aainstitutetext:  School of Physics and Electronics Science,
Hunan Provincial Key Laboratory of High-Energy Scale Physics and Applications,
Hunan University, Changsha 410082, P. R. Chinabbinstitutetext: Lanzhou Center
for Theoretical Physics, Key Laboratory of Theoretical Physics of Gansu
Province, School of Physical Science and Technology, Lanzhou University,
Lanzhou 730000, Chinaccinstitutetext: School of Physical Science and
Technology, Southwest University, Chongqing 400715, China

# Thick brane in Rastall gravity

Yi Zhong c Ke Yang b,1 and Yu-Xiao Liu111Corresponding author.
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In this work, thick branes in Rastall gravity are investigated. Three types of
maximally symmetric thick brane models are constructed and the linear tensor
perturbation is analyzed. In the flat brane model, the tensor modes of the
perturbation are either unstable or nonlocalizable for a nonvanishing Rastall
parameter. In the de Sitter brane model, only the ground state of the tensor
mode is localized. In the anti-de Sitter brane model, the number of the bound
tensor states is infinity. For both the de Sitter and anti-de Sitter brane
models, the condition of stability for the Rastall parameter is obtained.

##  1 Introduction

It has been an attractive idea for the last two decades that our spacetime may
have extra dimensions and the observable universe could be a brane in a
higher-dimensional spacetime. In the braneworld scenario, it is possible to
explain the hierarchy between the Planck scale and the electroweak scale
ArkaniHamed:1998rs ; Randall:1999ee ; Randall:1999vf . The braneworlds in
these models are geometrically thin. However, in a viable model, a braneworld
should have a thickness. Inspired by the domain wall in Ref. Rubakov:1983bb .
The thick braneworld generated by a background scalar field was proposed
Gremm:1999pj ; Csaki:2000fc ; PhysRevD.62.044017 ; Kobayashi:2001jd ;
PhysRevD.66.024024 ; Bazeia:2002sd ; Andrianov:2005hm ; Afonso:2006gi ;
Neupane:2010ey . Furthermore, since general relativity suffers from both
phenomenological and theoretical problems such as dark energy problem and
singularity problem, various modified theories of gravity have been proposed,
and braneworlds in various modified gravities have been investigated
Liu:2017gcn . It was found that the branes may have inner structure in some
theories such as $f(T)$ gravity Yang:2012hu  and mimetic gravity Zhong:2017uhn
. Pure geometric thick branes were found in modified gravities Arias:2002ew ;
Barbosa-Cendejas:2005vog ; Barbosa-Cendejas:2006cic ; Zhong:2015pta  as well.
See Refs. Giovannini:2001ta ; Cho:2001nf ; Heydari-Fard:2007dos ;
Nozari:2008ny ; Parry:2005eb ; Afonso:2007gc ; Dzhunushaliev:2009dt ;
Koivisto:2005yk ; Bazeia:2013oha ; Chakraborty:2015taq ; Liu:2012rc ;
Yang:2017puy ; Bogdanos:2006qw ; Farakos:2007ua ; Fiorini:2013hva  for more
recent works of braneworld in modified gravities and Ref. Liu:2017gcn  for a
review.

In general relativity and many modified gravities, matter fields are minimally
coupled to gravity, and the energy-momentum tensor conserves as a result of
the Noether theorem. However, the energy-momentum conservation law is violated
in the particle creation process in cosmology. Accordingly, it is reasonable
to relax the conservation condition of the energy-momentum tensor. P. Rastall
proposed such kind of modified gravity PhysRevD.6.3357 . In this theory, the
covariant divergence of the energy-momentum tensor is assumed to be
proportional to the scalar curvature, i.e.
$\nabla^{\mu}T_{\mu\nu}=\lambda\nabla_{\nu}R$. Therefore, the energy-momentum
conservation law is recovered in a flat spacetime. This theory has attracted
much attention recently, and cosmology and compact stars have been
investigated in Rastall gravity BezerradeMello:2014okn ; Heydarzade:2017wxu ;
Darabi:2017tay ; Xu:2017vse ; Darabi:2017coc ; Das:2018dzp ; Tang:2019dsk ;
Li:2019jkv ; Khyllep:2019odd ; Ghosh:2021byh ; Haghani:2022lsk ;
Shahidi:2021lxt . Constrains of the Rastall parameter $\lambda$ from the
galaxy-scale strong gravitational lensing and rotation curves of low surface
brightness galaxies were given in Refs. Tang:2019dsk ; Li:2019jkv .

It was shown that Rastall gravity and its generalization is equivalent to the
$f(R,T)$ gravity with the Lagrangian given by $f(R,T)=R+\alpha T$
Shabani:2020wja , where $R$ is the Ricci scalar and $T$ the trace of stress-
energy tensor. The thick brane model in $f(R,T)$ gravity has been considered
in Ref. Gu:2016nyo , and the full linear perturbations of the model were
analyzed. It was shown that only the scalar mode of the perturbation differs
from the standard case in general relativity. In this work, we will consider
the thick branes in Rastall gravity and investigate the localization of
gravity and the stability against tensor perturbation. Three types of
maximally symmetric thick branes will be considered. We will show that the
thick braneworlds in Rastall gravity are quite different from that in $f(R,T)$
gravity in the context of tensor perturbation.

This work is organized as follows. In Sec. 2, we solve the thick brane
generated by a scalar field in Rastall gravity. In Sec. 3, the linear
perturbation of the background metric is analyzed, and its Kaluza-Klein modes
are solved when $|\lambda|\ll 1$. Finally, we give out the conclusion and
discussion.

##  2 The Model

We start with a scalar filed $\phi$ of which the energy-momentum tensor
$T_{MN}=\partial_{M}\phi\partial_{N}\phi-
g_{MN}\left(\frac{1}{2}\partial^{M}\phi\partial_{M}\phi+V(\phi)\right)$ is not
conserved, and the covariant divergence is related to the scalar curvature $R$
by

$\displaystyle\nabla^{M}T_{MN}=\lambda\nabla_{N}R,$ (1)

where the constant $\lambda$ is a measure of the tendency of the nonminimal
coupling of the matter field and gravity. It is reasonable to assume that
$|\lambda|\ll 1$. Throughout this paper, we take the geometrized units in
which the gravitational constant $\kappa^{2}=1$, and the indices
$M,N\cdots=0,1,2,3,5$ denote the coordinates in the bulk, and
$\mu,\nu\cdots=0,1,2,3$ denote the ones on the brane. The field equation of
the generalized Rastall gravity in $5$-dimensional spacetime is

$\displaystyle G_{MN}+\lambda Rg_{MN}=T_{MN},$ (2)

or

$\displaystyle R_{MN}=T_{MN}+\alpha Tg_{MN},$ (3)

where $\alpha=\frac{1-2\lambda}{10\lambda-3}$. Note that
$\lambda\neq\frac{3}{10}$ as the trace of Eq. (2) gives
$(5\lambda-\frac{3}{2})R=T$. The metric is assumed as

$\displaystyle ds^{2}=a^{2}(z)(\gamma_{\mu\nu}dx^{\mu}dx^{\nu}+dz^{2}),$ (4)

where $a(z)=\text{e}^{A(z)}$ is the warp factor, and $\gamma_{\mu\nu}$ is the
metric of four-dimensional maximally symmetric spacetime, i.e

$\displaystyle\gamma_{\mu\nu}dx^{\mu}dx^{\nu}=\begin{cases}\eta_{\mu\nu}dx^{\mu}dx^{\nu},&\mbox{flat
brane},\\\ -dt^{2}+\text{e}^{2kt}(dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}),&\mbox{de
Sitter brane},\\\
\text{e}^{2kx_{3}}(-dt^{2}+dx_{1}^{2}+dx_{2}^{2})+dx_{3}^{2},&\mbox{anti-de
Sitter brane}.\end{cases}$ (5)

