# Absorption and scattering by a temporally switched lossy layer: Going beyond
the Rozanov bound

Chen Firestein and Amir Shlivinski<EMAIL_ADDRESS>School of Electrical
and Computer Engineering, Ben-Gurion University of the Negev, Beer Sheva,
84105, Israel  Yakir Hadad<EMAIL_ADDRESS>School of Electrical
Engineering, Tel-Aviv University, Ramat-Aviv, Tel-Aviv, Israel, 69978

###### Abstract

In this paper we study the electromagnetic scattering, absorption, and bounds
for short time modulated pulses that impinge on a time-varying lossy layer
that is sandwiched between vacuum and a perfect electric conductor. The
electric characteristics of the layer, namely, the conductivity, permittivity,
and permeability are assumed to change _abruptly_ in time. The analysis is
carried out using the equivalent transmission line (TL) model of the actual
electromagnetic system. Thus, it naturally take into account the effect of
polarization (TE and TM) as well as the effect of the angle of incidence, by
simply introducing the proper TL characteristics. After formulating the one-
dimensional TL problem in the complex temporal frequency variable $s$, we
apply the inverse Laplace transform to find the signals in time. Specifically,
we show that the spectral Green’s function in the $s$ plane possesses no
branch point singularities but exhibits only simple pole singularities, and
thus we use this fact to substantially accelerate the calculation of the
inverse Laplace transform. As an important application we demonstrate
numerically that a time-varying absorbing layer that undergoes temporal
switching of its _conductance only_ can absorb the power of a modulated,
ultrawideband, as well as a quasi-monochromatic, pulsed wave _beyond_ what is
dictated by the Rozanov bound when integrating over the _whole_ frequency
spectrum.

##  I Introduction

Wave absorbers find important applications in various wave based systems, in
acoustics and electromagnetics, where they are used for the minimization of
scattering and emission from certain domains, as for example radar absorbers
or in electromagnetic compatibility applications [1, 2, 3, 4, 5, 6]. The
typical absorber is based on a proper spatial arrangement of linear and time-
invariant (LTI) lossy materials. For LTI absorbers that consist of a lossy
layer backed by an ideal conductor, Rozanov [7] has developed a fundamental
tradeoff between the absorber thickness and the absorption bandwidth. However,
since this bound, as several other bounds in wave theory, has been developed
under the LTI assumption, it does not necessarily apply in non-LTI wave
systems. In fact, time-varying wave devices have been shown in recent years to
allow an additional degree of freedom in compare to conventional LTI designs
and therefore in many cases they provide a potential to achieve better
performances [8, 9, 10, 11, 12, 13]. Interestingly, the time-variation is
considered as a means for wave manipulations leading to additional peculiar
functionalities [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27]. As
spatial variations, also temporal variations can take different forms, such as
periodic modulation and temporal discontinuity. The former with better suited
for narrowband applications [9, 28, 29], whereas the latter fits better pulsed
wave applications. Specifically, it has been used in [8] to achieve a better
impedance matching between TL and reactive loads for short-time pulses, by
[27] for the purpose of anti-reflection coating, and by [11] to improve the
reflection bandwidth of a Dallenbach screen. Furthermore, the problem of
reflection and transmission, as well as energy balance, in the presence of
abrupt and gradual temporal variation of the guiding medium were explored in
[30, 31, 32, 33, 34, 35, 36, 37, 38, 39]. Scattering and radiation by time-
modulated small obstacles that can be described using the dipole approximation
have been explored in [40, 41, 42, 43]. Such description may be helpful for
the development of homogenization methodologies for time varying composite
media and may be used to practically emulate effective bulk metamaterials with
various time-varying characteristics.

In this paper we study the electromagnetic wave scattering and absorption of
short time pulses that impinge on a time-varying lossy dielectric layer that
is sandwiched between half-space of vacuum and a perfect electric conductor.
The electric characteristics of the layer, namely, its conductivity,
permittivity, and permeability are assumed to change _abruptly_ in time. To
make the analysis as general as possible we study the equivalent transmission
line problem of the actual electromagnetic system. Thus, we naturally take
into account the effect of polarization and the effect of possible oblique
incidence by introducing the proper transmission-line characteristics [44,
45]. We solve the temporally switched TL problem in the complex frequency
plane $s$, and specifically show that the spectral Green’s function possesses
only pole singularities and no branch cut singularities [46]. This feature
enables us to substantially accelerate the numerical calculations as discussed
in the paper. Then, as an important application we first augment the Rozanov
bound for ultrawideband short-time pulsed signals followed by a demonstration
of how it may be bypassed using a temporally-switched layered wave absorber.
While our time-domain absorption analysis has a broader applicability,
specifically to bypass the LTI bound we apply temporal switching of only the
layer’s conductivity which may simplify practical realizations.

##  II Description of the Problem

An electromagnetic pulsed wave with electric and magnetic fields,
$\vec{E}(\mathbf{r},t)$ and $\vec{H}(\mathbf{r},t)$, propagates in vacuum with
normalized wavevector
$\hat{\mathbf{k}}=\hat{\mathbf{x}}\cos\theta+\hat{\mathbf{y}}\sin\theta$ that
lies on the $xy$ plane. At $t=t_{0}$ the pulsed wave impinges a dielectric
layer with permittivity, permeability, conductivity and magnetic conductivity
($\epsilon_{1},\mu_{1},\sigma_{1},\sigma_{m1}$) that is normal to the $x$
axis. The layer thickness is $d$, and it is sandwitched between vacuum and an
ideal electric conductor. See Fig. 1(a) for illustration.

Figure 1:  Problem illustration; (a) A short pulse is propagating towards a
radar absorber that is composed of magneto-dielectric layer(green) attached to
a Perfect Electric Conductor (PEC) surface(brown). The figure illustrates the
$(x,y)$ plane, which is the plane of incidence.
Temporal description of the system: (b) At $t=t_{0}$ the incident wavefront
approaches the boundary. (c) At time $t=t_{s}$ the wave is completely included
within the layer and an abrupt switching is preformed. (d) For times $t>t_{s}$
the modified electric and magnetic properties may lead to dispersive reflected
and a transmitted waves.

With respect to the plane of incidence, the incident wave may be decomposed
into transverse electric (TE) and magnetic (TM) field polarizations which
enables to treat them separately by an equivalent TL model. Following standard
textbooks such as [44, 45], the parameters of the TL model are given in
Appendix A. For convenience, but with no loss of generality, the initial
electric and magnetic properties of the layer are selected to satisfy the non-
distortion property as follows [47],

$\frac{R_{1}}{L_{1}}=\frac{G_{1}}{C_{1}},$ (1)

where $\\{L_{1},C_{1}\\}$ [$\\{R_{1},G_{1}\\}$] are the per unit length
inductance and capacitance [series resistance and parallel admittance] in an
equivalent TL model. This condition renders lossy, albeit, non-dispersive
layer. Due to the spatial discontinuity at $x=0$ some of the impinging wave
will be back reflected upon incidence. Due to the non-distortion condition,
this reflection is non-dispersive, i.e., the transmitted and back reflected
pulses do not change their shape. This reflection has a simple reflection form
of $\Gamma_{s}=(Z_{1}-Z_{0})/(Z_{1}+Z_{0})$, where $Z_{0}=\sqrt{L_{0}/C_{0}}$
is the characteristic TL impedance modeling the host medium (vacuum) (see
Appendix A), and $Z_{1}=\sqrt{L_{1}/C_{1}}$ is the characteristic TL impedance
of the layer.

At time $t_{s}>t_{0}$, when the pulse is completely contained within the layer
(see Fig. 1.(b)-(d)), the TLs’ parameters are abruptly switched into a new set
of parameters $(\epsilon_{2},\mu_{2},\sigma_{2},\sigma_{m2})$ which translates
into $\\{L_{2},C_{2},R_{2},G_{2}\\}$ in the TL model. Thus forming a temporal
boundary separating between two TL problems, with initial conditions set at
$t=t_{s}^{+}$ along the TL. In general, this switching process ignites two
waves that counter-propagating inside the temporally discontinuous layer [48]
These two waves maintain the general pulse shape, however their temporal width
may be different than that of the pulse at $t<t_{s}$, giving rise to pulse
compression or expansion. In association with that, the switching process may
consume or pump energy to the system [8, 31].

##  III Mathematical Formulation

In this paper we strive to calculate the overall reflection by the temporally
switched layer. This includes, of course, the reflection by the spatial
discontinuity at $x=0$, but also it includes the reflection by the discussed
temporal switching process. In order to solve this scattering and absorption
problem, in the next section we develop a time-domain Green’s function
formalism for the complete analysis of the fields at times $t>t_{s}$. The wave
solution for $t>t_{s}$ is then preformed by using Green’s function propagator
within a superposition integral with the weighings that are equal to the
initial condition signals at $t=t_{s}^{+}$ for the pulsed waveform along the
layer $0<x<d$.

