e^{\lambda}}\left[\frac{2}{r}-\frac{\lambda^{\prime}}{e^{\lambda}-1}\right]\left[\frac{\nu^{\prime}e^{\nu}}{2rA_{2}^{2}}-1\right]$
(26)

Recently, many authors have also presented anisotropic compact star models in
the background of embedding class I space-time (Singh & Pant, 2016a; Singh et
al., 2016, 2016; Bhar et al., 2017a; Ortiz et al., 2019; Maurya et al., 2019c;
Prasad et al., 2019; Bhar, 2019; Sarkar et al., 2019). In Table 15 we have
given some metrics which have been used to study exact solutions of the
Einstein field equations by considering spacetime of embedding class one.

Table 15:  Solutions obtained by considering spacetime of embedding class one. name [ref.] | metric
---|---
M-G-S-R (Maurya et al., 2016a) |  $ds^{2}=\left[A+\frac{B\sqrt{(a-b)(1+br^{2})}}{b}\right]^{2}dt^{2}-\left[1-\frac{(a-b)r^{2}}{1+br^{2}}\right]dr^{2}-r^{2}d\Omega^{2}$, $a\neq b$
S-P I (Singh & Pant, 2016a) |  $\left[A+\frac{\sqrt{a}B\log{(br^{2}+1)}}{2b}\right]^{2}dt^{2}-\left[1+{a^{2}r^{2}}{(1+br^{2})^{2}}\right]dr^{2}-r^{2}d\Omega^{2}$; $\frac{\sqrt{2}}{a}<\frac{B}{A}\leq 2\sqrt{a}$.
S-P II (Singh & Pant, 2016b) |  $\left[A+\frac{\sqrt{a}B(br^{2}+1)^{\frac{n}{2}+1}}{b(n+2)}\right]^{2}dt^{2}-\left[1+{a^{2}r^{2}}{(1+br^{2})^{n}}\right]dr^{2}-r^{2}d\Omega^{2}$, $n\neq-2$
S-P-P (Singh et al., 2016) | $ds^{2}=\left[A-\frac{\sqrt{3}B}{\sqrt{a(2-ar^{2})}}\right]^{2}dt^{2}-\frac{2(1+ar^{2})}{2-ar^{2}}dr^{2}-r^{2}d\Omega^{2}$
M-M (Maurya & Maharaj, 2017) | $ds^{2}=A^{2}[B+\tan^{-1}\sinh(ar^{2}+b)]dt^{2}-\frac{1+2cr^{2}+cosh[2(ar^{2}+b)]}{1+cosh[2(ar^{2}+b)}dr^{2}-r^{2}d\Omega^{2}$
M-R-G (Maurya et al., 2017b) |  $ds^{2}=\left[A+B\left(e^{(ar^{2}+b)}-e^{(ar^{2}+b)}\right)\right]^{2}dt^{2}-\frac{4+Cr^{2}[e^{(ar^{2}+b)}-e^{(ar^{2}+b)}]^{2}}{4}dr^{2}+r^{2}d\Omega^{2}$; $a,b,C>0$
B-S-M (Bhar et al., 2017a) | $ds^{2}=\left[A-\frac{aB}{2b(1+br^{2})}\right]^{2}dt^{2}-\left[1+\frac{a^{2}r^{2}}{(1+br^{2})^{4}}\right]dr^{2}-r^{2}d\Omega^{2}$
O-M-E-N-D (Ortiz et al., 2019) |  $ds^{2}=\left[A+\frac{\sqrt{2a}B\cos(br^{2}+c)F(r)f_{1}(r)f_{2}(r)}{b(n+1)\sqrt{1-\sin(br^{2}+c)}}\right]^{2}dt^{2}-\frac{1}{1+ar^{2}[1+\sin(br^{2}+c)]^{n}}dr^{2}-r^{2}d\Omega^{2}$, where,
|
$F(r)=2F\left[\frac{1+n}{2},\frac{1}{2},\frac{3+n}{2},\sin^{2}(\frac{2c+\pi+2br^{2}}{4})\right]$
represents a Gauss hypergeometric function,
|  $f_{1}(r)=\left[\frac{\cos(c+br^{2})+\sin(c+br^{2})}{2}\right]^{-n}$ $\&$
$f_{2}(r)=[1+\sin(c+br^{2})]^{n}$.
M-D-R-K (Maurya et al., 2019c) |  $ds^{2}=\left[B(1-Ar^{2})\right]^{n}dt^{2}-[1+n^{2}A^{2}BKr^{2}(1-Ar^{2})^{n-2}]dr^{2}-r^{2}d\Omega^{2}$; $n\leq-3$
P-K-M-D (Prasad et al., 2019) | $ds^{2}=\left[A+\frac{Ba}{2b}\log{\frac{e^{br^{2}+c}-1}{e^{br^{2}+c}+1}}\right]^{2}dt^{2}-[1+a^{2}r^{2}\cosh^{2}(br^{2}+c)]dr^{2}-r^{2}d\Omega^{2}$
Bhar (Bhar, 2019) | $ds^{2}=\left[A-\frac{B\sqrt{1-A_{1}r^{2}}}{\sqrt{A_{1}}}\right]^{2}dt^{2}-[1+a^{2}r^{2}(1+b^{5}r^{5})^{-n}]dr^{2}-r^{2}d\Omega^{2}$
S-S-S-R (Sarkar et al., 2019) | $ds^{2}=\frac{1}{4a^{2}\pi}\left[2aA\sqrt{\pi}+B\sqrt{c}e^{-(1+ar^{2})^{2}}+\sqrt{\pi}(1+ar^{2})\\{erf[1+ar^{2}]\\}\right]^{2}dt^{2}-[1+cr^{2}\\{erf[1+ar^{2}]\\}^{2}]dr^{2}$
|  $-r^{2}d\Omega^{2}$, where, error function $erf[x]$ is defined as
$erf[x]=\frac{1}{\pi}\int_{-x}^{x}e^{-t^{2}}dt$.

Working Rule:
Maurya et al. (Maurya et al., 2016a): The authors have generated new uncharged
anisotropic compact star models by using the class one condition. They have
considered one of the metric functions as
$e^{\lambda}=1+\frac{(a-b)r^{2}}{1+br^{2}}$ and constructed the expression for
anisotropy factor by using the pressure anisotropy condition. Thereafter they
have obtained energy density and pressures. These physical parameters are well
behaved inside the star and satisfy all the required physical conditions.
Also, all the physical parameters of the models are dependent on the
anisotropy factor, which is a very interesting feature. The mass and radius of
the compact star models are shown to be compatible with the observational
astrophysical compact stellar objects like SAX J1808.4-3658, SAX J1808.4-3658,
Her X-1, and RXJ 1856-37.
Singh $\&$ Pant (Singh & Pant, 2016a): In this work, the authors have obtained
a family of exact solutions for relativistic uncharged anisotropic stellar
objects by considering a 4-D spacetime embedded in a 5-D pseudo-Euclidean
space. They have generated the solutions by assuming a new metric potential
given by $e^{\lambda}=1+ar^{2}(1+br^{2})^{n}$. The resulted solutions are well
behaved in all respects and satisfy all energy conditions. The resulting
compactness parameter is within the Buchdahl limit. The well-behaved nature of
the solutions for a given star candidate is completely dependent on the index
$n$. The authors have given a detailed discussion on the solutions for the
neutron star XTE J1739-285. For this particular star, the solution is well
behaved in all respects for $8\leq n\leq 20$. However, the solutions with
$n<8$ possess an increasing trend for the velocity of sound, and the solutions
belonging to $n>20$ violate the causality condition. Also, the well-behaved
nature of the solutions for PSR J0348+0432, EXO 1785-248, and Her X-1 are
specified by the index n with limits $24\leq n\leq 54$, $1.5\leq n\leq 4$, and
$0.8\leq n\leq 2.7$, respectively.
Singh et al. (Singh et al., 2016): The authors have presented a solution of
embedding class I that describes the interior of a spherically symmetric
charged anisotropic stellar configuration. The exact analytic solution has
been obtained by considering Buchdahl (Buchdahl, 1959) type metric potential.
Using this solution, the authors have discussed various physical aspects of a
compact star. The solution is free from central singularities. The counter-
balancing of different forces acting on the fluid system has been represented
by TOV equation, which signifies that the solution represents a stable
hydrostatic equilibrium. The solution also satisfies the energy conditions.
The calculated masses and radii of compact stars RX J1856.5-3754, XTE
J1739-285, PSR B0943+10, and SAX J1808.4-3658 have been compared with their
observational values.
Maurya $\&$ Maharaj (Maurya & Maharaj, 2017): By analyzing the Karmarkar
embedding condition, the authors have generated anisotropic solutions for
spherically symmetric spacetime. They have constructed a suitable form of one
of the gravitational potentials as
$e^{\lambda}=\frac{1+2cr^{2}+cosh[2ar^{2}+b]}{1+cosh[2ar^{2}+b]}$ to obtain a
closed-form solution. The resulting solution is well behaved, which has been
utilized to construct realistic static fluid sphere models. The physical
validity of the model depends on the parameter values of a, b and c. Also, by
using observational data set values, they have estimated the masses and radii
for compact stars PSR J1903+327, EXO 1785-248, LMC X-4, and 4U 1820-30, which
show that this model is suitable for representing fluid spheres to a very good
degree of accuracy.
Maurya et al. (Maurya et al., 2017b): The authors have presented a class of
new solutions describing neutral anisotropic stellar configuration satisfying
Karmarkar’s condition. This compact star model is physically well-behaved and
meets all the physical requirements for a stable configuration in hydrostatic
equilibrium. This model is compatible with compact stars 4U1608-52 and Vela
X-1 to a very good approximation.
Bhar et al. (Bhar et al., 2017a): In this paper, the authors have constructed
a new relativistic anisotropic compact star model having a spherically
symmetric metric of embedding class one. They have assumed
$e^{\lambda}=1+\frac{a^{2}r^{2}}{(1+br^{2})^{4}}$ and solved the Einstein’s
field equations with the help of Karmarkar condition for an anisotropic matter
distribution. A detailed discussion of the physical properties of the model
(pressure, density, mass function, surface red-shift, gravitational redshift)
are investigated and the stability of the stellar configuration are given. The
model is free from central singularities, satisfies all the energy conditions,
and satisfy the Harrison–Zeldovich–Novikov stability criterion for
$0\leq\rho_{c}\leq 4.04\texttimes 1017\ g/cm^{3}$, i.e., the region is stable.
However, for $\rho_{c}\geq 4.04\texttimes 1017\ g/cm^{3}$, the region is
unstable.
Ortiz et al. (Ortiz et al., 2019): In this work, the authors have presented a
completely new class of well-behaved solutions to field equations describing
uncharged anisotropic matter distribution in the embedding class one spacetime
framework using Karmarkar’s condition. They have performed this analysis by
proposing a new metric potential $e^{\lambda}=1+ar^{2}[1+sin(br^{2}+c)]^{n}$
which plays a key role in generating a realistic model. They have represented
the physical features of the solution analytically as well as graphically for
the strange star SAX J1808.4-3658, with $0.5\leq n\leq 3.4$. The resulted
solutions are free from singularities, satisfy Abreu’s criterion (Abreu et
al., 2007), causality condition as well as relativistic adiabatic index, and
exhibit well-behaved nature. The equilibrium condition, as well as the energy
conditions, are well-defined. These features confirm the physical
acceptability of the model. The authors have shown that the obtained moment of
inertia is as per the Bejger-Haensel concept (Bejger & Haensel , 2002), and
could provide a precise tool to the matching rigidity of the state equation
due to different values of $n$.
Maurya et al. (Maurya et al., 2019c): In this manuscript, the authors have
discussed a generalized model for neutral anisotropic compact stars in the
presence of highly dense and ultra-relativistic matter distribution. After
embedding the 4D Riemannian space locally and isometrically into a 5D pseudo-
Euclidean space, they have solved the field equations by employing a class of
physically acceptable metric functions $e^{\nu}=B(1-Ar^{2})^{n}$, $n\leq-3$.
The same form of metric potential was previously used by Maurya et al. (Maurya
et al., 2017c) for $n>2$, and Maurya and Govender (Maurya & Govender, 2017)
for $n\leq-1$ to model charged isotropic compact stars arising from the
Karmarkar condition. In the obtained model, the anisotropy is zero at the
center and maximum at the surface. The usual energy conditions are satisfied
and that the compact structures are shown to be stable, based on several
criteria, viz., the equilibrium of forces, Herrera cracking concept, and
adiabatic index. This stellar model also satisfies the Buchdahl condition.
Finally, the values of the numerous constants and physical parameters are
determined for the compact star LMC X-4, which has been considered as a
representative of the compact stars to present the analysis of the obtained
results. The authors have claimed that this model can justify most of the
compact stars including white dwarfs and ultra-dense compact stars for
suitable parametric values of $n$.
Prasad et al. (Prasad et al., 2019): In this paper, the authors have presented
a family of anisotropic compact stars by solving Einstein’s field equations.
The field equations have been solved by a particular choice for the metric
potential $\lambda$ as $e^{\lambda}=1+a^{2}r^{2}\csc h^{2}[br^{2}+c]$ and
embedding class one condition. The physical analysis of this model indicates
that the obtained model for relativistic anisotropic stellar structure is
physically viable for a compact star whose energy density of the order 1015
g/cm3. They have explored the hydrostatic equilibrium and the stability of the
compact stars Vela X-1, 4U 1820-30, EXO 1785-248, PSR J1903+327, 4U 1538-52,
SMC X-1, Her X-1, PSR J1614-2230, SAX J1808.4-3658, 4U 1608-52, LMC X-4, RX
J1856-37, and Cen X-3.
Bhar (Bhar, 2019): In this article, finite, regular, and exact class one
solutions of Einstein’s field equations have been presented. To generate the
model, the authors have introduced a completely new metric potential as
$1+cr^{2}\\{erf[1+ar^{2}]\\}^{2}$, with $c>0,a\neq 0$ and
$erf[x]=\frac{1}{\pi}\int_{-x}^{x}e^{-t^{2}}dt$. Various physical aspects of
the model such as energy density, pressure, anisotropy, equilibrium, energy
conditions, stability, mass, compactness parameter, surface red-shift,
gravitational red-shift, and their graphical representations have been
investigated, which ensure that the obtained solutions are well-behaved and
therefore, represent physically acceptable models for anisotropic fluid
spheres. The models has also satisfied the causality condition. It is shown to
be stable by satisfying the TOV equation, Herrera cracking concept, Adiabatic
index bound, and Harrison–Zeldovich–Novikov condition. The solutions are
representing Cen X-3, EXO 1785-248, Vela X-1, and LMC X-4. From the solutions,
the M–R graph has been generated and it matches the ranges of masses and radii
for the considered compact stars. In this work, the authors have also
estimated the approximate moment of inertia for the mentioned compact stars.

