# A novel method to test particle ordering and final state alignment in
helicity formalism

Mengzhen Wang  Center for High Energy Physics, Tsinghua University, Beijing,
China  Yi Jiang  University of Chinese Academy of Sciences, Beijing, China
Yinrui Liu  University of Chinese Academy of Sciences, Beijing, China  Wenbin
Qian  University of Chinese Academy of Sciences, Beijing, China  Xiaorui Lyu
University of Chinese Academy of Sciences, Beijing, China  Liming Zhang
Corresponding author<EMAIL_ADDRESS>Center for High Energy Physics,
Tsinghua University, Beijing, China

In this article, the importance is demonstrated of a proper choice of
reference particles for decay angle definitions, when constructing partial-
wave amplitude of multi-body decays using helicity formalism. This issue is
often ignored in the standard use case of the helicity formalism. A new
technique is proposed to determine the correct particle ordering, and can also
be used as a generalized method to calculate the rotation operators used for
the final-state alignment between different decay chains. Numerical
validations are also performed to support the arguments and new technique
proposed in this article.

##  1 Introduction

The partial-wave amplitude analysis is widely-used in experimental particle
physics to understand the resonant structures in multi-body decays. By using
the multi-dimensional phase-space variables, it gets more sensitivity to the
properties of resonant states than a one-dimensional mass-spectrum analysis,
and thus has become one of the most important techniques to explore exotic
hadrons and disentangle contributions from conventional contributions.
Helicity formalism [1] is one of the most popular techniques for constructing
partial-wave amplitude. It sets a guideline to construct the angle-dependent
amplitudes of two-body decays, which are further combined to form the
amplitude of a decay chain (the cascade decay series made up of several two-
body decays). The amplitudes of all the decay chains are combined to form the
total amplitude of multi-body decays. When particles with non-zero spins are
involved in the final state, a proper alignment of their spin axis should be
made between different chains before the combination [2], so that the final
state of different decay chains are defined consistently.

The helicity formalism has been used in the amplitude analysis of the
$\Lambda_{b}^{0}\to\psi pK^{-}$ decay in the LHCb experiment, where the first
observation of pentaquarks were made [2]. An alternative method for the
amplitude construction of a three-body decay, named as the Dalitz-Plot-
Decomposition (DPD) approach, is proposed in Ref. [3], and has been proved to
be equivalent to the one used in the LHCb analysis [3]. However, numerical
comparisons show an unexpected dependency between the the consistency of these
two formalisms and the choice of the reference particles when definiting the
two-body decay angles, denoted the “particle ordering issue”. Inspired by this
observation, further investigations are made on the general rule for the decay
amplitude construction using the helicity formalism.

The rest part of the article will be organised as follows: Sec. 2 briefly
introduces puzzles in the standard helisity formalism and the influence on the
decay amplitude; Sec. 3 proposes a systematic method, based on the final-state
alignment of different chains, to derive the correct helicity amplitude; Sec.
4 shows an example use case on the $\Lambda_{b}^{0}\to\psi pK^{-}$ amplitude
analysis. For a better clarification, the $\Lambda_{b}^{0}\to\psi pK^{-}$
three-body decay is used as an example through the text, but the discussions
and conclusions should validate for any other multi-body decays.

##  2 Puzzles in the standard helicity formalism

The total amplitude of the $\Lambda_{b}^{0}\to\psi pK^{-}$ decay is a sum of
the amplitudes for the $\Lambda^{*}$ chain, namely
$\Lambda_{b}^{0}\to\psi\Lambda^{*},\Lambda^{*}\to pK^{-}$, and the $P_{c}$
chain, namely $\Lambda_{b}^{0}\to P_{c}K^{-},P_{c}\to\psi p$ [2]. In the
standard helicity formalism [1], the amplitude for any two-body decay listed
above, denoted as $0\to 12$, where 0, 1 and 2 indicate different particles,
can be written as

$A_{0\to 12}(m_{0},\theta_{1},\phi_{1})=H^{0\to
12}_{\lambda_{1},\lambda_{2}}D^{J_{0}}_{\lambda_{0},\lambda_{1}-\lambda_{2}}(\phi_{1},\theta_{1},0)R(m_{0}),$
(1)

where $m_{0}$ denotes the invariant mass of the two-body system, $J_{0}$ is
the spin of particle 0. The angles $\theta_{1}$ and $\phi_{1}$ stand for the
polar and azimuthal angles of the momentum of particle 1 defined in the rest
frame of particle 0, as displayed in Fig. 1, and also named as the helicity
angles of the $0\to 12$ two-body decay. Particle 1, whose momentum is used to
define the helicity angles, is denoted hereafter the “reference particle” in
this two-body decay. The label $\lambda_{i}$ stands for the helicity of the
particle $i$, which is defined as its spin projection onto the direction of
the momentum. The angle-dependent part of the amplitude is described using the
Wigner D function
$D^{J_{0}}_{\lambda_{0},\lambda_{1}-\lambda_{2}}(\phi_{1},\theta_{1},0)$ [4],
corresponding to a rotation operator transfering the spin axis from the
initial stage111Usually defined as the direction of the momentum of particle 0
in the rest frame of its originating particle, or in the lab frame is particle
0 is the starting particle of the entire decay chain, to point to the momentum
direction of particle 1. The mass-dependency is denoted as a line-shape
function $R(m_{0})$, and a helicity coupling $H^{0\to
12}_{\lambda_{1},\lambda_{2}}$, which is a constant complex number, is used to
describe the decay dynamics.

(a)(b)

Figure 1: The helicity angles of the $0\to 12$ decay (a) calculated based on
the momentum of the particle 1, labelled as $(\theta_{1},\phi_{1})$ or (b)
calculated based on the momentum of the particle 2, labelled as
$(\theta_{2},\phi_{2})$.

The total amplitude of the $\Lambda_{b}^{0}\to\psi pK^{-}$ in Ref. [2] is
built following the standard procedure of the helicity formalism construction.
The decay amplitude of the $\Lambda^{*}$ chain is the product of the
amplitudes of the $\Lambda_{b}^{0}\to\psi\Lambda^{*}$ and $\Lambda^{*}\to
pK^{-}$ decays, and the decay amplitude of the $P_{c}$ chain is the product of
the amplitudes of the $\Lambda_{b}^{0}\to P_{c}K^{-}$ and $P_{c}\to\psi p$
decays. An alignment term, corresponding to a rotation within the decay plane,
is added in the $P_{c}$ chain amplitude before the combination of the two
decay chains. There are two potential defects remaining in this standard
procedure. The first one, denoted the “particle ordering issue”, is how to
make a correct choice of reference particles in each two-body decay. And the
second one, denoted the “particle-two factor issue”, is about a properly
description of the evolution on the spin axis, when the decay chain is
connected using particles not taken as the reference of a two-body decay.
These two issues are briefly mentioned in some other papers, such as Refs. [3,
5]. In this article, more discussions are given to highlight the importance of
these issues, in both analytic and numerical approaches.

###  2.1 Particle ordering issue

In the standard helicity formalism, there is in principle no preference on the
choice of the reference particles. For the $0\to 12$ decay, one should be able
to take either particle 1 or particle 2 as the reference. If particle 2 is
chosen, the angle-dependent part of the decay amplitude becomes
$D^{J_{0}}_{\lambda_{0},\lambda_{2}-\lambda_{1}}(\phi_{2},\theta_{2},0)$. As
shown in Fig. 1, we have $\theta_{1}+\theta_{2}=\pi$, and
$\phi_{2}=\phi_{2}^{\pm}=\phi_{1}\pm\pi$. The choice between $\phi_{2}^{+}$
and $\phi_{2}^{-}$ depends on how the range of the $\phi$ angles is defined.
The most natural choice is to define both $\phi_{1}$ and $\phi_{2}$ in the
same region, for example $[-\pi,\pi)$, and then $\phi_{2}^{+}$ is taken when
$\phi_{1}<0$, while $\phi_{2}^{-}$ is taken when $\phi_{1}>0$. Given the
properties of the Wigner D functions, we have

$D^{J_{0}}_{\lambda_{0},\lambda_{1}-\lambda_{2}}(\phi_{1},\theta_{1},0)=(-1)^{J_{0}+\lambda_{0}\pm\lambda_{0}}D^{J_{0}}_{\lambda_{0},\lambda_{2}-\lambda_{1}}(\phi_{2},\theta_{2},0),$
(2)

and the difference of the angle-dependent amplitude when taking different
reference particles is a factor of
$f^{\pm}=(-1)^{J_{0}+\lambda_{0}\pm\lambda_{0}}$, where $f^{+}$ is taken for
decays with $\phi_{1}<0$, and $f^{-}$ is taken for decays with $\phi_{1}>0$.
If particle 0 is a meson, $f^{\pm}={(-1)}^{J_{0}}$ is just a global factor
which can be absorbed by redefinition of helicity couplings. However, when
particle 0 is a baryon, we have $f^{+}=-f^{-}$. The value of $f^{\pm}$ is
different for events with $\phi_{1}>0$ or $\phi_{1}<0$. This minus sign has no
effect on the module square of the amplitude of one decay chain. However, it
becomes non-negligible when multiple decay chains are considered, by directly
influencing the behaviour of the interference terms, and it is impossible to
eliminate this phase-space-dependent factor by introducing any global terms
shared by all the events.