###  III.1 The TL model and its formal solution

We reduce the general plane wave scattering problem into a $1$D-TL problem
that is governed by the TLs equations

$\displaystyle\frac{\partial V(x,t)}{\partial x}+L\frac{\partial
I(x,t)}{\partial t}+RI(x,t)=0$ (2a) $\displaystyle\frac{\partial
I(x,t)}{\partial x}+C\frac{\partial V(x,t)}{\partial t}+GV(x,t)=0,$ (2b)

where $V(x,t)$ and $I(x,t)$ denote the voltage and current along the line,
respectively, and are related to the field in each domain following Appendix
A. As a first step, we transform the time-domain system in Eq. (2) into the
complex frequency (Laplace) domain

$\displaystyle\frac{\partial V(x,s)}{\partial x}+(sL+R)I(x,s)-LI_{0}(x)=0$
(3a) $\displaystyle\frac{\partial I(x,s)}{\partial
x}+(sC+G)V(x,s)-CV_{0}(x)=0.$ (3b)

where $V_{0}(x)$ and $I_{0}(x)$ are the initial voltage and current at
$t=t_{s}^{+}$ respectively, and $s$ denotes the, complex frequency, spectral
variable. Taking the $x$ derivative of (3a) followed by the substitution of
(3b), gives the Helmholtz equation for a lossy medium,

$\displaystyle\frac{\partial^{2}V(x,s)}{\partial
x^{2}}-(sL+R)(sC+G)V(x,s)=f_{s}(x,s)$ (4a) $\displaystyle
f_{s}(x,s)=L\frac{\partial I_{0}(x)}{\partial x}-C(sL+R)V_{0}(x),$ (4b)

with impedance boundary conditions at $x=0$, and short circuit due to the
ideal conductor at $x=d$. We define spectral green’s function for the voltage,
$g_{v}(x,x^{\prime},s)$, such that the solution for the spectral voltage
$V(x,s)$ can be expressed as the following superposition integral

$V(x,s)=\int_{-\infty}^{\infty}g_{v}(x,x^{\prime},s)f_{s}(x^{\prime},s)dx^{\prime}.$
(5)

From this definition, clearly, $g_{v}(x,x^{\prime},s)$ is obtained as a
solution of

$\frac{\partial^{2}g_{v}(x,x^{\prime},s)}{\partial
x^{2}}-\gamma^{2}g_{v}(x,x^{\prime},s)=\delta(x-x^{\prime}),$ (6)

where $\gamma=\sqrt{(sL+R)(sC+G)}$, with $\mbox{Re\\{$\gamma$ \\}}>0$, and
$\delta$ denotes the Dirac delta function. Once the voltage $V(x,s)$ is
solved, the time domain voltage is obtained by the inverse Laplace transform,

$V(x,t)=\frac{1}{2\pi j}\int_{\kappa-j\infty}^{\kappa+j\infty}V(x,s)e^{st}ds,$
(7)

where $-\sigma_{R}<\kappa<0$ in the region of convergence (ROC) as shown in
Fig. 3. In light of its realizability, our time-varying electromagnetic system
is causal. Moreover, since it involves a single temporal switching, as opposed
to periodic variation, it is necessarily stable. As a result, all the pole
singularities of $V(x,s)$ must lie in the left side of the complex frequency
plane ($\mbox{Re}\\{s\\}<0$).

###  III.2 The spectral Green’s function

The spectral Greens function $g_{v}(x,x^{\prime},s)$ is solved by following a
standard procedure. Using [49], the solution to Eq.(6) reads,

$\displaystyle g_{v}(x,x^{\prime},s)$ $\displaystyle=$
$\displaystyle\frac{F(x,x^{\prime},s)}{W(s)},$ $\displaystyle
F(x,x^{\prime},s)$ $\displaystyle=$
$\displaystyle\overleftarrow{u}(x^{<})\overrightarrow{u}(x^{>})$ (8)

where $\overleftarrow{u}(x)$ [$\overrightarrow{u}(x)$] is the solution of the
homogeneous equation corresponding to Eq. (6) that satisfies the boundary
conditions at left i.e., at $x=0$ [right, i.e., at $x=d$] (see Fig. 2),
$x^{<}=\min(x,x^{\prime})$ and $x^{>}=\max(x,x^{\prime})$. Moreover, $W(s)$,
the Wronskian that is responsible to tune the discontinuity between the left
and right solutions, $\overleftarrow{u}(x)$ and $\overrightarrow{u}(x)$, at
the source location $x^{\prime}$ reads,

$W\left[\overleftarrow{u},\overrightarrow{u}\right]=\overleftarrow{u}(x)\frac{d}{dx}\overrightarrow{u}(x)-\overrightarrow{u}(x)\frac{d}{dx}\overleftarrow{u}(x).$
(9)

Figure 2: Two half sided infinite transmission line problems terminated on
their other hand. By solving the two simple problems Green’s function is
obtained.

The homogenous, left and right solutions, i.e., $\overleftarrow{u}(x)$ and
$\overrightarrow{u}(x)$ are expressed by simple Fresnel coefficients,

$\displaystyle\overleftarrow{u}(x)$ $\displaystyle=$ $\displaystyle e^{\gamma
x}+\Gamma_{L}e^{-\gamma x},$ $\displaystyle\overrightarrow{u}(x)$
$\displaystyle=$ $\displaystyle e^{-\gamma(x-d)}+\Gamma_{R}e^{\gamma(x-d)},$
(10)

where $\Gamma_{L}(s)=[Z_{0}-Z_{2}(s)]/[Z_{0}+Z_{2}(s)]$ is the reflection
coefficient at $x=0$, $Z_{2}(s)=\sqrt{(sL_{2}+R_{2})/(sC_{2}+G_{2})}$ is the
characteristic impedance of the TL model of the layer _after_ the temporal
switching. Note that the square root should be selected such that
$\mbox{Re}\\{Z_{2}\\}>0$, and $\Gamma_{R}=-1$. Next, by using Eq. (III.2) in
Eq. (9), the Wronskian is evaluated as follows

$W(s)=-2\gamma(1+\Gamma_{L}e^{-2\gamma d}).$ (11)

By substituting Eqs. (III.2) and (11) into Eq.(III.2) the spectral domain
Green’s function is expressed as a sum of four wave contributions, as follows

$\displaystyle
g_{v}(x,x^{\prime},s)=\left[e^{-\gamma|x-x^{\prime}|}-e^{\gamma(x+x^{\prime}-2d)}\right.$
$\displaystyle\left.+\Gamma_{L}e^{-\gamma(x+x^{\prime})}-\Gamma_{L}e^{\gamma(|x-x^{\prime}|-2d)}\right]/W(s).$
(12)

Inserting Eq. (III.2) into Eq. (5) gives the spectral domain voltage $V(x,s)$.
Since the source term in Eq. (5), $f_{s}(x^{\prime},s)$, is an analytic
function, $g_{v}(x,x^{\prime},s)$ and $V(x,s)$ share the same singular points
in the complex $s$ plane. Then, applying Eq. (5) with Eq. (III.2) into Eq.(7)
provides the time domain voltage at any point inside the TL model of the
layer,

$V(x,t)=\frac{1}{2\pi
j}\int_{-j\infty}^{+j\infty}\frac{e^{st}}{W(s)}\int_{-\infty}^{\infty}F(x,x^{\prime},s)f_{s}(x^{\prime},s)dx^{\prime}ds.$
(13)

Next, we turn to evaluate the inverse Laplace transform in Eq. (13). This will
be carried out in an expedite fashion, by summing over the residues of pole
singularities. To that end we will show in the following that the original
integration path may be replaced, using Jordan’s lemma and analytic
continuation, with integration around all the singular points in the left side
of the complex plane $\mbox{Re}\\{s\\}<0$. In general these involve poles and
branch point singularities, however, the latter do not contribute as we show
below.

###  III.3 The inverse laplace transform

The integration in the inverse Laplace transform Eq. (13), should lie inside
the transform’s ROC. However, once this is formally done, it is possible to
close the integration path by adding the infinite left half circle, as shown
in Fig. 3. In light of Jordan’s lemma, since the integrand decays faster than
$1/s$ as $s\rightarrow\infty$ with $\mbox{Re}\\{s\\}<0$, this implies that the
addition to the integration path does not change the integration result.
Inside the close contour, there are several singular points. Branch points and
poles. Following the expression for $\gamma$ that is given right after Eq.
(6), two branch point singularities appear at $s_{bp_{1}}=-{G_{2}}/{C_{2}}$
and $s_{bp_{2}}=-{R_{2}}/{L_{2}}$, and their corresponding branch cuts, a
result of the choice $\mbox{Re}\\{s\\}>0$, are shown in Fig. 3. Nevertheless,
these singularities do not contribute to the field on the layer because
$g_{v}(x,x^{\prime},s)$ is an even function of $\gamma$ and as a result it is
analytic through the branch cut of $\gamma$. Moreover, as previously
mentioned, the spectral properties of $g_{v}(x,x^{\prime},s)$ and $V(x,s)$ are
the same, therefore $V(x,s)$ is also analytic through the branch cut. To be
more rigor, in appendix B we mathematically prove that the branch points in
our problem are in fact removable singularities and therefore the analyticity
through them. Consequently, the outer integration in Eq. (13) can be replaced
by a summation over multiple residues of simple poles without branch cut
contribution,

Figure 3: Time domain voltage is evaluated by preforming an integration over
imaginary axis which states for an inverse Laplace transformation.
$\sigma_{R}$ is the real part of the most right singularity of $V(x,s)$.
Alternatively, twisting the integration path by closing a semi-circle enables
to evaluate V(x,t) by residues summation.

