EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-EP-2020-242 LHCb-PAPER-2020-040 2 March 2021

Measurement of $C\\!P$ violation

in the decay ${{B}^{+}}\\!\rightarrow{{K}^{+}}{{\pi}^{0}}$

LHCb collaboration†††Full author list given at the end of the Letter.

A measurement of direct $C\\!P$ violation in the decay
${{B}^{+}}\\!\rightarrow{{K}^{+}}{{\pi}^{0}}$ is reported using a data sample
corresponding to an integrated luminosity of 5.4$\text{\,fb}^{-1}$, collected
with the LHCb experiment at a center-of-mass energy of $\sqrt{s}=13$ TeV. The
$C\\!P$ asymmetry is measured to be $0.025\pm 0.015\pm 0.006\pm 0.003$, where
the uncertainties are statistical, systematic, and due to an external input,
respectively. This is the most precise measurement of the $C\\!P$ asymmetry in
the decay ${{B}^{+}}\\!\rightarrow{{K}^{+}}{{\pi}^{0}}$ and exceeds the
precision on the current world average. This direct $C\\!P$ asymmetry is a key
input to studies of a long-standing anomaly in $B$ meson decays, known as the
$K\pi$-puzzle. The asymmetry is consistent with the previous measurements of
this quantity, confirming and significantly enhancing the observed anomalous
difference between the direct $C\\!P$ asymmetries of the $B^{0}\rightarrow
K^{+}\pi^{-}$ and ${{B}^{+}}\\!\rightarrow{{K}^{+}}{{\pi}^{0}}$ decays.

Published in Phys. Rev. Lett. 126 (2021) 091802

© 2024 CERN for the benefit of the LHCb collaboration. CC BY 4.0 licence.

Rare decays of heavy flavored hadrons that primarily proceed through loop-
level transitions are powerful probes of the effects of new physics (NP)
beyond the Standard Model (SM). The family of ${B}\\!\rightarrow{K}{\pi}$
decays is dominated by hadronic loop amplitudes in the SM, but include
contributions from suppressed tree-level processes, as well as electroweak
loop-level processes through which NP may affect the decay [1, 2, 3, 4].
Studies of the decay $B^{0}\rightarrow K^{+}\pi^{-}$ at the $B$-factory
experiments led to the first observation of direct $C\\!P$ violating
asymmetries in the $B$ system [5, 6] resulting from the interference of two
decay amplitudes where both the relative strong and weak phases are nonzero.
The observed asymmetries in these modes are a result of the interference
between tree- and loop-level amplitudes. Further studies at the $B$-factory
and Tevatron experiments, and at LHCb have provided measurements of the
branching fractions and $C\\!P$ asymmetries of the complete set of
$B\rightarrow K\pi$ decays: $B^{0}\rightarrow K^{+}\pi^{-}$ [7, 8, 9, 10],
$B^{+}\rightarrow K^{+}\pi^{0}$ [11, 8], $B^{0}\rightarrow K^{0}\pi^{0}$ [12,
13] and $B^{+}\rightarrow K^{0}\pi^{+}$ [14, 8, 15], where the inclusion of
charge-conjugated processes is implied throughout this Letter, except where
asymmetries are discussed. The amplitudes in the SM are expected to obey
relations imposed by isospin symmetry [1, 2, 3, 4, 16, 17, 18, 19, 20, 21].
However, measurements have revealed inconsistencies with this expectation. The
largest observed discrepancy is between the measured direct $C\\!P$
asymmetries of the decays ${{B}^{0}}\\!\rightarrow{{K}^{+}}{{\pi}^{-}}$ and
${{{B}^{+}}}\\!\rightarrow{{K}^{+}}{{\pi}^{0}}$. The difference between
$A_{{C\\!P}}(B^{0}\rightarrow K^{+}\pi^{-})=-0.084\pm 0.004$ and
$A_{{C\\!P}}(B^{+}\rightarrow K^{+}\pi^{0})=0.040\pm 0.021$ is nonzero at 5.5
standard deviations ($\sigma$), whereas equal asymmetries are expected based
on isospin arguments. A more accurate examination of this anomaly, known as
the $K$ $\pi$ puzzle, is through the sum rule

$A_{{C\\!P}}({{K}^{+}}{{\pi}^{-}})+A_{{C\\!P}}({{K}^{0}}{{\pi}^{+}})\frac{{\mathcal{B}}({{K}^{0}}{{\pi}^{+}})}{{\mathcal{B}}({{K}^{+}}{{\pi}^{-}})}\frac{\tau_{0}}{\tau_{+}}=A_{{C\\!P}}({{K}^{+}}{{\pi}^{0}})\frac{2{\mathcal{B}}({{K}^{+}}{{\pi}^{0}})}{{\mathcal{B}}({{K}^{+}}{{\pi}^{-}})}\frac{\tau_{0}}{\tau_{+}}+A_{{C\\!P}}({{K}^{0}}{{\pi}^{0}})\frac{2{\mathcal{B}}({{K}^{0}}{{\pi}^{0}})}{{\mathcal{B}}({{K}^{+}}{{\pi}^{-}})}$
(1)

proposed in Ref. [19], where $A_{CP}({K}\pi)$ and ${\mathcal{B}}({K}\pi)$ are
the $C\\!P$ asymmetries and the branching fractions of the
$B\rightarrow{K}{\pi}$ decays and $\tau_{0}/\tau_{+}$ is the ratio of the
${B}^{0}$ and ${{B}^{+}}$ lifetimes. This sum rule predicts a nonzero direct
asymmetry of $A_{{C\\!P}}({{B}^{0}}\rightarrow{{K}^{0}}{{\pi}^{0}})=-0.150\pm
0.032$ using current world averages for the other quantities [22]. The current
measurement of this quantity is $0.01\pm 0.10$[22]. The $K$ $\pi$ puzzle has
been the subject of significant theoretical attention, which includes more
complete examination of the SM predictions as well as potential NP sources of
the discrepancies [1, 2, 3, 4, 16, 18, 19, 20, 21].

This Letter presents a measurement of direct $C\\!P$ asymmetry in the decay
$B^{+}\rightarrow K^{+}\pi^{0}$,

$A_{{C\\!P}}=\frac{\Gamma({{{B}^{-}}}\rightarrow{{K}^{-}}{{\pi}^{0}})-\Gamma({{{B}^{+}}}\rightarrow{{K}^{+}}{{\pi}^{0}})}{\Gamma({{{B}^{-}}}\rightarrow{{K}^{-}}{{\pi}^{0}})+\Gamma({{{B}^{+}}}\rightarrow{{K}^{+}}{{\pi}^{0}})}$
(2)

where $\Gamma(B^{\pm}\rightarrow K^{\pm}{{\pi}^{0}})$ refers to the rate of
$B^{\pm}\rightarrow K^{\pm}{{\pi}^{0}}$ decays, using data recorded with the
LHCb detector at the CERN Large Hadron Collider. The data sample corresponds
to an integrated luminosity of 5.4$\text{\,fb}^{-1}$ collected at a center-of-
mass energy of $13\text{\,Te\kern-1.00006ptV}$ between 2016 and 2018.

The LHCb detector [23, 24] is a single-arm forward spectrometer covering the
pseudorapidity range $2<\eta<5$. The detector includes a high-precision
tracking system consisting of a silicon-strip vertex detector surrounding the
$pp$ interaction region, a large-area silicon-strip detector located upstream
of a dipole magnet with a bending power of about $4{\rm\,Tm}$, and three
stations of silicon-strip detectors and straw drift tubes placed downstream of
the magnet. The tracking system provides a measurement of momentum, $p$, of
charged particles with a relative uncertainty that varies from 0.5% at low
momentum to 1.0% at 200$\text{\,Ge\kern-1.00006ptV\\!/}c$. The minimum
distance of a track to a primary $pp$ collision vertex (PV), the impact
parameter (IP), is measured with a resolution of
$(15+29/p_{\mathrm{T}})\,\upmu\text{m}$, where $p_{\mathrm{T}}$ is the
component of $p$ transverse to the beam, in
$\text{\,Ge\kern-1.00006ptV\\!/}c$. Different types of charged hadrons are
distinguished using information from two ring-imaging Cherenkov detectors
(RICH). Photons, electrons and hadrons are identified by a calorimeter system
consisting of scintillating pad and preshower detectors, an electromagnetic
and a hadronic calorimeter. Charged and neutral clusters in the
electromagnetic calorimeter (ECAL) are separated by extrapolating the tracks
reconstructed by the tracking system to the calorimeter plane, while photons
and neutral pions are distinguished by cluster shape and energy distributions.

Simulated events are used to model the effects of the detector acceptance and
the imposed selection requirements. In the simulation, $pp$ collisions are
generated using Pythia [25, *Sjostrand:2007gs] with a specific LHCb
configuration [27]. Decays of unstable particles are described by EvtGen [28],
in which final-state radiation is generated using Photos [29]. The interaction
of the generated particles with the detector and its response are implemented
using the Geant4 toolkit [30, *Agostinelli:2002hh] as described in Ref. [32].

The decay topology $B^{+}\rightarrow h^{+}\pi^{0}$, where $h^{+}$ is a charged
hadron, presents a unique challenge in the proton-proton collision environment
of the LHC. These decays comprise a single charged track and lack a
reconstructible displaced vertex, a signature typically used to identify the
decays of $b$ hadrons. The candidate selection for
${{B}^{+}}\\!\rightarrow{{K}^{+}}{{\pi}^{0}}$ candidates instead relies on
identifying a charged kaon that is inconsistent with originating from any PV
but consistent with originating from the $B$-meson trajectory. That trajectory
is determined by adding the momenta of the ${K}^{+}$ and ${\pi}^{0}$
candidates, where the ${\pi}^{0}$ momentum is defined as pointing from the
LHCb interaction point to the coordinate of the energy deposited by the
${\pi}^{0}$ candidate in the calorimeter.