As an example, for the $\Lambda_{b}^{0}\to\psi pK^{-}$ amplitude analysis, the
reference particle of the $\Lambda^{*}\to pK^{-}$ decay is taken as the
$K^{-}$ particle in Ref. [2], and there is no particular reason of not taking
the proton as the reference. Two possible settings of reference particles
involved in the $\Lambda_{b}^{0}\to\psi pK^{-}$ amplitude analysis are listed
in Table. 1, where “ordering 1” corresponds to the choice of the LHCb analysis
[2], and “ordering 2” stands for the nominal choice used in the DPD paper
which makes sure the decay planes for all the decay chains are identical in
the “aligned center-of-momentum frame” [3]. The only difference between these
two orderings appears in decay angle definition of the $\Lambda^{*}\to pK^{-}$
decay, and they cannot be both correct, as discussed above, without
introducing corrections which are handled differently when
$\phi_{\Lambda^{*}\to pK^{-}}$ is larger or smaller than zero, where
$\phi_{\Lambda^{*}\to pK^{-}}$ stands for the azimuth decay angle of this two-
body decay. The technique for implementing this kind of correction is
discussed later in Sec. 5.

The contributions of the estimated interference term between the amplitudes of
the $P_{c}$ and $\Lambda^{*}$ chains, as a function of $\phi_{\Lambda^{*}\to
pK^{-}}$, calculated using the standard helicity formalism under ordering 1
and ordering 2, are shown in Fig. 2. When ordering 1 is used, an unphysical
discontinuity is seen at $\phi_{\Lambda^{*}\to pK^{-}}=0$, indicating a
potential problematic issue here, and the opposite behaviour of the
interference term when $\phi_{\Lambda^{*}\to pK^{-}}>0$ and
$\phi_{\Lambda^{*}\to pK^{-}}<0$ leads to a significant cancellation of the
interference contribution when integrating the full $\phi_{\Lambda^{*}\to
pK^{-}}$ regions, which can explain the zero interference contribution between
the $P_{c}$ and $\Lambda^{*}$ chains found in the ordering 1 configuration
used in the LHCb analysis [2].

Figure 2: The contribution from the interference term between the
$\Lambda^{*}$ and $P_{c}$ chains as a function of $\phi_{\Lambda^{*}\to
pK^{-}}$, obtained using simulated $\Lambda_{b}^{0}\to\psi pK^{-}$ events. The
left and right figures are based on particle ordering 1 and ordering 2,
respectively.

In the DPD formula [3], the $\phi$-angle related terms are shared by all the
decay chains, so it does not suffer from the potential non-global minus sign
in the interference between different chains. A numerical comparison between
the DPD formula and the standard helicity formalism helps to understand which
ordering is correct. The decay amplitudes using both the two formulas are
calculated for each simulated event. As shown in Fig. 3, these two formulas
are equivalent when ordering 2 is used. It also indicates that ordering 1
needs further treatment.

Table 1: The choice of the reference particles for each two-body decay involved in the $\Lambda_{b}^{0}\to\psi pK^{-}$ amplitude analysis. Ordering 1 and ordering 2 correspond to the choices in Ref. [2] and Ref. [3], respectively. | Ordering 1 | Ordering 2
---|---|---
$\Lambda_{b}^{0}\to\psi\Lambda^{*}$ | $\Lambda^{*}$ | $\Lambda^{*}$
$\Lambda^{*}\to pK^{-}$ | $K^{-}$ | $p$
$\Lambda_{b}^{0}\to P_{c}K^{-}$ | $P_{c}$ | $P_{c}$
$P_{c}\to\psi p$ | $\psi$ | $\psi$
Figure 3: Two dimensional distributions of the amplitude module square of the
simulated $\Lambda_{b}^{0}\to\psi pK^{-}$ decays based on ordering 2. The
x-axis stands for the amplitude module square calculated using the formula in
Ref. [2], while the y-axis is calculated using the DPD formula [3]. The mass-
dependent terms and the helicity couplings are generated randomly and is
shared in the two formulas.

###  2.2 Particle-two factor issue

In the DPD paper [3], the “Jacob-Wick particle-2 phase convention” is
introduced. In this section, the particle-2 phase convention is explained in
an alternative way, by introducing a “particle-two factor” to properly
describe the transition of the spin axis. The effect on the particle parity
determinations is also demonstrated to support the necessity of including this
factor in the decay amplitude.

The rotation operator indicated in Eq. 1 results in a spin axis along the same
direction as the momentum of particle 1. However, to consider the further
decay of particle 2, the usually used initial spin axis is along the direction
of the momentum of particle 2, which calls for an additional term between the
amplitudes of $0\to 12$ process and the further particle 2 decay. This
additional term is named as the particle-two factor, and is generally ignored
in the standard procedure to construct the helicity amplitude.

To get the exact form of particle-two factor, let’s define a Cartesian
coordinate system $(x_{1},y_{1},z_{1})$: as shown in Fig. 1, $\vec{z}_{1}$ and
$\vec{z}_{2}$ are the directions of the momentum of particle 1 and 2,
respectively, $\vec{y}_{1}$ is the normal vector of the $0\to 12$ decay plane,
and $\vec{x}_{1}=\vec{y}_{1}\times\vec{z}_{1}$. A rotation operator,
corresponding to an angle of $\pi$, along any axis in the $x_{1}-y_{1}$ plane
should be able to transfer spin axis from $\vec{z}_{1}$ to $\vec{z}_{2}$.
Difference choices of the rotation axis result in different $\vec{x}_{2}$ and
$\vec{y}_{2}$ axis, leading to different definitions of the helicity angles of
the particle 2 decay, which are calculated using the directions of
$\vec{z}_{2}$, $\vec{x}_{2}$ and $\vec{y}_{2}$ as inputs [2]. A Wigner D
function corresponds to this rotation operator should be taken as the
particle-two factor.

The most natural choice of the rotation axis is the $\vec{x}_{1}$ or
$\vec{y}_{1}$ vector, both of which should be correct if the technique
proposed later in Sec. 3 is used. In Ref. [2], the $\psi$ meson is not the
reference particle in the $\Lambda_{b}^{0}\to\Lambda^{*}\psi$ decay, and a
rotation along the $\vec{x}_{1}$ axis is considered before calculating the
$\psi\to\mu^{+}\mu^{-}$ decay angles. The corresponding particle-two factor,

$<J_{2},-\lambda_{2}|R_{x}(\pi)|J_{2},\lambda_{2}>=(-1)^{J_{2}},$ (3)

is a constant parameter and can be absorbed into the definition of the
helicity couplings. In this article, we suggest always taking the
$\vec{y}_{1}$ axis as the rotation axis, which can avoid rolling over the
decay planes to complicate the picture of helicity amplitude construction .
The corresponding particle-two factor is

$<J_{2},-\lambda_{2}|R_{y}(\pi)|J_{2},\lambda_{2}>=d_{-\lambda_{2},\lambda_{2}}^{J_{2}}=(-1)^{J_{2}-\lambda_{2}},$
(4)

where $J_{2}$ and $\lambda_{2}$ are the spin and helicity of particle 2,
respectively.

The particle-two factor is important to properly associate the helicity
couplings and their $LS$ representations, which is discussed in Ref. [3], and
is also essential for the parity determination of the resonant states, as
shown below. When particle 2 is a meson, this factor can be absorbed in the
definition of the helicity coupling, namely $H^{\prime 0\to
12}_{\lambda_{1},\lambda_{2}}=(-1)^{J_{2}-\lambda_{2}}H^{0\to
12}_{\lambda_{1},\lambda_{2}}$, and generates no visible effect in the
amplitude analysis. If particle 2 is a baryon, the effect becomes non-
negligible. The parity conservation requires that

$H^{0\to
12}_{-\lambda_{1},-\lambda_{2}}=P_{0}P_{1}P_{2}(-1)^{J_{1}+J_{2}-J_{0}}H^{0\to
12}_{\lambda_{1},\lambda_{2}},$ (5)

where $P$ stands for the parity of the particles [2]. Under the parity
transformation, the particle-two factor varies from $(-1)^{J_{2}-\lambda_{2}}$
to $(-1)^{J_{2}+\lambda_{2}}$, generating an additional minus sign. One cannot
absorb the particle-two factor of a baryon into the definition of the
corresponding helicity coupling, whose behaviour under parity transformation
would be otherwise modified, resulting in wrong determinations of particle
parities.

If the further decay of particle 2 is not considered in the amplitude
analysis, we suggest also adding the particle-two factor after the $0\to 12$
decay amplitude, to manage all the two-body decay amplitudes in a consistent
way. Otherwise, it acts as an additional term for the final-state alignment
between different chains.

##  3 A technique for reference particle determination

As demonstrated in the previous section, the choice of the reference particle
is not arbitrary, when fermions are involved in the decay process. It calls
for a guideline to make the correct choice, or to add an additional term to
the decay amplitude to cancel the non-global effect caused by switching the
reference particles. Both of these two features are not well-described in the
standard helicity formalism. A comparison between the standard helicity
formalism and the DPD formula [3] is a possible approach, but it works only
for three-body decays, and does not uncover the nature of this issue. In this
section, we try to solve this puzzle by investigating the final-state
alignment between different decay chains.