$V(x,t)=\sum_{i=-\infty}^{\infty}\mathrm{Res}\left[\frac{\int_{-\infty}^{\infty}F(x,x^{\prime},s_{p,i})f_{s}(x^{\prime},s_{p,i})dx^{\prime}}{W(s_{p,i})}e^{s_{p,i}t}\right].$
(14)

where $i$ denotes the index of the $i^{th}$ pole. Since the poles of in Eq.
(13) are only a result of the zeros of the Wronskina function, using standard
residue evaluation, Eq. (13) then reads,

$V(x,t)=\sum_{i=-\infty}^{\infty}\frac{\int_{-\infty}^{\infty}F(x,x^{\prime},s_{p,i})f_{s}(x^{\prime},s_{p,i})dx^{\prime}}{W^{\prime}(s_{p,i})}e^{s_{p,i}t}$
(15)

where prime denotes derivative by the argument,

$\begin{split}W^{\prime}(s)=&-2\gamma^{\prime}(s)[1+\Gamma(s)e^{-2\gamma(s)d}]-2\gamma(s)\\\
&[\Gamma^{\prime}(s)e^{-2\gamma(s)d}-2d\gamma^{\prime}(s)\Gamma(s)e^{-2\gamma(s)d}]\\\
\end{split}$ (16)

and,

$\displaystyle\gamma^{\prime}(s)=L_{2}C_{2}\frac{s+\alpha}{\gamma},\hskip
5.69046pt\alpha=\frac{1}{2}\left(\frac{G_{2}}{C_{2}}+\frac{R_{2}}{L_{2}}\right)$
(17a)
$\displaystyle\Gamma^{\prime}(s)=\frac{Z_{0}C_{2}(sL_{2}+R_{2})-Z_{0}L_{2}(sC_{2}+G_{2})}{\gamma(Z_{0}\sqrt{sC_{2}+G_{2}}+\sqrt{sL_{2}+R_{2}})^{2}}.$
(17b)

A few notes. (i) The summation must converge for any selection of parameters
due to the physical nature of the system (TL subject to initial conditions).
(ii) In practice only a finite number of poles is used. (iii) The above
description is valid for cases where $Z_{2}\neq Z_{0}$. If $Z_{2}=Z_{0}$, the
branch cut of $\gamma$ appears in $g_{v}(x,x^{\prime},s)$ and contributes to
the integral evaluation. In addition, in this case there are no poles to the
integrand. Therefore, the integration should be preformed either brute force
in the ROC or by deforming the integration path around the branch cut. (iv)
Eq. (15) represents a semi-analytic result, since the locations of the complex
poles has to be calculated for each set of parameters. These poles are the
zeros of the Wronskian function as defined in Eq. (11) Unfortunately, this
equation has no analytic solution for the general lossy case, and therefore a
efficient numerical search procedure is required as discussed in the next
section.

###  III.4 Pole search algorithm

To find the poles we apply an iterative Newton-Rapshon approach. In light of
the Wronskian’s expression in Eq. (11), we denote
$f(s)=\Gamma(s)+e^{2\gamma(s)d}$, and the iterative equation for the $i^{th}$
pole is given by

$s_{i,n+1}=s_{i,n}-a\frac{f(s_{i,n})}{f^{\prime}(s_{i,n})},\quad
f^{\prime}(s)=\Gamma^{\prime}(s)+2d\gamma^{\prime}(s)e^{2\gamma d}$ (18)

where $n$ subscript denotes the $n^{th}$ iterative calculation step and $a$ is
a parameter which controls on the step size and is taken here as $a=0.1$.
Considering a larger $a$ may lead to a faster convergence, however may cause
some inaccuracy since the algorithm might skip some of the solutions [50]. The
algorithm convergence criterion is $|s_{n+1}-s_{n}|<\mathrm{Tol}$, where
$\mathrm{Tol}$ denotes the tolerance of the iterative solution that has been
taken as $10^{-5}$.

We note that the slabs width (TL length), $d$, dictates the density of poles,
which implies that poles will be spectrally closely spaced for a ticker layer.
Furthermore, the TL parameters $\\{L_{2},C_{2}\\}$ play a similar role, since
a slow wave system is similar to a long TL in the manner of long propagation
time from end to end. Also, the time domain voltage distribution is real and
therefore, poles appear in conjugate pairs so we can limit our algorithm to
the upper half plane only $\mbox{Im}\\{s\\}>0$.

The search algorithm requires an initial guess. To that end we use the
lossless case scenario for which an analytical solution exists and is given by

$2\sqrt{L_{2}C_{2}}ds+j2\pi
m-j\pi=\ln\left(\frac{Z_{0}-\sqrt{{L_{2}}/{C_{2}}}}{Z_{0}+\sqrt{{L_{2}}/{C_{2}}}}\right)$
(19)

where the initial guess of the first pole $s_{1,0}$ is obtained by setting
$m=0$ into Eq.(19). By applying Newton-Raphson iterative algorithm, the first
pole is evaluated. To find the next pole a new initial guess should be taken,
therefore we use the previous solution with an additional increment of
$s_{m+1}-s_{m}$ where $s_{m}$ is the $m^{th}$ pole of the lossless system as
previously presented in Eq. (19). The increment is purely imaginary and is
expressed by the actual parameters of the problem as

$\displaystyle\Delta$ $\displaystyle=$ $\displaystyle
2j\mbox{Im}\\{{s_{1}}\\}H(Z_{0}-\sqrt{{L_{2}}/{C_{2}}})$ (20) $\displaystyle+$
$\displaystyle j\mbox{Im}\\{{s_{1}}\\}H(-(Z_{0}-\sqrt{{L_{2}}/{C_{2}}})),$

where $H(x)$ is the unit step function and Im{$s_{1}$} is the imaginary part
of the first pole of a lossless system. The process repeats itself $N/2$
times, where $N$ is the total amount of poles that are considered in the
process. Lastly, we note that additional purely real valued poles may appear
in the lossy system, therefore an additional step should be taken - searching
over the real negative axis.

###  III.5 Numerical demonstration

In this section we apply the numerical procedure discussed above for the
solution of pulsed wave scattering by a temporally switched layer. Here we
assume that the layer is initially lossless and matched to the surrounding
($\epsilon_{1}=\epsilon_{0}$ and $\mu_{1}=\mu_{0}$), so the incident wave is
completely transmitted into the layer without an initial reflection due to the
discontinuity at $x=0$. We select $t_{s}=0$ as the switching moment where the
pulse is located at the center of the layer. The excited voltage and current
for the TL problem at $t=0^{-}$ were considered in the form of a modulated
Blackman type function [51], and presented as follows

$\displaystyle V_{0}(x)=\frac{1}{w}\sum_{k=0}^{3}a_{k}\cos\left(\frac{2\pi
k\left(x-\frac{d}{2}\right)}{w}\right)\pi_{w}\left(x-\frac{d}{2}\right)$ (21)
$\displaystyle\hskip 76.82234pt\cos\left(\frac{2\pi
f_{0}\left(x-\frac{d}{2}\right)}{v_{p}}\right)$ $\displaystyle
I_{0}(x)=\sqrt{\frac{C}{L}}V_{0}(x),$

where $w$ is the pulse width within the layer, $v_{p}$ is the wavefront
propagation speed, $\\{a_{k}\\}$ are Blackman’s weighting coefficients,
$f_{0}$ is the carrier frequency and
$\pi_{w}(x)=H(x-\frac{w}{2})-H(x+\frac{w}{2})$. After the switching, the layer
has the following electrical characteristics: $\epsilon_{2}=2\epsilon_{0}$,
$\mu_{2}=1.5\mu_{0}$, electric conductance $\sigma=0.000707[{S}/{m}]$ (i.e.,
${G}/{\omega_{0}C}=0.0064$) and magnetic conductance
$\sigma_{m}=37.7[{\Omega}/{m}]$ (i.e.,${R}/{\omega_{0}L}=0.0032$). The
magnetic conductivity is used to show the applicability of our analysis also
for this case which may be relevant using metamaterial implementation [52].

As suggested in [8, 31], the temporal switching process leads to reflected and
transmitted wave (analogous to spatial boundary) as follows,

$\tau=\frac{1}{2}\frac{v_{2}}{v_{1}}\left(\frac{Z_{2}}{Z_{1}}+1\right),\quad\Gamma=\frac{1}{2}\frac{v_{2}}{v_{1}}\left(\frac{Z_{2}}{Z_{1}}-1\right)$
(22)

where $v_{1,2}$ and $Z_{1,2}$ are the wave velocity and characteristic
impedance before and after the switching, respectively. To evaluate the
reflected voltage (at $x=0$), a superposition integration is preformed between
Green’s function and the initial conditions, followed by an inverse Laplace
transform according to Eq.(13). For the simulation we considered the following
set of parameters: $d=1.5[m]$, $T_{p}=4.33$[ns] (temporal pulse width after
the temporal switching), $f_{0}=1$ [GHz]. The results are presented in Fig. 4.

(a)

(b)

(c)

Figure 4: (a) Incident voltage distribution along $\mathrm{x}$ axis at
$t_{s}=0$. (b) The reflected voltage is evaluated at $\mathrm{x}=0$. The
figure shows a comparison between the residue summation approach and a brute
force integration inside the ROC. (c) Incident, reflected and absorbed energy
spectral density.

An excellent agreement with relative energy error of $9.03*10^{-10}$ is
observed with $N=200$ poles (since the poles are in complex conjugate pairs we
had to search for only 100). Regarding numerical efficiency. Let us denote the
computation time by brute force integration [pole residues summation] for the
inverse Laplace with $t_{\mbox{\scriptsize direct}}$ [$t_{\mbox{\scriptsize
res}}$]. We denote by $t_{\mbox{\scriptsize suppos}}$ the duration of time it
took to evaluate the superposition integration in $x$ as presented in Eq.
(13). A comparison between the execution time of both approaches is presented
in Table 1. Since the simulation time is short (amount of seconds for a
standard PC), the numeric results which are provided in Table 1 may slightly
vary from machine to machine. However, they provide the order of magnitude of
the running time. From Table 1 we can conclude that residues summation
approach provides much faster results in compare to numerical integration to
get the same accuracy. Furthermore, the superposition time is comparable with
the optimization time, which means that the poles search procedure is not a
computation bottleneck. To verify our analysis, we have examined conservation
of energy under a lossless case $\left(\sigma=\sigma_{m}=0\right)$. As
expected, energy is conserved so the energy of the incident wave is equal to
the reflected energy combined with the absorbed energy due to the switching
process.

Table 1: Running time comparison N |  $t_{\mbox{\scriptsize direct}}/t_{\mbox{\scriptsize res}}$ |  $t_{\mbox{\scriptsize suppos}}/t_{\mbox{\scriptsize res}}$
---|---|---
100 |  37.9 |  0.20
200 |  27.1 |  0.23
300 |  19.5 |  0.27

##  IV Going beyond the Rozanov bound

We use the proposed time-variant approach to overcome Rozanov’s bound for LTI
systems [7] with an ultrawideband signal, i.e., a modulated short time pulse.
To that end the bound is formulated as,

$|\ln\rho_{0}|\left(\lambda_{\max}-\lambda_{\min}\right)\leq 2\pi^{2}\mu_{s}d$
(23)

where $\rho_{0}$ stands for the maximal reflection within the operating
wavelength band $\left[\lambda_{\min},\lambda_{\max}\right]$ and $\mu_{s}$
denotes the static permeability.

