. Flip-n-
Slide also improves performance results for underrepresented classes even with
a basic loss function and model architecture.

##  5 Conclusions

In this paper, we addressed a problem with earlier tiling augmentation
strategies—namely that their overlap strategy caused data redundancies, which
ultimately reduce a model’s ability to generalize effectively. However, tile
overlapping is necessary to preserve spatial context, which is important in
segmentation tasks for Earth observation use-cases. To solve this, we argued
that tile overlap when combined with distinct permutations of the data may not
only eliminate this redundancy but also expose the model to an expanded
training dataset with more spatio-contextual views for OoI. To maintain the
fidelity of this context, we emphasized the necessity of realistic
augmentation choices for use in scientific segmentation tasks. We introduced a
new preprocessing strategy, Flip-n-Slide, built for tiling large Earth
observation images. Flip-n-Slide maintains physically realistic
transformations of the input data and does not degrade the spatio-contextual
information available for small OoI in the overall training set. We show that
Flip-n-Slide outperforms the previous benchmarked approach at scale for tiled
augmentations in all evaluation metrics, especially in cases of class
imbalance, improving the detection of rare phenomena even in large imagery.

Acknowledgements The majority of this work was conducted on the unceded
territories of the xučyun, the ancestral land of the Chochenyo speaking
Muwekma Ohlone people. We have benefited, and continue to benefit, from the
use of this land. We recognize the importance of taking actions in support of
American Indian and Indigenous peoples who are living, flourishing members of
our communities today. Testing and development of this paper was done on the
CryoCloud cloud-based JupyterHub (Snow et al., 2023) that is funded by the
NASA Transform to Open Science Program and ICESat-2 Science Team (grant
numbers 80NSSC23K0002 and 80NSSC22K1877), and on the Jupyter Meets the Earth
(JMTE) cloud hub, an NSF EarthCube funded project (grant numbers 1928406 and
1928374). EA gratefully acknowledges support from a Two Sigma PhD Fellowship.
We also gratefully acknowledge funding support from the NASA Cryospheric
Science Program (grant number 80NSSC22K0385).