The LHCb trigger system [33] consists of a hardware stage, based on
information from the calorimeter and muon systems, followed by a software
stage, in which a full event reconstruction is applied. Events are required to
pass a hardware trigger that selects a neutral pion or photon with a high
transverse energy based on energy deposits in the calorimeter. Because of the
limited ECAL position resolution a significant fraction of high
$p_{\mathrm{T}}$ ${{\pi}^{0}}\rightarrow\gamma\gamma$ decays have their
photons merged into a single cluster. Only this ${\pi}^{0}$ category is used
in this analysis, as neutral pions with the photons resolved suffer from a
large background of randomly combined clusters. Further selection relies on a
dedicated software trigger developed for this analysis [34]. A
${{B}^{+}}\\!\rightarrow{{K}^{+}}{{\pi}^{0}}$ candidate is formed by adding
the four-momenta of the neutral pion and a charged track identified as a kaon
using information from the RICH detectors. The charged kaon is required to
have $p>12\text{\,Ge\kern-1.00006ptV\\!/}c$,
$p_{\mathrm{T}}>1.2\text{\,Ge\kern-1.00006ptV\\!/}c$, and a significant IP
with respect to any PV. The neutral pion is required to have
$p_{\mathrm{T}}>3.5\text{\,Ge\kern-1.00006ptV\\!/}c$ and the scalar sum of the
${K}^{+}$ and ${\pi}^{0}$ $p_{\mathrm{T}}$ must exceed
$6.5\text{\,Ge\kern-1.00006ptV\\!/}c$. The $B^{+}$ candidate is required to
have a ${{K}^{+}}{{\pi}^{0}}$ invariant mass in the range $4\leq
m({{K}^{+}}{{\pi}^{0}})\leq 6.2\text{\,Ge\kern-1.00006ptV\\!/}c^{2}$, and
$p_{\mathrm{T}}>5\text{\,Ge\kern-1.00006ptV\\!/}c$. Finally, the $B^{+}$
candidate trajectory is obtained by fixing its momentum vector to the PV with
the smallest kaon IP. The significance of the distance of closest approach
between the ${K}^{+}$ candidate and this trajectory is denoted as
DOCA-$\chi^{2}$. In order to identify ${K}^{+}$ candidates consistent with
production via $B$-meson decay, the DOCA-$\chi^{2}$ is required to be small.
In the offline reconstruction, a stricter set of particle identification
requirements are applied to the ${K}^{+}$ candidates.

Further candidate selection is based on variables characterizing how well
isolated a candidate is from other tracks in the event. Vertex-isolation
variables are calculated by combining each track in the event with the
${K}^{+}$ candidate individually to form a two-track secondary vertex. Three
related variables are calculated: the smallest $\chi^{2}$ of the vertex fit
between the ${K}^{+}$ and any other track, the smallest change in $\chi^{2}$
when one more track is added to that vertex, and the multiplicity of vertices
having small $\chi^{2}$. The isolation of the candidate is also measured with
the $p_{\mathrm{T}}$ asymmetry

${\cal
A}(p_{\mathrm{T}})=\frac{p_{{T}_{{B}}}-p_{{T}_{\text{cone}}}}{p_{{T}_{{B}}}+p_{{T}_{\text{cone}}}},$
(3)

comparing the transverse momentum of the ${B}^{+}$ candidate ($p_{{T}_{{B}}}$)
to a scalar sum of additional charged particles nearby
($p_{{T}_{\text{cone}}}$). Particles are considered in a cone around the
reconstructed ${B}^{+}$ trajectory with a radius in the $\eta$–$\phi$ plane of
$\Delta R\equiv\sqrt{(\Delta\phi)^{2}+(\Delta\eta)^{2}}=1.7$, where
$\Delta\phi$ is the difference in radians between the azimuthal angles of the
momentum of the reconstructed ${B}^{+}$ candidate and the track, and
$\Delta\eta$ is the difference between their pseudorapidities. To ensure that
the simulated distributions of these variables for the signal decays are
consistent with data, candidate weights are generated from the ratios of these
distributions between simulated ${{B}^{0}}\\!\rightarrow{{K}^{+}}{{\pi}^{-}}$
decays and a control sample of ${{B}^{0}}\\!\rightarrow{{K}^{+}}{{\pi}^{-}}$
candidates selected in the same data set. A gradient boosted reweighter (GBR)
[35] technique is used to determine the weights, which are subsequently
applied to the $B^{+}\rightarrow K^{+}\pi^{0}$ simulations.

The final candidate selection is performed using boosted decision tree (BDT)
classifiers with the isolation variables, the DOCA-$\chi^{2}$, the smallest
change in $\chi^{2}$ of the PV when including the ${K}^{+}$ track in the
vertex fit, the $p_{\mathrm{T}}$ of the ${B}^{+}$ and the ${K}^{+}$
candidates, and the momentum of the ${\pi}^{0}$ candidate as inputs. These
variables are chosen to provide discriminatory power between signal and
background without biasing the $m({{K}^{+}}{{\pi}^{0}})$ distribution.

Two pairs of BDTs are trained and tested using data to represent background
and simulated ${{B}^{+}}\\!\rightarrow{{K}^{+}}{{\pi}^{0}}$ decays, corrected
as described above, to represent signal. One pair of BDTs is trained on
background data with candidate invariant mass
$m({{K}^{+}}{{\pi}^{0}})<4860\text{\,Me\kern-1.00006ptV\\!/}c^{2}$, which is
dominated by partially reconstructed $b$-hadron decays. Another pair of BDTs
is trained on background data with
$m({{K}^{+}}{{\pi}^{0}})>5700\text{\,Me\kern-1.00006ptV\\!/}c^{2}$, which are
primarily random ${K}^{+}$ ${\pi}^{0}$ combinations (combinatorial
background). In each of these categories, the data sample is split randomly, a
BDT classifier is trained and tested on each half, and then used to assign a
score to the candidates in the other half [36, 37]. This avoids biases due to
artifacts in the training samples, while taking advantage of the full set of
data available. The optimal requirements on the classifier response variables
are found for the data set by maximizing $\epsilon/\sqrt{N}$, where $\epsilon$
is the signal selection efficiency, evaluated on simulated events, and $N$ is
the total number of candidates observed in a region of approximately 3 times
the observed ${{B}^{+}}\\!\rightarrow{{K}^{+}}{{\pi}^{0}}$ resolution around
the expected ${B}^{+}$ mass.

Kaon candidates with $p_{\mathrm{T}}>17\text{\,Ge\kern-1.00006ptV\\!/}c$ or
$p>250\text{\,Ge\kern-1.00006ptV\\!/}c$ are removed from the sample after BDT
selection because of insufficient coverage in the
$B^{+}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}K^{+}$ control sample
described below. They account for only $3\%$ of the candidates after final
selection.

The $m({{K}^{+}}{{\pi}^{0}})$ distribution of the selected
${{B}^{+}}\\!\rightarrow{{K}^{+}}{{\pi}^{0}}$ candidates, separated by the
charge of the $B$ meson, is shown in Fig. 1 along with the results of a fit to
the data. In the fit, the signal is modeled by the sum of a Crystal Ball
function [38] and a Gaussian function with an exponential tail describing the
high-mass region. The Crystal Ball and the Gaussian functions share a common
mean and width that varies freely in the fit, and their tail shape parameters
are fixed from simulation. Combinatorial background is modeled by an
exponential function, with the exponent parameter allowed to vary freely in
the fit. The tail of a Gaussian function is used to model the partially
reconstructed background in the low-mass region, with mean and width allowed
to vary freely in the fit. The rate of ${{\pi}^{+}}\rightarrow{{K}^{+}}$
misidentification is measured in $D^{0}\rightarrow K^{-}\pi^{+}$ decays as a
function of pion momentum and pseudorapidity, with the same particle
identification requirements as signal events [39]. The contribution of the
misidentified $B^{+}\rightarrow\pi^{+}\pi^{0}$ background is inferred from its
branching fraction [40] and the misidentification rate to be $2.4\%$ of the
${{B}^{+}}\\!\rightarrow{{K}^{+}}{{\pi}^{0}}$ yield. The
$B^{+}\rightarrow\pi^{+}\pi^{0}$ background component is modeled by a Gaussian
with mean and resolution fixed to values determined from simulated events and
a yield fixed to the expectation. There is assumed to be no asymmetry in this
background.

An additional class of background candidates arises from decays such as
$B^{+}\rightarrow(K^{*+}\rightarrow K^{+}\pi^{0})\pi^{0}$,
$B^{0}\rightarrow(K^{*0}\rightarrow K^{+}\pi^{-})\pi^{0}$, and
$B^{0}\rightarrow K^{+}(\rho^{-}\rightarrow\pi^{-}\pi^{0})$ where a pion from
the $K^{*}$ or $\rho^{-}$ decay is not reconstructed. The polarization of the
$K^{*}$ or $\rho^{-}$ meson results in a double peaked
$m({{K}^{+}}{{\pi}^{0}})$ distribution, where the higher-mass peak is close to
the expected $B$ mass. This type of background is modeled with a parabolic
function convolved with a Gaussian resolution function following the method
described in Ref. [41]. The width of the resolution function is fixed to that
of the signal resolution and the end points are fixed to the kinematic end
points, allowing for a shift between the fitted and the known ${B}^{+}$
masses[40]. The lower-mass peak contributes below the
$m({{K}^{+}}{{\pi}^{0}})$ range considered, and so its relative height is
fixed to a value determined from simulation.

Other background sources include: $B^{+}\rightarrow(K^{*+}\rightarrow
K^{+}\pi^{0})\gamma$ decays where the $\gamma$ is misidentified as a
$\pi^{0}$; $B^{+}\rightarrow(f^{0}(980)\rightarrow\pi^{0}\pi^{0})K^{+}$ decays
where one $\pi^{0}$ is not reconstructed; and
$B^{0}\rightarrow(\bar{D}^{0}\rightarrow K^{+}\pi^{-})\pi^{0}$,
$B^{0}\rightarrow(K^{*}_{0}(1430)\rightarrow K^{+}\pi^{-}){{\pi}^{0}}$ and
$B^{0}_{s}\rightarrow K^{+}\pi^{0}\pi^{-}$ decays where the $\pi^{-}$ is not
reconstructed. Simulation studies have shown that these background
contributions either have $m({{K}^{+}}{{\pi}^{0}})$ distributions
indistinguishable from the partially reconstructed samples described by a
Gaussian tail, or in the case of $B^{0}_{s}\rightarrow K^{+}\pi^{0}\pi^{-}$
has a branching fraction too small to give an observable contribution.

The data are fitted separately in four categories. In order to reduce
uncertainties due to non-uniformity of the detector, candidates are separated
according to whether the LHCb dipole magnetic field is aligned vertically
upward (Magnet Up) or downward (Magnet Down) in the experiment. Candidates are
further separated by $B$-meson charge in order to measure the $C\\!P$
asymmetry. The yield and asymmetry of each fit component are allowed to vary
freely while the shape parameters are the same for the $B^{+}$ and $B^{-}$
candidates. The total yield of $B^{\pm}\rightarrow K^{\pm}{{\pi}^{0}}$ decays
is measured to be $8310\pm 255$ in the Magnet Up data set and $8373\pm 253$ in
the Magnet Down data set. The raw asymmetry, $A_{\textrm{raw}}$, between the
$B^{-}$ and $B^{+}$ signal yields is found to be $0.019\pm 0.021$ for Magnet
Down and $0.005\pm 0.022$ for Magnet Up. The results are consistent when
candidates are separated by data-taking year as well as when shape parameters
are allowed to vary independently for all four data categories.