In the $\Lambda_{b}^{0}\to\psi pK^{-}$ amplitude analysis [2], before the
combination of the amplitude of the $\Lambda^{*}$ and $P_{c}$ chains, the spin
state of the proton and muons should be properly aligned to be the same in
different chains. The technique to correctly define the spin state of the
decay products is raised in Ref. [5], where the final spin states are obtained
by carefully considering how they are associated to the initial spin state
shared by all the decay chains. In this article, we focus on the operators
connecting the initial and the final spin states, and propose an equation to
define a correct final-state alignment in a mathematical way, which can be
used to validate whether the alignment is handled properly when ordering 1 or
2 is taken. It can also be used as a generalized method to determine the
correct alignment term in the decay amplitude, once the choice of the
reference particles is fixed.

The target for the spin-axis alignment is to make sure that the combined
$\Lambda^{*}$ chain and $P_{c}$ chain amplitudes are expressed indicating the
same initial and final states. With a proper alignment, the rotation
operators, which connect the initial and final spin states, should be
identical in the $\Lambda^{*}$ and $P_{c}$ chains, and a mathematical
description is

$R_{{\rm Euler},\Lambda^{*}}=B_{P_{c}}B^{-1}_{\Lambda^{*}}R_{{\rm
align},P_{c}}R_{{\rm Euler},P_{c}}$ (6)

where the subscripts $\Lambda^{*}$ and $P_{c}$ are used to denote two decay
chains, $R_{\rm Euler}$ stands for the cascade process of the Euler rotations
associated to each two-body decays in the corresponding decay chain, with the
Euler angles exactly taken from the decay angles in the amplitude. The symbol
$R_{\rm align}$ stands for the alignment rotation for a consistent final state
definition between $P_{c}$ and $\Lambda^{*}$ chains. Two boost operators,
$B_{P_{c}}$ and $B^{-1}_{\Lambda^{*}}$, are involved to consider the
difference between the reference frames under which the direction of the final
spin axis is defined. The operator $B$ stands for the boost from the
$\Lambda_{b}^{0}$ rest frame , to the rest frame of the final-state particles,
through the intermediate resonances involved in each decay chain.

Once the particle ordering is determined, the decay angles of all the two-body
decays can be calculated [1, 2], based on which $R_{\rm Euler}$ can be
obtained. The boost operators can be calculated using the four-momentum
information, and the alignment rotation becomes the only unknown part of Eq.
6. By solving Eq. 6 in either an analytic or a numerical way, one could
determine the correct alignment rotations. If the alignment angle has been
determined using other approaches [2], Eq. 6 can also be used to validate
whether the alignment is properly performed.

As all the angles in Eq. 6 are directly taken from the decay amplitude, it
should be sensitive to the switch of the reference particles, and should
generate visible effect once a proper representation is assigned on the
rotation operators. For the $\Lambda_{b}^{0}\to\psi
pK^{-},\psi\to\mu^{+}\mu^{-}$ amplitude analysis, the final-state particles
related to the puzzles in Sec. 2 are all spin-half states. So, in this
article, the two-dimensional representation of the SU(2) group is used to
describe the rotation operators. As the Wigner D function is the $jm$
representation of the SU(2) group, this should be a good choice to visualize
the properties in Eq. 2. The rotations along the z-axis, y-axis and an
arbitary axis labelled as $\vec{a}$ are expressed using :

$R_{z}(\alpha)=\left(\begin{array}[]{cc}e^{-i\alpha/2}&0\\\ 0&e^{i\alpha/2}\\\
\end{array}\right),$ (7)

$R_{y}(\alpha)=\left(\begin{array}[]{cc}\cos(\alpha/2)&-\sin(\alpha/2)\\\
\sin(\alpha/2)&\cos(\alpha/2)\\\ \end{array}\right),$ (8)

and

$R(\alpha,\vec{a})=R_{z}(\phi_{a})R_{y}(\theta_{a})R_{z}(\alpha)R_{y}(-\theta_{a})R_{z}(-\phi_{a}),$
(9)

respectively, where $\alpha$ stands for the rotation angle, with $\theta_{a}$
and $\phi_{a}$ denoting the polar and azimuthal angles of vector $\vec{a}$.
The corresponding representation of the boost operators, inspired by the
$(\frac{1}{2},0)$ representation of the Lorentz group [6], with a rapidity of
$\gamma$, along the z-axis and along any axis labelled as $\vec{b}$ are

$B_{z}(\gamma)=\left(\begin{array}[]{cc}e^{-\gamma/2}&0\\\ 0&e^{\gamma/2}\\\
\end{array}\right),$ (10)

and

$B(\gamma,\vec{b})=R_{z}(\phi_{b})R_{y}(\theta_{b})B_{z}(\gamma)R_{y}(-\theta_{b})R_{z}(-\phi_{b}),$
(11)

respectively, where $\theta_{b}$ and $\phi_{b}$ are the polar and azimuthal
angles of vector $\vec{b}$.

##  4 Validate different orderings for $\Lambda_{b}^{0}\to\psi pK^{-}$
amplitude analysis

An example is shown in this section for the validation of alignment for the
proton spin axis between the $P_{c}$ and $\Lambda^{*}$ chains. Following the
methodology of calculating both the decay angles and the alignment angles in
Ref. [2], the two choices of reference particles listed in Table. 1 are
validated by checking if Eq. 6 is satisfied.

###  4.1 Rotations when using ordering 1

If ordering 1 is used, the corresponding proton-related Euler rotation for the
$\Lambda^{*}$ chain is

$R_{\rm
Euler,\Lambda^{*}}=R(\pi,\vec{p}_{\Lambda^{*}}^{\Lambda_{b}^{0}}\times\vec{p}_{K}^{\Lambda^{*}})R(\theta_{\Lambda^{*}},\vec{p}_{\Lambda^{*}}^{\Lambda_{b}^{0}}\times\vec{p}_{K}^{\Lambda^{*}})R(\phi_{K},\vec{p}_{\Lambda^{*}}^{\Lambda_{b}^{0}})R(\theta_{\Lambda_{b}^{0}},\vec{p}_{\Lambda^{*}}^{\Lambda_{b}^{0}}\times\vec{p}_{\Lambda_{b}^{0}}^{lab})R(\phi_{\Lambda^{*}},\vec{p}_{\Lambda_{b}^{0}}^{lab}),$
(12)

where $\theta_{\Lambda_{b}^{0}}$ and $\phi_{\Lambda^{*}}$ are the helicity
angles of the $\Lambda_{b}^{0}\to\Lambda^{*}\psi$ decay,
$\theta_{\Lambda^{*}}$ and $\phi_{K}$ are the helicity angles of the
$\Lambda^{*}\to pK^{-}$ decay [2], and the symbol $\vec{p}_{a}^{b}$ is used to
denote the direction of the momentum of particle $a$ in the $b$ rest frame.
The rotation operator
$R(\theta_{\Lambda_{b}^{0}},\vec{p}_{\Lambda^{*}}^{\Lambda_{b}^{0}}\times\vec{p}_{\Lambda_{b}^{0}}^{lab})R(\phi_{\Lambda^{*}},\vec{p}_{\Lambda_{b}^{0}}^{lab})$
corresponds to the helicity amplitude of the
$\Lambda_{b}^{0}\to\Lambda^{*}\psi$ decay, and it transfers the spin axis from
the direction of the $\Lambda_{b}^{0}$ momentum in the lab frame to that of
the $\Lambda^{*}$ momentum in the $\Lambda_{b}^{0}$ rest frame. The operator
$R(\pi,\vec{p}_{K}^{\Lambda^{*}})R(\theta_{\Lambda^{*}},\vec{p}_{\Lambda^{*}}^{\Lambda_{b}^{0}}\times\vec{p}_{K}^{\Lambda^{*}})$
corresponds to the helicity amplitude of the $\Lambda^{*}\to pK^{-}$ decay,
and the direction spin axis is transfered along the direction of the $K^{-}$
momentum in the $\Lambda^{*}$ rest frame. An operator
$R(\pi,\vec{p}_{\Lambda^{*}}^{\Lambda_{b}^{0}}\times\vec{p}_{K}^{\Lambda^{*}})$
is added, corresponds to the particle-two factor described in Sec. 2.2, as
proton is not the reference particle for the $\Lambda^{*}\to pK^{-}$ decay
angle definition. Similarly, for the $P_{c}$ chain, the Euler rotation is

$R_{\rm
Euler,P_{c}}=R(\pi,\vec{p}_{\psi}^{P_{c}}\times\vec{p}_{K}^{P_{c}})R(\theta_{P_{c}},\vec{p}_{\psi}^{P_{c}}\times\vec{p}_{K}^{P_{c}})\\\
R(\phi_{\psi}^{P_{c}},\vec{p}_{P_{c}}^{\Lambda_{b}^{0}})R(\theta_{\Lambda_{b}^{0}}^{P_{c}},\vec{p}_{P_{c}}^{\Lambda_{b}^{0}}\times\vec{p}_{\Lambda_{b}^{0}}^{lab})R(\phi_{P_{c}},\vec{p}_{\Lambda_{b}^{0}}^{lab}),$
(13)

where
$R(\theta_{\Lambda_{b}^{0}}^{P_{c}},\vec{p}_{P_{c}}^{\Lambda_{b}^{0}}\times\vec{p}_{\Lambda_{b}^{0}}^{lab})R(\phi_{P_{c}},\vec{p}_{\Lambda_{b}^{0}}^{lab})$
and  $R(\theta_{P_{c}},\vec{p}_{\psi}^{P_{c}}\times\vec{p}_{K}^{P_{c}})\\\
R(\phi_{\psi}^{P_{c}},\vec{p}_{P_{c}}^{\Lambda_{b}^{0}})$ correspond to the
helicity amplitude of the $\Lambda_{b}^{0}\to P_{c}K^{-}$ and $P_{c}\to\psi p$
decays [2], respectively, and the spin axis is transfered from the same
initial stage as that in the $\Lambda^{*}$ chain, along the direction of the
$\psi$ momentum in the $P_{c}$ rest frame. The operator
$R(\pi,\vec{p}_{\psi}^{P_{c}}\times\vec{p}_{K}^{P_{c}})$ corresponds to the
particle-two factor. The alignment rotation is expressed in