##  6 Conclusion

Stellar models consisting of spherically symmetric distribution of charged as
well as uncharged anisotropic matter in gravitational fields have been
extensively considered in the frame of general relativity. To investigate
these structures, we generate exact models via solving the Einstein-Maxwell
(or Einstein) system of equations. Apart from the imposition of symmetry, the
fall-off behavior of the pressure and density is considered to obtain a
physical model. The vanishing radial pressure at the boundary has received
considerable recognition. The produced values of redshifts, luminosity, and
mass for the stars in the presence of charged matter are different from the
case of neutral matter. To solve the system of equations governing the
gravitational and thermodynamical behavior of bounded compact objects, several
techniques were employed by researchers such as ad-hoc assumptions of the
gravitational potentials, specific choices of pressure, density, or the
anisotropy factor, imposition of the equation of state, etc. Some of them have
assumed the spacetime to be in conformal motion. A lot of models have been
generated by considering spacetime of embedding class one, obeying the
Karmarkar condition. In this article, we have characterized the interior
solutions of the Einstein-Maxwell/Einstein system of equations for the
spherically symmetric distribution of anisotropic matter based on the
technique used to generate it.

There are two main ways to generate an anisotropic compact star model
irrespective of it being charged or uncharged, where the neutral case can be
considered as the charged one, with a given charge function zero. One can
consider that the components of the energy-momentum tensor are mutually
independent. Consecutively the system of field equation is solved by assuming
any three of the six fluid characteristics as known one. Alternatively, we can
impose some additional constraints on the system of field equations, assumming
an equation of state, spacetime admitting conformal motion, or higher-
dimensional spacetime (spacetime of embedding class one in particular). These
constraints establish relation between the fluid characteristics and thus,
these parameters remain no more independent to each other.

Generally, in the second approach after imposing one constraint on the system,
two more fluid characteristics should be taken as given one, so that the
resulting system becomes solvable. However, there are solutions present in the
literature in which more than one constraint have been imposed. One can assume
the equation of state along with the assumption that spacetime is of embedding
class one, or the equation of state in the spacetime assenting conformal
motion (Mak & Harko, 2004; Esculpi & Aloma, 2010), or the spacetime of
embedding class one validating conformal motion. Futhermore, one fluid
characteristic must be known in all mentioned cases to obtain the solutions.
To solve the Einstein-Maxwell/Einstein system of equations one can also
consider the spacetime of embedding class one, admitting conformal motion and
an equation of state to exist. In this approach, no additional information is
required to generate the model.

To sum it up, we have outlined the different ways to solve the system of
Einstein-Maxwell (or Einstein) equations for compact stars having anisotropic
matter distribution. Our goal was to present a comprehensive but still
thorough review of anisotropic fluid solutions in general relativity to
provide a basis for beginners and a reference guide for the keen ones. Even
though we have tried to cover all aspects, no review can match the actual
study of the literature itself. It is possible that certain readings might
have been omitted, or analyzed less rigorously and, therefore, the reader can
resort to the original contents if necessary.

## Declarations

  * •

Funding: Science and Engineering Research Board (SERB), DST, New Delhi.

  * •

Conflict of interest: The authors declare that they have no conflict of
interest.

  * •

Ethics approval: The authors have adhered to the accepted ethical standards of
a genuine research study.

  * •

Consent to participate: The authors give their concent to participate in this
work.

  * •

Consent for publication: The authors give their consent for publication.

  * •

Availability of data and materials: Not applicable

  * •

Code availability: Not applicable

  * •

Authors’ contributions: J. Kumar: study concept, critical revision of the
paper, study supervision. P. Bharti: literature survey, designing and drafting
of the paper. All the authors have read and approved the final version of
paper.