Figure 1: Invariant-mass distribution of the selected candidates with fit
projections overlayed. The data set is divided by the charge of the $B$ meson,
with $B^{+}\rightarrow K^{+}\pi^{0}$ shown on the left and $B^{-}\rightarrow
K^{-}\pi^{0}$ on the right.

As the measured asymmetry receives contributions from a number of nuisance
asymmetries, the $C\\!P$ asymmetry can be expressed as

$A_{{C\\!P}}(\mbox{${{B}^{+}}\\!\rightarrow{{K}^{+}}{{\pi}^{0}}$})=A_{\textrm{raw}}(\mbox{${{B}^{+}}\\!\rightarrow{{K}^{+}}{{\pi}^{0}}$})-A^{B}_{\textrm{prod.}}-A^{K}_{\textrm{det.}},$
(4)

where $A^{B}_{\textrm{prod.}}$ is the production asymmetry of $B^{\pm}$ mesons
and $A^{K}_{\textrm{det.}}$ is the combined asymmetry in detection,
triggering, and reconstruction of $K^{\pm}$ mesons. These effects must be
corrected for in order to extract $A_{{C\\!P}}$ from $A_{\textrm{raw}}$. The
combined effect of the nuisance asymmetries is measured with a control sample
of $B^{+}\rightarrow(J/\psi\rightarrow\mu^{+}\mu^{-})K^{+}$ decays, using the
same data sample as the signal channel.

In the hardware trigger, events with a
$B^{+}\rightarrow({{J\mskip-3.0mu/\mskip-2.0mu\psi}}\rightarrow\mu^{+}\mu^{-})K^{+}$
decay are required to trigger on particles other than the kaon, in order to
avoid introducing additional trigger asymmetries. At the software stage the
event must trigger on the kaon in the same manner as signal events. The
offline selection requires that the $B$-meson lifetime be greater than 0.1 ps
and that the kaon and muons have a significant IP with respect to all PVs.
Additional requirements on the momentum of the kaon and $B$ candidates as well
as kaon particle identification are imposed to match the signal selection. The
momentum distributions of the $B^{+}$ and $K^{+}$ candidates are weighted to
match those of the signal candidates using the GBR technique [35], as the
production and detection asymmetries may depend on kinematics of the decay.

The raw asymmetry in the
$B^{+}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}K^{+}$ signal yields is
determined via an unbinned maximum-likelihood fit in which the invariant-mass
distribution of the $B^{+}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}K^{+}$
candidates is modeled by the sum of two Gaussian functions sharing a common
mean, while the combinatorial background is modeled by an exponential
distribution. The total yield of
$B^{+}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}K^{+}$ decays is measured
to be $372874\pm 776$ for Magnet Down and $306821\pm 699$ for Magnet Up data
samples with a purity of approximately 99%. The raw asymmetry is found to be
$-0.009\pm 0.002$ for Magnet Up, and $-0.012\pm 0.002$ for Magnet Down
samples. The $C\\!P$ asymmetry for the decay
${B}^{+}\rightarrow({{J\mskip-3.0mu/\mskip-2.0mu\psi}}\rightarrow\mu^{+}\mu^{-})K^{+}$
is taken to be
$A_{{C\\!P}}(B^{+}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}K^{+})=0.002\pm
0.003$ from Ref. [40]. After subtracting $A_{{C\\!P}}$, the remaining
asymmetry is attributed to the combination of production, detection,
reconstruction, and triggering effects, which can then be determined from

$A^{B}_{\textrm{prod.}}+A^{K}_{\textrm{det.}}=A_{\textrm{raw}}(B^{+}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}K^{+})-A_{{C\\!P}}(B^{+}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}K^{+}).$
(5)

This estimate of the nuisance asymmetry is then used in Eq. 4 to determine
$A_{{C\\!P}}({{{B}^{+}}}\rightarrow{{K}^{+}}{{\pi}^{0}})$.This is done
separately for the Magnet Up and Magnet Down data. By averaging the Magnet Up
and Magnet Down results, the direct $C\\!P$ asymmetry is determined to be
$A_{{C\\!P}}(\mbox{${{B}^{+}}\\!\rightarrow{{K}^{+}}{{\pi}^{0}}$})=0.025\pm
0.015$, where the uncertainty is statistical only.

Table 1: Systematic uncertainties on $A_{{C\\!P}}(\mbox{${{B}^{+}}\\!\rightarrow{{K}^{+}}{{\pi}^{0}}$})$. Systematic | Value ($\times 10^{-3}$)
---|---
Signal modeling shape | $4.3$
Combinatorial background shape | $1.3$
Partial reco. background shape | $1.3$
Peaking partial reco. background shape | $1.2$
Peaking partial reco. background offset | $1.3$
Peaking partial reco. background resolution | $1.4$
${{{B}^{+}}}\rightarrow{{\pi}^{+}}{{\pi}^{0}}$ yield | $1.3$
${{{B}^{+}}}\rightarrow{{\pi}^{+}}{{\pi}^{0}}$ $C\\!P$ asymmetry | $1.5$
Multiple candidates | $1.3$
Production/detection asymmetry stat. | $2.1$
Production/detection asymmetry weights | $0.5$
Sum in quadrature | $6.1$

To assess the systematic uncertainty due to mismodeling of the signal and
background line shapes, pseudoexperiments are generated for variations of the
$m({{K}^{+}}{{\pi}^{0}})$ fit model. The leading source of systematic
uncertainty is from modeling the signal component in the fit. This uncertainty
is assessed by replacing the default model with a single Gaussian
distribution. Systematic uncertainties are assessed for numerous fit
variations: replacing the exponential distribution for the combinatorial
background with a linear function, individually replacing each low-mass
background model with an Argus function, allowing the position and resolution
of the peaking low-mass background to vary freely and independently of the
signal distribution, and varying the yield and asymmetry of
$B^{+}\rightarrow\pi^{+}{{\pi}^{0}}$ background. Pseudoexperiments are also
generated to assess the systematic uncertainty due to including events with
multiple candidates in the base analysis.

The statistical uncertainty on the determination of the raw
$B^{+}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}K^{+}$ asymmetry is also
considered as a systematic uncertainty, and is the subdominant source of
systematic uncertainty. Additionally, the difference between the nuisance
asymmetries with and without applying the GBR weights is taken to be a
systematic uncertainty. The estimated values for all systematic uncertainties
are shown in Table 1, where the common value of $0.0013$ is from the
statistical uncertainty of the pseudoexperiments generated. The
$A_{{C\\!P}}(B^{+}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}K^{+})$
precision of 0.003 is considered separately as an external-input uncertainty.

In conclusion, the direct $C\\!P$ asymmetry of the decay
${{B}^{+}}\\!\rightarrow{{K}^{+}}{{\pi}^{0}}$ has been measured with the LHCb
detector using a data sample corresponding to a luminosity of 5.4
$\text{\,fb}^{-1}$. It is found to be

$A_{{C\\!P}}(\mbox{${{B}^{+}}\\!\rightarrow{{K}^{+}}{{\pi}^{0}}$})=0.025\pm
0.015\pm 0.006\pm 0.003,$

where the first uncertainty is statistical, the second is systematic and the
third due to external inputs, exceeding the precision of the current world
average [22]. This result is consistent with the world average and consistent
with zero at approximately $1.5\,\sigma$. The $C\\!P$ asymmetry difference,
$\Delta A_{{C\\!P}}({K}{\pi})\equiv
A_{{C\\!P}}(\mbox{${{B}^{+}}\\!\rightarrow{{K}^{+}}{{\pi}^{0}}$}{})-A_{{C\\!P}}({{B}^{0}}\\!\rightarrow{{K}^{+}}{{\pi}^{-}})$,
is found to be $0.108\pm 0.017$, where
$A_{{C\\!P}}({{B}^{0}}\\!\rightarrow{{K}^{+}}{{\pi}^{-}})$ is taken from Ref.
[22]111The world average includes the LHCb measurement, and a small
correlation between the LHCb measurements of
$A_{{C\\!P}}({{B}^{0}}\\!\rightarrow{{K}^{+}}{{\pi}^{-}})$ and
$A_{{C\\!P}}(\mbox{${{B}^{+}}\\!\rightarrow{{K}^{+}}{{\pi}^{0}}$})$ due to the
charged kaon detection asymmetry has been neglected. Including the result
presented in this Letter, the new world average of
$A_{{C\\!P}}(\mbox{${{B}^{+}}\\!\rightarrow{{K}^{+}}{{\pi}^{0}}$}{})$ is found
to be $0.031\pm 0.013$. This corresponds to $\Delta
A_{{C\\!P}}({K}{\pi})=0.115\pm 0.014$, which is nonzero with a significance of
more than 8 standard deviations, substantially enhanced over the results prior
to this measurement. The updated sum rule prediction for
$A_{{C\\!P}}({{B}^{0}}\rightarrow{{K}^{0}}{{\pi}^{0}})$, shown in Eq. 1, is
found to be $-0.138\pm 0.025$, departing from zero with a significance of
approximately $5.5\,\sigma$.

## Acknowledgements

We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at the LHCb institutes. We acknowledge support from CERN
and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); MOST
and NSFC (China); CNRS/IN2P3 (France); BMBF, DFG and MPG (Germany); INFN
(Italy); NWO (Netherlands); MNiSW and NCN (Poland); MEN/IFA (Romania); MSHE
(Russia); MICINN (Spain); SNSF and SER (Switzerland); NASU (Ukraine); STFC
(United Kingdom); DOE NP and NSF (USA). We acknowledge the computing resources
that are provided by CERN, IN2P3 (France), KIT and DESY (Germany), INFN
(Italy), SURF (Netherlands), PIC (Spain), GridPP (United Kingdom), RRCKI and
Yandex LLC (Russia), CSCS (Switzerland), IFIN-HH (Romania), CBPF (Brazil), PL-
GRID (Poland) and OSC (USA). We are indebted to the communities behind the
multiple open-source software packages on which we depend. Individual groups
or members have received support from AvH Foundation (Germany); EPLANET, Marie
Skłodowska-Curie Actions and ERC (European Union); A*MIDEX, ANR, Labex P2IO
and OCEVU, and Région Auvergne-Rhône-Alpes (France); Key Research Program of
Frontier Sciences of CAS, CAS PIFI, CAS CCEPP, Fundamental Research Funds for
Central Universities, and Sci. & Tech. Program of Guangzhou (China); RFBR, RSF
and Yandex LLC (Russia); GVA, XuntaGal and GENCAT (Spain); the Royal Society
and the Leverhulme Trust (United Kingdom).