$R_{\rm
align,P_{c}}=R(\pi,\vec{p}_{K}^{\Lambda^{*}})R(\theta_{p},\vec{p}_{\psi}^{P_{c}}\times\vec{p}_{K}^{P_{c}}),$
(14)

corresponding to the alignment term between the $\Lambda^{*}$ and $P_{c}$
chains used in Ref. [2], where the alignment angle $\theta_{p}$ is defined as
the angle between the $K^{-}$ and $\psi$ particles in the proton rest frame
and the value is restricted in $\theta_{p}\in(0,\pi)$ . When ordering 1 is
used, the normal directions of the decay planes are opposite in the
$\Lambda^{*}$ and $P_{c}$ chains, so an additional rotation
$R(\pi,\vec{p}_{K}^{\Lambda^{*}})$, with an angle of $\pi$ along the spin
axis, is introduced to eliminate this difference.

###  4.2 Rotations when using ordering 2

If ordering 2 is used, the Euler rotation for the $\Lambda^{*}$ chain becomes

$R_{\rm
Euler,\Lambda^{*}}=R(\theta_{\Lambda^{*}},\vec{p}_{\Lambda^{*}}^{\Lambda_{b}^{0}}\times\vec{p}_{p}^{\Lambda^{*}})R(\phi_{p},\vec{p}_{\Lambda^{*}}^{\Lambda_{b}^{0}})R(\theta_{\Lambda_{b}^{0}},\vec{p}_{\Lambda^{*}}^{\Lambda_{b}^{0}}\times\vec{p}_{\Lambda_{b}^{0}}^{lab})R(\phi_{\Lambda^{*}},\vec{p}_{\Lambda_{b}^{0}}^{lab}).$
(15)

The Euler rotation for the $P_{c}$ chain becomes

$R_{\rm
Euler,P_{c}}=R(\pi,\vec{p}_{\psi}^{P_{c}}\times\vec{p}_{K}^{P_{c}})R(\theta_{P_{c}},\vec{p}_{\psi}^{P_{c}}\times\vec{p}_{K}^{P_{c}})R(\phi_{\psi}^{P_{c}},\vec{p}_{P_{c}}^{\Lambda_{b}^{0}})R(\theta_{\Lambda_{b}^{0}}^{P_{c}},\vec{p}_{P_{c}}^{\Lambda_{b}^{0}}\times\vec{p}_{\Lambda_{b}^{0}}^{lab})R(\phi_{P_{c}},\vec{p}_{\Lambda_{b}^{0}}^{lab}),$
(16)

and the alignment rotation $R_{\rm align,P_{c}}$ is

$R_{\rm
align,P_{c}}=R(\theta_{p},\vec{p}_{\psi}^{P_{c}}\times\vec{p}_{K}^{P_{c}}).$
(17)

The definition of all the angles are almost the same as that for ordering 1,
except that $\theta_{\Lambda^{*}}$ and $\phi_{p}$, which are the
$\Lambda^{*}\to pK^{-}$ decay angles defined using the proton, rather than the
$K^{-}$ particle as the reference. The rotation operator
$R(\pi,\vec{p}_{\psi}^{P_{c}}\times\vec{p}_{K}^{P_{c}})$ corresponds to the
particle-two factor mentioned in Sec. 2.2.

###  4.3 Boost operators

The boost operators are defined in the same way for both ordering 1 and
ordering 2. The operator $B_{\rm\Lambda^{*}}$ first boosts the
$\Lambda_{b}^{0}$ rest frame to the $\Lambda^{*}$ rest frame, and then to the
proton rest frame, namely

$B_{\rm\Lambda^{*}}=B(-y_{p}^{\Lambda^{*}},\vec{p}_{p}^{\Lambda^{*}})B(-y_{\Lambda^{*}}^{\Lambda_{b}^{0}},\vec{p}_{\Lambda^{*}}^{\Lambda_{b}^{0}}),$
(18)

where the symbol $y_{a}^{b}$ stands for the rapidity of particle a,

$y=\frac{1}{2}\ln(\frac{E_{a}+p_{a}}{E_{a}-p_{a}}),$ (19)

defined in the rest frame of particle b. Similarly $B_{\rm P_{c}}$ first boost
the $\Lambda_{b}^{0}$ rest frame to the $P_{c}$ rest frame, and then to the
proton rest frame, namely

$B_{\rm
P_{c}}=B(-y_{p}^{P_{c}},\vec{p}_{p}^{P_{c}})B(-y_{P_{c}}^{\Lambda_{b}^{0}},\vec{p}_{P_{c}}^{\Lambda_{b}^{0}}),$
(20)

###  4.4 Validate the alignment equation

For a better visibility of validation of Eq. 6, the distance between the
matrices in its left and right sides is defined as

$D=\sum_{i,j}|L_{i,j}-R_{i,j}|^{2},$ (21)

where $L$ stands for the left side, namely $R_{\rm Euler,\Lambda^{*}}$, while
$R$ for the right side, namely $B_{P_{c}}B_{\Lambda^{*}}^{-1}R_{\rm
align,P_{c}}R_{\rm Euler,P_{c}}$, and the subscripts $i$ and $j$ are the row
and column indexes for $L$ or $R$ matrices, respectively. A distance $D$ of
zero indicates that $L$ equals to $R$. As both $L$ and $R$ are $2\times 2$
unitary matrices, when $L=-R$ the distance becomes $D=8$. Figure 4 shows the
distribution of $D$ when all the helicity and alignment angles are calculated
using the method proposed in Ref. [2]. As shown in Fig. 4, the alignment is
performed properly only for half of the events with $\phi_{\Lambda^{*}\to
pK^{-}}>0$ if ordering 1 is taken, and for all the events if ordering 2 is
used. This is consistent with the numerical calculations discussed in Section
2.1, and demonstrate why ordering 2 is the correct choice.

Figure 4: The distribution of $D$ obtained with the helicity and alignment
angles obtained using the method proposed in Ref. [2]. The figure in the left
is obtained based on ordering 1, where half of the candidates have a perfect
alignment between the two chains, while the other half $D=8$, corresponding to
a minus sign difference between the rotation matrices of the two chains. The
figure in the right is obtained ordering 2, where all the rotation matrices of
the two chains are perfectly aligned for all the generated events.

##  5 Another use case: The correct ordering-1 based decay amplitude

In the above discussions in this section, the alignment angle $\theta_{p}$ is
fixed to the angle between the $K^{-}$ and $\psi$ momenta in the proton rest
frame, and the validation is made on the particle orderings. There is another
use case where the reference particles are determined, and one can get the
correct alignment rotation formula by solving Eq. 6. For example, if ordering
1 is used for the $\Lambda_{b}^{0}\to\psi pK^{-}$ amplitude analysis, the
correct alignment rotation is exactly the one in Eq. 14 for events with
$\phi_{\Lambda^{*}\to pK^{-}}>0$, as shown in Fig. 4. For the rest events with
$D=8$, Eq. 14 is almost the correct solution, except for an additional minus
sign. This can be solved by changing $\theta_{p}$ to $\theta_{p}+2\pi$ for
events with $\phi_{\Lambda^{*}\to pK^{-}}<0$. This non-global $2\pi$ factor
can recover the non-global minus sign introduced by a improper particle
ordering mentioned in Sec. 2.1.