For the sake of demonstration we use the modulated signal that is given in Eq.
(21) with $T_{p}=w/v_{p}=1.6[ns]$, and we vary the carrier frequency
$f_{0}=1/T_{0}$ with $T_{0}$ being the time period of the carrier signal.
Using these notations, $T_{p}/T_{0}$ is a measure of the signal’s fractional
bandwidth. To clarify our notations, note that with $(i)$ $T_{p}/T_{0}\to 0$
the signal is practically a baseband signal while with $(ii)$
$T_{p}/T_{0}\to\infty$ the signal is practically time-harmonic. In the range
in-between, the signal goes from being an ultra-wideband for
$T_{p}/T_{0}\lesssim O(1)$ to quasi-monochromatic and to narrowband for
$T_{p}/T_{0}\gtrsim O(1)$.

The physical layout is that of a layer with $d=0.4[m]$ that initially is
characterized by $\epsilon_{1}=1.5\epsilon_{0}$, $\mu_{1}=\mu_{0}$ and
$\sigma_{1}=0$. At time $t=T_{s}$, while the pulse is tightly contained within
the layer (i.e., $w\approx d$) the _conductivity is switched_ to $\sigma_{2}$.
We stress that the dielectric and magnetic parameters remain _unchanged_
($\epsilon_{2}=1.5\epsilon_{0}$ and $\mu_{2}=\mu_{0}$), and consequently the
_energy of the wave system is conserved_ during the switching process, and
moreover the frequency content is unaltered.

Based on these observations that are valid only when the time-variation is
applied solely on the conductivity, we evaluate, using the reflected energy,
the ratio between the _absorbed energy_ in the layer to the incident energy
for both time-variant and invariant layers, and for different values of
$T_{p}/T_{0}$. The relation between the incident, reflected and absorbed
energies are given by

$\frac{E_{\rm ref}}{E_{\rm
inc}}=\frac{\int_{-\infty}^{\infty}|\rho_{0}(f)v_{0}(f)|^{2}df}{\int_{-\infty}^{\infty}|v_{0}(f)|^{2}df},\quad\frac{E_{\rm
abs}}{E_{\rm inc}}=1-\frac{E_{\rm ref}}{E_{\rm inc}}.$ (24)

where $|v_{0}(f)|$ is the magnitude of the fourier transform of the signal
($|v_{0}(f)|=|FT\\{V_{0}(v_{p}t)\\}|$), $E_{\rm inc}$, $E_{\rm ref}$ and
$E_{\rm abs}$ denote the incident, reflected and absorbed energies,
respectively. $\rho_{0}$ is taken to maximize the left hand side of Eq. (23)
within the operating band

$\rho_{0}=\begin{cases}e^{\frac{-2\pi^{2}\mu_{s}d}{\Delta\lambda}}&\mbox{if
}\lambda\in\left[\lambda_{\min},\lambda_{\max}\right]\\\ 1&\mbox{else
}\end{cases}$ (25)

where $\Delta\lambda=\lambda_{\max}-\lambda_{\min}$. $\rho_{0}$ is a function
of $\Delta\lambda$ which can be expressed by two frequency related parameters:
a center frequency $f_{0}$ and the bandwidth $BW$ as follows:
$\Delta\lambda={c_{0}BW}/\left[{(f_{0}-{BW}/{2})(f_{0}+{BW}/{2})}\right]$. In
order to minimize the reflection within the operating band, we preform a sweep
along the parameters $f_{0}$ and $BW$ to maximize the ratio $E_{\rm
abs}/E_{\rm inc}$ according to Eq. (24).

The absorption optimization results are presented in Fig. 5 as a function of
the conductivity $\sigma_{2}$, for several values of $T_{p}/T_{0}\in(0,3)$
covering the range of extreme ultrawideband to quasi-monchromatic. In Fig. 5,
the blue lines represent a LTI system absorption response, the red lines
represent the proper Rozanov’s bound, and the black dashed lines are used to
depict the absorption with the time-varying scheme. It can be observed by Fig.
5 that for any given value of $T_{p}/T_{0}$ (signal bandwidth) it is possible
to determine a range of conductivities $\sigma_{2}$ such that the time-varying
scheme exhibits better, or at least equal, performance in comparison to
Rozanov’s bound. In particular, for certain bandwidth regimes, an enhanced
performances of the time-varying scheme occur for $\sigma\gtrsim 0.03$ (Figs.
5(a), 5(c) 5(e) 5(f)) while for other regimes it asymptotically occurs for
relatively larger values of the conductivity. Obviously, for the LTI case,
shown in blue lines, the performance is always below the bound.

A complementary view of the enhanced performances of the LTV scheme over
Rozanov’s bound in certain bandwidth regimes (“windows”) is demonstrated in
Fig. 6 that depicts the LTI, LTV and Rozanov’s bounds in (blue, dashed black
and red lines, respectively) for
$\sigma_{2}=0.03,0.036,0.054,0.09,0.18,0.3,1.002[\rm S/m]$ as a function of
the relative bandwidth $T_{p}/T_{0}$. It can be easily noted that for
$\sigma_{2}\gtrsim 0.036[\rm S/m]$ the LTV scheme becomes better than
Rozanov’s bounds in some bandwidth regimes of the excitation signal (
$\sigma_{2}\approx 0.036[\rm S/m]$ can be identified as the minimal required
value of loss for the LTV scheme in order to bypass the Rozanov’s bound for
ultrawideband signal excitation). This enhanced performance in terms of better
absorption and extended bandwidth improves as $\sigma_{2}$ is moderately
increased.

Finally, in Fig. 7 we draw a contour plot (“isolines”) of the ratio between
the absorbed energy of the LTV scheme and that predicted by the Rozanov’s
bound, i.e.,

$\frac{(E_{\rm{abs}})\big{|}_{\rm{LTV}}}{(E_{\rm{abs}})\big{|}_{\rm{Rozanov}}}.$

Note that the results presented in Fig. 5 and Fig. 6 are obtained by
horizontal and vertical “cuts” of Fig. 7 along the cartesian grids.
Furthermore the extent of the frequency bandwidth “windows” for which the LTV
scheme bypass the Rozanov bound are easily spotted by observing the regions in
which the contour line hight is larger the unity, i.e.,
($(E_{\rm{abs}})\big{|}_{\rm{LTV}}/(E_{\rm{abs}})\big{|}_{\rm{Rozanov}}>1$),
for $\sigma_{2}\gtrsim 0.036$ and $T_{p}/T_{0}<1.6$ while for
$T_{p}/T_{0}>1.6$ it approaches one asymptotically.

To conclude the discussion, this example demonstrates that Rozanov’s
absorption bound is valid only for LTI systems, and can be bypassed for
certain cases by utilizing mechanisms due to time variation, moreover the
example demonstrates the appealing characteristics of time-varying scheme that
involves _only conductivity (material loss) switching_ , for ultra-wideband
and quasi-monochromatic as well as narrowband (time harmonic) signals. In
order to reduce the “break-even” points for which the Rozanov’s bound bypasses
the LTV scheme in Fig. 5, and to further lower the required minimal
$\sigma_{2}$ values, it may be beneficial to use the more complex time-
variation schemes that involve, besides conductivity switching, also the
switching of the material permeability or permittivity. In These cases the
temporal switching itself may be lossy, namely, it may be used to sink _energy
out of the wave system_ [8].

The bounds demonstrated in this discussion were evaluated using the absorbed
energy spectrum of the signal along the _entire frequency band_ (note that
$f\in(-\infty,\infty)$ in Eq. (24)). It should be recalled that as opposed to
the mere switching of the conductivity, switching the medium parameters,
permeability and permittivity, changes the pulse temporal width and
consequently its frequency spectrum [8]. Therefore, if the interest is only
for example the minimization of the power reflection in the frequency band of
the incident wave, while discarding the reflection outside this band, then the
time-switching of the permittivity can be applied such that by the switching,
parts of the frequency spectrum are disperse to other bands outside that of
the incident wave. Such an effect is utilized and nicely discussed in [11] for
quasi-monochromatic signals.

(a) $T_{p}/T_{0}=0$

(b) $T_{p}/T_{0}=0.256$

(c) $T_{p}/T_{0}=0.48$

(d) $T_{p}/T_{0}=0.736$

(e) $T_{p}/T_{0}=0.992$

(f) $T_{p}/T_{0}=1.472$

(g) $T_{p}/T_{0}=1.984$

(h) $T_{p}/T_{0}=2.88$

Figure 5: Absorption caused by time invariant and time variant system compared
to Rozanov’s bound as a function of the conductivity and for some values of
the signal’s frequency bandwidth.

(a) $\sigma=0.03[\rm S/m]$

(b) $\sigma=0.036[\rm S/m]$

(c) $\sigma=0.054[\rm S/m]$

(d) $\sigma=0.090[\rm S/m]$

(e) $\sigma=0.18[\rm S/m]$

(f) $\sigma=0.3[\rm S/m]$

(g) $\sigma=1.002[\rm S/m]$

Figure 6: Absorption caused by time invariant and time variant schemes
compared to Rozanov’s bound as a function of the signal’s bandwidth and for
some values of the conductivity. Figure 7: A contour plot (“isolines”) of the
ratio of the absorbed energy of the LTV scheme to that predicted by Rozanov’s
bound, i.e.,
$(E_{\rm{abs}})\big{|}_{\rm{LTV}}/(E_{\rm{abs}})\big{|}_{\rm{Rozanov}}$, (see
in Eq. (24))

##  V Summary and Conclusion

In this manuscript, we consider the problem of a short pulse propagating and
scattering from electromagnetic wave absorber with time varying properties. A
transmission line equivalent is expressed such as to formulate an initial
boundary problem. The problem is solved by finding the spectral (on the
$s$-plane) Green’s function, followed by an inverse Laplace transform that is
implemented by summation over pole residues only (no branch point
singularities contribute, as we proved) to achieve expedite calculations. This
formalism is used to obtain the time domain reflected voltage from which all
other required quantities are calculated. Using this numerical procedure we
have solved the problem of reflection by a dielectric layer absorber in which
only the conductance is switched in time. We farther demonstrate that
Rozanov’s LTI absorbtion bound, when calculated over the _whole spectrum
range_ , can be bypassed for modulated pulsed signals, ultra-wideband, quasi-
monochromatic, and narrowband by such conductance abrupt switching.