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LHCb collaboration

R. Aaij32, C. Abellán Beteta50, T. Ackernley60, B. Adeva46, M. Adinolfi54, H.
Afsharnia9, C.A. Aidala85, S. Aiola25, Z. Ajaltouni9, S. Akar65, J.
Albrecht15, F. Alessio48, M. Alexander59, A. Alfonso Albero45, Z. Aliouche62,
G. Alkhazov38, P. Alvarez Cartelle55, S. Amato2, Y. Amhis11, L. An48, L.
Anderlini22, A. Andreianov38, M. Andreotti21, J.E. Andrews66, F. Archilli17,
A. Artamonov44, M. Artuso68, K. Arzymatov42, E. Aslanides10, M. Atzeni50, B.
Audurier12, S. Bachmann17, M. Bachmayer49, J.J. Back56, S. Baker61, P.
Baladron Rodriguez46, V. Balagura12, W. Baldini21,48, J. Baptista Leite1, R.J.
Barlow62, S. Barsuk11, W. Barter61, M. Bartolini24,g, F. Baryshnikov82, J.M.
Basels14, G. Bassi29, B. Batsukh68, A. Battig15, A. Bay49, M. Becker15, F.
Bedeschi29, I. Bediaga1, A. Beiter68, V. Belavin42, S. Belin27, V. Bellee49,
K. Belous44, I. Belov40, I. Belyaev41, G. Bencivenni23, E. Ben-Haim13, A.
Berezhnoy40, R. Bernet50, D. Berninghoff17, H.C. Bernstein68, C. Bertella48,
E. Bertholet13, A. Bertolin28, C. Betancourt50, F. Betti20,c, Ia. Bezshyiko50,
S. Bhasin54, J. Bhom35, L. Bian73, M.S. Bieker15, S. Bifani53, P. Billoir13,
M. Birch61, F.C.R. Bishop55, A. Bizzeti22,j, M. Bjørn63, M.P. Blago48, T.
Blake56, F. Blanc49, S. Blusk68, D. Bobulska59, J.A. Boelhauve15, O. Boente
Garcia46, T. Boettcher64, A. Boldyrev81, A. Bondar43, N. Bondar38,48, S.
Borghi62, M. Borisyak42, M. Borsato17, J.T. Borsuk35, S.A. Bouchiba49, T.J.V.
Bowcock60, A. Boyer48, C. Bozzi21, M.J. Bradley61, S. Braun66, A. Brea
Rodriguez46, M. Brodski48, J. Brodzicka35, A. Brossa Gonzalo56, D. Brundu27,
A. Buonaura50, C. Burr48, A. Bursche27, A. Butkevich39, J.S. Butter32, J.
Buytaert48, W. Byczynski48, S. Cadeddu27, H. Cai73, R. Calabrese21,e, L.
Calefice15,13, L. Calero Diaz23, S. Cali23, R. Calladine53, M. Calvi26,i, M.
Calvo Gomez84, P. Camargo Magalhaes54, A. Camboni45,84, P. Campana23, A.F.
Campoverde Quezada6, S. Capelli26,i, L. Capriotti20,c, A. Carbone20,c, G.
Carboni31, R. Cardinale24,g, A. Cardini27, I. Carli4, P. Carniti26,i, L.
Carus14, K. Carvalho Akiba32, A. Casais Vidal46, G. Casse60, M. Cattaneo48, G.
Cavallero48, S. Celani49, J. Cerasoli10, A.J. Chadwick60, M.G. Chapman54, M.
Charles13, Ph. Charpentier48, G. Chatzikonstantinidis53, C.A. Chavez
Barajas60, M. Chefdeville8, C. Chen3, S. Chen27, A. Chernov35, S.-G. Chitic48,
V. Chobanova46, S. Cholak49, M. Chrzaszcz35, A. Chubykin38, V. Chulikov38, P.
Ciambrone23, M.F. Cicala56, X. Cid Vidal46, G. Ciezarek48, P.E.L. Clarke58, M.
Clemencic48, H.V. Cliff55, J. Closier48, J.L. Cobbledick62, V. Coco48, J.A.B.
Coelho11, J. Cogan10, E. Cogneras9, L. Cojocariu37, P. Collins48, T.
Colombo48, L. Congedo19,b, A. Contu27, N. Cooke53, G. Coombs59, G. Corti48,
C.M. Costa Sobral56, B. Couturier48, D.C. Craik64, J. Crkovská67, M. Cruz
Torres1, R. Currie58, C.L. Da Silva67, E. Dall’Occo15, J. Dalseno46, C.
D’Ambrosio48, A. Danilina41, P. d’Argent48, A. Davis62, O. De Aguiar
Francisco62, K. De Bruyn78, S. De Capua62, M. De Cian49, J.M. De Miranda1, L.
De Paula2, M. De Serio19,b, D. De Simone50, P. De Simone23, J.A. de Vries79,
C.T. Dean67, W. Dean85, D. Decamp8, L. Del Buono13, B. Delaney55, H.-P.
Dembinski15, A. Dendek34, V. Denysenko50, D. Derkach81, O. Deschamps9, F.
Desse11, F. Dettori27,d, B. Dey73, P. Di Nezza23, S. Didenko82, L. Dieste
Maronas46, H. Dijkstra48, V. Dobishuk52, A.M. Donohoe18, F. Dordei27, A.C. dos
Reis1, L. Douglas59, A. Dovbnya51, A.G. Downes8, K. Dreimanis60, M.W. Dudek35,
L. Dufour48, V. Duk77, P. Durante48, J.M. Durham67, D. Dutta62, M.
Dziewiecki17, A. Dziurda35, A. Dzyuba38, S. Easo57, U. Egede69, V.
Egorychev41, S. Eidelman43,u, S. Eisenhardt58, S. Ek-In49, L. Eklund59, S.
Ely68, A. Ene37, E. Epple67, S. Escher14, J. Eschle50, S. Esen32, T. Evans48,
A. Falabella20, J. Fan3, Y. Fan6, B. Fang73, N. Farley53, S. Farry60, D.
Fazzini26,i, P. Fedin41, M. Féo48, P. Fernandez Declara48, A. Fernandez
Prieto46, J.M. Fernandez-tenllado Arribas45, F. Ferrari20,c, L. Ferreira
Lopes49, F. Ferreira Rodrigues2, S. Ferreres Sole32, M. Ferrillo50, M. Ferro-
Luzzi48, S. Filippov39, R.A. Fini19, M. Fiorini21,e, M. Firlej34, K.M.
Fischer63, C. Fitzpatrick62, T. Fiutowski34, F. Fleuret12, M. Fontana13, F.
Fontanelli24,g, R. Forty48, V. Franco Lima60, M. Franco Sevilla66, M. Frank48,
E. Franzoso21, G. Frau17, C. Frei48, D.A. Friday59, J. Fu25, Q. Fuehring15, W.
Funk48, E. Gabriel32, T. Gaintseva42, A. Gallas Torreira46, D. Galli20,c, S.
Gambetta58,48, Y. Gan3, M. Gandelman2, P. Gandini25, Y. Gao5, M. Garau27, L.M.
Garcia Martin56, P. Garcia Moreno45, J. García Pardiñas26, B. Garcia Plana46,
F.A. Garcia Rosales12, L. Garrido45, C. Gaspar48, R.E. Geertsema32, D.
Gerick17, L.L. Gerken15, E. Gersabeck62, M. Gersabeck62, T. Gershon56, D.
Gerstel10, Ph. Ghez8, V. Gibson55, M. Giovannetti23,o, A. Gioventù46, P.
Gironella Gironell45, L. Giubega37, C. Giugliano21,e,48, K. Gizdov58, E.L.
Gkougkousis48, V.V. Gligorov13, C. Göbel70, E. Golobardes84, D. Golubkov41, A.
Golutvin61,82, A. Gomes1,a, S. Gomez Fernandez45, F. Goncalves Abrantes70, M.
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Granado Cardoso48, E. Graugés45, E. Graverini49, G. Graziani22, A. Grecu37,
L.M. Greeven32, P. Griffith21,e, L. Grillo62, S. Gromov82, B.R. Gruberg
Cazon63, C. Gu3, M. Guarise21, P. A. Günther17, E. Gushchin39, A. Guth14, Y.
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Hatch48, J. He6, M. Hecker61, K. Heijhoff32, K. Heinicke15, A.M. Hennequin48,
K. Hennessy60, L. Henry25,47, J. Heuel14, A. Hicheur2, D. Hill49, M. Hilton62,
S.E. Hollitt15, J. Hu17, J. Hu72, W. Hu7, W. Huang6, X. Huang73, W.
Hulsbergen32, R.J. Hunter56, M. Hushchyn81, D. Hutchcroft60, D. Hynds32, P.
Ibis15, M. Idzik34, D. Ilin38, P. Ilten65, A. Inglessi38, A. Ishteev82, K.
Ivshin38, R. Jacobsson48, S. Jakobsen48, E. Jans32, B.K. Jashal47, A.
Jawahery66, V. Jevtic15, M. Jezabek35, F. Jiang3, M. John63, D. Johnson48,
C.R. Jones55, T.P. Jones56, B. Jost48, N. Jurik48, S. Kandybei51, Y. Kang3, M.
Karacson48, M. Karpov81, N. Kazeev81, F. Keizer55,48, M. Kenzie56, T. Ketel33,
B. Khanji15, A. Kharisova83, S. Kholodenko44, K.E. Kim68, T. Kirn14, V.S.
Kirsebom49, O. Kitouni64, S. Klaver32, K. Klimaszewski36, S. Koliiev52, A.
Kondybayeva82, A. Konoplyannikov41, P. Kopciewicz34, R. Kopecna17, P.
Koppenburg32, M. Korolev40, I. Kostiuk32,52, O. Kot52, S. Kotriakhova38,30, P.
Kravchenko38, L. Kravchuk39, R.D. Krawczyk48, M. Kreps56, F. Kress61, S.
Kretzschmar14, P. Krokovny43,u, W. Krupa34, W. Krzemien36, W. Kucewicz35,s, M.
Kucharczyk35, V. Kudryavtsev43,u, H.S. Kuindersma32, G.J. Kunde67, T.
Kvaratskheliya41, D. Lacarrere48, G. Lafferty62, A. Lai27, A. Lampis27, D.
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Langer15, O. Lantwin50,82, T. Latham56, F. Lazzari29,p, R. Le Gac10, S.H.
Lee85, R. Lefèvre9, A. Leflat40, S. Legotin82, O. Leroy10, T. Lesiak35, B.
Leverington17, H. Li72, L. Li63, P. Li17, Y. Li4, Y. Li4, Z. Li68, X. Liang68,
T. Lin61, R. Lindner48, V. Lisovskyi15, R. Litvinov27, G. Liu72, H. Liu6, S.
Liu4, X. Liu3, A. Loi27, J. Lomba Castro46, I. Longstaff59, J.H. Lopes2, G.
Loustau50, G.H. Lovell55, Y. Lu4, D. Lucchesi28,k, S. Luchuk39, M. Lucio
Martinez32, V. Lukashenko32, Y. Luo3, A. Lupato62, E. Luppi21,e, O. Lupton56,
A. Lusiani29,l, X. Lyu6, L. Ma4, R. Ma6, S. Maccolini20,c, F. Machefert11, F.
Maciuc37, V. Macko49, P. Mackowiak15, S. Maddrell-Mander54, O. Madejczyk34,
L.R. Madhan Mohan54, O. Maev38, A. Maevskiy81, D. Maisuzenko38, M.W.
Majewski34, J.J. Malczewski35, S. Malde63, B. Malecki48, A. Malinin80, T.
Maltsev43,u, H. Malygina17, G. Manca27,d, G. Mancinelli10, R. Manera
Escalero45, D. Manuzzi20,c, D. Marangotto25,h, J. Maratas9,r, J.F. Marchand8,
U. Marconi20, S. Mariani22,f,48, C. Marin Benito11, M. Marinangeli49, P.
Marino49,l, J. Marks17, P.J. Marshall60, G. Martellotti30, L.
Martinazzoli48,i, M. Martinelli26,i, D. Martinez Santos46, F. Martinez
Vidal47, A. Massafferri1, M. Materok14, R. Matev48, A. Mathad50, Z. Mathe48,
V. Matiunin41, C. Matteuzzi26, K.R. Mattioli85, A. Mauri32, E. Maurice12, J.
Mauricio45, M. Mazurek36, M. McCann61, L. Mcconnell18, T.H. Mcgrath62, A.
McNab62, R. McNulty18, J.V. Mead60, B. Meadows65, C. Meaux10, G. Meier15, N.
Meinert76, D. Melnychuk36, S. Meloni26,i, M. Merk32,79, A. Merli25, L. Meyer