For decays with more than three final-state particles, the alignment rotation
obtained from Eq. 6 can be more complicated, which needs to be expressed using
both the polar and azimuthal angles. The rotation matrix should be transfered
into the Euler style under a corresponding Cartesian coordinate system, which
makes it easier to get correct alignment term in the decay amplitude. Let’s
take Fig. 1 as an example, and consider the alignment of particle 1. The Euler
angles should be calculated in the $(x_{1},y_{1},z_{1})$ system, and the Euler
rotation for the spin-axis alignment can be written as

$R_{z_{1}}(\alpha)R_{y_{1}}(\beta)R_{z_{1}}(\gamma),$ (22)

The direction of $\vec{x}_{1},\vec{y}_{1}$ and $\vec{z}_{1}$ can be derived
using the four momentum information [2], based on which the Euler rotation
connecting the $(x_{1},y_{1},z_{1})$ and $(x,y,z)$ frames can be obtained,
labelled as $R_{\rm Euler,trans}$. Then we have

$R_{z_{1}}(\alpha)R_{y_{1}}(\beta)R_{z_{1}}(\gamma)=R_{\rm
Euler,trans}R_{z}(\alpha)R_{y}(\beta)R_{z}(\gamma)R^{-1}_{\rm Euler,trans}$
(23)

and

$R_{z}(\alpha)R_{y}(\beta)R_{z}(\gamma)=\left(\begin{array}[]{cc}e^{-i(\alpha+\gamma)/2}\cos(\beta/2)&-e^{-i(\alpha-\gamma)/2}\sin(\beta/2)\\\
e^{i(\alpha-\gamma)/2}\sin(\beta/2)&e^{i(\alpha+\gamma)/2}\cos(\beta/2)\\\
\end{array}\right).$ (24)

If the alignment rotation obtained from Eq. 6 is

$R_{\rm Euler}=\left(\begin{array}[]{cc}a&b\\\ c&d\\\ \end{array}\right),$
(25)

we can translate it into

$R^{-1}_{\rm Euler,trans}R_{\rm Euler}R_{\rm
Euler,trans}=\left(\begin{array}[]{cc}a^{\prime}&b^{\prime}\\\
c^{\prime}&d^{\prime}\\\ \end{array}\right),$ (26)

and the Euler angles for the alignment term can be determined using the
following relations

$\begin{split}\cos(\beta/2)=|a^{\prime}|&,\sin(\beta/2)=|b^{\prime}|,\\\
\alpha+\gamma&=-2{\rm Arg}(a^{\prime}),\\\ \alpha-\gamma&=2{\rm
Arg}(c^{\prime}).\end{split}$ (27)

##  6 Suggestions for the $\Lambda_{b}^{0}\to\psi pK^{-}$ amplitude analysis

Given the discussions above, we suggest several modifications on the
$\Lambda_{b}^{0}\to\psi pK^{-}$ decay amplitude with respect to the one used
in Ref. [2], based on either ordering 1 or ordering 2.

If ordering 2 is used, the modifications are:

  * •

For the $\Lambda^{*}\to pK^{-}$ decay, use the proton as the reference
particle

  * •

Add the particle-two factors in the decay amplitude, including a term of
$(-1)^{J_{p}-\lambda_{p}}$ to the $P_{c}$ chain’s amplitude, and a term of
$(-1)^{J_{\psi}-\lambda_{\psi}}$ to the $\Lambda^{*}$ chain’s amplitude. As
pointed out in Sec. 2.2, the suggested rotation axis to consider the particle-
two factor in this article differs from the one used in Ref. [2], so the decay
angles of the $\psi\to\mu^{+}\mu^{-}$ process in the $\Lambda^{*}$ chain
should be modified accordingly, where the $\phi$ angle should be changed to
$\phi+\pi$ when $\phi<0$, and $\phi-\pi$ when $\phi>0$. The modification of
the $\phi$ angle definition is just for consistency, but will not influence
the major results of the amplitude analysis, as the additional $\pm\pi$ phase
just contributes to a global minus sign to the $\Lambda^{*}$ chain’s amplitude
[2].

If ordering 1 is used, the modifications are:

  * •

For events with $\phi_{\Lambda^{*}\to pK^{-}}<0$, change the alignment angle
$\theta_{p}$ to $\theta_{p}+2\pi$

  * •

The operator $R(\pi,\vec{p}_{K}^{\Lambda^{*}})$ in Eq. 14 should also be
considered as part of the alignment rotation. The alignment term between the
$P_{c}$ and $\Lambda^{*}$ chains should be changed from
$d_{\lambda_{p},\lambda^{\prime}_{p}}^{\frac{1}{2}}(\theta_{p})$ to
$D_{\lambda_{p},\lambda^{\prime}_{p}}^{\frac{1}{2}}(\pi,\theta_{p},0)$.

##  7 Conclusion

The partial-wave amplitude analysis plays an important role for investigating
the properties of the resonant structures in multi-body decays. The helicity
formalism, as a widely-used technique to construct the decay amplitude, has
been adopted to successfully discover or precisely measure the properties of
both exotic and conventional resonant states. However, the principle of
choosing the reference decay product when calculating the helicity angles of
two-body decays, namely the particle ordering issue, is not often discussed in
the traditional usage of the helicity formalism.

In this article, we have first demonstrate the necessity of carefully
considering the particle ordering issue, especially for the decays involving
spin-half-integral particles where the choice of the reference particles has a
non-negligible influence on the interference term. Then a new technique has
been proposed to validate whether the decay amplitude is correctly written
under a dedicated particle ordering. This technique checks if the rotation
operators involved in different decay chains properly align the spin axes of
final-state particles. A dedicated representation for the operators has been
proposed, which could help experimentalists to do event-by-event check of the
final-state alignment in a numerical way. Using this new technique, a proper
final-state alignment can be reached with any given particle orderings, and
the inconsistency between different orderings can be cancelled by assigning
different alignment rotation operators. Numerical calculations using the
simulated $\Lambda_{b}^{0}\to\psi pK^{-}$ decays [2] have also been shown as
an example. The technique proposed in this article will help the ongoing and
future particle-wave amplitude analysis of decays with baryons, for example
$\Lambda_{c}^{+}\to pK^{-}\pi^{+}$, $\Lambda_{b}^{0}\to D^{0}p\pi^{-}$,
$B^{0}\to D^{0}p\bar{p}$ and $\psi(2S)\to\eta p\bar{p}$, to construct the
decay amplitude in a correct way.

## Acknowledgements

We thank Mikhail Mikhasenko and Tomasz Skwarnicki for the valuable discussions
about the Dalitz-plot decomposition formalism, which inspires the observation
of the particle ordering issue and motivate us to further investigate the
helicity formalism, and also for their highlighting to the Jacob-Wick
particle-2 phase convention in the amplitude analysis. We thank Andy Beiter,
Chen Chen, Mikhail Mikhasenko, Alessandro Pilloni, César Fernández Ramírez,
Adam Szczepaniak, Tomasz Skwarnicki, Zhihong Shen and Zehua Xu for the nice
discussions about the helicity formalism.

## References

  * [1] J. D. Richman, _An Experimenter’s Guide to the Helicity Formalism_ ,
  * [2] LHCb, R. Aaij et al., _Observation of $J/\psi p$ Resonances Consistent with Pentaquark States in $\Lambda_{b}^{0}\to J/\psi K^{-}p$ Decays_, Phys. Rev. Lett. 115 (2015) 072001, arXiv:1507.03414
  * [3] JPAC, M. Mikhasenko et al., _Dalitz-plot decomposition for three-body decays_ , Phys. Rev. D 101 (2020) 034033, arXiv:1910.04566
  * [4] Particle Data Group, P. A. Zyla et al., _Review of Particle Physics_ , PTEP 2020 (2020) 083C01
  * [5] D. Marangotto, _Helicity amplitudes for generic multi-body particle decays featuring multiple decay chains_ , arXiv:1911.10025
  * [6] M. Srednicki, _Quantum field theory. Part 2. Spin one half_ , arXiv:hep-th/0409036


# Bicubic++: Slim, Slimmer, Slimmest
Designing an Industry-Grade Super-Resolution Network

Bahri Batuhan Bilecen Mustafa Ayazoglu
Aselsan Research
Ankara, Türkiye
<EMAIL_ADDRESS>
###### Abstract

We propose a real-time and lightweight single-image super-resolution (SR)
network named Bicubic++. Despite using spatial dimensions of the input image
across the whole network, Bicubic++ first learns quick reversible downgraded
and lower resolution features of the image in order to decrease the number of
computations. We also construct a training pipeline, where we apply an end-to-
end global structured pruning of convolutional layers without using metrics
like magnitude and gradient norms, and focus on optimizing the pruned
network’s PSNR on the validation set. Furthermore, we have experimentally
shown that the bias terms take considerable amount of the runtime while
increasing PSNR marginally, hence we have also applied bias removal to the
convolutional layers. Our method adds $\sim$1dB on Bicubic upscaling PSNR for
all tested SR datasets and runs with $\sim$1.17ms on RTX3090 and $\sim$2.9ms
on RTX3070, for 720p inputs and 4K outputs, both in FP16 precision. Bicubic++
won NTIRE 2023 RTSR Track 2 $\times$3 SR competition and is the fastest among
all competitive methods. Being almost as fast as the standard Bicubic
upsampling method, we believe that Bicubic++ can set a new industry standard.

Portion of LR | Bicubic (1 ms) | Ours (2.9 ms) |  ESPCN (9.6 ms) [29]
---|---|---|---
|  |  |
|  |  |
XCAT (11.7 ms) [4] |  ABPN (15.8 ms) [10] |  XLSR (25.3 ms) [3] |  FSRCNN (33.8 ms) [9]

Figure 1: Image 70 from the NTIRE 2023 RTSR Challenge [7] Track 2 $\times$3
test set [36], where the data is JPEG Q=90 degraded. Our practical method runs
much faster ($<$ 3ms) and yields nearly identical results to the other
comparable ($\sim$10-30ms) models on RTX3070.