## Acknowledgment

C. F. would like to thank to the Darom Scholarships and High-tech, Bio-tech
and Chemo-tech Scholarships and to Yaakov ben-Yitzhak Hacohen excellence
Scholarship. This research was supported by the Israel Science Foundation
(grant No. 1353/19).

##  Appendix A TL equivalent model for plane waves in plane stratified media

In this appendix we present the transmission line modeling for both TE and TM
plane wave illumination cases. A short pulse propagates in $\hat{k}$ direction
in (x,y) plane (see Fig. 8).

Figure 8: The total electromagnetic field is decomposed into two orthogonal
polarizations (TE and TM), as presented in (a) and (b) respectively. Such
modeling is applicable for both regions ($1$ and $2$).

Such modeling enables to translate a multi dimension vector problem into a
more simple form. In each medium the total electric and magnetic fields can be
expressed as

$\displaystyle\bar{E}=E_{\bot}f(x,y,t)\hat{z}+E_{||}f(x,y,t)\hat{z}\times\hat{k}$
(26a)
$\displaystyle\bar{H}=H_{\bot}f(x,y,t)\hat{k}\times\hat{z}+H_{||}f(x,y,t)\hat{z},$
(26b)

where $\hat{k}=k_{x}\hat{x}+k_{y}\hat{y}$, $f(x,y,t)=f(k_{x}x+k_{y}y-v_{p}t)$
and $v_{p}$ is the phase velocity. For the TE case the electric and magnetic
fields components can be written as

$\displaystyle E_{z}=E_{\bot}f(x,y,t)$ (27a) $\displaystyle
H_{y}=-\frac{E_{\bot}}{\eta}\cos\theta f(x,y,t)$ (27c) $\displaystyle
H_{x}=\frac{E_{\bot}}{\eta}\sin\theta f(x,y,t).$

The analog transmission line voltage, current and impedance are

$\displaystyle V(x,y,t)=E_{\bot}f(x,y,t)$ (28a) $\displaystyle
I(x,y,t)=\frac{V(x,y,t)}{Z_{TE}}$ (28b) $\displaystyle
Z_{TE}=\frac{|E_{z}|}{|H_{y}|}=\frac{\eta}{\cos\theta}.$ (28c)

Similarly, the field components of TM polarization are

$\displaystyle E_{y}=E_{||}\cos\theta f(x,y,t)$ (29a) $\displaystyle
E_{x}=-E_{||}\sin\theta f(x,y,t)$ (29b) $\displaystyle
H_{z}=\frac{E_{||}}{\eta\cos\theta}f(x,y,t),$ (29c)

where the transmission line voltage, current and impedance are expressed as

$\displaystyle V(x,y,t)=E_{||}\cos\theta f(x,y,t)$ (30a) $\displaystyle
I(x,y,t)=\frac{V(x,y,t)}{Z_{TM}}$ (30b) $\displaystyle
Z_{TM}=\frac{|E_{y}|}{|H_{z}|}=\eta\cos\theta.$ (30c)

The equivalent impedance can be described through wave ($\epsilon$,$\mu$ and
$\sigma$) and through transmission line terminologies ($L$,$C$,$R$ and $G$)
for a complex frequency($s$) as follows

$Z_{TE/TM}=\sqrt{\frac{s\mu+\sigma_{m}}{s\epsilon+\sigma}}A(\theta)=\sqrt{\frac{sL+R}{sC+G}}A(\theta),$
(31)

where $A(\theta)$ is defined according to (28c) and (30c) as

$A(\theta)=\begin{cases}\frac{1}{\cos\theta}&\mbox{TE }\\\
\cos\theta&\mbox{TM}\end{cases}$ (32)

##  Appendix B Analyticity of the spectral Green’s function through the branch
point singularities

In section $3$ we have shown that the spectral Green’s function is analytic
through the branch cut of $\gamma(s)$. Here, we prove that the spectral domain
Green’s function is analytic at the branch points of $\gamma(s)$ as well.
Recall Eq. (III.2), with the spectral variable $\gamma(s)$ as follows,

$g_{v}(x,x^{\prime},\gamma(s))=\frac{F(x,x^{\prime},\gamma(s))}{W(\gamma(s))}$
(33)

The $\gamma(s)$ branch points are the roots of $\gamma(s)$. One may note that
$\Gamma(s)$ is regular at $\gamma(s)=0$

$\Gamma(\gamma(s)=0)=\left\\{\begin{array}[]{lr}1,&\text{for
}s=-\frac{L}{R}\\\ -1,&\text{for }s=-\frac{G}{C}.\end{array}\right\\}$ (34)

Approximating $F(x,x^{\prime},\gamma(s))$ by first order Tylor series around
$\gamma(s)=0$ provides

$F(x,x^{\prime},\gamma(s))\sim-\gamma\left[1+\Gamma(\gamma)]\right(|x-x^{\prime}|+(x+x^{\prime})-2d).$
(35)

According to Riemann’s theorem for removable singularities [53], Green’s
function is regular at $\gamma=0$ since the following limit exists

$\lim_{\gamma\to
0}g_{v}(x,x^{\prime},\gamma)=\frac{|x-x^{\prime}|+x+x^{\prime}-2d}{2}.$ (36)

As previously explained, the spectral properties of $g_{v}(x,x^{\prime},s)$
and $V(x,s)$ are identical, therefore the spectral voltage $V(x,s)$ is also
analytic through the branch points.

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# PatchFormer: An Efficient Point Transformer with Patch Attention

Cheng Zhang1, Haocheng Wan1∗, Xinyi Shen2, Zizhao Wu1
1Hangzhou Dianzi University, Hangzhou China
2University College London, London UK
<EMAIL_ADDRESS><EMAIL_ADDRESS>These authors contributed equally.Corresponding author<EMAIL_ADDRESS>
###### Abstract

The point cloud learning community witnesses a modeling shift from CNNs to
Transformers, where pure Transformer architectures have achieved top accuracy
on the major learning benchmarks. However, existing point Transformers are
computationally expensive since they need to generate a large attention map,
which has quadratic complexity (both in space and time) with respect to input
size. To solve this shortcoming, we introduce Patch ATtention (PAT) to
adaptively learn a much smaller set of bases upon which the attention maps are
computed. By a weighted summation upon these bases, PAT not only captures the
global shape context but also achieves linear complexity to input size. In
addition, we propose a lightweight Multi-Scale aTtention (MST) block to build
attentions among features of different scales, providing the model with multi-
scale features. Equipped with the PAT and MST, we construct our neural
architecture called PatchFormer that integrates both modules into a joint
framework for point cloud learning. Extensive experiments demonstrate that our
network achieves comparable accuracy on general point cloud learning tasks
with 9.2$\times$ speed-up than previous point Transformers.

##  1 Introduction

Transformer has recently drawn great attention in natural language processing
[7, 34] and 2D vision [8, 21, 38, 33] because of its superior capability in
capturing long-range dependencies. Self-Attention (SA), the core of
Transformer, obtains an attention map by computing the affinities between self
$queries$ and self $keys$, generating a new feature map by weighting the self
$values$ with this attention map. Benefitting from SA module, Transformer is
capable of modeling the relationship of tokens in a sequence, which is also
important to many point cloud learning tasks. Hence, plenty of researches have
been done to explore Transformer-based point cloud learning architectures.

Figure 1: A large indoor scene often consists of small instances (e.g., chair
and typewriter) and large objects (e.g., table and blackboard), building the
relationships among them requires a multi-scale attention mechanism.

Recently, Nico et al. proposed PT1 [9] to extract global features by
introducing the standard SA mechanism, which aims to capture spatial point
relations and shape information. Guo et al. proposed offset-attention (PCT
[10]) to calculate the offset difference between the SA features and the input
features by element-wise subtraction. Lately, more and more researchers have
applied SA module to various point cloud learning tasks and achieved
significant performance such as [56, 24]. However, existing point Transformers
are computationally expensive because the original SA module needs to generate
a large attention map, which has high computational complexity and occupies a
huge number of GPU memory. This bottleneck lies in that both the generation of
attention map and its usage require the computation with respect to all
points.

Towards this issue, we propose a novel lightweight attention mechanism, namely
PAT which calculates the attention map via low-rank approximation [52, 17].
Our key observation is that a 3D shape is composed of its local parts and thus
the features of points in the same part should have similar semantics. Based
on this observation, we first exploit the intrinsic geometry similarity,
cluster local points on a 3D shape as one patch and estimate a base by
aggregating the features of all points in the same patch. Then we use a
product of self $queries$ and self bases to approximate the global attention
map, which can be obtained by computing self $queries$ and self $keys$.
Notably, the representation of such product is low-rank and discards noisy
information from the input.

In addition, to aggregate local neighborhood information, Zhao et al. [56]
proposed PT2 to build local vector attention in neighborhood point sets, Guo
et al. (PCT[10]) proposed to use a neighbor embedding strategy to improve
point embedding. Though PT2 and PCT have achieved significant progress, there
exist problems that restrict their efficiency and performance. First, they
wastes a high percentage of the total time on structuring the irregular data,
which becomes the efficiency bottleneck [23]. Second, they fails to build the
attentions among features of different scales which is very important to 3D
visual tasks. As shown in Fig 1, a large indoor scene often contains small
instances (e.g., chair and lamp) and large objects (e.g., table), building the
relationships among them required a multi-scale attention mechanism. However,
the input sequence of PT2 and PCT is generated from equal-sized points, so
only one single scale feature will be preserved in the same layer.