Since the brane is maximally symmetric, the scalar field can be assumed as
$\phi=\phi(z)$. With the metric assumption of (4), Eqs. (2) and (1) read

$\displaystyle 3A^{\prime 2}-\phi^{\prime 2}-3A^{\prime\prime}-3\epsilon
k^{2}$ $\displaystyle=$ $\displaystyle 0,$ (6) $\displaystyle
2\text{e}^{2A}V(\phi)+3(3+8\lambda)(A^{\prime 2}-\epsilon
k^{2})+(3+16\lambda)A^{\prime\prime}$ $\displaystyle=$ $\displaystyle 0,$ (7)
$\displaystyle-24\lambda A^{\prime 3}+8\lambda
A^{\prime\prime\prime}+4A^{\prime}\left(2\lambda A^{\prime\prime}+\phi^{\prime
2}+6\epsilon k^{2}\right)-\text{e}^{2A}\phi^{\prime}V^{\prime}(\phi)$
$\displaystyle=$ $\displaystyle 0,$ (8)

where the primes denote the derivatives with respect to $z$, and the constant
$\epsilon=0,1,-1$ for the flat, de Sitter and anti-de Sitter branes,
respectively. Note that there are only two independent equations in Eqs.
(6)-(8), we can construct the brane model by specifying the parameter
$\lambda$ and assuming the warp factor $a(z)$. The potential is solved as

$\displaystyle V(\phi)=-\frac{1}{2}\text{e}^{-2A}\left[3(3+8\lambda)(A^{\prime
2}-\epsilon k^{2})+(3+16\lambda)A^{\prime\prime}\right].$ (9)

###  2.1 Case I: Flat Brane

Firstly, we consider the flat brane. Let us assume the warp factor as

$\displaystyle a(z)=\frac{1}{\sqrt{\beta^{2}z^{2}+1}}.$ (10)

The scalar field can be solved with Eq. (6):

$\displaystyle\phi(z)=\sqrt{3}\text{arctan}(\beta z).$ (11)

Since Eq. (6) is the same as that of the brane model in general relativity,
the nonconservation of the scalar field only has an effect on the potential.
The potential $V(\phi)$ is given by

$\displaystyle
V(\phi)=\frac{1}{4}\beta^{2}\left[(56\lambda+15)\cos\left(\frac{2\phi}{\sqrt{3}}\right)-3(8\lambda+3)\right],$
(12)

which is almost the same as that of the brane solution in general relativity
except for a constant term.

(a) The warp factor (b) The scalar field

Figure 1: The shapes of the warp factor $a(z)$ and the scalar field $\phi(z)$
in the flat brane model.

###  2.2 Case II: de Sitter Brane

Similarly, we assume the following warp factor of a de sitter brane,

$\displaystyle a(z)=\text{sech}^{p}(\beta z),$ (13)

with the scalar field solved as

$\displaystyle\phi(z)=\phi_{0}\text{arcsin}\left(\tanh(\beta z)\right),$ (14)

where $\phi_{0}=\left[3p(p-1)\right]^{\frac{1}{2}}$, $\beta=\frac{k}{p}$ and
$0<p<1$. The corresponding potential of the scalar field is

$\displaystyle
V(\phi)=\frac{1}{2}\beta^{2}p\left(16\lambda+3(8\lambda+3)p+3\right)\cos^{2(1-p)}\left(\frac{\phi}{\phi_{0}}\right).$
(15)

(a) The warp factor (b) The scalar field

Figure 2: The shapes of the warp factor $a(z)$ and the scalar field $\phi(z)$
in the de Sitter brane model. The parameter is set as $p=0.5$ for the thin
black lines, $p=0.7$ for the thick red lines, and $p=0.9$ for the dashed blue
lines.

###  2.3 Case III: Anti-de Sitter Brane

At last, the warp factor of the anti-de Sitter brane is assumed as follows
PhysRevD.84.044033 ,

$\displaystyle a(z)=\cos^{-p}(\beta z).$ (16)

The scalar field is solved as

$\displaystyle\phi(z)=\phi_{0}\text{arcsinh}\left(\tan(\beta z)\right),$ (17)

where $\phi_{0}=\left[3p(1-p)\right]^{\frac{1}{2}}$ and $\beta=\frac{k}{p}$.
The parameter $p$ satisfies $p<0$. The range of the extra dimension is
$-z_{b}<z<z_{b}$ with $z_{b}=\frac{p\pi}{2k}$. In this model, the metric has a
naked singularity and the scalar field is divergent at $\pm z_{b}$. However,
it is demonstrated that these problems are acceptable since they can be solved
by lifting the bulk to ten dimensions or by string theory PhysRevD.62.044017 ;
PhysRevD.84.044033 .

Correspondingly, the potential of the scalar field is given by

$\displaystyle
V(\phi)=-\frac{1}{2}\beta^{2}p(16\lambda+3(8\lambda+3)p+3)\cosh^{2(1-p)}\left(\frac{\phi}{\phi_{0}}\right).$
(18)

(a) The warp factor (b) The scalar field

Figure 3: The shapes of the warp factor $a(z)$ and the scalar field $\phi(z)$
of the flat brane model. The parameter is set as $p=1.5$ for the thin black
lines, $p=2$ for the thick red lines, and $p=3$ for the dashed blue lines.

##  3 Tensor perturbation

In this section, we investigate the tensor perturbation of the Rastall thick
brane to analyze the stability and the localization of gravity. The perturbed
metric of the background is given by

$\displaystyle\tilde{ds}^{2}=a^{2}(z)\left[(\gamma_{\mu\nu}+h_{\mu\nu}(x^{\mu},z))dx^{\mu}dx^{\nu}+dz^{2}\right],$
(19)

where the tensor mode $h_{\mu\nu}$ satisfies the transverse and traceless
condition $\eta^{\mu\nu}\partial_{\mu}h_{\alpha\nu}=0$ and
$\eta^{\mu\nu}h_{\mu\nu}=0$. The corresponding perturbation of the Ricci
tensor $\delta R_{MN}$ is given by

$\displaystyle\delta R_{\mu\nu}=$
$\displaystyle-\frac{1}{2}\Box^{(4)}h_{\mu\nu}-\frac{1}{2}h_{\mu\nu}^{\prime\prime}-\frac{3}{2}A^{\prime}h_{\mu\nu}^{\prime}-(3A^{\prime
2}+A^{\prime\prime}-4\epsilon k^{2})h_{\mu\nu},$ (20)

where $\Box^{(4)}\equiv\eta^{\mu\nu}\partial_{\mu}\partial_{\nu}$ denotes the
four-dimensional d’Alember operator, and the primes denote the derivatives
with respect to the extra dimension coordinate $z$. Substituting Eq. (20) into
the perturbation of Eq. (2), we can obtain the equation of the tensor
perturbation $h_{\mu\nu}$,

$\displaystyle-\frac{1}{2}\Box^{(4)}h_{\mu\nu}-\frac{1}{2}h_{\mu\nu}^{\prime\prime}-\frac{3}{2}A^{\prime}h_{\mu\nu}^{\prime}+\left[\epsilon
k^{2}+8\lambda(3\epsilon k^{2}-3A^{\prime
2}-2A^{\prime\prime})\right]h_{\mu\nu}=0.$ (21)

To eliminate the first-order term in Eq. (21), we define
$h_{\mu\nu}=a^{-\frac{3}{2}}\bar{h}_{\mu\nu}$ and the above equation turns to