## References

  * Abramowicz (1983) Abramowicz M. A., Acta Astronomica. 33, 313 (1983)
  * Abreu et al. (2007) Abreu H., Hernandez H., Nunez L. A., Classical and Quantum Gravity 24, 4631 (2007)
  * Adhav et al. (2015) Adhav K. S., Dawande M. V., Purandare M. A., Bulg. J. Phys. 42, 20 (2015)
  * Alcock et al. (1986) Alcock C., Farhi E., Olinto A., APJ. 310, 261 (1986)
  * Alford et al. (1999) Alford M., Rajagopal K., Wilczek F., Nucl. Phys. B 537, 443 (1999)
  * Alford et al. (2001) Alford M., Rajagopal K., Reddy S., Wilczek F., Phys. Rev. D 64, 074017 (2001)
  * Ananda & Bruni (2006) Ananda K., Bruni M., Phys. Rev. D 74, 023523 (2006)
  * Anderson (1929) Anderson W., Zeitschrift für Physik 56, 851 (1929)
  * Azam et al. (2016a) Azam M., Mardan S. A., Noureen I., Rehman M. A., Eur. Phys. J. C 76, 315 (2016)
  * Azam et al. (2016b) Azam M., Mardan S. A., Noureen I., Rehman M. A., Eur. Phys. J. C 76, 510 (2016)
  * Baade & Zwicky (1934a) Baade W., Zwicky F., Proceedings of the National Academy of Sciences 20, 259 (1934)
  * Baade & Zwicky (1934b) Baade W., Zwicky F., Physical Review 46, 76 (1934)
  * Barreto & Rojas (1992) Barreto W., Rojas S., Astrophys. Space Sci. 193, 201 (1992)
  * Barreto (1993) Barreto W., Astrophys. Space Sci. 201, 191 (1993)
  * Bayin (1982) Bayin S., Physical Review D 26, 1262 (1982)
  * Bejger & Haensel (2002) Bejger M., Haensel P., A $\&$ A 396, 917 (2002)
  * Bombaci (1996) Bombaci I., Astronomy and Astrophysics. 305, 871 (1996)
  * Bondi (1999) Bondi H., Mon. Not. R. Astron. Soc. 302, 337 (1999)
  * Bondi (1992) Bondi H., Mon. Not. Roy. Astr. Soc. 259, 365 (1992)
  * Böhmer & Harko (2006) Böhmer C. G., Harko T., Class. Quantum Gravity 23, 6479 (2006)
  * Böhmer & Harko (2007) Böhmer C. G., Harko T., Mon. Not. R. Astron. Soc. 379, 393 (2007)
  * Bonolis (2017) Bonolis L., The European Physical Journal H 42, 311 (2017)
  * Bowers & Liang (1974) Bowers R. L., Liang E. P. T., Class. Astrophys. J. 188, 657 (1974)
  * Benvenuto & Horvath (1989) Benvenuto O. G., Horvath J. E., Mon. Not. R. Astron. Soc. 241, 43–50 (1989)
  * Bibi et al. (2016) Bibi R., Feroze T., Siddiqui A. A., Can. J. Phys. 94, 758 (2016)
  * Biswas et al. (2019) Biswas S., Ghosh S., Guha B. K., Rahaman F., Ray S., Ann. Phys. 401, 1 (2019)
  * Bhar (2014) Bhar P., Astrophys. Space Sci. Astrophysics and Space Science 354 (2), 457 (2014)
  * Bhar (2015a) Bhar P., Astrophys Space Sci. 357, 46 (2015)
  * Bhar (2015b) Bhar P., Eur. Phys. J. C 75, 123 (2015)
  * Bhar (2015c) Bhar P., Astrophysics and Space Science. 359, 41 (2015)
  * Bhar et al. (2015) Bhar P., Murad M. H., Pant N., Astrophys. Space Sci. 359, 13 (2015)
  * Bhar et al. (2017a) Bhar P., Singh K. N., Manna T., International Journal of Modern Physics D 26 (09), 1750090 (2017)
  * Bhar et al. (2017b) Bhar P., Singh K. N., Pant N., Indian J Phys. 91, 701 (2017)
  * Bhar et al. (2018) Bhar P., Govender M., Sharma R., Pramana – J. Phys. 90, 5 (2018)
  * Bhar (2019) Bhar P., Eur. Phys. J. C 79, 138 (2019)
  * Brilenkov et al. (2013) Brilenkov M., Eingorn M., Jenkovszky L., Zhuk A., J Cosmol Astropart Phys. 8, 002 (2013)
  * Buchdahl (1959) Buchdahl H. A., Phys. Rev. D 116, 1027 (1959)
  * Canuto (1973) Canuto V., Solvay Conf. on Astrophysics and Gravitation (Brussels, Sept., 1973)
  * Capozziello et al. (2002) Capozziello S., Martino S. D., Falanga M., Phys. Lett. A 299, 494 (2002)
  * Capozziello et al. (2003) Capozziello S., Carloni S., Troisi A., arXiv:astro-ph/0303041 (2003)
  * Capozziello et al. (2006) Capozziello S., Cardone V. F., Elizalde E., Nojiri S., Odintsov S. D., Phys. Rev. D 73, 043512 (2006)
  * Cook (1994) Cook G. B., Shapiro S. L., Teukolsky S. A., Astrophys. J. 422, 227 (1994)
  * Coley & Tupper (1994) Coley A. A., Tupper B. O. J., Class. Quantum Gravity 11, 2553 (1994)
  * Cosenza et al. (1981) Cosenza M., Herrera L., Esculpi M., Witten L., J. Math. Phys. 22, 118 (1981)
  * Cottam et al. (2002) Cottam J., Paerels F., Mendez M., Nature 420, 51 (2002)
  * Chaisi & Maharaj (2005) Chaisi M., Maharaj S. D., Gen. Relativ. Gravit. 37, 1177 (2005)
  * Chaisi & Maharaj (2006) Chaisi M., Maharaj S. D., Pramana J. Phys. 66, 609 (2006)
  * Chan et al. (2003) Chan R., da Silva M. F. A., da Rocha J. F. V., J. Math. Phys. 12, 347 (2003)
  * Chandrasekhar (1931a) Chandrasekhar S., Astrophysical Journal 74, 81 (1931)
  * Chandrasekhar (1931b) Chandrasekhar S., The London, Edinburgh, Dublin Philosophical Magazine and Journal of Science 11, 592 (1931)
  * Chandrasekhar (1931c) Chandrasekhar S., Monthly Notices of the Royal Astronomical Society 91, 456 (1931)
  * Chandrasekhar (1932) Chandrasekhar S., Zeitschrift fur Astrophysik 5, 321 (1932)
  * Chandrasekhar (1934a) Chandrasekhar S., The Observatory 57, 93 (1934)
  * Chandrasekhar (1934b) Chandrasekhar S., The Observatory 57, 373 (1934)
  * Chandrasekhar (1935a) Chandrasekhar S., Monthly Notices of the Royal Astronomical Society 95, 207 (1935)
  * Chandrasekhar (1935b) Chandrasekhar S., Monthly Notices of the Royal Astronomical Society 95, 226 (1935)
  * Chandrasekhar (1935c) Chandrasekhar S., Monthly Notices of the Royal Astronomical Society 95, 676 (1935)
  * Chandrasekhar (1939) Chandrasekhar S., An introduction to the study of stellar structure, University of Chicago, Chicago, (1939)
  * Chavanis (2013) Chavanis P. H., Journal of Gravity, 682451 (2013)
  * Chavanis (2015) Chavanis P. H., Universe. 1 (3), 357 (2015)
  * Deb et al. (2017) Deb D., Chowdhury S. R., Ray S., Rahaman F., Guha B. K., Ann. Phys. 387, 239 (2017)
  * Dev & Gleiser (2002) Dev K., Gleiser M., Gen. Relativ. Gravit. 34, 1793 (2002)
  * Dev & Gleiser (2003) Dev K., Gleiser M., Gen. Relativ. Gravit. 35, 1435 (2003)
  * Dev & Gleiser (2004) Dev K., Gleiser M., Int. J. Mod. Phys. D 13, 1389 (2004)
  * Dey et al. (1998) Dey M., Bombaci I., Dey J., Ray S., Samanta B. C., Phys. Lett. B. 438, 123–128 (1998)
  * Eriguchi (1994) Eriguchi Y., Hachisu I., Nomoto K., Monthly Notices of the Royal Astronomical Society 266, 179 (1994)
  * Esculpi et al. (2007) Esculpi M., Malaver M., Aloma E., Gen. Relativ. Gravit. 39, 633 (2007)
  * Esculpi & Aloma (2010) Esculpi M., Aloma E., Eur. Phys. J. C 67, 521 (2010)
  * Feroze & Siddiqui (2011) Feroze T., Siddiqui A. A., Gen. Relativ. Gravit. 43, 1025 (2011)
  * Finch & Skea (1989) Finch M. R., Skea J. E. F., Class. Quantum Gravity 6, 467 (1989)
  * Frieman & Olinto (1989) Frieman J. A., Olinto A., Nature. 341, 63 (1989)
  * Florides (1974) Florides P. S., Proc. Roy. Soc. A 337, 529 (1974)
  * Fowler (1926a) Fowler R. H., Proceedings of the Royal Society A 113, 432 (1926)
  * Fowler (1926b) Fowler R. H., On Dense Matter. Monthly Notices of the Royal Astronomical Society 87, 114 (1926)
  * Friedman (1987) Friedman J. L., Ipser J. R., Astrophys. J. 314, 594 (1987)
  * Fuzfa (2001) Fuzfa A., Gerard J. M., Lambert D., e-print, gr-qc/0109097 (2001)
  * Gokhroo & Mehra (1994) Gokhroo M. K., Mehra A. L., Gen. Rel. Grav. 26, 75 (1994)
  * Goswami et al. (2020) Goswami K. B., Saha A., Chattopadhyay P. K., Astrophys Space Sci 365, 141 (2020)
  * Haensel & Zdunik (1989) Haensel P., Zdunik J. L., Nature. 340, 617 (1989)
  * Hillebrandt & Steinmetz (1976) Hillebrandt W., Steinmetz K. O., Astronomy and Astrophysics 53, 283 (1976)
  * Harko & Cheng (2000) Harko T., Cheng K. S., Phys. Lett. A 266, 249 (2000)
  * Harko & Mak (2000) Harko T., Mak M. K., J. Math. Phys. 41, 4752 (2000)
  * Harko & Mak (2002) Harko T., Mak M. K., Annalen der Phys. 11, 3 (2002)
  * Harrison et al. (1965) Harrison B. K., et al. Gravitational Theory and Gravitational collapse (Chicago: University of Chicago Press-1965)
  * Heintzmann & Hillebrandt (1975) Heintzmann H., Hillebrandt W., Astron. Astrophys. 38, 51 (1975)
  * Herrera & Santos (1995) Herrera L., Santos N. O., Astrophys. J 438, 308 (1995)
  * Herrera & Santos (1997) Herrera L., Santos N. O., Phys. Report. 286, 53 (1997)
  * Hernández & Núñez (2004) Hernández H., Núñez L. A., Canadian journal of physics, 82 (1), 29 (2004)
  * Hernández et al. (1999) Hernández H., Núñez L. A., Percoco U., Class. Quantum Gravity 16, 871 (1999)
  * Herrera et al. (1984) Herrera L., Jimenez J., Leal L., de Leon J. P., Esculpi M., Galino V., J. Math. Phys. 25, 3274 (1984)
  * Herrera et al. (2001) Herrera L., Prisco A. D., Ospino J., Fuenmayor E., J. Math. Phys. 42, 2129 (2001)
  * Herrera et al. (2008a) Herrera L., Ospino J., Prisco A. D., Phys. Rev. D 77, 027502 (2008)
  * Herrera et al. (2008b) Herrera L., Santos N. O., Wang A., Phys. Rev. D 78, 084026 (2008)
  * Herrera (2020) Herrera L., Phys. Rev. D 101, 104024, (2020)
  * Herrera & León (1985a) Herrera L., de León J. P., J. Math. Phys. 26, 2302 (1985)
  * Herrera & León (1985b) Herrera L., de León J. P., J. Math. Phys. 26, 2018 (1985)
  * Isayev (2018) Isayev A. A., Phys. Rev. D 96, 083007 (2018)
  * Islam et al. (2015) Islam S., Rahaman F., Sardar I. H., Astrophys. Space Sci. 356, 293 (2015)
  * Islam et al. (2019) Islam S., Datta S., Das T. K., Pramana J. Phys. 92, 43 (2019)
  * Ivanov (2002) Ivanov B. V., Phys. Rev. D 65, 104011 (2002)
  * Ivanov (2010) Ivanov B. V., Int. J. Theor. Phys. 49, 1236 (2010)
  * Jasim et al. (2018) Jasim M. K., Gupta Y. K., Ray S., Deb D., Chowdhury S. R., The European Physical Journal C 78 (7), 1 (2018)
  * Jasim et al. (2021) Jasim M. K., Maurya S. K., Ray S., Shee D., Deb D., Rahaman F., Results in Physics. 20, 103648 (2021)
  * Kalam et al. (2013) Kalam M., Usmani A. A., Rahaman F., Hossein S. K. M., Karar I., Sharma R., Int. J. Theor. Phys. 52, 3319 (2013)
  * Khadekar & Tade (2007) Khadekar G., Tade S., Astrophys. Space. Sci. 310, 41 (2007)
  * Karmarkar (1948) Karmarkar K. R., Proc. Ind. Acad. Sci. 27, 56 (1948)
  * Karmakar et al. (2007) Karmakar S., Mukherjee S., Sharma R., Maharaj S. D., Pramana, 68 (6), 881 (2007)
  * Karttunen et al. (1987) Karttunen H., Kröger P., Oja H., Poutanen M., Donner K. J., Springer Study Edition, Springer, New York, NY (1987)
  * Kippenhahn & Weigert (1990) Kippenhahn R., Weigert A., Stellar Structure and Evolution (Springer, Berlin, 1990)
  * Komathiraj & Sharma (2020) Komathiraj K., Sharma R., Journal of Science-FAS-SEUSL 1, 22 (2020)
  * Komatsu et al. (1989) Komatsu H., Eriguchi Y., Hachisu I., RAS 237, 2 (1989)
  * Kovetz (1968) Kovetz A., Astrophys. J. 154, 999 (1968)
  * Krori et al. (1984) Krori K. D., Borgohain P., Devi R., Can. J. Phys. 62, 239 (1984)
  * Krori & Barua (1975) Krori K. D., Barua J., J. Phys. A.: Math. Gen. 8, 508 (1975)
  * Kuchowicz (1968) Kuchowicz B., Acta Phys. Pol. 33, 541 (1968)
  * Kumar & Bharti (2021) Kumar J., Bharti P., Physical Review D 104, 083009 (2021)
  * Lake (2004) Lake K., Phys. Rev. Lett. 92, 051101 (2004)
  * Lake (2009) Lake K., Phys. Rev. D 80, 064039 (2009)
  * Lemaitre (1933) Lemaitre G., Ann. Soc. Sci. Bruxelles A 53, 51 (1933)
  * Letelier (1980) Letelier P., Physical Review D 22, 807 (1980)
  * Lobo (2007) Lobo F. S. N., Phys. Rev. D 75, 024023 (2007)
  * Lobo (2006) Lobo F. S. N., Class. Quantum Grav. 23, 1525 (2006)
  * Lugones & Horvath (2003) Lugones G., Horvath J., A&A. 403 (1), 173 (2003)
  * Maartens & Maharaj (1990) Maartens R., Maharaj M. S., J. Math. Phys. 31, 151 (1990)
  * Maartens et al. (1995) Maartens R., Maharaj S. D., Tupper B. O. J., Class. Quantum Gravity 12, 2577 (1995)
  * Maartens et al. (1996) Maartens R., Maharaj S. D., Tupper B. O. J., Class. Quantum Gravity 13, 317 (1996)
  * Maharaj & Chaisi (2006) Maharaj S. D., Chaisi M., General Relativity and Gravitation. 38, 1723 (2006)
  * Maharaj & Maartens (1989) Maharaj S. D., Maartens R., Gen. Relativ. Gravit. 21, 9 (1989)
  * Maharaj & Takisa (2012) Maharaj S. D., Takisa P. M., General Relativity and Gravitation. 44 (6), 1419 (2012)
  * Mak & Harko (2003) Mak M. K., Harko T., Proc. R. Soc. Lond. A 459, 393 (2003)
  * Mak & Harko (2002) Mak M. K., Harko T., Chin. J. Astron. Astrophys. 2 (3), 248 (2002)
  * Mak & Harko (2004) Mak M. K., Harko T., Int. J. Mod. Phys. D 13, 149 (2004)