To solve these issues, we present a lightweight Multi-Scale aTtention (MST)
block for point cloud learning, which consists of two steps. In the first
step, our MST block transforms point cloud into voxel grids, sampling boxes
with multiple convolution kernels of different scales and then concatenates
these grids as one embedding (see Fig 4). Specifically, we propose to use the
depth-width convolution (DWConv [11]) on boxes sampling beacuse of both few
parameters and FLOPs. In the second step, we incorporate 3D relative position
bias and build attentions to non-overlapping local 3D window, providing our
model with strong multi-scale features at a low computational cost.

Based on these proposed blocks, we construct our neural architecture called
PatchFormer (see Fig 2). Speciﬁcally, we perform the classiﬁcation task on the
ModelNet40 and achieve the strong accuracy of 93.5% (no voting) with
9.2$\times$ faster than previous point Transformers. On ShapeNet and S3DIS
datasets, our model also obtains strong performance with 86.5% and 68.1% mIoU,
respectively.

The main contributions are summarized as following:

  * •

We present PatchFormer for efficient point cloud learning. Experiments show
that our network achieves strong performance with 9.2$\times$ speed-up than
prior point Transformers.

  * •

We propose PAT, the first linear attention mechanism in the point cloud
analysis paradigm.

  * •

We present a lightweight voxel-based MST block, which compensates for previous
architectures’ disability of building multi-scale relationship.

##  2 Related works

###  2.1 Transformer for 2D Vision

Motivated by the success of Transformers in NLP [34, 29, 48, 6, 14],
researchers designed visual Transformers for vision tasks to take advantage of
their great attention mechanism. In particular, Vision Transformer (ViT) [8]
is the first such example of a Transformer-based approach to match or even
surpass convolution neural networks (CNNs) for image classification. Later,
Wang et al. [37] proposed pyramid structure into transformers, named PVT,
greatly decreasing the number of patches in the later layers of the model. Liu
et al. [21] proposed Swin Transformer whose representation is computed with
non-overlapping local windows. Subsequently, Wang et al. [38] and Chen et al.
[2] proposed CrossFormer and CrossViT to study how to learn multi-scale
features in Transformers.

Inspired by the cross-scale attention used in CrossFormer and CrossViT for
image analysis, we present a voxel-based MST block for point cloud learning
that combines voxel grids of different sizes to learn stronger local features.

###  2.2 Point Cloud Learning

Most existing point cloud learning methods could be classified into two
categories in terms of data representations: the _voxel_ -based models and the
_point_ -based models. The _voxel_ -based models generally rasterize point
clouds onto regular grids and apply 3D convolution for feature learning [57,
27, 40, 13, 15]. These models are computationally efficient due to their
excellent memory locality, but suffer from the inevitable information degrades
on the fine-grained localization accuracy [31, 23, 51]. Instead of
voxelization, developing a neutral network that consumes directly on point
clouds is possible [26, 54, 12, 30, 47, 35, 39, 53, 50, 20, 44]. Although
these _point_ -based models naturally preserve accuracy of point location,
they are usually computationally intensive.

Generally, the voxel-based models have regular data locality and can
efficiently encode coarse-grained features, while the point-based networks
preserve accuracy of location information and can effectively aggregate fine-
grained features. In this paper, we propose PatchFormer to incorporate both
the advantages from the two models mentioned above.

Figure 2: The architecture of PatchFormer: PatchFormer is comprised of three
stages and each stage contains two blocks: MST block and PAT bolck. A
specialized head (e.g., the classification head) is followed by the final
stage for the specific task. MST block: It first voxelizes point cloud into
voxel grids, aggregating multi-scale features, and then conducts 3D window-
based SA (W-SA) to capture local information. Finally, MST block transforms
voxel grids to points and feed them into PAT block. Numbers in MST represent
the size of kernels used DWConv, R denotes the relative position bias and W-SA
refers to 3D window-based self-attention.

###  2.3 Point Transformers

Powered by Transformer [34] and its variants [21, 8], the point-based models
have recently applied SA to extract features from point clouds and improve
performance significantly [10, 9, 51, 16, 25, 49]. In particular, PT1 is the
first such example of a Transformer-based approach for point cloud learning.
Later, Guo et al. and Zhao et al. proposed PCT and PT2 to construct SA
networks for general 3D recognition tasks.

However, they suffer from the fact that as the size of the feature map
increases, the computing and memory overheads of the original SA increase
quadratically. To address this issue, we propose PAT to compute the relation
between self $queries$ and a much smaller bases, yet captures the global
context of a point cloud as well.

##  3 Overview

An overview of the PatchFormer architecture is presented in Fig 2. Our method
first embeds a point cloud $\mathcal{P}$ into a D dimensional space
$F\in\mathbb{R}^{N\times D}$ using a shared MLP, where $N$ is the number of
points. We empirically set D = 128, a relatively small value for computational
efficiency. Late, we split our model into three stages and each stage is
comprised of two blocks: Multi-Scale aTtention (MST) block and Patch-ATtention
(PAT) block.

As illustrated in Fig 2, the MST block first voxelizes a point cloud into
regular voxel grids and then feeds them into a multi-scale aggregating module.
In this module, we sample boxes using three DWConv kernels of different size
and concatenate them as one embedding. After that, we limit SA computation to
non-overlapping local boxes in order to alleviate the quadratic complexity of
the original SA. Note that a LayerNorm (LN) layer is applied before W-SA
module and MLP module, and a residual connection is applied after each module.
Eventually, we leverage the trilinear interpolation to transform the voxel
grids to points.

Like ViT [8], the PAT block treats each point as a “token” and aggregates
global feature by using patch attention. It receives MST block’s output as
input, estimates a much more compact bases and generates global attention map
upon these bases. Note that, our method reduces the complexity (both in space
and time) from $\mathcal{O}(N^{2})$ of the original SA to $\mathcal{O}(MN)$
($M<<N$) where $M$ is the number of bases. As a result, the proposed patch
attention can conveniently replace the backbone networks in existing point
Transformers for various point cloud learning tasks.

Throughout the next sections we use the following notations: the original
point cloud with $N$ points is denoted by
$\mathcal{P}=\\{p_{i}\\}_{i=1}^{N}\subseteq\mathbb{R}^{C}$. In the simplest
setting of $C$ = 3, each point contains 3D coordinates.
$F=\\{f_{i}\\}_{i=1}^{N}\subseteq\mathbb{R}^{D}$ is the input embedding
feature.

##  4 Method

In this section, we first analysis the original SA mechanism, then we detail
our novel way to define attention: patch attention. And finally, we discuss
the design of MST block in detail.

Figure 3: Architecture of PAT block (patch attention). PAT can be seen as an
variant to the original self-attention to approximate global map at a lower
computational cost.

###  4.1 Self-Attention

We first revisit the self-attention (SA) mechanism. The standard SA, also
called scalar dot-product attention, is a mechanism that calculates semantic
affinities among different elements within a sequence of data. Following the
terminology in [34], let Q, K, V be the $query,key$ and $value$ matrices,
respectively, generated by linear transformations of the input features
$F\in\mathbb{R}^{N\times D}$ as follows

$\displaystyle(Q,K,V)$ $\displaystyle=(W_{q},W_{k},W_{v})F,$ (1)
$\displaystyle Q,K,V$ $\displaystyle\in\mathbb{R}^{N\times D},$ (2)

where $W_{q},W_{k}$ and $W_{v}$ are the shared learnable linear transformation
as illustrated in Fig 3.

Using the pairwise dot product $QK^{T}\in\mathbb{R}^{N\times N}$, then SA can
be formulated as:

$\displaystyle A=(\alpha_{i,j})$ $\displaystyle=softmax(QK^{T}),$ (3)
$\displaystyle F_{out}$ $\displaystyle=AV,$ (4)

where $A\in\mathbb{R}^{N\times N}$ is the attention map and $\alpha_{i,j}$ is
the pair-wise affinity between (similarity of) the $i$-th and $j$-th elements.
It is apparent that the output $F_{output}$ is a weighted sum of $V$, where a
value gets more weight if the similarity between the keys and values yields a
higher attention weighting score.

However, the high computational complexity of $\mathcal{O}(N^{2}D)$ presents a
significant drawback to use of SA. The quadratic complexity in the number of
input points makes it infeasible to apply SA to point cloud directly.

###  4.2 Patch Attention

In view of the high computational complexity of the attention mechanism and
limitations, we first propose the PAT, which is an augmented version of SA.
Unlike prior point Transformers obtain an attention map by computing
affinities between self $queries$ and self $keys$, our patch attention (PAT)
computes the relation between self $queries$ and a much smaller bases, yet
captures the global context of a point cloud.

For simplicity, we consider an input point cloud $\mathcal{P}$ and its
corresponding feature map $F$ of size $N\times D$, our proposed PAT is
illustrated in Fig 3 which consists of two steps, including _base estimation_
and _data re-estimation_.

Base Estimation. In this step, we estimate a compact basis set
$B\in\mathbb{R}^{M\times D}$ where $M$ is the number of bases. In particular,
we introduce the concept of patch-instance base. For each point cloud
$\mathcal{P}$ in the dataset, we over-segment it into $M$ patches ($M<<N$) and
based on which, we create $M$ patch-instance bases. In this way, the global
shape can be approximated by the set of each patch-instance base, which have a
less total number. For simplicity, we use the K-Means algorithm to segment
$\mathcal{P}$ into M patches $\\{S_{1},S_{2},...,S_{M}\\}$, M=96, by default
in classification task. We define each base as $b_{m}$ by aggregating the
representations of all the points in the $S_{m}$, it can be described as:

$\displaystyle b_{m}$ $\displaystyle=\sum_{f_{i}\in
S_{m}}w_{i}(\varphi(f_{i})),$ (5) $\displaystyle B$
$\displaystyle=\\{b_{m}\\}_{m=1}^{M}\subseteq\mathbb{R}^{D}.$ (6)

Here, $f_{i}$ is the representation of point $p_{i}$, the transformation
function $\varphi(\cdot)$ is an MLP with one linear layer and one ReLU
nonlinearity, $w_{i}$ is the normalized degree for $f_{i}$ belonging to the
$S_{m}$. We use spatial softmax to normalize each patch.