Garcia2, M. Mikhasenko48, D.A. Milanes74, E. Millard56, M. Milovanovic48,
M.-N. Minard8, L. Minzoni21,e, S.E. Mitchell58, B. Mitreska62, D.S. Mitzel48,
A. Mödden 15, R.A. Mohammed63, R.D. Moise61, T. Mombächer15, I.A. Monroy74, S.
Monteil9, M. Morandin28, G. Morello23, M.J. Morello29,l, J. Moron34, A.B.
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J.A. Rodriguez Lopez74, A. Rollings63, P. Roloff48, V. Romanovskiy44, M.
Romero Lamas46, A. Romero Vidal46, J.D. Roth85, M. Rotondo23, M.S. Rudolph68,
T. Ruf48, J. Ruiz Vidal47, A. Ryzhikov81, J. Ryzka34, J.J. Saborido Silva46,
N. Sagidova38, N. Sahoo56, B. Saitta27,d, D. Sanchez Gonzalo45, C. Sanchez
Gras32, R. Santacesaria30, C. Santamarina Rios46, M. Santimaria23, E.
Santovetti31,o, D. Saranin82, G. Sarpis59, M. Sarpis75, A. Sarti30, C.
Satriano30,n, A. Satta31, M. Saur15, D. Savrina41,40, H. Sazak9, L.G.
Scantlebury Smead63, S. Schael14, M. Schellenberg15, M. Schiller59, H.
Schindler48, M. Schmelling16, B. Schmidt48, O. Schneider49, A. Schopper48, M.
Schubiger32, S. Schulte49, M.H. Schune11, R. Schwemmer48, B. Sciascia23, A.
Sciubba23, S. Sellam46, A. Semennikov41, M. Senghi Soares33, A. Sergi53,48, N.
Serra50, L. Sestini28, A. Seuthe15, P. Seyfert48, D.M. Shangase85, M.
Shapkin44, I. Shchemerov82, L. Shchutska49, T. Shears60, L. Shekhtman43,u, Z.
Shen5, V. Shevchenko80, E.B. Shields26,i, E. Shmanin82, J.D. Shupperd68, B.G.
Siddi21, R. Silva Coutinho50, G. Simi28, S. Simone19,b, I. Skiba21,e, N.
Skidmore62, T. Skwarnicki68, M.W. Slater53, J.C. Smallwood63, J.G. Smeaton55,
A. Smetkina41, E. Smith14, M. Smith61, A. Snoch32, M. Soares20, L. Soares
Lavra9, M.D. Sokoloff65, F.J.P. Soler59, A. Solovev38, I. Solovyev38, F.L.
Souza De Almeida2, B. Souza De Paula2, B. Spaan15, E. Spadaro Norella25,h, P.
Spradlin59, F. Stagni48, M. Stahl65, S. Stahl48, P. Stefko49, O.
Steinkamp50,82, S. Stemmle17, O. Stenyakin44, H. Stevens15, S. Stone68, M.E.
Stramaglia49, M. Straticiuc37, D. Strekalina82, S. Strokov83, F. Suljik63, J.
Sun27, L. Sun73, Y. Sun66, P. Svihra62, P.N. Swallow53, K. Swientek34, A.
Szabelski36, T. Szumlak34, M. Szymanski48, S. Taneja62, F. Teubert48, E.
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N. Tuning32, A. Ukleja36, D.J. Unverzagt17, E. Ursov82, A. Usachov32, A.
Ustyuzhanin42,81, U. Uwer17, A. Vagner83, V. Vagnoni20, A. Valassi48, G.
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Regueiro46, C. Vázquez Sierra48, S. Vecchi21, J.J. Velthuis54, M. Veltri22,q,
A. Venkateswaran68, M. Veronesi32, M. Vesterinen56, D. Vieira65, M. Vieites
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1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4Institute Of High Energy Physics (IHEP), Beijing, China
5School of Physics State Key Laboratory of Nuclear Physics and Technology,
Peking University, Beijing, China
6University of Chinese Academy of Sciences, Beijing, China
7Institute of Particle Physics, Central China Normal University, Wuhan, Hubei,
China
8Univ. Grenoble Alpes, Univ. Savoie Mont Blanc, CNRS, IN2P3-LAPP, Annecy,
France
9Université Clermont Auvergne, CNRS/IN2P3, LPC, Clermont-Ferrand, France
10Aix Marseille Univ, CNRS/IN2P3, CPPM, Marseille, France
11Université Paris-Saclay, CNRS/IN2P3, IJCLab, Orsay, France
12Laboratoire Leprince-ringuet (llr), Palaiseau, France
13LPNHE, Sorbonne Université, Paris Diderot Sorbonne Paris Cité, CNRS/IN2P3,
Paris, France
14I. Physikalisches Institut, RWTH Aachen University, Aachen, Germany
15Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
16Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
17Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
18School of Physics, University College Dublin, Dublin, Ireland
19INFN Sezione di Bari, Bari, Italy
20INFN Sezione di Bologna, Bologna, Italy
21INFN Sezione di Ferrara, Ferrara, Italy
22INFN Sezione di Firenze, Firenze, Italy
23INFN Laboratori Nazionali di Frascati, Frascati, Italy
24INFN Sezione di Genova, Genova, Italy
25INFN Sezione di Milano, Milano, Italy
26INFN Sezione di Milano-Bicocca, Milano, Italy
27INFN Sezione di Cagliari, Monserrato, Italy
28Universita degli Studi di Padova, Universita e INFN, Padova, Padova, Italy
29INFN Sezione di Pisa, Pisa, Italy
30INFN Sezione di Roma La Sapienza, Roma, Italy
31INFN Sezione di Roma Tor Vergata, Roma, Italy
32Nikhef National Institute for Subatomic Physics, Amsterdam, Netherlands
33Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, Netherlands
34AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
35Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
36National Center for Nuclear Research (NCBJ), Warsaw, Poland
37Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
38Petersburg Nuclear Physics Institute NRC Kurchatov Institute (PNPI NRC KI),
Gatchina, Russia
39Institute for Nuclear Research of the Russian Academy of Sciences (INR RAS),
Moscow, Russia
40Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
41Institute of Theoretical and Experimental Physics NRC Kurchatov Institute
(ITEP NRC KI), Moscow, Russia
42Yandex School of Data Analysis, Moscow, Russia
43Budker Institute of Nuclear Physics (SB RAS), Novosibirsk, Russia
44Institute for High Energy Physics NRC Kurchatov Institute (IHEP NRC KI),
Protvino, Russia, Protvino, Russia
45ICCUB, Universitat de Barcelona, Barcelona, Spain
46Instituto Galego de Física de Altas Enerxías (IGFAE), Universidade de
Santiago de Compostela, Santiago de Compostela, Spain
47Instituto de Fisica Corpuscular, Centro Mixto Universidad de Valencia -
CSIC, Valencia, Spain
48European Organization for Nuclear Research (CERN), Geneva, Switzerland
49Institute of Physics, Ecole Polytechnique Fédérale de Lausanne (EPFL),
Lausanne, Switzerland
50Physik-Institut, Universität Zürich, Zürich, Switzerland
51NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
52Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
53University of Birmingham, Birmingham, United Kingdom
54H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
55Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
56Department of Physics, University of Warwick, Coventry, United Kingdom
57STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
58School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
59School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
60Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
61Imperial College London, London, United Kingdom
62Department of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
63Department of Physics, University of Oxford, Oxford, United Kingdom
64Massachusetts Institute of Technology, Cambridge, MA, United States
65University of Cincinnati, Cincinnati, OH, United States
66University of Maryland, College Park, MD, United States
67Los Alamos National Laboratory (LANL), Los Alamos, United States
68Syracuse University, Syracuse, NY, United States
69School of Physics and Astronomy, Monash University, Melbourne, Australia,
associated to 56
70Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
71Physics and Micro Electronic College, Hunan University, Changsha City,
China, associated to 7
72Guangdong Provencial Key Laboratory of Nuclear Science, Institute of Quantum
Matter, South China Normal University, Guangzhou, China, associated to 3
73School of Physics and Technology, Wuhan University, Wuhan, China, associated
to 3
74Departamento de Fisica , Universidad Nacional de Colombia, Bogota, Colombia,
associated to 13
75Universität Bonn - Helmholtz-Institut für Strahlen und Kernphysik, Bonn,
Germany, associated to 17
76Institut für Physik, Universität Rostock, Rostock, Germany, associated to 17
77INFN Sezione di Perugia, Perugia, Italy, associated to 21
78Van Swinderen Institute, University of Groningen, Groningen, Netherlands,
associated to 32
79Universiteit Maastricht, Maastricht, Netherlands, associated to 32
80National Research Centre Kurchatov Institute, Moscow, Russia, associated to
41
81National Research University Higher School of Economics, Moscow, Russia,
associated to 42
82National University of Science and Technology “MISIS”, Moscow, Russia,
associated to 41
83National Research Tomsk Polytechnic University, Tomsk, Russia, associated to
41
84DS4DS, La Salle, Universitat Ramon Llull, Barcelona, Spain, associated to 45
85University of Michigan, Ann Arbor, United States, associated to 68