$\displaystyle-\bar{h}_{\mu\nu}^{\prime\prime}+V_{t}(z)\bar{h}_{\mu\nu}=(\Box^{(4)}-48\epsilon\lambda
k^{2}-2\epsilon k^{2})\bar{h}_{\mu\nu},$ (22)

where

$\displaystyle
V_{t}(z)=\Big{(}1-\frac{64}{3}\lambda\Big{)}\Big{(}\frac{9}{4}A^{\prime
2}+\frac{3}{2}A^{\prime\prime}\Big{)}.$ (23)

Applying the Kaluza-Klein decomposition
$\bar{h}_{\mu\nu}=\epsilon_{\mu\nu}(x^{\alpha})t(z)$, where the four-
dimensional part $\epsilon_{\mu\nu}(x^{\alpha})$ satisfies
$(\Box^{(4)}-48\epsilon\lambda k^{2}-2\epsilon
k^{2})\epsilon_{\mu\nu}(x^{\alpha})=\tilde{m}^{2}t\epsilon_{\mu\nu}(x^{\alpha})$,
we obtain the Schrodinger-like equation of the tensor mode

$\displaystyle-t^{\prime\prime}(z)+V_{t}(z)t(z)=\tilde{m}^{2}t(z),$ (24)

with the effective potential given by Eq. (23). This Schrodinger-like equation
of the tensor mode is similar to the one of the thick brane in general
relativity, except that the effective potential is modified by a factor
$(1-\frac{64}{3}\lambda)$. When $\lambda=0$, the theory reduces to general
relativity and Eq. (24) can be factorized as

$\displaystyle\Big{(}\\!-\partial_{z}+\frac{3}{2}A^{\prime}\Big{)}\Big{(}\partial_{z}+\frac{3}{2}A^{\prime}\Big{)}t(z)=m^{2}t(z),$
(25)

of which the solution of the localized mode is

$\displaystyle t_{0}(z)=\text{e}^{\frac{3}{2}A},$ (26)

and the corresponding mass square is $m_{0}^{2}=0$.

In the Rastall gravity, Eq. (24) can not be factorized, which makes it
difficult to obtain the solutions of localized modes. Nevertheless, it is
possible to deduce the feature of the tensor mode by making use of the
knowledge in quantum mechanism. Comparing Eq. (24) with a stationary
Schrodinger equation, one can obtain the following Hamiltonian,

$\displaystyle\tilde{H}=H-\frac{64}{3}\lambda V$ (27)

where $H$ and $V$ are given by

$\displaystyle H$ $\displaystyle=$ $\displaystyle-\frac{d^{2}}{dz^{2}}+V,$
(28) $\displaystyle V$ $\displaystyle=$ $\displaystyle\frac{9}{4}A^{\prime
2}+\frac{3}{2}A^{\prime\prime}.$ (29)

The Hellmann-Feynman theorem states

$\displaystyle\frac{\partial\tilde{m}_{n}^{2}}{\partial\lambda}$
$\displaystyle=$ $\displaystyle\frac{\langle
t_{n}|\frac{\partial\tilde{H}}{\partial\lambda}|t_{n}\rangle}{\langle
t_{n}|t_{n}\rangle}$ (30) $\displaystyle=$
$\displaystyle-\frac{64}{3}\frac{\langle t_{n}|V|t_{n}\rangle}{\langle
t_{n}|t_{n}\rangle}$ $\displaystyle\equiv$
$\displaystyle-\frac{64}{3}\overline{V}_{n}$

for an arbitrary bound state $|t_{n}\rangle$. Since the Rastall parameter
satisfies $|\lambda|<<1$, the mass spectrum of the tensor modes is similar
with the ones in the general relativity, except for a shift, i.e.

$\displaystyle\tilde{m}_{n}^{2}\approx
m_{n}^{2}-\frac{64}{3}\lambda\overline{V_{n}}.$ (31)

Note that the prerequisite for the above conclusion is that the bound states
of Eq. (24) exist.

Next, we analyze the tensor perturbation further in the cases of flat, de
Sitter and anti-de Sitter branes, respectively.

###  3.1 Case I: Flat Brane

In the flat brane model, the warp factor is given by Eq. (10), and the
corresponding effective potential $V_{t}(z)$ reads

$\displaystyle
V_{t}(z)=-\frac{\beta^{2}(64\lambda-3)\left(5\beta^{2}z^{2}-2\right)}{4\left(\beta^{2}z^{2}+1\right)^{2}},$
(32)

which is plotted in Fig. 4. When $\lambda=0$, Eq. (24) has and only has one
bound tensor mode with eigenvalue $m^{2}=0$. Thus, Eq. (33) implies that for
$|\lambda|\ll 1$, the mass square of the tensor zero mode is

$\displaystyle\tilde{m}_{0}^{2}\approx-\frac{64}{3}\lambda\overline{V_{0}},$
(33)

where $\overline{V_{0}}$ is the potential expectation value of the zero mode
for $\lambda=0$. Since the corresponding kinetic energy expectation value is
positive, we have $\overline{V_{0}}<0$. To prevent the ghost instability, the
mass square must satisfy $\tilde{m}_{0}^{2}\geq 0$, which implies that
$\lambda\geq 0$.

On the other hand, it can be seen that $V_{t}(\infty)=0$, therefore, the mass
square $\tilde{m}_{0}^{2}$ of the localized mode satisfies $m_{0}^{2}\leq 0$.
Accordingly for a viable flat brane model, we must demand
$\tilde{m}_{0}^{2}=0$, which is not possible for nonvanishing $\lambda$. In
conclusion, in the flat brane of Rastall gravity, the constraints from the
stability and localization yield $\lambda=0$, which just reduce to the case of
general relativity.

Figure 4:  The shapes of the effective potential $V_{t}(z)$ of the flat brane
model. The parameter is set as $\lambda=0$ for the thin black line,
$\lambda=-0.04$ for the thick red line, and $\lambda=0.02$ for the dashed blue
line. The other parameter is set as $\beta=1.$

###  3.2 Case II: de Sitter Brane

In the de Sitter brane model, with the warp factor (13), the effective
potential (23) turns to

$\displaystyle
V_{t}(z)=\frac{3}{8}p\beta^{2}\Big{(}1-\frac{64}{3}\lambda\Big{)}\left[-(6p+4)\text{sech}^{2}(\beta
z)+6p\right]$ (34)

Since it is difficult to obtain the analytical solution of Eq. (24) with the
above potential, we consider the case that $|\lambda|\ll 1$ and solve this
equation by perturbation method.

When $\lambda=0$, it is shown in PhysRevD.66.024024  that the only localized
mode is the zero mode, which can be solved by using the factorizing method,

$\displaystyle t(z)=\text{sech}^{\frac{3p}{2}}(\beta z).$ (35)

Then we turn to the case of $\lambda\neq 0$. Since $|\lambda|\ll 1$, the mass
square of the bound state is

$\displaystyle\tilde{m}_{0}^{2}\approx-\frac{64}{3}\lambda\overline{V_{0}},$
(36)

with $\overline{V_{0}}$ given by

$\displaystyle\overline{V_{0}}$ $\displaystyle=$
$\displaystyle\frac{\left\langle
t_{0}\right|\tilde{H}\left|t_{0}\right\rangle}{\left\langle
t_{0}|t_{0}\right\rangle}$ (37) $\displaystyle=$
$\displaystyle\frac{\int^{\infty}_{-\infty}t_{0}(z)^{2}V(z)dz}{\int^{\infty}_{-\infty}t_{0}(z)^{2}dz}$
$\displaystyle=$
$\displaystyle\frac{3\beta^{2}p\Gamma\left(\frac{3p}{2}+\frac{1}{2}\right)\left(-4\,_{2}F_{1}\left(1,-\frac{3p}{2}-1;\frac{3p}{2}+1;-1\right)+\frac{3\sqrt{\pi}p\Gamma\left(\frac{3p}{2}+1\right)}{\Gamma\left(\frac{3(p+1)}{2}\right)}+4\right)}{8\sqrt{\pi}\Gamma\left(\frac{3p}{2}\right)},$

which is plotted in Fig. 6. Using the perturbation method, the corresponding
solution of the tensor mode $\tilde{t}(z)$ can be solved, which is still given
by (35). It can been seen that $\overline{V_{0}}<0$, therefore the Rastall
parameter must satisfies $\lambda\geq 0$ to ensure that $\tilde{m}_{0}^{2}\geq
0$.