  * Martínez et al. (1994) Martínez J., Pavón D., Nuñez L. A., Mon. Not. R. Astron. Soc. 271, 463 (1994)
  * Maurya et al. (2015) Maurya S. K., Gupta Y. K., Ray S., Dayanandan B., Eur. Phys. J. C 75, 225 (2015)
  * Maurya et al. (2016a) Maurya S. K., Gupta Y. K., Smitha T. T., Rahaman F., Eur. Phys. J. A 52, 191 (2016)
  * Maurya et al. (2016b) Maurya S. K., Gupta Y. K., Ray S., Deb D., Eur. Phys. J. C 76, 693 (2016)
  * Maurya et al. (2017a) Maurya S. K., Gupta Y. K., Ray S., Eur. Phys. J. C 77, 360 (2017)
  * Maurya et al. (2017b) Maurya S. K., Ratanpal B. S., Govender M., Ann. Phys. 382, 36-49 (2017)
  * Maurya et al. (2017c) Maurya S. K., Gupta Y. K., Ray S., Deb D., Eur. Phys. J. C 77, 45 (2017)
  * Maurya et al. (2018a) Maurya S. K., Ray S., Ghosh S., Manna S., Smitha T., Annals of Physics, 395, 152 (2018)
  * Maurya et al. (2018b) Maurya S. K., Banerjee A., Hansraj S., Physical Review D 97 (4), 044022 (2018)
  * Maurya et al. (2019a) Maurya S. K., Banerjee A., Jasim M. K., Kumar J., Prasad A. K., A. Pradhan, Physical Review D 99 (4), 044029 (2019)
  * Maurya et al. (2019b) Maurya S. K., Maharaj S. D., Kumar J., Prasad A. K., General Relativity and Gravitation. 51, 86 (2019).
  * Maurya et al. (2019c) Maurya S. K., Deb D., Ray S., Kuhfittig P. K. F., Int. J. Mod. Phys. D 28, 1950116 (2019)
  * Maurya et al. (2019d) Maurya S. K., Maharaj S. D., Deb D., Eur. Phys. J. C 79, 170 (2019)
  * Maurya & Govender (2017) Maurya S. K., Govender M., Eur. Phys. J. C 77, 347 (2017)
  * Maurya & Maharaj (2017) Maurya S. K., Maharaj S. D., Eur. Phys. J. C 77, 328 (2017)
  * Malaver (2013a) Malaver M., World Applied Programming. 3, 309 (2013)
  * Malaver (2013b) Malaver M., American Journal of Astronomy and Astrophysics. 1, 41 (2013)
  * Malaver (2014) Malaver M., FMIA. 1, 9 (2014)
  * Malaver (2017) Malaver M., International Journal of Astronomy, Astrophysics and Space Science, 4 (3), 12 (2017)
  * Malaver & Kasmaei (2019) Malaver M., Kasmaei H. D., International Journal of Astrophysics and Space Science 7 (5), 58 (2019)
  * Malaver & Kasmaei (2020) Malaver M., Kasmaei H. D., International Journal of Mathematical modelling and Computation. 10 (2), 111 (2020)
  * Malaver & Kasmaei (2021) Malaver M., Kasmaei H. D., Int J Phys Stud Res. 3(1), 83 (2021)
  * Malaver & Wirel (2018) Malaver M., Wirel. Sensor Netw. 92, 327 (2018)
  * Manjonjo et al. (2018) Manjonjo A. M., Maharaj S. D., Moopanar S., Class. Quantum Gravity 35, 205007 (2018)
  * Mardan et al. (2020) Mardan S. A., Rehman M., Noureen I., Jamil R. N., Eur. Phys. J. C. 80, 119 (2020)
  * Matondo et al. (2018) Matondo D. K., Maharaj S. D., Ray S., Eur. Phys. J. C 78, 437 (2018)
  * Matondo & Maharaj (2016) Matondo D. K., Maharaj S. D., Astrophys. Space Sci. 361, 221 (2016)
  * Mazzali et al. (2017) Mazzali P. A., Ropke F. K., Benetti S., Hillebrandt W., Science (PDF) 315, 825 (2017)
  * Milne (1929) Milne E. A., Monthly Notices of the Royal Astronomical Society 90, 17 (1929)
  * Minkowski (1939) Minkowski R., Astrophysical Journal 89, 156 (1939)
  * Misner & Sharp (1964) Misner C. W., Sharp D. H., Physical Review B 136, 571 (1964)
  * Morales & Ortiz (2018) Morales E., Ortiz F. T., Eur. Phys. J. 78, 841 (2018)
  * Moustakidis (2017) Moustakidis Ch. C., General Relativity and Gravitation 49, 68 (2017)
  * Nasim & Azam (2018a) Nasim A., Azam M., Astrophys. Space Sci. 363, 132 (2018)
  * Nasim & Azam (2018b) Nasim A., Azam M., Eur. Phys. J. C 78, 34 (2018)
  * Ngubelanga et al. (2015) Ngubelanga S. A., Maharaj S. D., Ray S., Astrophys. Sp. Sci. 357, 74 (2015)
  * Ngubelanga & Maharaj (2015) Ngubelanga S. A., Maharaj S. D., Eur. Phys. J. Plus. 130, 211 (2015)
  * Ngubelanga & Maharaj (2017) Ngubelanga S. A., Maharaj S. D., Astrophys. Space Sci. 362, 43 (2017)
  * Nojiri & Odintsov (2005) Nojiri S., Odintsov S. D., Phys. Rev. D 72, 023003 (2005)
  * Noureen et al. (2019) Noureen I., Mardan S. A., Azam M., Shahzad W., Khalid S., Eur. Phys. J. C. 79, 302 (2019)
  * O’Connor & Robertson (1996) O’Connor J. J., Robertson E. F., General relativity, Mathematical Physics index, School of Mathematics and Statistics, University of St. Andrews, Scotland (1996)
  * Oppenheimer & Serber (1938) Oppenheimer J. R., Serber R., Physical Review 54,540 (1938)
  * Oppenheimer & Snyder (1939) Oppenheimer J. R., Snyder H., Physical Review 56, 455 (1939)
  * Oppenheimer & Volkoff (1939) Oppenheimer J. R., Volkoff G. M., Physical Review 55, 374 (1939)
  * Opic (1938) Opik E., Publication of the Tartu University Observatory 30, 1 (1938)
  * Orsaria et al. (2007) Orsaria M., Rodrigues H., Duarte S. B., Braz. J. Phys. 37 (1a) (2007)
  * Ortiz et al. (2019) Ortiz F. T., Maurya S. K., Errehymy A., Singh K. N., Daoud M., Eur. Phys. J. C 79, 885 (2019)
  * Ortiz et al. (2020) Ortiz F. T., Malaver M., Rincón A., Leyton Y. G., Eur. Phys. J. C 80,371 (2020)
  * Özel (2006) Özel F., Nature. 441, 1115 (2006)
  * Panda (2015) Panda N. R., Mohanta K. K., Sahu P. K., J. Phys. Conf. Ser. 599, 012036 (2015)
  * Pandey et al. (1991) Pandey S. C., Durgapal M. C., Pandey A. K., Astrophys. Space Sci. 180, 75 (1991)
  * Pandey & Sharma (1981) Pandey S. N., Sharma S. P., Gen. Relativ. Gravit. 14, 113 (1981)
  * Pandya (2015) Pandya D. M., Thomas V. O., Sharma R., Astrophys. Space Sci. 356, 285 (2015)
  * Pant et al. (2020) Pant N., Gedela S., Pant R. P., Upreti J., Bisht R. K., Eur. Phys. J. Plus. 135, 180 (2020)
  * Pant & Fuloria (2021) Pant K., Fuloria P., New Astron. 84, 101509 (2021)
  * Paschalidis & Stergioulas (2017) Paschalidis V., Stergioulas N., Living Reviews in Relativity 20, 7 (2017)
  * Patel & Mehta (1995) Patel L. K., Mehta N. P., Austr. J. Phys. 48, 635 (1995)
  * Prasad et al. (2019) Prasad A. K., Kumar J., Maurya S. K., Dayanandan B., Astrophysics and Space Science 364, 66 (2019)
  * Prasad et al. (2021) Prasad A. K., Kumar J., Sarkar A., arXiv:2104.13004 (2021)
  * Ragel & Thirukkanesh (2019) Ragel F. C., Thirukkanesh S., Eur. Phys. J. C 79, 306 (2019)
  * Rajagopal et al. (2001) Rajagopal K., Wilczek F., Paris M. W., Reddy S., Phys. Rev. Lett. 86, 3492 (2001)
  * Rahaman et al. (2010a) Rahaman F., Ray S., Jafry A. K., Chakraborty K., Phys. Rev. D 82, 10405 (2010)
  * Rahaman et al. (2010b) Rahaman F., Jamil M., Sharma R., Chakraborty K., Astrophys. Space Sci. 330, 249 (2010)
  * Rahaman et al. (2010c) Rahaman F., Jamil M., Kalam M., Chakraborty K., Ghosh A., Astrophys. Space Sci. 137, 325 (2010)
  * Rahaman et al. (2012) Rahaman F., Sharma R., Ray S., Eur. Phys. J. C. 72, 2071 (2012)
  * Rahman et al. (2015) Rahaman F., Pradhan A., Ahmed N., Ray S., Saha B., Rahaman M., International Journal of Modern Physics D 24 (07), 1550049 (2015)
  * Rahman et al. (2017) Rahaman F., Maharaj S. D., Sardar I. H., Chakraborty K., Mod. Phys. Lett. A 32, 1750053 (2017)
  * Rahaman & Jamil (2011) Rahaman F., Jamil M., Chakraborty K., Astrophysics and Space Science. 331 (1), 191 (2011)
  * Ray & Das (2007) Ray S., Das B., Gravit. Cosmol. 13, 224 (2007)
  * Rocha et al. (2019) Rocha L. S., Bernardo A., de Avellar M. G. B., Horvath J. E., Astron. Nachr. 340, 180 (2019)
  * Rocha et al. (2020) Rocha L. S., Bernardo A., De Avellar M. G. B., Horvath J. E., International Journal of Modern Physics D. 29 (7), 2050044 (2020)
  * Rodrigues et al. (2011) Rodrigues H., Duarte S. B., de Oliveira J. C. T., Astrophys. J. 730, 31 (2011)
  * Rosinska et al. (200) Rosinska D. G., Bulik T., Zdunik L., Gourgoulhon E., Ray S., Dey J., Dey M., A$\&$A 363, 1005 (2000)
  * Ruderman (1972) Ruderman R., Reviews of Astronomy and Astrophysics 10, 427 (1972)
  * Sagert et al. (2006) Sagret I., Hempel M., Greiner C., Bielich J. S., European Journal of Physics 27, 577 (2006)
  * Sarkar et al. (2019) Sarkar N., Singh K. N., Sarkar S., Rahaman F., Eur. Phys. J. C 79, 516 (2019)
  * Sawyer (1972) Sawyer R. F., Physical Review Letter 29, 382 (1972)
  * Schlai (1871) Schlai L., Ann. di Mat. 5, 170, (1871)
  * Schunck & Mielke (2003) Schunck F. E., Mielke E. W., Class. Quantum Gravity 20, R301 (2003)
  * Schwarzschild (1916) Schwarzschild K., Phys.-Math. Klasse 189 (1916)
  * Shapiro & Teukolsky (1983) Shapiro L. S., Teukolsky S. A., BlackHoles, White Dwarfs and Neutron Stars, Wiley, New York, (1983)
  * Sharma et al. (2020) Sharma R., Das S., Govender M., Pandya D. M., Ann. Phys. 414, 168079 (2020)
  * Sharma & Mukherjee (2002) Sharma R., Mukherjee S., Mod. Phys. Lett. A 17, 2535 (2002)
  * Sharma & Ratanpal (2013) Sharma R., Ratanpal B. S., Int. J. Mod. Phys. D 22, 1350074 (2013)
  * Sharma & Maharaj (2007) Sharma R., Maharaj S. D., Mon. Not. R. Astron. Soc. 375, 1265 (2007)
  * Shee et al. (2016) Shee D., Rahaman F., Guha B. K., Ray S., Astrophys. Space Sci. 361, 167 (2016)
  * Singh et al. (1995) Singh T., Singh G. P., Helmi A. M., Il Nuovo Cim. B110, 387 (1995)
  * Singh et al. (2016) Singh K. N., Pant N., Pradhan N., Astrophysics and Space Science 361, 173 (2016)
  * Singh et al. (2021) Singh K. N., Das S., Bhar P., Rahaman M., Rahaman F., International Journal of Modern Physics A. DOI.org/10.1142/S0217751X2150192X (2021)
  * Singh & Pant (2016a) Singh K. N., Pant N., Astrophysics and Space Science 361, 177 (2016)
  * Singh & Pant (2016b) Singh K. N., Pant N., Eur. Phys. C 76, 524 (2016)
  * Sokolov (1980) Sokolov A. I., JETP 79, 1137 (1980)
  * Sotani et al. (2004) Sotani H., Kohri K., Harada T., Phys. Rev. D 69, 084008 (2004)
  * Stewart (1982) Stewart B. W., J. Phys. A 15, 2419 (1982)
  * Stoner (1929) Stoner C. E., Philosophical Magazine 7, 63 (1929)
  * Stoner (1930) Stoner C. E., Philosophical Magazine 9, 944 (1930)
  * Sunzu & Thomas (2018) Sunzu J. M., Thomas M., Pramana-J. Phys. 91, 75 (2018)
  * Sunzu (2019) Sunzu J. M., Mathias A. K., Maharaj S. D., J. Astrophys. Astr. 40, 8 (2019)
  * Sunzu & Danford (2017) Sunzu J. M., Danford P., Pramana – J. Phys. 89, 44 (2017)
  * Sunzu & Mahali (2018) Sunzu J. M., Mahali K. A., Global journal of science frontier research. 18, 19 (2018)
  * Takisa et al. (2019) Takisa P. M., Maharaj S. D., Mulangu C., Pramana, 92, 40 (2019)
  * Takisa & Maharaj (2013) Takisa P. M., Maharaj S. D., Astrophys. Space Sci. 343, 569 (2013)
  * Takisa & Maharaj (2016) Takisa P. M., Maharaj S. D., Astrophys. Space Sci. 361, 262 (2016)
  * Tikekar & Thomas (2005) Tikekar R., Thomas V. O., Pramana J. Phys. 64, 5 (2005)
  * Thirukkanesh et al. (2020) Thirukkanesh S., Kaisavelu A., Govender M., Eur. Phys. J. C 80, 214 (2020)
  * Thirukkanesh et al. (2021) Thirukkanesh S., Bogadi R. S., Govender M., Moyo S., Eur. Phys. J. C 81, 62 (2021)
  * Thirukkanesh & Maharaj (2008) Thirukkanesh S., Maharaj S. D., Class. Quantum Grav. 25, 235001 (2008)
  * Thirukkanesh & Ragel (2012) Thirukkanesh S., Ragel F. C., Pramana 89, 687 (2012)
  * Thirukkanesh & Ragel (2014) Thirukkanesh S., Ragel F. C., Astrophys. Space Sci. 354, 415 (2014)
  * Thomas et al. (2005) Thomas V. O., Ratanpal B. S., Vinodkumar P. C., Int. J. Mod. Phys. D 14, 85 (2005)
  * Thomas & Ratanpal (2007) Thomas V. O., Ratanpal B. S., Int. J. Mod. Phys. D 16, 1497 (2007)
  * Tolman (1930) Tolman R., Physical Reviews 35, 875 (1930)
  * Tolman (1939) Tolman R. C., Phys. Rev. 55, 364 (1939)
  * Tupper et al. (2012) Tupper B. O. J., Keane A. J., Carot J., Class. Quantum Gravity 29, 145016 (2012)
  * Usmani et al. (2011) Usmani A. A., Rahaman F., Ray S., Nandi K. K., Kuhfittig P. K. F., Rakib S. K. A., Hasan Z., Phys. Lett. B 701, 388 (2011)
  * Vaidya & Tikekar (1982) Vaidya P. C., Tikekar R., J. Astrophys. Astron. 3, 325 (1982)
  * Weber (1999) Weber F., Pulsars as Astrophysical Observatories for Nuclear and Particle Physics (IOP Publishing, Bristol, 1999)
  * Whittaker (1935) Whittaker E. T., Proceedings of Royal Society of London A 149, 384 (1935)
  * Zeldovich & Novikov (1971) Zeldovich Ya. B., Novikov I. D., Relativistic Astrophysics Vol 1 : Stars and Relativity (Chicago: University of Chicago Press-1971)
  * Zwicky (1938a) Zwicky F., Physical Review 54, 242 (1938)
  * Zwicky (1938b) Zwicky F., Astrophysical Journal 88, 522 (1938)