Generally, our base estimation method can adaptively adjust the contribution
of all points in the same patch to the base via a data-driven way. Such
adaptive adjusting facilitates to fit the intrinsic geometry submanifold.

Data Re-estimation. After estimating the bases $B$, we can replace $K$
matrices with $B$ and re-formulate Eq 3 as:

$\displaystyle A=softmax(QB^{T}),$ (7)

where $A\in\mathbb{R}^{N\times M}$ is the attention map constructed from a
compact basis set. After that, the final bases $B$ and attention map $A$ are
used to re-estimate the inputs $F$. We formulate a new equation to re-estimate
the $F$ using $\tilde{F}$ as follows:

$\displaystyle\tilde{f_{i}}$ $\displaystyle=\sum_{m=1}^{M}A_{i}^{m}b_{m},$ (8)
$\displaystyle\tilde{F}$
$\displaystyle=\\{\tilde{f_{i}}\\}_{i=1}^{N}\subseteq\mathbb{R}^{D}.$ (9)

As $\tilde{F}\in\mathbb{R}^{N\times D}$ is constructed from a compact basis
set $B$, it has the low-rank property compared with the input $F$.

Inspired by PCT [10], we calculate the difference between the estimated
features $\tilde{F}$ and the input features $F$ by element-wise subtraction.
Finally, we feed the difference into MLP layer and adopt residual connection
strategy to help propagate information to higher layers. This step can be
formulated as:

$\displaystyle F_{output}=\phi(\tilde{F}-F)+F,$ (10)

where $F_{ouput}\in\mathbb{R}^{N\times D}$ is the outputs of our PAT block and
$\phi(\cdot)$ is an MLP with one linear layer and one ReLU nonlinearity.

Complexity Analysis. Compared with the standard SA module, our PAT finds a
representative set of bases for points of a point cloud, which reduces the
complexity from $\mathcal{O}(N^{2})$ to $\mathcal{O}(MN)$ ($M<<N$), where $M$
and $N$ are the number of bases and points, respectively. Moreover, we only
need to calculate K-Means algorithm once on the original point cloud
$\mathcal{P}$ that can be accelerated in parallel by CUDA. Although the
K-Means optimization has an asymptotic complexity $\mathcal{O}(NMC)$, it can
be ignored in our network because $M$ is fixed and $C=3$.

###  4.3 Multi-Scale Attention

In this subsection, we detail how our MST block learns multi-scale feature
representations in attention models. This block consists of two steps,
including _Multi-scale feature aggregating_ and _Attention building_.

$3\times 3\times 3$$5\times 5\times 5$$7\times 7\times 7$32 $dims$32 $dims$64
$dims$

Figure 4: Illustration of multi-scale feature aggregating in MST block on
Stage-1. We note that this is a 2D example and can be easily extended to 3D
cases. The input voxel grids is sampled by three DWConv kernels (i.e.,
$3\times 3\times 3$, $5\times 5\times 5$, $7\times 7\times 7$) with stride
$1\times 1\times 1$. Each embedding is constructed by projecting and
concatenating the three 3D boxes.

Multi-scale Feature Aggregating. This step is used to generate multi-scale
features for each stage. Fig 4 illustrates the first MST block, which is ahead
of the Stage-1, as an example. we receive voxel grids as input, sampling boxes
using three kernels of different size. The strides of three kernels are kept
the same so that they generate the same number of embeddings. As can be seen
in Fig 4, every three corresponding boxes own the same center but locate at
different scales. These three boxes will be projected and concatenated as one
embedding. In practice, the process of sampling and projecting can be
implemented through three DWConv layers. Note that we use a lower dimension
for large kernels while a higher dimension for small kernels. Fig 4 provides
the specific allocation rule in its subtable, where a 128 dimensional example
is given. Compared with allocating the dimension equally, our scheme reduces
computational cost while maintaining the model’s high performance. The MST
blocks in other stages work in a similar way. As shown in Fig 2, MST blocks in
Stage-2/3 use two kernels ($3\times 3\times 3$ and $5\times 5\times 5$). The
strides are set as $1\times 1\times 1$. For computing efficiency, DWConv with
kernel sizes larger than $5\times 5\times 5$ is implemented by stacking
multiple convolutions with kernel size $3\times 3\times 3$ and $5\times
5\times 5$.

Attention Building. To build the attentions among features of different
scales. we attempt to conduct the standard SA on multi-scale feature map.
However, the computation complexity of the full SA mechanism is quadratic to
feature map size. Therefore, it will suffer from huge computation cost for 3D
vision tasks that take high resolution feature maps as input, such as semantic
segmentation.

To solve this shortcoming, our MST block limits the SA computation to non-
overlapping local 3D windows. In addition, we observe that numerous previous
works [22, 21, 41] have shown that it can be advantageous to include relative
position bias in SA computation. Thus we introduce 3D relative position bias
$R\in\mathbb{R}^{V^{3}\times V^{3}}$ as

$F_{output}=softmax(QK^{T}+R)V,$ (11)

where $Q,K,V\in\mathbb{R}^{V^{3}\times D}$ are the $query$, $key$ and $value$
matrices, and $V^{3}$ is the number of voxel grids in a local 3D window. Since
the relative position along each axis lies in the range $[-V+1,V-1]$, we
parameterize a smaller-sized bias matrix
$\hat{R}\in\mathbb{R}^{(2V-1)\times(2V-1)\times(2V-1)}$, and values in $R$ are
taken from $\hat{R}$.

For cross-window information interaction, existing works [37, 21, 33]
suggested to apply halo or shifted window to enlarge the receptive filed.
However, the elements within each Transformer block still has limited
attention area and requires stacking more blocks to achieve large receptive
filed. In our network, the local attention is building in multi-scale input
features. Thus, we don’t need to stack more attention layers for cross-window
connection or larger receptive filed.

##  5 Experiments

In this section, we evaluate the proposed PatchFormer for different tasks:
classiﬁcation, part segmentation, and scene semantic segmentation. Performance
is quantitatively evaluated using four metrics: mean class accuracy, overall
accuracy (OA), per-class intersection over union (IoU), and mean IoU (mIoU).
For fair comparison, we report the measured latency and model size on a RTX
2080 GPU to reﬂect the efﬁciency but evaluate other indicators on a RTX 3090
GPU.

Implementation details. We implement the PatchFormer in PyTorch. We use the
SGD optimizer with momentum 0.9 and weight decay 0.0001, respectively. For 3D
shape classification on ModelNet40 and 3D object part segmentation on
ShapeNetPart, we train for 250 epochs. The initial learning rate is set to
0.01 and is dropped until 0.0001 by using cosine annealing. For semantic
segmentation on S3DIS, we train for 120 epochs with initial learning rate 0.5,
dropped by 10$\times$ at 50 epochs and 80 epochs.

###  5.1 Shape Classification

Model | Input | OA | Latency
---|---|---|---
OA$<$92.5
PointNet [26] | 16$\times$1024 | 89.2 | 13.6ms
PointNet++ [28] | 16$\times$1024 | 91.9 | 35.3ms
SpiderCNN [45] | 8$\times$1024 | 92.4 | 82.6ms
PointCNN [18] | 16$\times$1024 | 92.2 | 221.2ms
PointWeb [55] | 16$\times$1024 | 92.3 | $-$
PVCNN [23] | 16$\times$1024 | 92.4 | 24.2ms
OA$>$92.5
KPConv [32] | 16$\times$6500 | 92.9 | 120.5ms
DGCNN [39] | 16$\times$1024 | 92.9 | 85.8ms
LDGCNN [53] | 16$\times$1024 | 92.7 | $-$
PointASNL [46] | 16$\times$1024 | 93.2 | 923.6ms
PT1 [9] | 16$\times$1024 | 92.8 | 320.6ms
PT2 [56] | 8$\times$1024 | 93.7 | 530.2ms
PCT [10] | 16$\times$1024 | 93.2 | 92.4ms
PatchFormer | 16$\times$1024 | 93.5 | 34.3ms
Table 1: Results on ModelNet40 [43]. Compared with previous Transformer-based
models, our PatchFormer achieves the promissing accuracy with 9.2$\times$
measured speed-up on average.

Data. We evaluate our model on the ModelNet40 [42] dataset. This dataset
contains 12,311 computer-aided design models from 40 man-made object
categories, in which 9,843 models are used for training and 2,468 models are
used for testing. We follow the experimental conﬁguration of Qi et al. [26]:
(1) we uniformly sample 1,024 points from the mesh faces for each model; (2)
the point cloud is re-scaled to ﬁt the unit sphere; and (3) the (x,y,z)
coordinates and the normal of the sampled points are used in the experiment.
During the training process, randomly scaling, translating and perturbing the
objects are adopted as the data augmentation strategy in our experiment.

Results. The results are presented in Table 1. The overall accuracy of
PatchFormer on ModelNet40 is 93.5%. It outperforms strong graph-based models
such as DGCNN, strong point-based models such as KPConv and excellent
attention-based networks such as PointASNL. Remarkably, compared with existing
Transformer-based models such as PT1,PT2 and PCT, our model is 9.2$\times$
faster while achieving comparable accuracy.