Figure 5:  The shapes of the effective potential $V_{t}(z)$ of the de Sitter
brane model. The parameter is set as $\lambda=0$ for the thin black line,
$\lambda=-0.04$ for the thick red line, and $\lambda=0.02$ for the dashed blue
line. Figure 6:  The potential expectation $\overline{V_{0}}$ as a function of
$p$ for the de Sitter brane model. The parameter is set as $\beta=1.$

###  3.3 Case III: Anti-de Sitter Brane

In the anti-de Sitter brane model, with the warp factor (16), the effective
potential (23) reads

$\displaystyle
V_{t}(z)=\Big{(}1-\frac{64}{3}\lambda\Big{)}\frac{3\beta^{2}}{4p}\left[(3p+2)\sec^{2}(\beta
z)-3p\right],$ (38)

which is plotted in Fig. 7. Since $|\lambda|\ll 1$,
$(1-\frac{64}{3}\lambda)>0$, thus the parameter $p$ must satisfy
$p<\frac{3}{2}$ to ensure the shape of the potential $V_{t}(z)$ open upwards.

Figure 7:  The shapes of the effective potential $V_{t}(z)$ of the anti-de
Sitter brane model. The parameter is set as $\lambda=0$ for the thin black
line, $\lambda=-0.04$ for the thick red line, and $\lambda=0.02$ for the
dashed blue line.

Again, we solve the tensor perturbation for the case of $\lambda=0$ and
obtained the solution PhysRevD.84.044033

$\displaystyle
t_{n}(z)=_{2}\\!\\!F_{1}\left[1-n,1+n+3p,\frac{3(1+p)}{2},\frac{1-\sin(\beta
z)}{2}\right]\cos^{1+\frac{3p}{2}}(\beta z),$ (39)

where $n=1,2,\cdots$, and ${}_{2}F_{1}(c_{1},c_{2},c_{3},z)$ is the
hypergeometric function. And the corresponding mass spetrum is
PhysRevD.84.044033

$\displaystyle m_{n}^{2}=\beta^{2}n(n+3p).$ (40)

Therefore, the mass spetrum for the case of $\lambda\ll 1$ is

$\displaystyle\tilde{m}_{n}^{2}\approx\beta^{2}n(n+3p)-\frac{64}{3}\lambda\overline{V_{n}}$
(41)

with $\overline{V_{n}}$ given by

$\displaystyle\overline{V_{n}}=\frac{\int^{z_{b}}_{-z_{b}}t_{n}(z)^{2}V(z)dz}{\int^{z_{b}}_{-z_{b}}t_{n}(z)^{2}dz}.$
(42)

The corresponding KK mode can by obtained by using the perturbation method

$\displaystyle\tilde{t}_{n}(z)\approx
t_{n}(z)+\sum_{i}\frac{-\frac{64}{3}\lambda\int^{z_{b}}_{-z_{b}}t_{n}(z)V(z)t_{i}(z)dz}{m_{n}-m_{i}}t_{i}(z).$
(43)

For the ground state of the tensor perturbation, Eq. (42) reads
$\overline{V_{1}}=\frac{9p^{2}}{12p+4}$. Thus the mass $m_{1}$ reads

$\displaystyle\tilde{m}_{1}^{2}\approx\beta^{2}\left[\left(3p+1\right)-\lambda\frac{3p(9p+4)}{12p+4}\right],$
(44)

and the solution of the ground state is

$\displaystyle\tilde{t}_{1}(z)\approx\sqrt{\beta}\pi^{-\frac{1}{4}}\cos^{\frac{3p}{2}+1}(\beta
z)\left[c_{1}-\lambda c_{2}\left((3p+4)\sin^{2}(\beta z)-1\right)\right],$
(45)

where

$\displaystyle c_{1}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{\Gamma\left(\frac{3p}{2}+2\right)}{\Gamma\left(\frac{3(p+1)}{2}\right)}},$
(46) $\displaystyle c_{2}$ $\displaystyle=$
$\displaystyle\frac{16\sqrt{2}p(3p+2)\sqrt{\frac{(p+2)\Gamma\left(\frac{3p}{2}+4\right)}{\left(3p^{2}+7p+4\right)\Gamma\left(\frac{3p}{2}+\frac{5}{2}\right)}}}{(3p+1)\left(\sqrt{3p+1}-3\sqrt{3}(p+1)\right)}.$
(47)

Since the ground state satisfies $m_{1}^{2}\geq 0$, the Rastall parameter must
satisfy $\lambda<\frac{4(3p+1)^{2}}{3p(9p+4)}$.

##  4 Conclusions and Discussions

In this paper, we investigated the thick brane model in the Rastall gravity.
In this theory, the thick brane is generated by a nonconserved scalar field,
of which the covariant divergence is proportional to the scalar curvature. The
flat, de Sitter, and anti-de Sitter brane solutions were obtained and the
tensor perturbation of the metric was analyzed for each model. We showed that
the tensor perturbation in this theory is quite different from that in the
general relativity and $f(R,T)$ gravity Gu:2016nyo , though the Rastall
gravity is equivalent to the $f(R,T)$ gravity with the Lagrangian given by
$f(R,T)=R+\alpha T$ Shabani:2020wja . The new features of the tensor
perturbation in the Rastall thick brane models come from that the effective
potential of the Schrodinger-like equation is multiplied by the factor
$1-\frac{64}{3}\lambda$.

In the flat brane model, for the case $\lambda\neq 0$, it was proved that the
tensor mode of the perturbation is either unstable or nonlocalizable. In the
de Sitter brane model, only the ground state of the tensor mode is localized,
and the mass and solution of the ground state were obtained by the
perturbation method. In the anti-de Sitter brane model, the number of the
bound state is infinity. For both the de Sitter and anti-de Sitter brane
models, the condition of stability for the parameter $\lambda$ was obtained.

## Acknowledgement

Y. Zhong acknowledges the support of the Fundamental Research Funds for the
Central Universities (Grants No.531118010195) and the National Natural Science
Foundation of China (Grant No. 12047501), he also thanks the generous
hospitality offered by the Center of Theoretical Physics at Lanzhou University
where part of this work was completed. K. Yang acknowledges the support of the
National Natural Science Foundation of China under Grant No. 12005174, and
Natural Science Foundation of Chongqing, China under Grant No. cstc2020jcyj-
msxmX0370.