# DifAugGAN: A Practical Diffusion-style Data Augmentation for GAN-based
Single Image Super-resolution

Axi Niu1, Kang Zhang2, Joshua Tian Jin Tee2, Trung X. Pham2, Jinqiu Sun1,
Chang D. Yoo2, In So Kweon2, Yanning Zhang1
1 Northwestern Polytechnical University
2 Korea Advanced Institute of Science Technology
<EMAIL_ADDRESS>sunjinqiu<EMAIL_ADDRESS>
2{zhangkang, joshuateetj, trungpx, cd_yoo<EMAIL_ADDRESS>
###### Abstract

It is well known the adversarial optimization of GAN-based image super-
resolution (SR) methods makes the preceding SR model generate unpleasant and
undesirable artifacts, leading to large distortion. We attribute the cause of
such distortions to the poor calibration of discriminator, which hampers its
ability to provide meaningful feedback to the generator for learning high-
quality images. To address this problem, we propose a simple but non-travel
diffusion-style data augmentation scheme for current GAN-based SR methods,
known as DifAugGAN. It involves adapting the diffusion process in generative
diffusion models for improving calibration of the discriminator during
training motivated by the successes of data augmentation schemes in the field
to achieve good calibration. Our DifAugGAN can be a Plug-and-Play strategy for
current GAN-based SISR methods to improve the calibration of the discriminator
and thus improve SR performance. Extensive experimental evaluations
demonstrate the superiority of DifAugGAN over state-of-the-art GAN-based SISR
methods across both synthetic and real-world datasets, showcasing notable
advancements in both qualitative and quantitative results.