Figure 2: Runtime versus $\Delta$PSNR wrt. Bicubic, and the effect of the
proposed training pipeline. Due to Eq. 1, Bicubic score is 0. Bic++ scores the
highest with $<$3ms runtime, achieving $\sim$+1dB on top of Bicubic on our
test set. A..I denote the models experimented. We follow
B$\rightarrow$B*$\rightarrow$C$\rightarrow$Bic++. B* is not in plot’s range.
Figure 3: General network structure of Bicubic++. DS denotes the downscaling
sub-network (Fig. 5), and D2S is the depth-to-space layer [29]. Residual
addition blocks are shown by $\mathbf{R}$, and the number of convolutional
layers inside $\mathbf{R}$ is $\mathbf{m}$. s and p denote the stride and
padding of the convolutional layers, respectively. ch is the global channel
number parameter. In the final proposed model, $\mathbf{R}$=1, $\mathbf{m}$=2,
ch=32, and DS is a 3$\times$3 convolutional layer with s=2, p=1.

##  1 Introduction

Single-image super-resolution (SR) is the task of upscaling a low-resolution
(LR) image to a high-resolution (HR) one. SR in an highly ill-posed inverse
problem, and researchers are still coming up with different approaches which
would yield reasonable and high-quality HR images under varying LR conditions
(noise, blur, camera motion, compression, _etc_.). Besides providing high-
quality HR’s, many SR networks also focus on runtime performance [15, 16]
since SR is also extensively utilized in real-life scenarios such as on the
cameras of unmanned vehicles and surveillance systems. Nonetheless, a lot of
industry-grade applications still consider Bicubic as the tried-and-true
upscaling method because of its speed and ease-of-use. Despite all, we believe
that a carefully constructed and trained neural network can be just as useful
as the traditional Bicubic upsampling, run in real-time, and even replace
Bicubic upsampling with visually more pleasing outputs.

Hence, to contribute to the ongoing development of real-time single-image
super-resolution, we present our work in the NTIRE 2023 Real-Time Super-
Resolution Challenge Track 2 ($\times$3 upscaling) [7, 36] and propose a
lightweight Bicubic upscaling alternative, Bicubic++. Bicubic++ first
downscales the image features by $\times$2 to reduce the number of operations
significantly, and applies $\times$6 upscaling at the end. We also train the
networks with our proposed three-stage training pipeline, where we first train
a network with convolutional layer channels greater than the ”hardware’s sweet
spot”. Then, we apply global structured layer pruning without conditioning on
weight or gradient norms while focusing on optimizing the PSNR, convolutional
layer bias removal, and fine-tuning operations to further decrease its runtime
speed without sacrificing much from its visual output quality. Bicubic++
ranked 1st on the NTIRE 2023 Real-Time Super-Resolution Challenge Track 2 and
is the fastest method among all competitive methods.

Throughout the paper, we emphasise on optimizing the score function of the
challenge given in Eq. 1:

$S(P,t)=\begin{cases}0&\text{if }P\leq P_{bic}\\\ \frac{{}2^{P-P_{bic}}\times
2}{0.1\times\sqrt{t}}&\text{else }\end{cases}$ (1)

where $P$ and $P_{bic}$ are the PSNR values of the network and the Bicubic
upsampling on the test set, respectively. $t$ is the runtime of the network
when a 720p LR image is given as an input and a 4K HR is obtained. In
addition, due to the real-time requirements, $t$ must to be smaller than 30ms.
This score formulation is provided by the competition holders and is
calculated with their given
code111https://github.com/eduardzamfir/NTIRE23-RTSR.

##  2 Related Works

Deep learning-based single-image super-resolution. Dong  _et al_. proposed
SRCNN [8] as the first deep learning based SR algorithm. Later on, FSRCNN [9]
was proposed, as a faster version for SRCNN. It replaced ReLU with a PReLU
activation layer, reformed SRCNN with smaller filter sizes with more mapping
layers, and proposed postponing the upscaling layer at the end of the network
which resulted in a great speed-up. ESPCN [29] introduced a sub-pixel
convolutional layer, often known as depth to space, which is currently used in
several efficient SR networks [10, 4, 3]. The development of deep learning-
based SR was carried on by VDSR [18], EDSR [23], and WDSR [35] by expanding
the number of parameters in exchange for accuracy and sacrificing speed.
Concepts like generative adversarial networks [20, 34, 30], recursive &
residual networks [2, 14, 5], and attention mechanism [22, 28] also rapidly
took place in SR network topologies with the recent advancements [27].

Efficient super-resolution. Just like the regular SR, efficient SR did not
only emerge with deep learning either, as there are notable methods using
sparse representations like Zeyde  _et al_.’s [37], ANR [32], and A+ [33].
However, thanks to the challenges and workshops held recently [16, 15, 21, 36,
7], there is an abundance of deep learning-based SR methods now, each more
efficient than the previous ones most of the time. XLSR [3] utilized clipped
ReLU at the end of the network to minimize the PSNR decrease in post-training
INT8 quantization for the first time. ABPN [10] proposed an anchor-based
residual network and applied quantization-aware training. SCSRN [11] applied a
similar idea of residual adding to ABPN but in the feature level, and
performed reparametrization of the convolutional layers to train with larger
number of parameters and reduce them in the inference time. RLFN [19] used
three convolutional layers for residual learning to simplify feature
aggregation, and stated to have achieved a balance between visual quality and
runtime. A CPU-inference based method named SR-LUT [17] proposed to train a
deep SR network to construct a lookup table, and matched the input LR image
patches to the output HR image patches utilizing the said table.

##  3 Methodology

Our proposed network structure is given in Fig. 3 in its most general form.
The architecture and the design choices are inspired from the efficient SR
methods [15, 16, 21], and the experiments conducted. Based on this
architecture, we tune its parameters and try different training strategies to
maximize the scoring function given in Eq. 1.

###  3.1 Design Procedure and Network Architecture

One initial observation of ours during the design procedure was that the speed
of the network does not directly depend on the number of parameters, but on
the amount of activations as stated in [21], which is a value proportional to
the volume of data passing through the network. Hence, processing the feature
tensors with lower width & height than those of the input image dimensions can
decrease the number of activations significantly. Furthermore, it has been
shown in [31] that the downscaling operation has positive effects when learned
together with the upscaling. Therefore, although it seems to complicate the
problem at first, we chose to ”downscale” the features ($\times 0.5$) by a
strided convolution and then apply super-resolution operation ($\times$6) in
order to speed up the network. We also experiment with other downscaling
approaches (Fig. 5), but go for a strided convolution at the end. In a sense,
this lower dimensional feature extraction approach can be thought as a way of
compressing the data spatially; however, this compression is reversible since
it is trained all together with the upscaling.

In our network, we wanted to use the ”optimum” number of channels for the
convolutional layers. We observed that decreasing the number of channels of
consecutive layers does not always reflect to a decrease in the runtime.
Hence, we decided to keep the same number of channels across the network
(unless necessary), even though some SR methods do the quite opposite by
squeeze and expand blocks [12] to extract features better. However, the
increase in runtime caused by the blocks with varying channels outweigh its
benefits considerably. To get a feeling of the optimum channel numbers, we
conducted a simple experiment and observed that runtime versus the number of
channels is not positively correlated all the time, and there exists some
”sweet spots” (Fig. 4). For our setup and under the real-time constraints, 56
channels used by RFDN [24] and 28 channels used by ABPN [10] in the literature
came out to be not the optimal points. Thus, among the options shown in Fig.
4, we opted for a channel number of 32 in our final model. However, as
detailed in Sec. 4.1, to get the most out of the model, we will start from a
non-optimal channel (34) and globally prune the model to achieve 32 channels.

Figure 4: The effect of the channel number on the runtime and normalized
runtime (runtime of ch=6 is mapped to the origin, and ch=65 to 1) of our
model, RTSRN [15], and RFDN [24]. Note the ”sweet spots” are on ch $\in$ {16,
20, 24, 28, 32, …}, and are observed in common among different models. Figure
5: Different DS structures in Fig. 3. S2D (a) denotes the space-to-depth layer
[29], and DWT (b) is the discrete wavelet transform. We use a strided
convolution (c) for DS in the end.

In addition to all these, we also noticed that the innocent-looking bias terms
of the convolutional layers are usually overlooked, and actually take a
considerable portion of the overall runtime. However, the drastic decrease in
the runtime caused by removing the bias terms actually overpower the marginal
decrease in PSNR, which is also clearly visible from the obtained score values
(Fig. 2). Hence, we wanted to eliminate the bias terms from the layers in our
final network structure as well.

To sum up those aforementioned observations and ideas:

  * •

Data volume processed is correlated with runtime rather than number of
parameters

  * •

Varying the number of channels among blocks decreases runtime performance

  * •

There exists some ”sweet spots” for the channel size for the hardware, and
reaching those ”sweet spots” can be done by training a larger network and
pruning afterwards

  * •

The bias terms may take a considerable amount of time

In the end, to apply these points to our proposed network, we constructed a
training pipeline. For each choice, we empirically proved its benefits. We
also detail the pipeline and empirical justifications in Sec. 3.2 and Sec.
4.1, respectively.

Figure 6: Global structured pruning mask generation structure.
$\mathcal{W}_{R\times T}$ is the matrix consisting of the convolutional
filters for a specific Conv2D layer with kernel size n$\times$n, and b is the
corresponding bias term of the layer. Each entry of $\mathcal{W}$ is an
n$\times$n kernel.

###  3.2 Three-Stage Training Pipeline

We construct a three-stage training pipeline: training with a larger number of
channels, global structured pruning of channels, and bias removal of
convolutional layers. For the last two stages, we fine-tune the network after
the modifications.