## References

  * Akyon et al. (2022) Fatih Cagatay Akyon, Sinan Onur Altinuc, and Alptekin Temizel.  Slicing Aided Hyper Inference and Fine-Tuning for Small Object Detection.  In _2022 IEEE International Conference on Image Processing (ICIP)_ , pp. 966–970, October 2022.  doi: 10.1109/ICIP46576.2022.9897990.
  * Amara et al. (2022) Kahina Amara, Ali Aouf, Hoceine Kennouche, A. Oualid Djekoune, Nadia Zenati, Oussama Kerdjidj, and Farid Ferguene.  COVIR: A virtual rendering of a novel NN architecture O-Net for COVID-19 Ct-scan automatic lung lesions segmentation.  _Comput Graph_ , 104:11–23, May 2022.  ISSN 0097-8493.  doi: 10.1016/j.cag.2022.03.003.
  * Chen et al. (2019) Xinlei Chen, Ross Girshick, Kaiming He, and Piotr Dollar.  TensorMask: A Foundation for Dense Object Segmentation.  In _2019 IEEE/CVF International Conference on Computer Vision (ICCV)_ , pp. 2061–2069, October 2019.  doi: 10.1109/ICCV.2019.00215.
  * Cubuk et al. (2019) Ekin D. Cubuk, Barret Zoph, Dandelion Mané, Vijay Vasudevan, and Quoc V. Le.  AutoAugment: Learning Augmentation Strategies From Data.  In _2019 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR)_ , pp. 113–123, June 2019.  doi: 10.1109/CVPR.2019.00020.
  * Cubuk et al. (2020) Ekin Dogus Cubuk, Barret Zoph, Jonathon Shlens, and Quoc Le.  RandAugment: Practical Automated Data Augmentation with a Reduced Search Space.  In Hugo Larochelle, Marc’Aurelio Ranzato, Raia Hadsell, Maria-Florina Balcan, and Hsuan-Tien Lin (eds.), _Advances in Neural Information Processing Systems 33: Annual Conference on Neural Information Processing Systems 2020, NeurIPS 2020, December 6-12, 2020, Virtual_ , 2020.
  * Dabouei et al. (2021) Ali Dabouei, Sobhan Soleymani, Fariborz Taherkhani, and Nasser M. Nasrabadi.  SuperMix: Supervising the Mixing Data Augmentation.  In _2021 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR)_ , pp. 13789–13798, June 2021.  doi: 10.1109/CVPR46437.2021.01358.
  * Earth Resources Observation And Science Center(2013) (EROS) Earth Resources Observation And Science (EROS) Center.  Collection-2 Landsat 8-9 OLI (Operational Land Imager) and TIRS (Thermal Infrared Sensor) Level-1 Data Products, 2013.
  * Ghosh et al. (2018) Arthita Ghosh, Max Ehrlich, Sohil Shah, Larry Davis, and Rama Chellappa.  Stacked U-Nets for Ground Material Segmentation in Remote Sensing Imagery.  In _2018 IEEE/CVF Conference on Computer Vision and Pattern Recognition Workshops (CVPRW)_ , pp. 252–2524, June 2018.  doi: 10.1109/CVPRW.2018.00047.
  * Guo et al. (2019) Sheng Guo, Weilin Huang, Xiao Zhang, Prasanna Srikhanta, Yin Cui, Yuan Li, Hartwig Adam, Matthew R. Scott, and Serge Belongie.  The iMaterialist Fashion Attribute Dataset.  In _2019 IEEE/CVF International Conference on Computer Vision Workshop (ICCVW)_ , pp. 3113–3116, October 2019.  doi: 10.1109/ICCVW.2019.00377.
  * Hestness et al. (2017) Joel Hestness, Sharan Narang, Newsha Ardalani, Gregory Diamos, Heewoo Jun, Hassan Kianinejad, Md Mostofa Ali Patwary, Yang Yang, and Yanqi Zhou.  Deep Learning Scaling is Predictable, Empirically.  _arXiv_ , arXiv:1712.00409, December 2017.  doi: 10.48550/arXiv.1712.00409.
  * Hong et al. (2021) Minui Hong, Jinwoo Choi, and Gunhee Kim.  StyleMix: Separating Content and Style for Enhanced Data Augmentation.  In _2021 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR)_ , pp. 14857–14865, June 2021.  doi: 10.1109/CVPR46437.2021.01462.
  * Huang et al. (2019) Bohao Huang, Daniel Reichman, Leslie M. Collins, Kyle Bradbury, and Jordan M. Malof.  Tiling and Stitching Segmentation Output for Remote Sensing: Basic Challenges and Recommendations.  _arXiv_ , arXiv:1805.12219, February 2019.  doi: 10.48550/arXiv.1805.12219.
  * Kingma & Ba (2017) Diederik P. Kingma and Jimmy Ba.  Adam: A Method for Stochastic Optimization.  _arXiv_ , arXiv:1412.6980, January 2017.  doi: 10.48550/arXiv.1412.6980.
  * Latifovic (2020) Rasim Latifovic.  2020 Land Cover of Canada - Open Government Portal.  https://open.canada.ca/data/en/dataset/ee1580ab-a23d-4f86-a09b-79763677eb47, 2020.
  * Li & Li (2023) Lujun Li and Anggeng Li.  A2-Aug: Adaptive Automated Data Augmentation.  In _2023 IEEE/CVF Conference on Computer Vision and Pattern Recognition Workshops (CVPRW)_ , pp. 2267–2274, June 2023.  doi: 10.1109/CVPRW59228.2023.00221.
  * Pereira & dos Santos (2021) Matheus Barros Pereira and Jefersson Alex dos Santos.  ChessMix: Spatial Context Data Augmentation for Remote Sensing Semantic Segmentation.  In _2021 34th SIBGRAPI Conference on Graphics, Patterns and Images (SIBGRAPI)_ , pp. 278–285, October 2021.  doi: 10.1109/SIBGRAPI54419.2021.00045.
  * Pinckaers & Litjens (2018) Hans Pinckaers and Geert Litjens.  Training convolutional neural networks with megapixel images.  _arXiv_ , arXiv:1804.05712, April 2018.  doi: 10.48550/arXiv.1804.05712.