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New Filters for Image Interpolation and Resizing

Amir Said
Media Technologies Laboratory
Hewlett-Packard Laboratories, Palo Alto, CA, USA

HPL-2007-179
Nov. 2, 2007***Internal Accession Data Only, Approved for External Publication
This report contains material published in: A. Said, “A new class of filters
for image interpolation and resizing,” Proc. IEEE Int. Conf. Image Processing,
2007.

interpolation kernels,
image processing,
image resizing

We propose a new class of kernels to simplify the design of filters for image
interpolation and resizing. Their properties are defined according to two
parameters, specifying the width of the transition band and the height of a
unique sidelobe. By varying these parameters it is possible to efficiently
explore the space with only the filters that are suitable for image
interpolation and resizing, and identify the filter that is best for a given
application. These two parameters are also sufficient to obtain very good
approximations of many commonly-used interpolation kernels. We also show that,
because the Fourier transforms of these kernels have very fast decay, these
filters produce better results when time-stretched for image downsizing.

###### Abstract

We propose a new class of kernels to simplify the design of filters for image
interpolation and resizing. Their properties are defined according to two
parameters, specifying the width of the transition band and the height of a
unique sidelobe. By varying these parameters it is possible to efficiently
explore the space with only the filters that are suitable for image
interpolation and resizing, and identify the filter that is best for a given
application. These two parameters are also sufficient to obtain very good
approximations of many commonly-used interpolation kernels. We also show that,
because the Fourier transforms of these kernels have very fast decay, these
filters produce better results when time-stretched for image downsizing.

##  1 Introduction

Image resizing and interpolation (e.g., for rotation) are two of the most
useful image processing operations, and consequently there is a great amount
of literature on the subject [1]–[8]. However, many imaging professionals find
the task of sorting out and implementing the most appropriate method quite
challenging, due to the great number of possibilities and conflicting
opinions. It is common to settle for some very simple approaches which were
once meant to reduce complexity, or adopt one type that was shown to be
excellent for one application, without knowing that it may be suboptimal for
other applications.

For instance, even in commercial products we find the mistake of using
interpolation kernels for downsampling without the necessary lowpass
filtering. A less serious, but also common mistake, is to use for downsampling
low-order filters which can be good for interpolation, but have much worse
properties when time-scaled for downsizing.

What is still missing is an approach that is more convenient and easy to use,
with less emphasis on computational complexity, and that yields high image
quality. For that purpose we propose a family of parametrized functions that
are simple, and are designed with enough versatility so that well-known
kernels can be very closely approximated by simply using the proper
parameters. This way it is easy to experiment and identify the parameters that
are best for a certain type of image, without having to understand and
implement several methods.

This paper is organized by first having a very brief review of sampling and
interpolation concepts, which allow us to establish the notation. Next, we
define the proposed family of functions, and some of their basic properties.
We present a set of features that are desirable for interpolation and resizing
kernels, and explain how well the proposed functions support those features.
Finally, we present some results of finding the approximation to some
commonly-used kernels, show how well these kernels are approximated, and
discuss some additional features of the proposed kernels.

##  2 Signal Sampling and Reconstruction

$s[0]$$s[1]$$s[2]$$s[3]$$s[4]$$s[5]$$t$$s(t)$$0$$T$$2T$$3T$$4T$$5T$ Figure 1:
A signal $s(t)$ that is sampled with period $T$.

Even though this work is about two-dimensional images, to simplify the
notation we assume that only separable filters are used [4], and thus consider
only the one-dimensional case. In practice separable filters are almost always
used because of their much lower computational complexity.

In our analysis of sampling and reconstructing signals, we consider a function
$s(t)$ with Fourier transform

$S(f)=\int_{-\infty}^{\infty}s(t)\,e^{-j2\pi ft}\>\mathrm{d}{t},$ (1)

that is sampled with period $T$, as shown in Figure 1, to generate the
sequence

$s[n]=s(nT),\quad n\in\mathbb{Z}.$ (2)

The Fourier transform of this sequence is

$S(e^{j\omega})=\sum_{n=-\infty}^{\infty}s[n]\,e^{-j\omega n}.$ (3)

Note that we use the notation of reference [3], using parenthesis for
functions of real variables and brackets for functions of integer variables.

From the sampling theorem [3, 4] we have, for $\omega=2\pi fT$,

$S(e^{j\omega})=\frac{1}{T}\sum_{n=-\infty}^{\infty}S\\!\left(\frac{\omega}{2\pi
T}-\frac{n}{T}\right),$ (4)

or, equivalently,

$S(e^{j2\pi{fT}})=\frac{1}{T}\sum_{n=-\infty}^{\infty}S\\!\left(f-\frac{n}{T}\right).$
(5)

Figure 2 shows in red an example of the bandlimited spectrum of a signal
$s(t)$, and the corresponding periodic spectrum of its sampled sequence. We
can note that in this example the bandwidth of the signal is larger than
$1/(2T)$, so there is overlap in the shifted versions of $S(f)$, i.e.,
aliasing [3].

$-2\pi$$-\pi$$0$$\pi$$2\pi$$3\pi$$4\pi$$5\pi$$6\pi$$\omega$$S(e^{j\omega})$$\frac{1}{T}S(\omega/[2\pi
T])$ Figure 2: A signal spectrum and the Fourier Transform of its sampled
sequence.

Given a reconstruction kernel $\phi(t)$ with Fourier transform $\Phi(f)$, we
define a reconstructed function

$r(t)=\sum_{k=-\infty}^{\infty}s[k]\,\phi\\!\left(\frac{t}{T}-k\right),$ (6)

which has Fourier transform

$\displaystyle R(f)$ $\displaystyle=$ $\displaystyle
T\,S(e^{j2\pi{fT}})\,\Phi(fT)$ (7) $\displaystyle=$
$\displaystyle\Phi(fT)\sum_{k=-\infty}^{\infty}S\\!\left(f-\frac{k}{T}\right).$
(8)

From this equation we can observe that if the signal $s(t)$ is strictly
bandlimited, i.e., if its spectrum satisfies

$S(f)=0,\quad\forall f\not\in\left(-\frac{1}{2T},\frac{1}{2T}\right)$ (9)

then we can have $R(f)=S(f)$, and consequently $r(t)=s(t)$, if we use as
$\phi(t)$ the box (ideal filter) kernel

$b(t)=\frac{\sin(\pi t)}{\pi t},$ (10)

which has Fourier transform

$B(f)=\left\\{\begin{array}[]{ll}1,&|f|<1/2,\\\ 1/2,&|f|=1/2,\\\
0,&|f|>1/2,\\\ \end{array}\right.$ (11)

If we resample $r(t)$ with period $\tau$, we have

$r[n]=r(n\tau)=\sum_{k=-\infty}^{\infty}s[k]\,\phi\\!\left(\frac{n\tau}{T}-k\right)$
(12)

and defining

$\gamma=\frac{T}{\tau},$ (13)

we obtain

$R(e^{j2\pi{f}})=\gamma\sum_{n=-\infty}^{\infty}\Phi(\gamma[f-n])S(e^{j2\pi{\gamma[f-n]}}).$
(14)

From this result it is clear that we can only have
$R(e^{j\omega})=S(e^{j\omega})$ when $T=\tau$ and $\phi(t)=b(t)$. However, if
$\tau\leq T$ and $s(t)$ satisfies (9), then we can preserve all information on
$s(t)$ using $\phi(t)=b(t)$. If, on the other hand, we have $\tau>T$, then
spectrum of $R(f)$ has to satisfy

$R(f)=0,\quad\forall f\not\in\left(-\frac{1}{2\tau},\frac{1}{2\tau}\right)$
(15)

to avoid aliasing. This can be done by using a time-stretched version
$\phi(t)=\gamma b(\gamma t)$ as the reconstruction kernel.

Figure 3 shows the two cases, when ideal (box) filters are used. When
$\tau\leq T$ (upsampling and interpolation) the filter response covers the
whole original bandwidth, and when $\tau>T$ (downsampling) the filter response
is reduced by a factor $T/\tau$ to avoid aliasing.