Training hyperparameters & computational issues: For each stage, we use Adam
optimizer with parameters $\beta_{1,2}$ 0.99 and 0.999, respectively. Models
in all stages are trained for 1000 epochs, each epoch consuming 800 randomly
cropped and rotated patches of dimension (108,108,3) (for LR) from JPEG Q=90
degraded DIV2K training [1] dataset. For the learning rate (LR), we initialize
with 5e-4 and use a decaying learning rate scheduler for all stages, where for
the first 500 epochs the LR stays constant and for the last 500 epochs the LR
decays linearly until 1e-8.

To be able to train such a tiny model by utilizing the training hardware to
its fullest, all the elements of the training pipeline need to be adjusted
properly. This is due to the model forward and backward passes being done in
almost no time, and memory accesses along with the validation steps becoming
the actual bottleneck. To speed-up the validation, we use 48 images with same
dimensions (680,452,3) (for LR) from JPEG Q=90 degraded DIV2K validation
dataset, and hence be able to pick a batch size of 8 instead of 1. To take
care of the memory access bottleneck, we pre-loaded both the training and the
validation images to RAM and avoided fetching the data from disk continuously
for each training and validation step. This way, we were able to complete a
training cycle of 1000 epochs in only 1 hour using a single Tesla V100. Note
that since the model’s capacity is small, opting for a comparably smaller
training dataset (like DIV2K) instead of favoring for a larger dataset is
usually viable, and does not hurt the validation score in the end generally.

Stage 1 - Slim: Training with a larger number of channels. We train the
network in Fig. 3, where ch is 34, m is 2, R is 1, and DS is the strided
convolutional layer in Fig. 5. All convolutional layer biases are enabled in
this stage. Respective network parameter choices are justified and explained
thoroughly in Sec. 4.1.

Stage 2 - Slimmer: Global structured pruning. We take the resulting checkpoint
from Stage 1 and perform the proposed pruning procedure given in Fig. 6. The
proposed method is a structured pruning (prune channel-wise instead of kernel
element-wise), in a global sense, since we prune the entire network
considering the entire network’s structure through sharing and transferring
pruning masks and using a global fitness measure (_i.e_. using validadion PSNR
of the pruned model), instead of focusing on individual elements of the
network and heuristic measures for fitness.

In detail; for the convolutional layers to be pruned, we first construct all
possible Mask #1’s in Fig. 6 ($\mathcal{M}_{1}$), each only masking a single
channel. After obtaining different $\mathcal{M}_{1}$’s, we calculate the PSNR
scores on the validation set for each mask applied individually, and unify two
$\mathcal{M}_{1}$’s (to reach the closest ”sweet spot”, ch=32) which cause the
most marginal PSNR drop as our best & final $\mathcal{M}_{1}$.

After constructing $\mathcal{M}_{1}$, we move onto the Mask #2
($\mathcal{M}_{2}$). We follow the same procedure as in the creation of
$\mathcal{M}_{1}$; however, now we do not start from the beginning and keep
the $\mathcal{M}_{1}$’s modifications (though starting from scratch resulted
the same set of masks). We do not perform any additional training while
$\mathcal{M}_{1}$ and $\mathcal{M}_{2}$ are being selected, but only after
$\mathcal{M}_{1}$ and $\mathcal{M}_{2}$ are all applied. We do not prune the
bias terms here, either. One observation we made here was the network’s
overall performance was a lot more sensitive to the channels in
$\mathcal{M}_{1}$ than those in $\mathcal{M}_{2}$.

Stage 3 - Slimmest: Bias removal of the convolutional layers. After applying
$\mathcal{M}_{1}$ & $\mathcal{M}_{2}$ and reducing ch down from 34 to 32, we
take the checkpoint from Stage 2, disable all bias from the convolutional
layers, and fine-tune the overall network again.

##  4 Experiments

We have performed extensive experiments trying to find the optimal parameters
for our general network structure in Fig. 3, with the given scoring function
in Eq. 1. In addition to network structure and the proposed training pipeline
in Sec. 3, we experiment with partial loading of model components to gradually
decrease the size and runtime of the network. All PSNR scores in the ablation
study are evaluated on our 48-image JPEG Q=90 degraded DIV2K validation set,
unless stated otherwise.

Exp | Models | DS | Act | ch | m | R | b |  PSNR (Y) (dB) |  Runtime (ms) | Score | Load | Operation |
---|---|---|---|---|---|---|---|---|---|---|---|---|---
#1: DS, b, and activation | Q | SC | ReLU | 24 | 2 | 1 | ✓ | 30.157 | 3.19 | 19.86 | ✗ | ✗ |
X | DWT | ReLU | 24 | 2 | 1 | ✓ | 30.219 | 3.59 | 19.55 | ✗ | ✗ |
Y | S2D | ReLU | 24 | 2 | 1 | ✓ | 30.172 | 3.33 | 19.64 | ✗ | ✗ |
Z | S2D | ReLU | 32 | 2 | 1 | ✓ | 30.269 | 3.72 | 19.88 | ✗ | ✗ |
W | DWT | ReLU | 32 | 2 | 1 | ✓ | 30.326 | 4.14 | 19.60 | ✗ | ✗ |
T | SC | ReLU | 32 | 2 | 1 | ✓ | 30.271 | 3.57 | 20.32 | ✗ | ✗ |
A | SC | LReLU | 32 | 2 | 1 | ✓ | 30.282 | 3.58 | 20.43 | ✗ | ✗ |
E | SC | LReLU | 32 | 2 | 1 | ✗ | 30.254 | 2.89 | 22.32 | ✗ | ✗ |
#2: m and R | E | SC | LReLU | 32 | 2 | 1 | ✗ | 30.254 | 2.89 | 22.32 | ✗ | ✗ |
1 | SC | LReLU | 32 | 2 | 2 | ✓ | 30.410 | 4.95 | 19.00 | ✗ | ✗ |
2 | SC | LReLU | 32 | 4 | 1 | ✓ | 30.433 | 4.84 | 19.53 | ✗ | ✗ |
3 | SC | LReLU | 32 | 4 | 1 | ✗ | 30.424 | 4.15 | 20.96 | ✗ | ✗ |
4 | SC | LReLU | 32 | 2 | 2 | ✗ | 30.390 | 4.30 | 20.11 | ✗ | ✗ |
#3: ch selection for pruning | P | SC | LReLU | 35 | 2 | 1 | ✓ | 30.293 | 6.50 | 15.29 | ✗ | ✗ |
P* | SC | LReLU | 32 | 2 | 1 | ✓ | 29.712 | 3.58 | 13.77 | P | Global structured pruning |
R | SC | LReLU | 33 | 2 | 1 | ✓ | 30.286 | 6.16 | 15.63 | ✗ | ✗ |
R* | SC | LReLU | 32 | 2 | 1 | ✓ | 30.113 | 3.58 | 18.44 | R | Global structured pruning |
B | SC | LReLU | 34 | 2 | 1 | ✓ | 30.324 | 6.60 | 15.50 | ✗ | ✗ |
B* | SC | LReLU | 32 | 2 | 1 | ✓ | 30.155 | 3.58 | 18.72 | B | Global structured pruning |
#4: Fine tuning and bias removal | C | SC | LReLU | 32 | 2 | 1 | ✓ | 30.320 | 3.58 | 20.99 | B* | Fine tuning (FT) |
(Bic++) | SC | LReLU | 32 | 2 | 1 | ✗ | 30.295 | 2.89 | 22.96 | C | Bias removal + FT |
F | SC | LReLU | 32 | 2 | 1 | ✓ | 30.301 | 3.58 | 20.72 | A | Fine tuning |
G | SC | LReLU | 32 | 2 | 1 | ✗ | 30.281 | 2.89 | 22.74 | A | Bias removal + FT |
H | SC | LReLU | 32 | 2 | 1 | ✗ | 30.277 | 2.89 | 22.68 | E | Fine tuning |
I | SC | LReLU | 32 | 2 | 1 | ✗ | 30.283 | 2.89 | 22.77 | F | Bias removal + FT |
| J | SC | LReLU | 32 | 2 | 1 | ✗ | 30.290 | 2.89 | 22.88 | H | Fine tuning |
#5 | 5 | SC | LReLU | 32 | 2 | 1 | ✓ | 30.210 | 3.58 | 19.45 | 2 |  Partial load $\mathbf{m}$’s + FT |
6 | SC | LReLU | 32 | 2 | 1 | ✓ | 30.281 | 3.58 | 20.43 | 1 |  Partial load $\mathbf{R}$’s + FT |
Other methods | Bicubic | - | - | - | - | - | - | 29.334 | 1.0 | 0 | - | - |
ESPCN [29] | - | - | - | - | - | - | 30.419 | 9.68 | 13.67 | - | - |
XCAT [4] | - | - | - | - | - | - | 30.435 | 11.68 | 12.58 | - | - |
ABPN [10] | - | - | - | - | - | - | 30.703 | 15.76 | 13.05 | - | - |
XLSR [3] | - | - | - | - | - | - | 30.637 | 25.25 | 9.84 | - | - |
FSRCNN [9] | - | - | - | - | - | - | 30.547 | 33.75 | 8.00 | - | - |
RFDN [24] | - | - | - | - | - | - | 30.921 | 159.3 | 4.77 | - | - |
Table 1: Quantitative ablation study to pick the best scoring model.
$\mathbf{b}$ denotes the bias of convolutional layers, and LReLU is the leaky
ReLU activation. DS configurations are given in Fig. 5. We follow B
$\rightarrow$ B* $\rightarrow$ C $\rightarrow$ Bic++ for the final model. PSNR
scores are evaluated on the 48-image DIV2K validation dataset described in
Sec. 3.2. Runtimes are measured on RTX3070.