###  5.2 Computational requirements analysis

Model | Params | FLOPs | SDA($\%$) | OA($\%$)
---|---|---|---|---
PointNet | 3.47M | 0.45G | 0.0 | 89.2
PointNet++(SSG) | 1.48M | 1.68G | 43.5 | 90.7
PointNet++(MSG) | 1.74M | 4.09G | 47.6 | 91.9
DGCNN | 1.81M | 2.43G | 57.2 | 92.9
PointASNL | 3.98M | 5.92G | 39.8 | 93.1
PT1 | 21.1M | 5.05G | 32.5 | 92.8
PT2 | 9.14M | 17.1G | 65.4 | 93.7
PCT | 2.88M | 2.17G | 24.6 | 93.2
PatchFormer | 2.45M | 1.62G | 6.3 | 93.5
Table 2: Computational resource requirements. SDA means the rate of total
runtime on structuring the sparse data.

We now consider the computational requirements of PatchFormer and several
other baselines by comparing the floating point operations required (FLOPs)
and number of parameters (Params) in Table 2. We evaluate these indicators on
ModelNet40 dataset. From Table 2, we can see that PatchFormer has the lowest
memory requirements with only 2.45M parameters and also puts a low load on the
processor of only 1.62G FLOPs, yet delivers comparable accurate results of
93.5%. Notably, we summarize Table 2 the PatchFormer only spend 6.3% of the
total runtime on structuring the irregular data, which is much lower than
previous point Transformers. Compare with its baselines, PatchFormer has not
only strong performance but also the lowest computational and memory
requirements. These characteristics make PatchFormer suitable for deployment
on a edge devices.

###  5.3 Object Segmentation

Figure 5:  Attention map and segmentation results on ShapeNet. From left to right: attention maps w.r.t. three selected entries in the bases, segmentation results. Figure 6: Visualization of semantic segmentation results on the S3DIS dataset. The input is in the top row, PatchFormer prediction is on the middle, the ground truth is on the bottom. Model | Input | mIoU | Latency
---|---|---|---
DGCNN [39] | 8$\times$2048 | 85.4 | 96.2ms
PT1 [9] | 6$\times$2048 | 85.9 | 360.4ms
PointCNN [18] | 8$\times$2048 | 86.1 | 145.6ms
PointASNL [46] | 6$\times$2048 | 86.1 | 1023.2ms
KPConv [32] | 6$\times$6500 | 86.4 | 127.8ms
PCT [10] | 8$\times$2048 | 86.4 | 101.1ms
PT2 [56] | 4$\times$2048 | 86.6 | 560.2ms
PatchFormer | 8$\times$2048 | 86.5 | 45.8ms
Table 3: Results of part segmentation on ShapeNet Part. Model | Input | mIoU | Latency
---|---|---|---
DGCNN | 8$\times$4096 | 47.1 | 178.1ms
PointCNN [18] | 4$\times$4096 | 57.3 | 282.3ms
PointASNL [46] | 4$\times$4096 | 62.6 | 1895.2ms
PT1 [9] | 4$\times$4096 | 63.1 | 1223.6ms
MinkowskiNet [5] | $-$ | 65.4 | $-$
KPConv [32] | 4$\times$6500 | 67.1 | 267.5ms
PatchFormer | 8$\times$4096 | 68.1 | 109.8ms
Table 4: Indoor scene segmentation results on S3DIS, evaluated on Area5. From
this table, we can see that PatchFormer outperforms most of previous models in
accuracy and efficiency.

Data. We use the large-scale 3D dataset ShapeNet Parts [19] as the experiment
bed. ShapeNet Parts contains 16,880 models (14,006 models are used for
training, and 2874 models are used for testing), each of which is annotated
with two to six parts, and the entire dataset has 50 different part labels. We
sample 2,048 points from each model as input, with a few point sets having six
labeled parts. We directly adopt the same train–test split strategy similar to
DGCNN [39] in our experiment.

Results and Visualization. From Table 3, we can see that with similar
accuracy, our PatchFormer is 12.4$\times$ faster than PT2 and 2.2$\times$
faster than PCT. Notably, with better accuracy, PatchFormer is 22.7$\times$
faster than PointASNL. In addition, we randomly select three entity from $B$
in the last layer of our network and show their corresponding attention score
of all points. As we can see, each basis corresponds to an abstract concept of
the point cloud and the learned attention maps focus on meaningful parts for
object segmentation as in Fig 5.

###  5.4 Indoor Scene Semantic Segmentation

Data. We evaluate our model on the S3DIS dataset [1], which contains 3D RGB
point clouds from six indoor areas of three different buildings. Each point is
marked with a semantic label from 13 categories (e.g., board, bookcase, chair,
ceiling, and beam) plus clutter. Following a common protocol [32, 26], we
divide and sample each room into 1 m $\times$ 1 m blocks, wherein each point
is represented by a 9D vector (XYZ, RGB, and normalized spatial coordinates).
In addition, the points in each block are sampled into a uniform number of
4,096 points during the training process, and all points are used in the test.

Results and Visualization. The results are presented in Tables 4. From this
table we can see that our PatchFormer attains mIoU of 68.1%, which outperforms
graph-based methods such as DGCNN [39], sparse convolutional networks such as
MinkowskiNet [5], continuous convolutional networks such as KPConv [32],
attention-based models such as PointASNL [46] and point Transformer such as
PT1. Remarkably, our PatchFormer also outperforms these powerful model by a
large margin in latency.

Fig 6 shows the PatchFormer’s predictions. We can see that the predictions are
very close to the ground truth. PatchFormer captures detailed multi-scale
features in complex 3D scenes, which is important in our network.

###  5.5 Ablation Studies

We now conduct a number of controlled experiments that examine specific
decisions in the PatchFormer design.

M | ModelNet40(OA) | ShapeNet(mIoU) | Latency
---|---|---|---
32 | 91.54 | 84.92 | 33.25ms
64 | 92.94 | 85.82 | 33.82ms
96 | 93.52 | 86.52 | 34.32ms
128 | 93.50 | 86.54 | 35.56ms
Table 5: Ablation study: number of bases M in our network. We report latency
on ModelNet40 dataset.

Number of Bases. We first investigate the setting of the number of bases. The
results are shown in Table 5. The best performance of classification task is
achieved when $M$ is set to 96. On the one hand, when the bases is smaller
($M$ = 32 or $M$ = 64), the model may not have sufficient context for its
predictions. On the other hand, increasing $M$ doesn not give PatchFormer much
accuracy benefit but incurs a raise on latency. This also demonstrates the
efficiency and effectiveness of our PAT.

Ablation | ModelNet40(OA) | ShapeNet(mIoU)
---|---|---
w/o MS feature | 92.85 | 85.22
MLP | 92.62 | 85.32
EdgeConv | 93.10 | 85.89
self-attention | 93.29 | 86.22
no rel. pos | 93.15 | 86.30
Ours | 93.52 | 86.52
Table 6: Ablation study on the multi-scale feature aggregation, PAT and relative bias on two benchmarks. w/o MS feature: all MST block without aggregate multi-scale features. MLP: replace PAT with MLP layer in our architecture. EdgeConv: replace PAT with EdgeConv layer in our architecture. self-attention: replace PAT with self-attention layer in our architecture. rel. pos: the default settings with an additional relative position bias term. Ablation | ModelNet40(OA) | Latency(ms)
---|---|---
$A^{2}$ Net [3] | 92.89 | 36.89
EMA Net [17] | 93.02 | 37.27
Linformer [36] | 93.14 | 40.22
Performer [4] | 93.22 | 35.46
Ours | 93.52 | 34.32
Table 7: We replace our PAT with other linear attention mechanisms. We collect
their public code and adapt them to 3D data.

Effect of Multi-scale feature aggregating. We conduct an ablation study on the
Multi-scale feature aggregating step. From Table 6, we can see the performance
without this step on ModelNet40 and ShapeNet are 92.85%/85.22%, in terms of
OA/mIoU. It is much lower than the performance with Multi-scale feature
(93.52%/86.52%). This suggests that the Multi-scale feature is essential in
this setting.

Impact of PAT. We investigate the impact of PAT used in the PAT block. From
Table 6, we can see that PAT is more effective than the no-attention baseline
(MLP). The performance gap between PAT and MLP baseline is significant: 93.52%
vs. 92.62% and 86.52% vs. 85.32%, an improvement of 0.9 and 1.2 absolute
percentage points. Compared with EdgeConv baseline, our PAT also achieves
improvements of 0.42 and 0.63 absolute percentage points. Notably, our PAT
outperforms the self-attention baseline with 0.23 and 0.30 absolute percentage
points. We also compare PAT with other linear attention mechanisms in Table 7
and find it achieves the best accuracy and running speed. PAT has two obvious
advantages. First, we only need to calculate K-Means once on the original
point cloud due to the intrinsic geometry similarity, which means that the
computational cost of the base estimation can be neglected. Second. PAT based
on residual learning is more robust to any rigid transformation of objects.

Effect of 3D Relative position bias. Finally, we investigate the effect of 3D
relative position bias used in the MAS block. Table 6 shows results. We can
see that the PatchFormer with relative position bias yields +0.37% OA/+0.47%
mIoU on ModelNet40 and ShapeNe in relation to those without position encoding
respectively, indicating the effectiveness of the relative position bias.

##  6 Conclusion and Future Work

In this work, we propose a new type of attention mechanism, namely Patch
ATtention (PAT) for point cloud learning, which computes a much smaller bases
by exploiting the geometric similarity of nearby points. The reconstructed
output of our PAT is low-rank and achieves linear time-space complexity to
input size. Further, we propose a lightweight MST block, building attentions
among features of different scales and providing our model with multi-scale
features. Based on these modules, we construct PatchFormer for various point
cloud learning task. Experiments show that our PatchFormer achieves comparable
accuracy and better speed than other point Transformers.

We hope that our work will provide empirical guidelines for new method design
and inspire further investigation of the properties of point Transformers. For
example, performing K-Means in the points, extracting a patch feature for each
cluster, directly reducing the number of tokens in the embedding stage.

##  7 Acknowledgements

This work was partially supported by the Zhejiang Provincial Natural Science
Foundation of China (LGF21F20012), the National Natural Science Foundation of
China (No.61602139), and the Graduate Scientific Research Foundation of
Hangzhou Dianzi University (CXJJ2021082, CXJJ2021083).

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