In conclusion, when it is known that a signal that has been sampled with
period $T$ needs to be resampled with period $\tau$, we can use the kernels

$\phi(t)=\beta\,b(\beta t),$ (16)

where

$\beta=\min(1,T/\tau).$ (17)

$-2\pi$$-\pi$$0$$\pi$$2\pi$$3\pi$$4\pi$$-\beta\pi$$\beta\pi$$\omega$$\frac{\displaystyle
S(\omega/[2\pi T])}{\displaystyle T}$$S(e^{j\omega})$Ideal filter for upsizing
and interpolationIdeal filter for downsizing Figure 3: Box filters used for
strictly bandlimited signals.

##  3 Kernels for Image Interpolation and Resizing

(a) (b)

(c) (d)

Figure 4: Examples of an image with text interpolated using different kernels.
(a) Ideal bandlimited filter; (b) Pixel replication kernel; (c) “Screen door”
effect caused by insufficient attenuation of repeated versions of the
spectrum; (d) cubic spline kernel.

The theory presented in Section 2 was developed for analyzing exact recovery
of sampled signals that are strictly bandlimited, but in imaging applications
we normally do not have signals that are truly bandlimited. One common example
is in images containing text, which define discontinuous signals. The
application of ideal filters to those images produces highly visible “ringing”
artifacts, as shown in Figure 4(a). Thus, other types of filters have been
used for images [1]–[8]. For upsizing by integer factors, the simplest
technique is to just replicate the pixel values, but as can be seen in Figure
4(b), the results are also visually quite bad.

Another problem occurs when the filters do not remove enough of the periodic
versions in the spectrum of the sampled signal. In this case we have the
“screen door” effect, as shown in Figure 4(c). Much better results are
obtained using some kernels defined for general function interpolation. For
instance, Figure 4(d) shows the image obtained with cubic-spline
interpolation.

$\bullet$$\bullet$$\bullet$$\bullet$$t$$n-1$$n$$n+1$$r(t)$$s[n-1]$$s[n]$$s[n+1]$
Figure 5: Notation used for interpolation and resizing.

Figure 5 shows the basic notation we use. We assume $T=1$, and for a sequence
of signal values $s[n]$, and an interpolation kernel function $h(t)$,222In
Section 2 we use $\phi(t)$ in the analysis of strictly bandlimited signals. In
the rest of this document we use $h(t)$ to identify the type of kernels that
are used in practice. the reconstructed value at point $t$ is defined as

$r(t)=\beta\,\sum_{i=-\infty}^{\infty}s[i]h(\beta[t-i]).$ (18)

where $\beta$ is the normalized bandwidth of the signal, as defined by (17).
When downsampling the image factor $\beta$ should be equal or smaller than the
reduction factor, which means that $0<\beta<1$. When the image is upsampled or
rotated, we have $\beta=1$.

This series can be also defined as

$r(t)=\beta\,\sum_{k\in S_{h}}s[k+\left\lfloor
t\right\rfloor]h(\beta[t-k-\left\lfloor t\right\rfloor]),$ (19)

where $S_{h}$ is a range of integers defined by the condition $h(\beta t)\neq
0$. In practice, we can use the condition $|h(\beta t)|<\epsilon$, where
$\epsilon$ is sufficiently small.

Using the new notation with interpolation kernels, if we resample $r(t)$ with
period $\tau$, we have the convolution

$r[n]=r(n\tau)=\beta\,\sum_{k=\infty}^{\infty}s[\eta_{n}-k]h_{\alpha_{n},\beta}[k],$
(20)

where

$\displaystyle\eta_{n}$ $\displaystyle=$ $\displaystyle\left\lfloor
n\tau\right\rfloor,$ (21) $\displaystyle\alpha_{n}$ $\displaystyle=$
$\displaystyle n\tau-\left\lfloor n\tau\right\rfloor,$ (22) $\displaystyle
h_{\alpha,\beta}[k]$ $\displaystyle=$ $\displaystyle h(\beta[k+\alpha]).$ (23)

This means that the new sequence has each sample defined by a convolution sum,
but the coefficients can be different in each case. When the ratio $T/\tau$ is
integer or rational, the sets of coefficients occur periodically, and we have
a multi-rate system.

For image applications it is commonly more efficient to pre-compute and store
sequentially all the coefficients of these convolutions in a single array (cf.
Figure 6), which are later repeatedly used when the images lines are
resampled. Only one set is needed for the vertical resizing, and another for
the horizontal resizing.

$h_{\alpha_{1},\beta}$$h_{\alpha_{2},\beta}$$h_{\alpha_{3},\beta}$$h_{\alpha_{4},\beta}$$h_{\alpha_{N},\beta}$$\cdots$
Figure 6: Arrays with sequence of filter coefficients that are pre-computed
for image resizing, and repeatedly used in the vertical (or horizontal)
resampling computations.

##  4 New Kernel for Interpolation and Resizing

The family of functions that we propose for interpolation and resizing has
only two parameters, $\chi$ and $\eta$, and is defined by

$h_{s}(t;\chi,\eta)=\mbox{sinc}\\!\left(t\right)\,\cosh\\!\left(\frac{\sqrt{2\eta}\pi\chi
t}{2-\eta}\right)e^{-[\pi\chi t/(2-\eta)]^{2}}$ (24)

where

$\mbox{sinc}\\!\left(t\right)=\left\\{\begin{array}[]{ll}1,&t=0,\\\ \sin(\pi
t)/(\pi t),&t\neq 0,\\\ \end{array}\right.$ (25)

The corresponding Fourier transforms are

$H_{s}(f;\chi,\eta)=P_{s}\\!\left(\frac{[2f+1][2-\eta]}{\sqrt{2}\chi};\eta\right)-P_{s}\\!\left(\frac{[2f-1][2-\eta]}{\sqrt{2}\chi};\eta\right)$
(26)

where

$P_{s}(f;\eta)=\frac{e^{\eta/2}}{\sqrt{2\pi}}\int_{0}^{f}e^{-\phi^{2}/2}\cos(\sqrt{\eta}\phi)\,\mathrm{d}\phi$
(27)

0123456$t$-60 dB-40 dB-20 dB0
dB$|h_{s}(t;\chi,\eta)|$$\chi=0.2,\eta=0$$\chi=0.2,\eta=0.3$$\chi=0.2,\eta=0.4$$\chi=0.2,\eta=0.55$$\chi=0.2,\eta=0.75$

0.000.250.500.751.001.251.50$f$-80 dB-60 dB-40 dB-20 dB0
dB$|H_{s}(f;\chi,\eta)|$$\chi=0.2,\eta=0$$\chi=0.2,\eta=0.3$$\chi=0.2,\eta=0.4$$\chi=0.2,\eta=0.55$$\chi=0.2,\eta=0.75$

Figure 7: Impulse and frequency responses for some selected parameters,
$\chi=0.2$.

0123456$t$-60 dB-40 dB-20 dB0
dB$|h_{s}(t;\chi,\eta)|$$\chi=0.3,\eta=0$$\chi=0.3,\eta=0.3$$\chi=0.3,\eta=0.4$$\chi=0.3,\eta=0.55$$\chi=0.3,\eta=0.75$

0.000.250.500.751.001.251.50$f$-80 dB-60 dB-40 dB-20 dB0
dB$|H_{s}(f;\chi,\eta)|$$\chi=0.3,\eta=0$$\chi=0.3,\eta=0.3$$\chi=0.3,\eta=0.4$$\chi=0.3,\eta=0.55$$\chi=0.3,\eta=0.75$

Figure 8: Impulse and frequency responses for some selected parameters,
$\chi=0.3$.