###  4.1 Ablation Studies

We provide an extensive ablation study in Tab. 1. We also divide the
experiments in Tab. 1 into 5 different sections for easy readibility and
comparability:

Downscaling (DS), bias (b), and activation selection. From Q-T, we observed
that using a strided convolution (SC) is superior compared to discrete wavelet
transform (DWT) and space-to-depth (S2D), even when the number of channels are
smaller (notice that Q&Z nearly got the same score despite Q having less
number of channels). T&A reveal that using leaky ReLU instead of ReLU can
provide a significant benefit in PSNR without nearly any increase in the
runtime. We can also see from A&E that disabling bias terms can be
advantageous to get higher scores.

Number of residual blocks (R) and convolutional layers (m). Results of 1-4 and
E suggest that setting m=2 and R=1 yields the best score among tried model
architectures. Hence, after the experiments done up to this stage and the
observation in Fig. 4, we decided that for our final model, ch is 32, DS is
SC, activations are leaky ReLU, and the bias terms should be disabled at some
point.

Selection of number of channels for pruning. This part includes the
experiments done for searching the ideal number of channels for reducing to
32. We mainly focus on ch$\in${33,34,35}, and apply our proposed pruning
method (Sec. 3.2) for P, R, B, and obtain P*, R*, and B*, respectively. The
results reveal that choosing ch=34 and applying the pruning obtains the
highest score, hence we move on from B*.

Fine tuning and bias removal. After training B and pruning to obtain B*, we
fine-tune the network by training it again, and obtain C. The positive effect
of pruning can be observed from the comparison of C&F.

Afterwards, we remove the bias terms from C, fine-tune it once more, and
obtain our proposed model, Bicubic++. I, cannot reach the final network’s
score, indicating the advantages of the proposed training pipeline. We also
show other possible training paths ($\mathbf{A\rightarrow
G},\mathbf{E\rightarrow H}$, $\mathbf{A\rightarrow F\rightarrow
I},\mathbf{A\rightarrow E\rightarrow H\rightarrow J}$) to further point out
that our training pipeline yields the best score. A visual comparison is also
provided for this experimental setup in Fig. 2.

Extra experiments regarding partial loading of R&m. We also experiment with
the residual blocks (R), the number of convolutional layers (m), and partial
loading models with different R&m and provide the results.

It is worthy to note that one can always increase the number of channels &
blocks and construct a model obtaining higher PSNR scores with slower runtime;
however, we aimed to stay under $<$3ms on RTX3070, hence decided to stick with
the choices made in this paper.

###  4.2 Comparative Results

We provide quantitative and qualitative comparative results in Tab. 2, Tab. 3,
Fig. 7, and Fig. 8. Bicubic++ achieves better results than the traditional
Bicubic upsampling and runs much faster compared to the other relevant
methods, making it a good candidate for real-time SR appplications.

Although relevant, we did not include the results of sparse coding methods
like A+ [33] and SR-LUT [17] since they are not compliant with the challenge
requirements, either due to their slower runtime or lack of implementation in
GPU.

|  Track 2 Score |  PSNR (RGB) | SSIM |  PSNR (Y) |  Runtime (ms)
---|---|---|---|---|---
Aselsan Research | 31.26 | 32.06 | 0.8344 | 34.56 | 1.17
Team OV | 29.63 | 32.17 | 0.8376 | 34.72 | 1.51
ALONG | 28.57 | 32.18 | 0.8367 | 34.66 | 1.66
RTVSR | 26.89 | 32.22 | 0.8372 | 34.77 | 1.96
Noah_TerminalVision | 26.68 | 32.65 | 0.8455 | 35.10 | 3.64
NJUST-RTSR | 23.51 | 32.25 | 0.8384 | 34.90 | 2.68
Antins_cv | 23.44 | 32.63 | 0.8457 | 35.21 | 4.60
DFCDN Team | 22.64 | 32.07 | 0.8371 | 34.63 | 2.25
Multimedia | 21.55 | 32.33 | 0.8398 | 34.83 | 3.56
z6 | 20.90 | 32.59 | 0.8446 | 35.05 | 5.47
R.I.P. ShopeeVideo | 15.67 | 32.84 | 0.8469 | 35.30 | 13.79
ECNU_SR | 15.39 | 32.64 | 0.8458 | 35.17 | 10.75
Touch_Fish | 11.55 | 32.67 | 0.8468 | 35.31 | 19.86
P.AI.R | 8.66 | 32.55 | 0.8441 | 35.04 | 30.03
SEU_CNII | 6.68 | 31.85 | 0.8326 | 34.52 | 19.05
diSRupt | 6.34 | 31.64 | 0.8292 | 34.25 | 16.00
Bicubic | 0.00 | 31.30 | 0.8246 | 33.82 | 0.5
Organizer’s baseline [7] | 14.01 | 31.74 | 0.8299 | 34.25 | 3.74
Table 2: All methods scoring above Bicubic in Track 2 [7] and the suggested
baselines. PSNR is evaluated on the competition’s test set, and runtime is
measured on RTX3090/3060 with FP16 precision.

##  5 Conclusion and Discussion

In this paper, we proposed our real-time SR network, Bicubic++. Our model
learns downscaled features of the input image to increase efficiency, and is
trained with our proposed three-stage pipeline where we apply global
structured pruning and bias removal. We believe that our experimental results,
observations, proposed architecture, and the training pipeline may help the
development of real-time SR methods. We believe that Bicubic++ sets a new
practical industry standard for upscaling tasks and it can be the contemporary
alternative of Bicubic upscaling.

For further research, a fusion of deep learning and lookup table along with
sparse representation methods can be investigated on GPUs. Furthermore, we
believe that the proposed training pipeline would benefit from including
reparametrization along with INT8 quantization.

|  Set5 [6] |  Set14 [38] |  BSD100 [25] |  Urban100 [13] |  Manga109 [26] |  DIV2K Val [1] | Runtime (ms)
---|---|---|---|---|---|---|---
| PSNR | SSIM | PSNR | SSIM | PSNR | SSIM | PSNR | SSIM | PSNR | SSIM | PSNR | SSIM
ESPCN [29] | 31.56 | 0.8737 | 28.53 | 0.7833 | 27.80 | 0.7454 | 25.66 | 0.7680 | 29.29 | 0.8768 | 30.37 | 0.8351 | 9.6
XCAT [4] | 31.50 | 0.8728 | 28.52 | 0.7843 | 27.81 | 0.7464 | 25.67 | 0.7697 | 29.33 | 0.8781 | 30.38 | 0.8360 | 11.7
ABPN [10] | 31.93 | 0.8812 | 28.76 | 0.7910 | 27.99 | 0.7523 | 26.13 | 0.7862 | 30.14 | 0.8928 | 30.65 | 0.8421 | 15.8
XLSR [3] | 31.84 | 0.8800 | 28.74 | 0.7900 | 27.95 | 0.7509 | 25.98 | 0.7812 | 29.94 | 0.8899 | 30.59 | 0.8406 | 25.3
FSRCNN [9] | 31.65 | 0.8762 | 28.61 | 0.7863 | 27.86 | 0.7477 | 25.81 | 0.7736 | 29.52 | 0.8814 | 30.48 | 0.8377 | 33.8
Bicubic | 29.98 | 0.8434 | 27.30 | 0.7529 | 26.99 | 0.7180 | 24.31 | 0.7196 | 26.54 | 0.8300 | 29.33 | 0.8127 | 1.0
Bicubic++ | 31.19 | 0.8656 | 28.35 | 0.7799 | 27.68 | 0.7431 | 25.49 | 0.7626 | 28.72 | 0.8653 | 30.24 | 0.8324 | 2.9
Table 3: Quantitative comparative results of our method and other comparable models for $\times$3 SR. All low-resolution images in the datasets are JPEG Q90 degraded, and all PSNR values are calculated for the Y channel. Runtime is measured on RTX3070 with FP16 precision. Portion of LR | Bicubic | Ours |  ESPCN [29] |  XCAT [4] |  ABPN [10] |  XLSR [3] |  FSRCNN [9]
---|---|---|---|---|---|---|---
|  |  |  |  |  |  |
|  |  |  |  |  |  |
|  |  |  |  |  |  |
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|  |  |  |  |  |  |
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|  |  |  |  |  |  |
|  |  |  |  |  |  |
Figure 7: Qualitative comparative results of Bicubic++ and other relevant methods on the datasets [13, 26, 25, 1]. Portion of LR | Bicubic | Ours |  ESPCN [29] |  XCAT [4] |  ABPN [10] |  XLSR [3] |  FSRCNN [9]
---|---|---|---|---|---|---|---
|  |  |  |  |  |  |
|  |  |  |  |  |  |
|  |  |  |  |  |  |
|  |  |  |  |  |  |
|  |  |  |  |  |  |
|  |  |  |  |  |  |
|  |  |  |  |  |  |
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|  |  |  |  |  |  |
Figure 8: Qualitative comparative results of Bicubic++ and other relevant
methods on the challenge test set [36].

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