aUniversidade Federal do Triângulo Mineiro (UFTM), Uberaba-MG, Brazil
bUniversità di Bari, Bari, Italy
cUniversità di Bologna, Bologna, Italy
dUniversità di Cagliari, Cagliari, Italy
eUniversità di Ferrara, Ferrara, Italy
fUniversità di Firenze, Firenze, Italy
gUniversità di Genova, Genova, Italy
hUniversità degli Studi di Milano, Milano, Italy
iUniversità di Milano Bicocca, Milano, Italy
jUniversità di Modena e Reggio Emilia, Modena, Italy
kUniversità di Padova, Padova, Italy
lScuola Normale Superiore, Pisa, Italy
mUniversità di Pisa, Pisa, Italy
nUniversità della Basilicata, Potenza, Italy
oUniversità di Roma Tor Vergata, Roma, Italy
pUniversità di Siena, Siena, Italy
qUniversità di Urbino, Urbino, Italy
rMSU - Iligan Institute of Technology (MSU-IIT), Iligan, Philippines
sAGH - University of Science and Technology, Faculty of Computer Science,
Electronics and Telecommunications, Kraków, Poland
tP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
uNovosibirsk State University, Novosibirsk, Russia
vHanoi University of Science, Hanoi, Vietnam


11institutetext: Indian Institute of Technology, Madras, India
22institutetext: Healthcare Technology Innovation Centre, IIT Madras, India

# A Study of Representational Properties of Unsupervised Anomaly Detection in
Brain MRI

Ayantika Das 11 Arun Palla 11 Keerthi Ram 22
Mohanasankar Sivaprakasam 1122

###### Abstract

Anomaly detection in MRI is of high clinical value in imaging and diagnosis.
Unsupervised methods for anomaly detection provide interesting formulations
based on reconstruction or latent embedding, offering a way to observe
properties related to factorization. We study four existing modeling methods,
and report our empirical observations using simple data science tools, to seek
outcomes from the perspective of factorization as it would be most relevant to
the task of unsupervised anomaly detection, considering the case of brain
structural MRI. Our study indicates that anomaly detection algorithms that
exhibit factorization related properties are well capacitated with delineatory
capabilities to distinguish between normal and anomaly data. We have validated
our observations in multiple anomaly and normal datasets. The code is
available at https://github.com/ayantikadas/Unsupervised_anomaly_detection.

##  1 Introduction

Brain MRI is valuable in diagnosis, monitoring, and surgical intervention
planning and guidance. Numerous signs and changes are visualized in brain MRI
images, which could be pathological or normal variations [6]. MRI is
versatile, and various sequences are available to observe visual features of
several structural anomalies, including localized discrete (e.g., tumor,
hemorrhage), diffuse (e.g., inflammation), regional and global (enlargement or
contraction of structures, midline shift) changes. We focus on the detection
of anomalies visible in the brain MRI, arising from patient related factors,
explainable as geometric and photometric changes when related to normal
appearance.

Normative brain atlases have been developed for quantitative neuroinformatics
[17] and are used in stereotaxic neurosurgery and electrode placement, along
with diagnostic decision assistance in clinical applications like stroke
management. For comparative analysis, the methods assume that anomaly causes
wider structural change than normal variations (we term this the compactness
property of normal variations). Population averaged atlases and multiatlas
methods [16] are used to cover multiple anatomical phenotypes and widen the
range of applicability. These methods use intensity based registration for
computing a warp or map between the brain scan and the atlas, to standardize
the data, and study differences. The standardization can be construed as
abstracting normal geometric variations through the warp and abstracting
photometric variations to standardize across scanners and imaging parameters,
leaving the apparent patient-related changes for comparison.

The resulting image can be ascribed to a simple additive model in 2D:

$image=g\\_normalcy(sliceno)+anomaly+inhomogeneities$ (1)

Where $g\\_normalcy$ is a generative process to compute a normative reference
slice. We assume that anomaly effects an additive change over the generative
process, co-existing in the same domain of intensities. For lesions that
distort surrounding brain parenchyma (such as intra-axial lesions), the
generative process includes local elastic deformation, following which the
intensity variations continue to be additive. When using a population atlas
such as MNI or Colin-27 [9], the generative process is trivially indexing of
an atlas slice. Other data-driven methods (including ones described in the
next section) built on normal data alone, can also be used for the generative
process. The last component is residual uncorrelated inhomogeneities to model
scanner and parameter variations left over beyond the standardization step.

Under this oversimplified model, the task of anomaly detection is reduced to
one of decomposing the image domain; this however entails problems analogous
to adaptive thresholding methods for detection – bias to local contrast,
affecting smaller and subtle anomalies, and also subtle global changes.

In this study, continuing with the premise of unsupervised comparative methods
for modeling anomaly, we analyze some selected methods, restricting them to
use only normal data and seek to elicit an understanding of their functioning
from the standpoint of factorization.

##  2 Approaches for modeling anomaly

Supervised learning techniques, known to have achieved high performance in
detection, localization, and delineation of pathologies in MRI [7], are
specific to a few types of lesions like tumor and stroke [11], and biased by
the available annotations at training time (which are laborious to obtain).

Contemporary studies [2] explore unsupervised techniques, towards building
disease-agnostic models for pathology detection and delineation [3]. Being
trained only with normal data, test time pathological data (out-of-
distribution) are expected to result in performance degradation in the task of
reconstruction, offering a means of flagging anomaly. Auto-encoders are a
well-known approach used in literature, formulated as jointly learning to
encode image $x\in{\rm I\\!R}^{m\times n}$(m and n are the height and width of
the image) into an embedding $z\in{\rm I\\!R}^{d}$, and learning to decode the
embedding to reconstruct the image $\hat{x}$. Notationally,
$z=f_{\theta}(x);\hat{x}=g_{\phi}(z)$, where $f_{\theta}$ is the encoder and
$g_{\phi}$ is the decoder.

Compatible with unsupervised learning, the detection step is possible at the
output (where a noticeable reconstruction error $\delta(\hat{x},x)$ is
expected for the out-of-distribution anomalous test data) or at the embedding
space (where embedding of the anomalous test data $z_{a}$ is expected to
deviate from the normative embedding distribution $P(z_{n})$).

###  2.1 Selected methods

We select the following methods for modeling normal brain data:

  1. 1.

VAE \- VAE forms a baseline for the unsupervised anomaly detection setting we
have considered. VAE imposes $P(z_{n})$ to be multivariate Gaussian with
learnable mean and variance (weak i.i.d assumption). The underlying assumption
is sufficiency of a simple Gaussian latent distribution for explaining normal
data. VAE is implemented by including a Kullback-Leibler (KL) divergence
regularization term along with a reconstruction term $\delta:|x-\hat{x}|_{1}$
in the network loss function and trained with data using routine optimization
methods like stochastic gradient descent (SGD);

  2. 2.

FactorVAE [12] \- FactorVAE also imposes $P(z_{n})$ to be a multivariate
Gaussian. Like VAE, the KL divergence and reconstruction term are included in
the loss function, but it also includes an additional Total Correlation (TC)
penalty for independence between variables in $z$. The independence in $z$
aids in the better encoding of meaningful normal features within the
dimensions of $z$.

  3. 3.

SSAE [4] \- A Laplacian scale-space Auto-encoder, which declaratively performs
image domain multiband factorization into high- and low-frequency content,
processing the Laplacian pyramid band-wise through independent sparse
convolutional Auto-encoders and forming the reconstruction image by combining
the multiresolution outputs back into a Gaussian pyramid;

  4. 4.

Glow [14] \- Generative Flow, learns a bijective mapping
$f_{\theta}:X\rightarrow Z$ from the data space into a multivariate standard
Gaussian distribution. The learning process is based on the _change of
variables_ formulation. The reversible transformation follows due to the
bijective nature, resulting from the architectural choice of layers. The
selected methods are described in Sec.7.

The latent space convention for the set of methods (VAE, FactorVAE, and SSAE)
which are derivatives of the Auto-encoder family, is to have embeddings
($z\in{\rm I\\!R}^{d}$) of much lower dimension as compared to the input image
space. But, GLOW is architecturally bound to have an embedding space dimension
equated to the total number of pixels in the image space ($d=m\times n$). The
conventional usage of the low dimensional embeddings in the former set of
methods is attributed to the fact that the latent space of the Auto-encoders
must have limited information capacity to prevent them from learning trivial
identity functions [10]. Additionally, reconstructions of VAE suffer from
image quality degradation if there are high levels of increment in the latent
dimensions. The latent dimensions of VAE can be increased only up to a certain
range, beyond which the generation process does not benefit from the
introduced variability [20]. Thus, we have carried out experimental
evaluations adopting the conventional architectural formulation of the
selected methods, allowing the inter-method dimensional variations in the
latent space.

###  2.2 Hierarchy of properties

Our chosen methods exhibit some properties closely relatable to factorization
at the latent embedding $z$ (via $f$ and loss terms), and the output $\hat{x}$
(via $g$ and loss terms). The specific properties of interest in increasing
the order of capability for solving the task of unsupervised anomaly detection
under consideration are discussed below:

  1. 1.

Compact organization \- proximal representation of normal variations;

  2. 2.

Separable representations \- distinctiveness within groups of variables in $z$
(or embedding subspaces) representing normalcy and anomaly, disentangling both
the groups;

  3. 3.

Controllable representations \- observe controlled changes in $\hat{x}$ via
perturbing $z$;

  4. 4.