0123456$t$-60 dB-40 dB-20 dB0
dB$|h_{s}(t;\chi,\eta)|$$\chi=0.4,\eta=0$$\chi=0.4,\eta=0.3$$\chi=0.4,\eta=0.4$$\chi=0.4,\eta=0.55$$\chi=0.4,\eta=0.75$

0.000.250.500.751.001.251.50$f$-80 dB-60 dB-40 dB-20 dB0
dB$|H_{s}(f;\chi,\eta)|$$\chi=0.4,\eta=0$$\chi=0.4,\eta=0.3$$\chi=0.4,\eta=0.4$$\chi=0.4,\eta=0.55$$\chi=0.4,\eta=0.75$

Figure 9: Impulse and frequency responses for some selected parameters,
$\chi=0.4$.

Figures 7, 8, and 9 show some examples. Note that all graphs have
$|h_{s}(t;\chi,\eta)|$ and $|H_{s}(f;\chi,\eta)|$ in decibels. We can observe
that $\chi$ basically controls the width of the transition band, and $\eta$
affects the height of the first sidelobe. Thus, when $\chi\rightarrow 0$ the
functions $H_{s}(f;\chi,\eta)$ converge to the ideal lowpass filter, i.e.,

$\lim_{\chi\rightarrow 0}H_{s}(f;\chi,\eta)=\begin{cases}1,&|f|<1/2,\\\
1/2,&|f|=1/2,\\\ 0,&|f|>1/2.\end{cases}$ (28)

While there is no closed-form expression for integral (27) (related to the
complex-valued error function), we used the time-frequency properties of the
functions, and found that it can approximated with absolute error smaller than
$10^{-16}$ using

$P_{s}(f;\eta)\approx\begin{cases}1/2,&f>8,\\\
\displaystyle\frac{f}{17}+\sum_{n=1}^{22}\frac{e^{-\omega^{2}n^{2}}}{\pi
n}\cosh(2\omega n\sqrt{\eta})\sin(2\omega nf),&|f|\leq 8,\\\
-1/2,&f<-8,\end{cases}$ (29)

where $\omega=\pi/17$.

##  5 Desirable Features

There are some important features—not all simultaneously achievable—that are
desirable for the kernel functions used for creating the discrete-time
filters.

###  5.1 Flexibility

It is necessary to recognize that different types of images (natural, medical,
synthetic, etc.) have different requirements. While we have a variety of
theoretical tools developed for the analysis and design of interpolation and
resizing filters, in most cases it is still essential to experiment several
filters, and visually inspect the results.

The proposed kernels are meant to allow imaging professionals to try different
filters more easily. In fact, to make their performance and visual quality
easier to predict, they can closely approximate other commonly used kernels.
This way, it is possible to start with those approximations, and see how the
image quality is altered after changes in the parameters.

Table 1 shows some sets of parameters that can be used for these
approximations, and Figures 12 and 13 show how good the approximation are.
(More details in Section 6.)

###  5.2 Intuitive Controls

While experimenting, it is desirable to be able to finely tune the filter’s
response. Some kernels provide very little control, being defined only for
some discrete parameters, like “order.” Others are defined by parameters
related to approximation theory, which may have limited relation to the
properties of natural images.

We defined the parameters of our kernel in a way that makes its equations
somewhat more complicated, but aiming to make them much more intuitive. The
parameter $\chi$ is defined to be the main control for achieving a compromise
between blurring, aliasing and ringing artifacts. If it it too small, we have
nearly ideal filters, which create ringing artifacts around edges. If it is
too large we some have blurring and the screen door effect. The amount of
aliasing depends also on the parameter $\eta$, which controls the height of
the sidelobe. These properties can be seen in Figures 7 to 9.

###  5.3 Symmetry and Exact Interpolation

For imaging applications it is necessary to use linear phase filters, and
commonly interpolation functions have even symmetry, i.e., $h(t)=h(-t)$. Under
the assumption that pixel values corresponds to samples of a strictly
bandlimited signal, we would like to not change the values that are already
know, and this is achieved when

$h(0)=1,\quad h(n)=0,\quad n=\pm 1,\pm 2,\ldots$ (30)

Image signals are certainly not strictly bandlimited, but the property is
still useful because it implies that for all $f$ we have

$H_{d}\\!\left(e^{j2\pi f}\right)=\sum_{n=-\infty}^{\infty}H(f-n)=1,$ (31)

i.e., we know that the gain for signal plus aliasing always adds to one. Our
kernels satisfy this property because they belong to the class of functions
created by multiplying sinc($t$) with another function.

###  5.4 Good Response with Small Spatial Support

When considering the filtering computational complexity, it is good to use
filters with small numbers of taps. Since in imaging applications we need to
avoid ringing resulting from lowpass filters with steep transition, good
interpolation results had been obtained with very short filters.

Our kernels were chosen such that $h_{s}(t;\chi,\eta)$ and
$H_{s}(f;\chi,\eta)$ have very fast asymptotic decay. In fact, there are
constants $a,b,c$, and $d$ such that

$\displaystyle|h_{s}(t;\chi,\eta)|$ $\displaystyle\leq$ $\displaystyle
a\,e^{-bt^{2}}$ $\displaystyle|H_{s}(f;\chi,\eta)|$ $\displaystyle\leq$
$\displaystyle c\,e^{-df^{2}}.$

Thus, while $h_{s}(t;\chi,\eta)$ strictly has infinite support, the very fast
decay makes it easy to find where to truncate the response without
significantly changing the filter’s performance (cf. Figures 7 to 9). This
approach tends to yield somewhat longer filter responses, but it is more
convenient for obtaining downsampling filters, which need to be more carefully
designed.

###  5.5 Good Performance For Both Interpolation and Downsampling

One of the most natural requirements in interpolation and resizing is that
when applied to an image with a constant pixel value, it should always create
another image with the same value. Thus, a kernel’s DC response, i.e., the
function obtained when $s[n]\equiv 1$ should be also identical to one. This is
possible only when we have an exact partition of unity:

$\sum_{k=-\infty}^{\infty}\beta\cdot h(\beta\cdot[t-k])=1,\quad t\in[0,1).$
(32)

In the frequency domain this corresponds to

$H(0)+2\sum_{n=1}^{\infty}H\\!\left(\frac{n}{\beta}\right)\,\cos(2\pi
nt)=1,\quad t\in[0,1).$ (33)

Interpolation kernels commonly satisfy this property exactly by having
$H(0)=1$, and $H(n)=0$, $n=1,2,\ldots$ Others provide very good approximation
with very small values of $|H(0)-1|$ and $|H(n)|$, $n=1,2,\ldots$

The problem of using interpolation functions for downsampling is that while
the condition above may be satisfied exactly for $\beta=1$, it may not be a
good approximation when $\beta<1$.

For example, let us consider the kernel for linear interpolation (tent
function)

$h_{l}(t)=\begin{cases}1-|t|,&|t|<1,\\\ 0,&|t|\geq 1,\end{cases}$ (34)

Its DC responses for $\beta=1$ and $\beta=0.7$ are shown in Figure 10. We can
observe that this kernel satisfies the partition of unity condition when
$\beta=1$, but is clearly inadequate when $\beta=0.7$. In fact, in the latter
case, if the resampling offset is zero, then the sampled response will
alternate between values that are about 10% too large, and 10% too small,
i.e., an oscillation of about 20%! Figure 11 shows how the DC response error
varies for other values of $\beta$.