##  1 Introduction

Single image super-resolution (SISR) stands as a pivotal pursuit within low-
level computer vision, aiming to reconstruct high-resolution (HR) images from
their low-resolution (LR) counterparts [58, 34]. Similar to various inverse
problems, SISR presents challenges due to the potential existence of multiple
HR images that align with a single LR input image with an unknown degradation
[38].

Since the pioneering work of SRCNN, which demonstrated notable performance in
super-resolution by introducing a straightforward CNN architecture, CNN-based
methods have gradually emerged as the predominant solution for image super-
resolution [24, 58, 9, 35]. These approaches implicitly tackle the SISR
problem by learning the intricate, nonlinear LR-to-HR mapping from numerous
LR-HR image pairs. While these CNN-based SISR methods optimize the training
process using basic image reconstruction losses (e.g., $L_{1}$ or $L_{2}$),
yielding impressive PSNR results, these losses have shown a tendency to drive
reconstructed results toward an average of potential SR predictions, often
leading to over-smoothing in the images [49, 45, 48, 51, 36].

Figure 1: From left to right: LR input, SR obtained using the artifacts
removing oriented method LDL [23], SR result employing our LDL+DifAugGAN, and
the ground truth. Our DifAugGAN notably enhances the performance of LDL,
generating more realistic outcomes while significantly reducing distortions.

To achieve SR results with better perceptual quality, GAN-based SR methods
emerge to meet a historic destiny [20, 43, 54, 55, 41]. GAN-based SISR methods
[43, 39, 41] often involve a generator and a discriminator in an adversarial
way to push the generator to generate realistic images. The generator can
generate an SR result for the input LR, and the discriminator aims to
distinguish if the generated SR is true. The training process is optimized by
combining the reconstruction loss and adversarial loss, which in fact helps
the GAN-based SR methods recover sharp images with rich details. However, this
true-or-false adversarial optimization is vulnerable to generating SR images
with a high degree of distortions, as shown in Fig. 1.

In this paper, we aim to analyze and solve the potential cause of such
distortions in the GAN training procedure that prevent it from achieving high-
quality SR images. We conjecture that the cause of such distortions is due to
the discriminator’s inability to send useful signals to the generator to learn
good-quality images. To elaborate, during training, we treat our discriminator
as a classifier, predicting real or fake images with outputs as the
probability of the images being real. As such, a good discriminator should
have outputs matching its true probability. For example, an ideal
discriminator should be one that has an accuracy of 80% when its prediction
output is 0.8. Interestingly, this coincides with the definition of
calibration in the field of conventional classifiers [15, 4, 13]. To analyze
such behaviors, we resort to measuring the expected calibration error (ECE)
[15] in GAN-based SR methods. Significantly, we’ve noted a pronounced level of
calibration in the discriminators throughout the training process, as depicted
in Fig. 2. Those poor calibration model is either over-confidence or under-
confidence which will amplify or weaken the learning gradient. Consequently,
these discriminators are unable to provide the proper gradient information
that is crucial for training the generators.

Figure 2: Confidence histograms (top) and reliability diagrams (bottom)
comparing LDL (left) with LDL+DifAugGAN (right) on URBAN100. The top row
illustrates the ratio of samples predicted with corresponding confidence,
while the bottom row visualizes the alignment of the model prediction with the
gap between an ideal calibration model. Evidently, the introduction of our
DifAugGAN noticeably diminishes the gap. This observation highlights that our
DifAugGAN effectively achieves enhanced calibration for the LDL model.

Inspired by the calibration field’s techniques that incorporate Gaussian noise
into various network components during training to enhance neural network
generalization and robustness [8, 33, 2, 15, 10, 3, 13], we leverage similar
data augmentation strategies to address the calibration issue within
discriminators in GAN-based SR models. Differing from prior endeavors, we
introduce Gaussian noise into the discriminator’s inputs using a diffusion-
style approach derived from diffusion models [17]. Through experimentation, we
confirm that this calibration procedure not only yields better-calibrated
discriminators, as depicted in Fig. 2, but also effectively resolves
distortions inherent in GAN-based approaches, achieving the desired outcomes.
In summary, our contributions are as follows:

  * •

We identify that discriminators in GAN-based SR methods, like conventional
classifiers, are poorly calibrated.

  * •

We propose a simple but non-trivial method for calibration of the
discriminator during training motivated by the successes of data augmentation
schemes in the field to achieve good calibration.

  * •

We show through extensive experiments that such an approach can be easily
stacked with existing GAN-based SR approaches to improve calibration and thus
improve SR performance.

  * •

Extensive experiments on synthetic and real-world SISR tasks demonstrate
demonstrates our DifAugGAN performs superior improvements against the state-
of-the-arts both quantitatively and qualitatively.

##  2 Related Work

In this section, we categorize current SISR methods into two main types: PSNR-
oriented and GAN-driven approaches. In addition, we also introduce the
development of technologies for obtaining well-calibrated neural networks.

###  2.1 PSNR-oriented SR methods

To establish the mapping between HR and LR images, lots of CNN-based methods
have emerged [20, 58, 30, 9, 28]. These methods focus on designing novel
architectures by adopting different network modules, such as residual blocks
[24], attention blocks [35], non-local blocks [31], and transformer layers
[22, 25]. For example, [20] employs the ResNet architecture from [16] and
solves the time and memory issues with good performance. Then [24] further
optimizes it by analyzing and removing unnecessary modules to simplify the
network architecture and produce better results. After them, RCAN [58] and
MCAN [30] adopt the attention mechanism [47] and design new residual dense
networks. Then MLRN [28] and BSRT [21] proposed multi-scale fusion or internal
and external features fusion architecture to solve the problem that the
existing SISR could not make full use of the characteristic information of the
middle network layer and internal features. In addition, SwinIR [22] and ESRT
[25] apply transformer technology to improve the performance further. For
optimizing the training process, they prefer to use the MAE or MSE loss
(_e.g_., $L_{1}$ or $L_{2}$) to optimize the architectures, which often leads
to over-smooth results because the above losses provide a straightforward
learning objective and optimize for the popular PSNR (peak signal-to-noise-
ratio) metric [7, 14, 46].

###  2.2 GAN-driven SR methods

With generative adversarial networks (GANs) of all types exhibiting high-
quality samples in a wide variety of data modalities, approaches based on the
deep generative model have become one of the mainstream for image super-
resolution [43, 39, 41], which have shown convincing image generation ability.
SRGAN [20] is the first GAN-based SISR method. It adopts the GAN technology to
push the generator to produce results with better Visual effects. Compared
with SRGAN, ESRAGN [43] trains the discriminator to predict the authenticity
of the generated image instead of predicting if the generated image is valid.
USRGAN [54] proposes to train an unfolding network, which integrates the
merits of traditional model-based methods and CNN-based ones. Then SPSR [29]
proposes a structure-preserving method to generate perceptual-pleasant
details, which achieves leading performance on synthetic data. BSRGAN [55] and
Real-ESRGAN [44] extend GAN-based models to real-world applications and obtain
promising results, demonstrating their immense potential to restore textures
for real-world images. However, GAN-based methods have an obvious drawback in
that they optimize the whole training process by the adversarial losses, which
often inevitably introduce undesirable artifacts, leading to large distortion,
limiting GAN-based SR methods’ further application [27, 46, 23, 37, 50].

###  2.3 Calibration

Modern neural networks are known to lack calibration [33, 15, 4]. The
calibration of a neural network shows that the confidence of the model matches
the true probability [3, 56, 13]. Moreover, the notion of miscalibration
typically involves the disparity between confidence and accuracy. This
discrepancy in modern neural networks prompts the use of metrics such as
Expected Calibration Error (ECE) [32, 15], which provides a scalar summary of
calibration that complements reliability diagrams, allowing for a more
convenient and comprehensive assessment of model calibration. One notion of
miscalibration is the difference in expectation between confidence and
accuracy,_i.e_.,

${\mathbb{E}_{k}}=\left[\left|\mathbb{P}\left(\hat{Y}=Y|\hat{P}=p\right)-p\right|\right].$
(1)

The reliability diagrams, depicted at the bottom of Figure 2, serve as a
common method to illustrate the calibration of a model prediction [32]. To
compute the Expected Calibration Error (ECE), a binary classifier (e.g.,
discriminator) prediction denoted as $f(x)$ is utilized. Here, the prediction
confidence for True is represented by $\sigma(f(x))$, while for False it is
$1-\sigma(f(x))$. To obtain the model’s predicted label, $y$, the argmax
function is applied to determine $y=\arg\max{\sigma(f(x)),1-\sigma(f(x))}$.
The process involves equally dividing the prediction confidence into $M$ bins
of equal spacing. A weighted average of the accuracy/confidence difference
within these bins is then computed. Consequently, the Expected Calibration
Error [32], or ECE, is calculated as:

$\mathrm{ECE}=\sum_{m=1}^{M}\frac{B_{m}}{n}\left|\mathrm{acc}(B_{m})-\mathrm{conf}(B_{m})\right|,$
(2)

where $n$ is the total number of samples, while $B_{m}$ denotes the samples
falling within the $m$th bin. The disparity between $\mathrm{acc}$ and
$\mathrm{conf}$ within a specific bin represents the calibration gap. For
further insights into this metric, refer to [32, 15] for comprehensive
analyses.