  * Reed et al. (2021) Colorado J Reed, Sean Metzger, Aravind Srinivas, Trevor Darrell, and Kurt Keutzer.  SelfAugment: Automatic Augmentation Policies for Self-Supervised Learning.  In _2021 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR)_ , pp. 2673–2682, June 2021.  doi: 10.1109/CVPR46437.2021.00270.
  * Reina et al. (2020) G. Anthony Reina, Ravi Panchumarthy, Siddhesh Pravin Thakur, Alexei Bastidas, and Spyridon Bakas.  Systematic Evaluation of Image Tiling Adverse Effects on Deep Learning Semantic Segmentation.  _Front Neurosci_ , 14:65, February 2020.  ISSN 1662-4548.  doi: 10.3389/fnins.2020.00065.
  * Ronneberger et al. (2015) Olaf Ronneberger, Philipp Fischer, and Thomas Brox.  U-Net: Convolutional Networks for Biomedical Image Segmentation.  In Nassir Navab, Joachim Hornegger, William M. Wells, and Alejandro F. Frangi (eds.), _Medical Image Computing and Computer-Assisted Intervention – MICCAI 2015_ , Lecture Notes in Computer Science, pp. 234–241, Cham, 2015. Springer International Publishing.  ISBN 978-3-319-24574-4.  doi: 10.1007/978-3-319-24574-4˙28.
  * Roy et al. (2014) D. P. Roy, M. A. Wulder, T. R. Loveland, Woodcock C.e., R. G. Allen, M. C. Anderson, D. Helder, J. R. Irons, D. M. Johnson, R. Kennedy, T. A. Scambos, C. B. Schaaf, J. R. Schott, Y. Sheng, E. F. Vermote, A. S. Belward, R. Bindschadler, W. B. Cohen, F. Gao, J. D. Hipple, P. Hostert, J. Huntington, C. O. Justice, A. Kilic, V. Kovalskyy, Z. P. Lee, L. Lymburner, J. G. Masek, J. McCorkel, Y. Shuai, R. Trezza, J. Vogelmann, R. H. Wynne, and Z. Zhu.  Landsat-8: Science and product vision for terrestrial global change research.  _Remote Sensing of Environment_ , 145:154–172, April 2014.  ISSN 0034-4257.  doi: 10.1016/j.rse.2014.02.001.
  * Shin et al. (2011) Hoo-Chang Shin, Matthew Orton, David J. Collins, Simon Doran, and Martin O. Leach.  Autoencoder in Time-Series Analysis for Unsupervised Tissues Characterisation in a Large Unlabelled Medical Image Dataset.  In _2011 10th International Conference on Machine Learning and Applications and Workshops_ , volume 1, pp. 259–264, December 2011.  doi: 10.1109/ICMLA.2011.38.
  * Snow et al. (2023) Tasha Snow, Joanna Millstein, Jessica Scheick, Wilson Sauthoff, Wei Ji Leong, James Colliander, Fernando Pérez, James Munroe, Denis Felikson, Tyler Sutterley, and Matthew Siegfried.  CryoCloud JupyterBook.  Zenodo, January 2023.
  * Szeliski (2022) Richard Szeliski.  _Computer Vision: Algorithms and Applications_.  Texts in Computer Science. Springer International Publishing, Cham, 2022.  ISBN 978-3-030-34371-2 978-3-030-34372-9.  doi: 10.1007/978-3-030-34372-9.
  * Tan et al. (2020) Mingxing Tan, Ruoming Pang, and Quoc V. Le.  EfficientDet: Scalable and Efficient Object Detection.  In _2020 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR)_ , pp. 10778–10787, June 2020.  doi: 10.1109/CVPR42600.2020.01079.
  * Ünel et al. (2019) F. Özge Ünel, Burak O. Özkalayci, and Cevahir Çiğla.  The Power of Tiling for Small Object Detection.  In _2019 IEEE/CVF Conference on Computer Vision and Pattern Recognition Workshops (CVPRW)_ , pp. 582–591, June 2019.  doi: 10.1109/CVPRW.2019.00084.
  * Van Dyk & Meng (2001) David A Van Dyk and Xiao-Li Meng.  The Art of Data Augmentation.  _Journal of Computational and Graphical Statistics_ , 10(1):1–50, March 2001.  ISSN 1061-8600, 1537-2715.  doi: 10.1198/10618600152418584.
  * Wang & Zhu (2023) Xuan Wang and Zhigang Zhu.  Context understanding in computer vision: A survey.  _Computer Vision and Image Understanding_ , 229:103646, March 2023.  ISSN 1077-3142.  doi: 10.1016/j.cviu.2023.103646.
  * Yang et al. (2018) Zhengyuan Yang, Yixuan Zhang, Jerry Yu, Junjie Cai, and Jiebo Luo.  End-to-end Multi-Modal Multi-Task Vehicle Control for Self-Driving Cars with Visual Perceptions.  In _2018 24th International Conference on Pattern Recognition (ICPR)_ , pp. 2289–2294, August 2018.  doi: 10.1109/ICPR.2018.8546189.
  * Yun et al. (2019) Sangdoo Yun, Dongyoon Han, Sanghyuk Chun, Seong Joon Oh, Youngjoon Yoo, and Junsuk Choe.  CutMix: Regularization Strategy to Train Strong Classifiers With Localizable Features.  In _2019 IEEE/CVF International Conference on Computer Vision (ICCV)_ , pp. 6022–6031, October 2019.  doi: 10.1109/ICCV.2019.00612.
  * Zeng & Zheng (2019) Guodong Zeng and Guoyan Zheng.  Holistic decomposition convolution for effective semantic segmentation of medical volume images.  _Medical Image Analysis_ , 57:149–164, October 2019.  ISSN 1361-8415.  doi: 10.1016/j.media.2019.07.003.
  * Zhang et al. (2017) Hongyi Zhang, Moustapha Cissé, Yann N. Dauphin, and David Lopez-Paz.  Mixup: Beyond Empirical Risk Minimization.  _CoRR_ , abs/1710.09412, 2017.