Identifiable features \- visual features of image space captured in embedding
subspaces.

As the properties are to be observed at $z$ and $\hat{x}$, we resort to the
following intuitive ways of evidencing them:

  * •

Studying the two dimensional spectral embeddings111http://surl.li/cirgr; in
order to analyze linkages between the latent vectors and comment upon their
proximities. The spectral embeddings are computed using a non-linear
dimensionality reduction technique which deploys spectral decomposition of
$z$.

  * •

Performing Local Outlier Factor (LOF)222http://surl.li/cirgo analysis on $z$
to examine the separability, in order to comment upon the distinctness of
representations between the normal and anomalous sets. LOF gives a Local
Outlier Detection (LOD) score, which is computed by estimating the local
density deviation measured in a certain neighbourhood.

  * •

Introducing variable perturbations on $z$ to investigate changes in $\hat{x}$
and infer upon the nature and extent of controllability in both the domains.

##  3 Experimental setup

We take MRI brain scans of only healthy individuals for the training phase and
include both healthy and pathological scans for the testing phase. The healthy
set consists of T2 weighted MRI scans from CamCAN [18], IXI [1] and HCP [19]
and the anomaly set consists of T2 weighted High Grade Glioma (HGG) data from
BraTS 2017 [15]. The training and testing datasets follow the regimes as
mentioned below:

  * •

Training set: We have considered a set of 100, 100, and 26 volumes from IXI,
CamCAN, and HCP datasets respectively; in order to impart better modeling
capability in the presence of data from multiple sources.

  * •

Test set: We have taken four sub-categories, in order to facilitate the
investigation of the factorization properties under consideration:

    * –

Normal: A set of 100 volumes, 50 each from IXI and CamCAN;

    * –

Noisy: A simulated pathological dataset generated by adding random patches of
Gaussian noise to the Normal set;

    * –

BraTS: A set 100 volumes from BraTS HGG dataset;

    * –

Only Pathology: Only the pathological extractions from the BraTS set, by
masking out the healthy regions.

A sequence of standardization protocols is applied to minimize the variations
within the images that are incurred during the acquisition process. We perform
affine registration of all the images to Colin-27, which transforms the
spacing of all axial slices to 0.5 mm with an in-plane resolution of 0.5
$\times$ 0.5 mm. We take corrective measures to remove inhomogeneities from
the scans through N4 bias correction. The brain mask for the Colin-27 atlas is
used to curate brain regions through the removal of the skull and other
unrelated anatomical regions. Also, the intensity profile of the images is
matched with the Colin-27 atlas by using histogram matching.

For modeling the Auto-encoder based approaches that we have selected (VAE,
FactorVAE, and SSAE), the generalised encoder-decoder architecture from [3] is
adopted. Although they share the same architecture, the representational
learning of latent space is defined by the training protocol, which is
different among them. VAE captures the latent space with a dimensionality
value $d$ equal to 128. The training specifications for VAE include; training
of the model for 80 epochs, usage of L1 loss for supervision of the
reconstruction term, and the weightage to the KL Divergence term is adjusted
using $\beta$-annealing methods till 10 epochs and then made consistent
throughout [8]. FactorVAE shares a similar training strategy as VAE but with
an additional Total Correlation (TC) penalty term. The weightage to the TC
term is linearly annealed from 0 to 0.35 for 30 epochs and then set to a
constant value throughout the training process of 60 epochs. The SSAE is
modeled with three Auto-encoders operating at different levels in the
Laplacian pyramid and is supervised with the L1 reconstruction loss at all
three levels with equal weightage to each of them. The latent space is
captured with dimensionality $d$ equal to 1024, 256, and 64 for all the three
Auto-encoders respectively. The model is trained for 50 epochs.

For modeling Glow, the architecture from [14] is adopted. The training is
carried out till the data is transformed into a fixed latent representation,
with its dimension $d$ equal to 16,384. The objective function is based on the
_change of variables_ formulation which consists of a Gaussian function and
determinant of Jacobian of the model. The former term penalizes the deviation
of the model’s latent representation from the fixed Gaussian and the latter
reflects a local change in the volume incurred by the model. Both the penalty
terms are weighted equally throughout the training process of 50 epochs. All
the methods use Adam optimizer with an initial learning rate of $10^{-4}$
[13].

(a) GLOW embeddings

(b) FactorVAE embeddings

(c) VAE embeddings

(d) SSAE embeddings

Figure 1: The two dimensional Spectral embeddings extracted from the high
dimensional latent vectors of the Normal, BraTS, Noisy and Only Pathology test
set categories as elaborated in Sec. 3 for all the four methods. The embedded
space clearly depicts that the latent representations of the Normal set (in-
distribution), from the (a) GLOW method, has a proximal and compact structure
with distinctiveness as compared to the other out-of-distribution sets (BraTS
+ Only pathology + Noisy). The embedding space of the (b) FactorVAE and (c)
VAE forms elliptical structures with less density. The distinctiveness
property is completely missing from (c) VAE but the (b) FactorVAE has pushed
away the only Pathology set to some extent. The embedding space from (d) SSAE
neither adheres to any certain shape nor forms a compact structure.

##  4 Observations

The observations pertaining to the four properties for the different test set
categories; Normal, out-of-distribution (BraTS + Noisy + Only Pathology) are
described below, highlighting comparative aspects between all the four
methods.

How compact and distinctive are the spectral embedding of the latent vectors?

The Fig.1 depicts the two dimensional spectral embeddings of the latent
vectors. The organization of the latent vectors in the lower dimensional
spectral embedding space is leveraged to carry out the analysis.

  1. 1.

GLOW: From Fig.1(a) it is evident that the GLOW method is able to learn
compact and proximal representations for the Normal set, which is surrounded
by the representations of the other three out-of-distribution sets, with
different degree of overlap. The degree of overlap in the Only Pathology set
is very minimal with most of the samples pushed away from the Normal set in a
less dense fashion. The BraTS and the Noisy set have relatively dense
representations as compared to the Only Pathology set and their degree of
overlap with the Normal set is high. Thus, GLOW is seen to achieve a compact
representation within the Normal set, with semantically relevant separation of
the embeddings of the out-of-distribution set, disentangling meaningfully
between the two sets.

  2. 2.

VAE: From Fig.1(c), it is seen that all the spectral embeddings of VAE are
contained in an elliptical structure, which is much lower in density when
compared to the GLOW embeddings. The Normal set does not uniquely hold the
notion of proximal representations, rather the embeddings from all the sets
are distributed in a similar manner. Thus, VAE presents a compact
representation of embeddings by entangling both Normal and out-of-distribution
sets within an elliptical structure, with no separation formed between any of
the sets.

  3. 3.

FactorVAE: Similar to the VAE, the embeddings of the FactorVAE approximates an
elliptical structural formulation as in Fig.1(b). Compared to VAE, FactorVAE
has relatively better proximity within the Normal set and distinctiveness
between different sets; the deviation of the Only Pathology set from the
concentrated region of the Normal set is visible. Thus, the notion of compact
representations in FactorVAE is similar to VAE but it highlights better
properties of separability.

  4. 4.

SSAE: The embeddings from SSAE are neither structurally bound nor proximal for
any of the sets, as shown in Fig.1(d). The samples from the Normal set have
formed certain smaller groups in different regions within the embedding space.
Thus, representations from SSAE are not compact for any of the sets, with
multiple separations within the Normal set, making separations amongst sets
less evident.

Which method shows delineation between Normal set and out-of-distribution set
based on LOD score?

The Fig.2 represents the histogram of the LOD scores, calculated over the
frequencies of brain slices for Normal and out-of-distribution sets. The
histogram plot from 2(a) highlights that samples from the out-of-distribution
set have a long tail on the left region of the plot and the Normal set is
densely populated and contained within the right half. This behaviour
showcases delineatory properties in GLOW, despite having certain overlapping
regions between the two distributions. The other histogram plots 2(b), 2(c)
and 2(d) does not exhibit delineatory properties in the latent space.

What is the nature of controllability in the reconstructed images ($\hat{x}$)
due to different settings of perturbations in the latent vectors?

We have introduced scalar multiplicative factors ($m$) to perturb the latent
space representations ($z*m$). The Fig.3 highlights the dependency of the
change in $\hat{x}$ due to the introduced perturbations in latent space. All
the adopted methods are sensitive and show monotonic increment to the
perturbations, but the degree of sensitivity is method specific. The GLOW
method exhibits the highest sensitivity to the perturbations as compared to
the other methods. The VAE, FactorVAE, and SSAE show much staggered changes to
the perturbations. Thus, GLOW allows highly sensitive and monotonic
controllability to these perturbations as compared to other methods.

We have experimented with another form of perturbation in the latent space, by
corrupting a set of consecutive coordinates of the latent vectors, leaving all
other coordinates undisturbed. The corruption is introduced in two folds Case
I and Case II. For both the cases, out of all dimensions d in a latent vector,
a certain set of consecutive coordinates p-q were changed and the remaining
d-(p-q) were unchanged. Also for both the cases, p coordinate values are
equal, but q coordinate values for Case II is greater than Case I, implying
that perturbations in Case II are extended to a relatively larger span. Table1
indicates the numerical values taken by p and q for all the methods in both
the cases and the effect of these perturbations in $\hat{x}$. In both Case I
and II, $\hat{x}$ from GLOW shows the effect of perturbation within certain
consecutive rows, while the other methods show dispersed effects throughout
the spatial extent of the images. For GLOW in Case II, when the perturbations
get extended to consecutive coordinates of z, the $\hat{x}$ also transmits
this to an extended set of consecutive rows. These sets of descriptive
qualifiers from the Table1 are pictorially depicted in Fig.4 for the GLOW
method. The orange and blue notations signify Case I and Case II respectively.
Thus, GLOW shows localized and proportional controllability to these
perturbations, while the other methods are only responsive to these changes
without any specific nature.

How does latent space perturbations affect the identifiable visual features in
the image space?

From the Table1, we can comment that SSAE, FactorVAE, and VAE disperse the
effect of perturbations (within a few latent dimensions), over the entire
image space (brain and background regions). But, GLOW is highlighting the
effect of perturbations within the brain boundaries, as shown in the residual
images of the Fig.4. Although regions within the brain boundaries are
affected, but not necessarily different identifiable and meaningfully relevant
compartments of the brain.