To improve the calibration of the model during the training, many kinds of
methods are proposed. One of the simple and effective ways is data
augmentation, e.g. MixUp [53] and ODS [40]. Those techniques can be applied in
a model architecture and task-agnostic way. MixUp regularizes the neural
network to favor simple linear behavior in between training examples [53],
which has been proven to improve calibration in [56]. ODS proposes a novel
sampling strategy that attempts to maximize diversity in the target model
outputs among the generated samples, which has been widely used in adversarial
attack tasks to improve the robustness of neural networks (NN). Furthermore,
[8, 2, 10] try to add Gaussian noise to the activation of NN, while uniform
noise injection adds uniform noise [33] has shown adding Gaussian noise to the
gradients during training can improve the generalization of neural network.
[3] adding Gaussian noise via shrinking and perturbing weights of neural
networks to improve retraining generalization. In addition, [13] has shown
adding Gaussian noise during training is effective in improving calibration.
These methods give us much inspiration that introducing noise to various
components of the network, including the input, targets, activation,
gradients, and the model itself during the training helps to explore more
effective neural network models.

Figure 3: Overall framework of DifAugGAN. It consists of three parts. An
upsampling generator to generate $I_{SR}$ for the input $I_{LR}$. The
diffusion process to add noise to the inputs of the discriminator. The
discriminator to recognize its input true or false.

##  3 Perliminaries

To avoid ambiguity, we start by presenting the problem of our interest,
_i.e_., GAN-based SR, and how it is performed in current literatures [23, 50].

GAN-based super resolution target to reconstruct a high perceptual quality SR
image $x^{SR}\in\mathcal{R}^{H\times W\times 3}$ from a given low-resolution
image $x^{LR}$. To accomplish this goal, a generator model $G_{\theta}$ is
usually trained with the following three sorts of losses [43, 44, 55]:

$\mathcal{L}_{G}=\mathcal{L}_{pixel}+\lambda_{1}\mathcal{L}_{percep}+\lambda_{2}\mathcal{L}_{adv},$
(3)

where $\mathcal{L}_{pixel}$ denotes a reconstruction loss such as $l_{1}$ or
$l_{2}$ distance. $\mathcal{L}_{percep}$ represents a perceptual loss [57]
measure the feature distance and $\mathcal{L}_{adv}$ is the adversarial loss.
Specifically, the adversarial loss involves the utilization of a discriminator
network $D_{\theta}$ that classifies a high-resolution image $x^{HR}$ as TRUE
and an image generated from generator $\hat{y}$ as FALSE:

$\left\\{\begin{aligned} D_{\theta}(x)\rightarrow 1,&~{}~{}\text{if
$x=x^{HR}$}\\\ D_{\theta}(x)\rightarrow 0,&~{}~{}\text{if $x=x^{LR}$}\\\
\end{aligned}\right..$ (4)

The training objective for the discriminator then is usually defined as
follows:

$\mathcal{L}_{D}=-\log(1-\sigma(D(x^{HR})))-\log(\sigma(D(\hat{y}))),$ (5)

where $\sigma(\cdot)$ is the sigmoid function. Furthermore, certain studies
[43, 23] have explored the utilization of relativistic GAN as their GAN loss,
which can be considered an extended case or a variant of the normal GAN loss.
However, a comprehensive discussion of this aspect is deferred to future
investigations.

Two phases training. To get the best performance, GAN-based SR typically
follows a two-step training process. Initially, the generator is trained using
$\mathcal{L}_{pixel}$ to optimize for Peak Signal-to-Noise Ratio (PSNR), thus
yielding a model tend to output over-smoothed image [43]. Subsequently, the
generator undergoes further fine-tuning by incorporating $\mathcal{L}_{G}$ and
$\mathcal{L}_{D}$ to encourage the model to generate realistic images.
However, the adversarial optimization in the second stage is unstable and
makes the preceding SR model generate unpleasant and undesirable artifacts,
leading to large distortion. [23, 50]. We attribute the cause of such
distortions to the discriminator’s poor calibration makes its prediction
either over confidence or under confidence. Thus the discriminator is unable
to send useful signals to the generator to learn good-quality images. In this
work, we focus on improving the second stage of GAN-based SR training, and the
first stage are keeping the same with existing works [43, 44].

##  4 Method

Building upon existing studies highlighting noise injection as a successful
method for improving calibration [10, 3, 13], we introduce a novel and dynamic
diffusion-style noise augmentation strategy aimed at addressing the prevalent
issue of subpar calibration in contemporary GAN-based SR training. Visualized
in Fig. 3, our strategy encompasses three essential components: 1) an up-
sampling generator denoted as $G_{up}$, 2) a diffusion processor characterized
by $\mathcal{F}$, and 3) a discriminator $D_{DN}$. In the following content,
we will introduce the whole workflow in detail.

Diffusion process on discriminator. The core idea of our method is adding
Gaussian noise on the input of the discriminator which we call it
discriminator diffusion process. $\mathcal{F}_{dis}$. In normal image
generative model [17] the diffusion process $\mathcal{F}$ is described as
gradually adding Gaussian noise into a data sample according to a noise
schedule $\\{\alpha_{1},...,\alpha_{T}\\}$ with a total of $T$ steps. Given a
data sample $x_{0}$, we define the processor at arbitrary time step $t$ as
follows:

$\mathcal{F}(x_{0},t)=\alpha_{t}\mathbf{x}_{0}+\sqrt{1-\alpha_{t}^{2}}\epsilon,\epsilon\sim\mathcal{N}(0,I).$
(6)

The noise schedule $\alpha_{t}$ is usually a monotonically decreasing function
where $\alpha_{0}=1,\alpha_{T}=0$. Thus, given a training dataset with a high-
resolution image and a low-resolution pair $(x,y)\sim\mathcal{X}$. We can
formulate our diffusion processor $\mathcal{F}$ which takes input of a
generator output or the corresponding low-resolution image. Specifically,
different from the diffusion process in the stand DDPM [17], we find that
making the mean value of the Gaussian noise added in each step has the mean
value of a bicubic-upsampled low-resolution image instead of zero mean helps
the model get better performance. Our discriminator diffusion process is
therefore formulated as follows:

$\mathcal{F}(x,t,x^{LR})=\alpha_{t}\mathbf{x}_{0}+\sqrt{1-\alpha_{t}^{2}}(\eta\epsilon+x^{LR}),\epsilon\sim\mathcal{N}(0,I),$
(7)

where $\eta$ downscaling the variance of the noise. Then our final
discriminator training can be formed as follows:

$\mathcal{L}_{DifAug}=-\log(1-\sigma(D(\mathcal{F}(x^{HR}))))-\log(\sigma(D(\mathcal{F}(\hat{y})))).$
(8)

Correspondingly, we can obtain a new adversarial loss for the generator can be
written as follows:

$\mathcal{L}_{adv}=-\log(1-\sigma(D(\mathcal{F}(\hat{y})))).$ (9)

As for the noise schedule for the diffusion process, in this paper, we mainly
consider the linear schedule proposed in DDPM [17]:

$\alpha_{t}=\exp{[-\frac{1}{4}t^{2}(\beta_{max}-\beta_{min})-\frac{1}{2}t\beta_{min}]},$
(10)

where $\beta_{min}=0.1$ and $\beta_{max}=20$ in all of our experiment. After
training with the new loss adversarial loss $\mathcal{L}_{adv}$, the
calibration of the discriminator will be optimized and then stabilize the
whole adversarial training, alleviating the distortions resulting from GANs’
unstable optimization.

Figure 4: Reliability graph for ESRGAN model and training with our DifAugGAN
on DIV2K100 and URBAN100 testing dataset. Both models are trained on the DIV2K
dataset. It shows the alignment of the model prediction with the gap between
the ideal calibration model. Table 1: Comparision different method in terms of
ECE [11, 15] and accuracy of the discriminator on the real-world scenario.

Dataset | Method | ECE $\downarrow$ (%) | ACC $\uparrow$ (%)
---|---|---|---
DIV2K | RealESRGAN | 8.8 | 97.0
+DifAugGAN | 7.4 (-1.4) | 98.8
RealESRGAN + LDL | 9.6 | 95.2
+DifAugGAN | 5.6 (-4.0) | 97.7
Urban100 | RealESRGAN | 6.9 | 97.5
+DifAugGAN | 2.8 (-4.2) | 97.5
RealESRGAN + LDL | 8.1 | 95.3
+DifAugGAN | 3.1 (-5.0) | 97.4

##  5 Experiments

Table 2: Quantitative comparison between the proposed DifAugGAN and SOTA GAN-
SR methods. bold +DifAugGAN represents our model, which takes the method in
its before column as its baseline. From this table, we can find whether it is
ESRGAN or LDL, with the addition of our DifAugGAN, the performance has been
significantly improved on different training sets. The best and second-best
results are highlighted in red and blue. $\uparrow$ and $\downarrow$ mean that
the larger or smaller score is better, respectively.