##  Appendix A Appendix

###  A.1 The Flip-n-Slide Algorithm

Algorithm 1 The Flip-n-Slide Algorithm.

1:Inputs: $I$: Large Image ($x\times y$)

2: $S$: Set of sliding tile windows

3: $(i\times j)\in(x\times y)$

4: $F$: Set of corresponding transformations

5: $(i\times j)\in(x\times y)$

6:Output: $T$: Set of augmented image tiles, $t\in T$

7:for each ($s_{i\times j}\in S$) $\underset{i\times j}{\cap}$ ($f_{i\times
j}\in F$) do:

8: $T_{t}=s_{ii\times j}(I)$

9: $T_{t}=f_{i\times j}$($s_{i\times j}$)

10:end for

11:return $T$

###  A.2 Computational Cost Analysis

In contrast to the previously accepted convention of overlap tiling, Flip-n-
slide simultaneously augments and tiles the input image at the preprocessing
stage, introducing a larger computational cost at this phase of the machine
learning pipeline. The timing differences between the three tiling approaches
used in this paper are detailed in Table 2. To find the run time, we run the
$10240\times 10240$ pixel image that we used for our segmentation task seven
times through each of the three tiling strategies and take the average of
these runs. We find that in both overlap approaches the $256\times 256$ tiles
are most efficient.

Although the Flip-n-Slide approach incurs a higher computational cost upfront,
it streamlines downstream processes and enhances model performance during
inference, particularly for underrepresented classes (Figure 2). By
integrating data augmentation at the tiling stage of the pipeline, Flip-n-
Slide also eliminates the need for additional augmentation at train time,
thereby reducing computational costs at the training stage. Furthermore, many
approaches that employ the previous overlap convention at training time
recommend using the same overlapping tiling approach for prediction averaging
at each pixel during test time Ünel et al. (2019); Zeng & Zheng (2019); Reina
et al. (2020); Akyon et al. (2022). However, this necessitates tracking pixel
locations from tile to tile and complicates the process of reconstructing test
tiles back into the overall input image, incurring further computational
overhead at the inference stage of the pipeline. Therefore, despite the
increased computational cost associated with the Flip-n-Slide strategy during
preprocessing, its performance improvements for underrepresented classes
justify the upstream computational investment. Depending on the choices that a
user makes throughout the rest of the pipeline, both at train and test time,
these initial costs may result in reduced overhead later on.

Tiling Runtime for $10240\times 10240$ Input Image
---
Method |  $64\times 64$ Tiles |  $128\times 128$ Tiles |  $256\times 256$ Tiles |  $512\times 512$ Tiles
No Overlap | 4.9s | 4.3s | 4.2s | 4.0s
50% Overlap | 16.4s | 14.9s | 14.9s | 16.3s
Flip-n-Slide | 71s | 63s | 59.3s | 61s
Table 2: Computational cost for simultaneously tiling and augmenting a large
input image using the Flip-n-Slide approach. Although Flip-n-Slide has an
increased upstream cost, its performance improvements for underrepresented
classes justify the upstream computational investment and these initial costs
may result in reduced overhead later on.

###  A.3 Ablation Studies

Figure 3: Ablation studies on varying tile size confirm that our strategy
leads to expected model behavior. Here we show mAP for each study, averaged
over three test runs with the standard deviation shown in grey. Flip-n-Slide
outperforms the conventional 50% overlap strategy even at smaller input tile
sizes.

Ablation studies are necessary to disambiguate between the role of various
fixed decisions on the model outcomes. We explore the impacts of varying tile
size choice on the model performance. Previous studies show that larger tiles
are more effective for segmentation tasks under traditional augmentation
strategies Zeng & Zheng (2019); Reina et al. (2020). We test the performance
of the Flip-n-Slide strategy across a range of two additional tile sizes:
$64\times 64$, $128\times 128$. Due to GPU size limitations we could not test
tiles of $512\times 512$ at the same batch size, so we do not include it here.
We reproduce the results of previous studies, finding that larger tile size
does lead to better performance. However, we find that even at smaller tile
sizes, our approach outperform previous strategies even when they are
implemented at larger sizes. When $128\times 128$ tile sizes are generated
with Flip-n-Slide they perform at a better score to $256\times 256$ tiles
generate by a 50% overlap strategy (Figure 3).

Figure 4: To minimize redundancy in the Flip-n-Slide strategy, each
overlapping tile is uniquely permuted with a distinct, physically-realistic
transformation, as shown here. Previous strategies have not employed overlap-
specific transformations; any augmentations have been applied across all tiles
or at random. Tiles are shown in false color to illustrate overlapping areas.
Transparency indicates areas that do not overlap with the blue tile shown
here. They overlap with neighboring blue tiles instead.