(a)

(b)

(c)

(d)

Figure 2: The histogram plot of the Local Outlier Detection (LOD) scores with the frequency of brain slices for the Normal set and out-of-distribution set (BraTS + Only pathology + Noisy). The LOD score is evaluated on the latent space components directly. The histogram plot from (a) GLOW highlights that the out-of-distribution set has a very heavy tail on the left hand side and the Normal set is mostly centered on the right side, showing delineatory properties. The histogram plot for (c) VAE, (b) FactorVAE, and (d) SSAE does not exhibit delineatory properties in the latent space. Table 1: Effect of perturbations in the latent space Methods | Dimensions of z (d) |  | Perturbation defined
---
from p to q coordinates
of latent z
| Location of the effect of
---
Perturbations in
reconstructed image
Case I (p-q) | Case II (p-q) | Case I | Case II
GLOW | 16,384 | 6500-7000 | 6500-8000 |  | Localized
---
within
Rows 75-95
| Localized
---
within
Rows 75-125
VAE | 128 | 50-90 | 50-120 |  Dispersed over the entire Image
FactorVAE | 128 | 50-90 | 50-120
SSAE | 1024 | 50-150 | 50-600
Figure 3: The dependency of latent space perturbations with change in the
reconstructed image space, captured through Mean Squared error (MSE).
Perturbations are in the form of $z*m$, where $z$ is the latent vector and $m$
is the multiplicative factor, with values ranging from 0 to 1. Effectively, an
increasing order of perturbation in the latent space arises from decreasing
values of $m$, implying $m=1$ has no change on $z$. MSE values are calculated
between reconstructed images (due to varying values of $m$) and the original
images. The GLOW has higher sensitivity to the perturbations followed by SSAE,
FactorVAE, and VAE in decremental order of their sensitivity nature.  Figure
4: The top row pictorially depicts the introduced perturbations for the GLOW
method with Case I and Case II, as in Table1. The second row highlights the
reconstructed images (left to right) for non-perturbed $z$, and perturbed $z$
for 50 and 1500 dimensions respectively. The third row shows the residue
(differences after subtracting) of the reconstructed images due to
perturbations w.r.t the non-perturbed image. They are reflecting localized and
proportional effects.

##  5 Inferences and Discussion

The relatability of the properties under consideration to the task of
unsupervised anomaly detection is discussed below, connecting to the
observations from our methods and their reasoning.

Relating compact and separable representations to the task: The separation
between the representations of the Normal and out-of-distribution set is quite
essential for modeling anomaly detection in the unsupervised setting we have
considered. This holds from the fact that our unsupervised setting assumes
representations of the out-of-distribution (test) set will be separated from
the representations of the Normal (training) set, which will result in
performance degradation in the former set indicating anomalies. Our
observations have made us conclusive of the fact that separability within the
representations is a consequence of the property of compact organization
within the Normal set.

  * •

GLOW is the only method that is able to show compact organization within the
Normal set and so it is able to disentangle and create separation between the
sets. GLOW is able to show this property because it has learned to map the
image space to a high-dimensional standard Gaussian, whose dimensionality is
equal to the image space. This has enabled it to capture attributes more
meaningfully and semantically relevant to the image space.

  * •

VAE is entangling representations for Normal and out-of-distribution sets,
enclosing representations from all these sets to an elliptical structure,
since it has learned to map the image space to a much lower dimensional
standard Gaussian. VAE needs to learn the reconstruction task to make the
representations meaningful, while this task constrains the assignment of
higher dimensions to $z$. This bottleneck entangles all the representations
without any separation between them.

  * •

FactorVAE also encloses representations within an elliptical structure since
it shares a common objective with VAE but it has better separability since an
additional term is introduced in the objective to ensure statistical
independence amongst vectors in the latent space.

  * •

SSAE is not capable of encoding compact and separable representation, since it
learns a transformation for the Laplacian of the images at different scale
spaces without any constraint in the formulation to the latent space.

Relating controllable representations to the task: It is essential to observe
controlled changes $\hat{x}$ while there are perturbations in $z$, for the
task of anomaly detection that we have considered. In order to indicate the
separation of the latent representations for Normal and out-of-distribution
sets in $\hat{x}$, it is crucial for the anomaly detection task to transmit
changes in $z$ to $\hat{x}$. The transmission of changes in $\hat{x}$ should
be sensitive to the extent and location of perturbation in $z$ to localize
anomalies.

  * •

GLOW method has the capability to control over $\hat{x}$, while there are
perturbations in $z$, with high sensitivity and localizability as compared to
other methods. These properties arise in GLOW due to the fact that it is
architecturally designed to be bijective in nature, having a dimensionality of
$z$ equal to $\hat{x}$.

Relating identifiable feature representations to the task: The property of
capturing the dimensions of $z$ with identifiable and visually meaningful
compartments of the brain is hierarchically at the highest level of capability
for our unsupervised anomaly detection task. This property can enable better
capturing of the features for variations of normal brain information so as to
reduce the chances of falsely detecting these normal variations as anomalies.
The methods under consideration could not potentially encode information
meaningfully about different compartments of the brain.

##  6 Conclusion

We have performed an empirical study on the nature of latent representations
of four existing models; GLOW, VAE, FactorVAE and SSAE by considering four
properties; compact organization, separable representation, controllable
features and identifiable visual features. We have visualized the latent
representations in a reduced dimensional space, inferred on LOD score and
introduced perturbations to compare the properties of the different methods.
We observed that the GLOW method is able to show compact representations with
meaningful separation and controllability. We have discussed explicitly how
the properties relate to the task of unsupervised anomaly detection. We are
able to conclude that GLOW is a potential candidate for the unsupervised
anomaly detection since it exhibits better properties of factorization, which
are closely related to the end task.

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##  7 Appendix

###  7.1 VAE

VAE [5] enforces the learning of marginal likelihood of the training data(x)
in a generative setting. This learning process has an objective of maximizing
the lower bound of the likelihood, written as:

$\log p_{\theta}(x|z)\geq L_{VAE}=\sum_{i=1}^{N}[E_{q_{\phi}(z|x)}[\log
p_{\theta}(x_{i}|z)]-D_{KL}(q(z|x_{i})||p(z))]$ (2)

The notations $\phi$, $\theta$ denote the parameterization of the encoder and
decoder in the VAE, z is the latent representation, N is the number of
training samples, $D_{KL}$( || ) is non-negative Kullback–Leibler(KL)
divergence, p(z) and $q_{\phi}$(z|x) are the prior and posterior distributions
respectively.

Although from the theoretical perspective, there seems to be no bound on the
prior and posterior distributions, but to make the optimization of the
objective function in 2 tractable both the distributions are approximated to
be Gaussians with diagonal covariance matrix. Specifically, the prior
distribution is set to standard Gaussians $P(Z)=\mathcal{N}(0,I)$. The encoder
in our VAE setup takes healthy brain image $x$ as input and then estimates the
mean and diagonal covariance matrix of the posterior for the given input
sample. These estimated parameters are used to sample the latent vector
corresponding to the input healthy scan through “reparameterization trick”.
The reparameterization technique is introduced to enable gradient calculation
with respect to the parameters of the encoder, which is generally not feasible
in this scenario. The sampled latent vector is passed through the decoder to
generate the reconstructed image $\hat{x}$.

###  7.2 FactorVAE

FactorVAE [12] closely shares modeling components and optimization objectives
with VAE along with some additional constraints on the latent space. The
trade-off between the two objective components in VAE does not allow the
posterior to achieve factorial nature similar to the prior. This factorial
nature would enhance the independence of the latent vectors and thus would add
more interpretability. The objective for FactorVAE is given as follows:

$L_{FactorVAE}=\sum_{i=1}^{N}[E_{q(z|x)}[\log
p(x|z)]-D_{KL}(q(z|x)||p(z))]-\gamma D_{KL}(q(z)||q\textquoteright(z))$ (3)

where, q(z) = $E_{p(x)}$[q(z|x)], $q\textquoteright(z)$:=product of (q($z_{j}$
)) (j=1: d), d is dimensionality of z. The objective 3 augments a new
component called Total Correlation (TC) to the objective function of VAE,
which minimizes the KL divergence between the marginal posterior q(z) and its
factorial representation $q\textquoteright(z)$, but these terms are generally
not tractable. So, a discriminator (D) is used to estimate the density ratio
that arises in the TC term. The discriminator basically classifies latent
vectors coming from each class q(z)/$q\textquoteright(z)$ while encouraging
the vectors to be factored.

###  7.3 GLOW

GLOW [14] learns an invertible nonlinear transformation between input
distribution and independent latent distribution. The invertibility is
facilitated by the fact that the layer is designed to have bijective
properties. Such bijective layers are cascaded to increase the
capacity/expressiveness of the model. This non-linear but invertible mapping,
together with the change-of-variables, would lead to a tractable and closed-
form solution of the marginal likelihood representing the healthy images as
in:

$\log p(x)=\log p(z)+\log|det(df_{\theta}(x)/dx)|$ (4)

where, p(z) is the prior, $f_{\theta}$ is the encoder,
det($df_{\theta}$(x)/dx) determinant of Jaccobian of the model.

This likelihood is decomposed in terms of 1) a fixed latent distribution,
which is a standard Gaussians $P(Z)=\mathcal{N}(0,I)$ and 2) Jacobian of the
model, constituting the training process. Since, GLOW learns an invertible
mapping, explicit training for the decoder is not required. The encoder itself
will uniquely determine the decoder. Due to bijective mapping, each point in
the input space is uniquely represented by a latent code. Similarly, distinct
latent codes are mapped to non-overlapping input images.

###  7.4 SSAE

SSAE [4] is an Auto-encoder based method that models normalcy without any
additional constraint in the latent space and forms the most widely used
baselines for pathology detection in unsupervised settings. SSAE disentangles
the high and low frequency of the input data by learning to compress and
reconstruct the laplacian pyramid of healthy MRI brain scans. Our application
requires laplacian pyramids at four levels (K=0,1,2,3). Images at each level k
are denoted as:

$I_{k}=d(g_{\sigma}(I_{k-1}))\qquad\forall\ 0<k\leq K,$ (5)

where $I_{0}$ is the input image x, $g_{\sigma}$ (·) is a gaussian kernel with
variance $\sigma$, d(·) is a downsampling operator. The high-frequency
residuals $H_{k}$ at each level k are denoted as:

$H_{k}=I_{k}-u(I_{k+1})\qquad\forall\ 0\leq k<K$ (6)

where u(·) is an upsampling operator. An image x is completely represented by
the low-resolution image $I_{K}$ after K downsamplings and the high frequency
residuals $H_{0}$, …$H_{K-1}$. the final reconstructed image is obtained via
the following:

$\hat{x}=\sum_{k=0}^{K-1}[u(I_{K-k})+H_{K-1-k}]$ (7)