00.511.52$t$00.511.5DC responses Figure 10: Sum of linear interpolation (tent
function) kernels when $s[n]\equiv 1$. Dashed red lines correspond to shifted
components and continuous red line to their sum when $\beta=1$. The blue lines
correspond to case $\beta=0.7$. 00.10.20.30.40.5$t$-15 %-10 %-5 %0 %5 %10 %15
%DC Response Error$\beta=0.85$$\beta=0.80$$\beta=0.75$$\beta=0.70$$\beta=0.67$
Figure 11: Error in the DC response for linear interpolation kernels.

Here we see one of the main advantages of kernel $h_{s}(t;\chi,\eta)$. With
the proper choice of $\chi$ and $\eta$, the very fast decay guarantees that
$|H_{s}(1/b;\chi,\eta)|$ is very small for $\beta<1$, which means that it is
also good as a downsampling filter. In the next section we present some
example of this property and comparisons of the DC response errors (e.g.,
Figures 14 and 15).

##  6 Approximation of Other Kernels

Many of the features of the new kernels can be observed by analyzing versions
that have parameters chosen to closely approximate kernels commonly used for
interpolation. Table 1 shows some sets of parameters that can be used for
these approximations, and Figures 12 and 12 show comparisons of the
corresponding Fourier transforms. In Figure 12 we can observe that with
$\chi=0.248$ and $\eta=0.48$ we have a response nearly identical to the
Blackman-Harris ($N=6$) kernel [1, 6]. Choosing $\chi=0.163$ and $\eta=1.2$
produces a response very similar to the Lanczos kernel ($M=2$) [5, § 3] up to
its first zero. After that, $H_{s}(f;\chi,\eta)$ produces a wider sidelobe
with roughly the same height, but with nearly monotonic decay, instead of
several sidelobes.

Table 1: Parameters that approximate popular interpolation kernels. Kernel | $\chi$ | $\eta$
---|---|---
Lanczos, $M=2$ | 0.414 | 0.61
Lanczos, $M=3$ | 0.284 | 0.64
Lanczos, $M=4$ | 0.212 | 0.65
Lanczos, $M=5$ | 0.170 | 0.65
Blackman-Harris, $N=6$ | 0.411 | 0.23
Cubic B-Spline | 0.310 | 0
Mitchell-Netravali, $B=C=1/3$ | 0.550 | 0.32
0.000.250.500.751.001.251.50$f$-80 dB-60 dB-40 dB-20 dB0
dB$|H(f)|$$\chi=0.411,\eta=0.23$Blackman-Harris,
$N=6$$\chi=0.414,\eta=0.61$Lanczos, $M=2$ Figure 12: Fourier transforms of
some commonly used kernels, and of their approximation with the proposed
parametrized kernels. First set. 0.000.250.500.751.001.251.50$f$-80 dB-60
dB-40 dB-20 dB0 dB$|H(f)|$$\chi=0.31,\eta=0$Cubic
B-Spline$\chi=0.212,\eta=0.65$Lanczos, $M=4$ Figure 13: Fourier transforms of
some commonly used kernels, and of their approximation with the proposed
parametrized kernels. Second set.

Figure 13 shows a comparison with another Lanczos kernel, with similar
results. In part, these results are not very surprising, since the Blackman-
Harris and Lanczos kernels are based on the sinc($t$) function. However, they
show that we can get remarkable control of the properties of
$H_{s}(f;\chi,\eta)$ by changing only its two parameters.

Furthermore, the comparison of $H_{s}(f;\chi,\eta)$ with the cubic b-spline
kernel in Figure 13 shows that we can have very good approximations for other
types of kernels too. One important difference, is that $H_{s}(f;\chi,\eta)$
has no sidelobes above –80 dB when $f>1$, which means that it has better
partition of unity when used for downsampling.

We also tested the Lanczos (LZ4), Blackman-Harris (BH6), and cubic b-spline
(CBS) kernels in the $2048\times 2560$ image “Bike” (used in the JPEG2000
tests), chosen because it has many details and test patterns. First, we
upsampled all images by factor 1.7. In all cases, the images are visually
indistinguishable, so we measured the differences between images. The results,
defined as PSNR in dB, are:

Kernel | LZ4 | BH6 | CBS
---|---|---|---
LZ4-Approx. | 51.3 | 42.6 | 49.4
BH6-Approx. | 43.4 | 58.3 | 48.4
CBS-Approx. | 47.9 | 47.3 | 58.1

The values in the main diagonal show that the images obtained with the kernels
and their approximations are indeed very close. The other values are also
large, indicating that all those kernels produce good results, but the
differences are also clear.

In the second experiment we downsampled the image by factor 0.6, and obtained
similar results

Kernel | LZ4 | BH6 | CBS
---|---|---|---
LZ4-Approx. | 51.4 | 38.9 | 45.6
BH6-Approx. | 39.9 | 56.1 | 45.2
CBS-Approx. | 45.2 | 44.0 | 57.0

As explained in Section 5.5, an advantage of the new kernels is that they
present very fast decays on the frequency response, which can make them
(depending on the parameters) better for downsizing because of the smaller
error on the DC response. Figures 14 and 15 show comparisons of the DC
response error for two kernel approximations. In Figure 14 we observe that
while the cubic b-spline—which has ideal DC response when $\beta=1$—produces
errors for other values of $\beta$. Its approximation, on the other hand, has
not sidelobe ($\eta=0$, cf. Figure 13), and consequently the very fast decay
produces very small DC response errors. In fact, the error has a nearly
constant it $-5\cdot 10^{-6}$ for all values of $0\leq t\leq 1$ and
$0.5\leq\beta\leq 1]$. Figure 15 shows a similar comparison, with the Lanczos
kernel. Note that is this case the DC response of the approximation is
similarly nearly independent of $t$ and $\beta$ (in fact, $\beta=1$ is
commonly the worst case). However, there are error because $\eta$ has a
relatively large value.

0.00.10.20.30.40.5$t$-2 %-1 %0 %1 %2 %DC Response ErrorCubic B-spline,
$\beta=0.85$Cubic B-spline, $\beta=0.80$Cubic B-spline, $\beta=0.75$Cubic
B-spline, $\beta=0.70$New kernel, $\chi=0.31,\eta=0$ Figure 14: Comparison of
DC response errors between the cubic b-spline kernel and its approximation.
The error of the new kernel has a nearly constant value of $-5\cdot 10^{-6}$
for all values of $t$ and $\beta$ in the graph. 0.00.10.20.30.40.5$t$-1.0
%-0.5 %0.0 %0.5 %DC Response ErrorLanczos, $M=3$, $\beta=1.0$Lanczos, $M=3$,
$\beta=0.9$Lanczos, $M=3$, $\beta=0.7$Lanczos, $M=3$, $\beta=0.6$New kernel,
$\chi=0.284,\eta=0.64,\beta=1.0$New kernel, $\chi=0.284,\eta=0.64,\beta=0.6$
Figure 15: Comparison of DC response errors between the Lanczos kernel and its
approximation.

##  7 Conclusions

We have shown that the proposed kernels for image interpolation and resizing
can be easily designed, since their two parameters provide direct control over
the most important features, which are the width of the transition band, and
the sidelobe height. We also explain that the kernels naturally satisfy many
desirable conditions. They yield exact interpolation, both the functions and
their Fourier transforms have very fast decays, and thus the same kernels can
produce good results for both interpolation and downsampling. We tested the
flexibility of the design by presenting sets of parameters that produce
kernels that are very good approximations of kernels that are well known for
their properties and superior image quality. The differences between the
original kernels and the approximations are evaluated by analyzing the
frequency response, and also measuring the difference between images resized
with those kernels. In conclusion, the new class of kernels provide a very
convenient way to test different kernels in order to identify those that
produce the best image quality.

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