Method | ESRGAN | +DifAugGAN | LDL | +DifAugGAN | ESRGAN | +DifAugGAN | USRGAN | SPSR | LDL | +DifAugGAN | SwinIR+LDL | +DifAugGAN
---|---|---|---|---|---|---|---|---|---|---|---|---
Metric | Dataset | DIV2K | DF2K
| Set5 | 30.095 | 30.769 | 30.985 | 31.416 | 30.438 | 31.430 | 30.911 | 30.397 | 31.033 | 31.475 | 31.028 | 31.733
| Set14 | 26.738 | 27.358 | 27.491 | 27.919 | 26.594 | 27.872 | 27.405 | 26.860 | 27.228 | 27.883 | 27.526 | 28.183
| Urban100 | 24.668 | 25.136 | 25.498 | 25.842 | 24.365 | 25.923 | 24.891 | 24.804 | 25.458 | 25.923 | 26.231 | 26.543
| DIV2K(val) | 28.132 | 28.787 | 28.951 | 29.373 | 28.175 | 29.456 | 28.787 | 28.182 | 28.818 | 29.437 | 29.117 | 29.820
PSNR$\uparrow$ | General100 | 29.489 | 30.023 | 30.232 | 30.206 | 29.426 | 30.775 | 30.002 | 29.423 | 30.289 | 30.783 | 30.441 | 31.014
| Set5 | 0.8415 | 0.8598 | 0.8626 | 0.8730 | 0\. 8513 | 0.8724 | 0.8658 | 0.8443 | 0.8611 | 0.8741 | 0.8611 | 0.8782
| Set14 | 0.7204 | 0.7442 | 0.7476 | 0.7603 | 0.7168 | 0.7587 | 0.7486 | 0.7254 | 0.7358 | 0.7585 | 0.7478 | 0.7706
| Urban100 | 0.7405 | 0.7551 | 0.7673 | 0.7770 | 0.7364 | 0.7802 | 0.7503 | 0.7474 | 0.7661 | 0.7795 | 0.7918 | 0.7998
| DIV2K(val) | 0.7722 | 0.7903 | 0.7951 | 0.8049 | 0.7759 | 0.8067 | 0.7941 | 0.7720 | 0.7897 | 0.8060 | 0.8011 | 0.8193
SSIM$\uparrow$ | General100 | 0.8103 | 0.8267 | 0.8277 | 0.8367 | 0.8095 | 0.8406 | 0.8241 | 0.8091 | 0.8280 | 0.8406 | 0.8347 | 0.8482
| Set5 | 0.0690 | 0.0708 | 0.0670 | 0.0763 | 0.0758 | 0.0795 | 0.0795 | 0.0647 | 0.0690 | 0.0809 | 0.0655 | 0.0654
| Set14 | 0.1236 | 0.1298 | 0.1207 | 0.1315 | 0.1241 | 0.1188 | 0.1347 | 0.1207 | 0.1132 | 0.1189 | 0.1091 | 0.1117
| Urban100 | 0.1231 | 0.1238 | 0.1096 | 0.1167 | 0.1229 | 0.1152 | 0.1330 | 0.1184 | 0.1084 | 0.1156 | 0.1021 | 0.1041
| DIV2K(val) | 0.1137 | 0.1117 | 0.1011 | 0.1085 | 0.1154 | 0.1074 | 0.1325 | 0.1099 | 0.0999 | 0.1074 | 0.0944 | 0.0969
LPIPS$\downarrow$ | General100 | 0.0876 | 0.0873 | 0.0790 | 0.0842 | 0.0879 | 0.0865 | 0.0937 | 0.0862 | 0.0796 | 0.0876 | 0.0740 | 0.0735
| Set5 | 0.0940 | 0.0987 | 0.0917 | 0.0976 | 0.0950 | 0.0990 | 0.1045 | 0.0922 | 0.0919 | 0.1029 | 0.0899 | 0.0912
| Set14 | 0.0977 | 0.0996 | 0.0935 | 0.0968 | 0.0951 | 0.0924 | 0.0997 | 0.0921 | 0.0866 | 0.0945 | 0.0869 | 0.0888
| Urban | 0.0898 | 0.0909 | 0.0822 | 0.0869 | 0.0880 | 0.0859 | 0.0975 | 0.0849 | 0.0793 | 0.0861 | 0.0800 | 0.0800
| DIV2K(val) | 0.0574 | 0.0576 | 0.0528 | 0.0585 | 0.0594 | 0.0588 | 0.0645 | 0.0546 | 0.0526 | 0.0588 | 0.0507 | 0.0520
DIST$\downarrow$ | General100 | 0.0896 | 0.0892 | 0.0827 | 0.0854 | 0.0874 | 0.0855 | 0.0931 | 0.0885 | 0.0802 | 0.0855 | 0.0794 | 0.0782

In this section, we conduct extensive experiments to verify the effectiveness
of the proposed DifAugGAN. We first introduce the experiment setups and then
conduct analyses of how DifAugGAN helps to achieve well-calibrated
discriminators for GAN-based SR methods on both synthetic and real-world
datasets. Finally, we report the qualitative and quantitative results obtained
by comparing DifAugGAN with SOTA methods.

###  5.1 Experiment setup

Implementation details. For the synthetic experiments, we adopt the DIV2K [1]
including 800 image pairs and DF2K [24, 42] including 3450 image pairs with
the scaling factor of 4$\times$ as the training dataset respectively. We
employ benchmark datasets including Set5 [6], Set14 [52], Urban100 [18], DIV2K
validation [1], and General100 [12] for validation. For real-world image
super-resolution, we train our model on DF2K+OST [44] with the scaling factor
of 4$\times$ and adopt DIVKRK [5], NTIRE20 [26], RealSR(V3) [9], RealSRset
[55], and RealSR [19] as the validation dataset. All the experiments are run
with 4 NVIDIA GTX A100 GPUs and the batch size is 16 per GPU for synthetic
experiments and 12 per GPU for real word experiments. Following [23], we use
Adam optimizer with learning rate 1e-4. The total training iterations are
$300K$. For real-world experiments, we use the same setting with Real-ESRGAN
[44]. For all experiments, we set the noise downscaling factor $\eta=0.05$ and
the total time step $T=1000$.

Compared methods. We employ ESRGAN [43] and LDL [23] as our baselines to
validate the effectiveness of the proposed DifAugGAN, resulting in our models
ESRGAN+DifAugGAN and LDL+DifAugGAN. In addition, we also use SwiniR [22] as
the generator model to take the place of the RRDB-based generator model in
LDL, resulting in a new baseline SwinIR+LDL for our DifAugGAN. In addition, we
also compare our model with USRGAN [54] and SPSR [29]. We further validate
DifAugGAN for conducting real-world SISR tasks by employing DifAugGAN to Real-
ESRGAN [44] and LDL [23], resulting in Real-ESRGAN+DifAugGAN and
LDL+DifAugGAN. Then we compare them with BSRGAN [55], Real-ESRGAN, and LDL
models.

###  5.2  Calibration Improvements Analysis

The introduced DifAugGAN employs a diffusion-style data augmentation scheme
which can enhance the calibration of the discriminator during training. This
strategy enables the discriminator to provide more accurate and informative
feedback to the generator, fostering the production of high-quality, realistic
images. To evaluate the effectiveness of our proposed method for calibration,
we mainly apply ECE metric to measure the confidence and accuracy.

Calibration on Synthetic Datasets. Our analysis, as depicted in Fig. 4,
demonstrates a remarkable reduction in ECE when DifAugGAN is integrated on top
of the ESRGAN baseline. Specifically, the ECE witnesses a halving on both the
DIV2K validation dataset and the URBAN100 dataset, using 1000 random crops
from high-resolution and super-resolved images. Concurrently, there is an
improvement in discriminator accuracy. The improved calibration, denoted by
lower ECE values, enables the discriminator to furnish more accurate and
informative feedback to the generator. This calibration refinement, coupled
with a well-calibrated discriminator, augments the generation of high-quality,
photorealistic images. Notably, the ESRGAN model used in this analysis is
trained by us, employing normal GAN loss as opposed to the relativistic GAN
loss stipulated in the original paper.

Calibration on Real-World Dataset. Assessing our method in a real-world
scenario involves comparison within setups like RealESRGAN and RealESRGAN+LDL.
The testing dataset, DIV2K100 and URBAN100, undergoes the same processing
strategy as RealESRGAN when fed into the discriminator model. The comparison
results illustrated in Tab. 1 reveal a consistent and significant decrease in
ECE across both DIV2K and Urban100 testing datasets with the incorporation of
DifAugGAN compared to baseline methods. This reduction indicates an enhanced
alignment between predicted confidence and actual accuracy, validating the
effectiveness of DifAugGAN in enhancing calibration. While maintaining or
slightly improving discriminator accuracy in comparison to baseline methods,
DifAugGAN notably enhances calibration. This enhancement is crucial as it
ensures that the confidence estimates provided by the discriminator are more
dependable, guiding the generator toward producing higher-quality images.

Figure 5: Visual comparison to state-of-the-art GAN-SR methods. LR images are
from Urban100 and General100. The bold +DifAugGAN represents our method, which
employs our DifAugGAN to its baseline in front of it. All methods are trained
on DIV2K. More visual comparisons can be found in the supplementary materials.
($\times$4 scale. Zoomed in for a better view.)  Figure 6: Visual comparison
to state-of-the-art GAN-SR methods. LR images are from Urban100, DIV2K100, and
General100, respectively. The bold +DifAugGAN represents our method, which
employs our DifAugGAN to its baseline in front of it. All methods are trained
on DF2K. More visual comparisons can be found in the supplementary materials.
($\times$4 scale. Zoomed in for a better view.)

###  5.3 Validation on synthetic datasets

Qualitative comparison. Fig. 5 and Fig. 6 illustrate the visual comparisons
among the proposed DifAugGAN and its baselines and other SOTA GAN-based SR
methods. Obviously, for the baselines,  _i.e_. ESRGAN+DifAugGAN,
LDL+DifAugGAN, after adopting our DifAugGAN, the artifacts in the outputs have
been significantly weakened, especially in regions where there is a lot of
regional duplication (_e.g_. buildings, windows, and hair.). Furthermore,
superfluous details have also been alleviated, such as the small flowers in
the green plants. When compared with BSRGAN and USRGAN, our models still
perform superior. This effectiveness makes DifAugGAN a practical and effective
framework to further improve GAN-based SR methods.

Quantitative comparison. We first report the performance of the proposed
DifAugGAN and the state-of-the-art GAN-based SR methods on synthetic datasets
in Tab. 2. We trained the methods on both DIV2K and DF2K respectively. The
bold +DifAugGAN represents our model which takes its before line as its
baseline. It is obvious that the proposed DifAugGAN can achieve a better
trade-off between the distortion quality metrics (PSNR, SSIM) and perceptual
quality metrics (LPIPS, DISTS,) on most benchmarks than the three baselines,
_i.e_., ESRGAN+DifAugGAN, LDL+DifAugGAN, and SwinIR+LDL+DifAugGAN, though our
DifAugGAN models perform a little worse on perceptual quality metrics LPIPS
+0.0003 $\sim$ 0.001 and DISTS +0.0004 $\sim$ 0.006 than their baselines. In
addition, among LDL, USRGAN, and SPSR, there also subtle fluctuations exist in
their perceptual quality metrics. We attribute these fluctuations to the
properties of GANs, which are famous for producing overly realistic image
details than ground truth, resulting in good perceptual quality metrics. After
performing the artifact elimination methods, the artifacts generated by GAN
are eliminated, along some of the details generated by GAN will inevitably be
eliminated, which will lead to worse perceptual quality metrics. In terms of
distortion quality metrics PSNR and SSIM, our DifAugGAN models obtain obvious
improvement with PSNR +0.4 $\sim$ 0.7dB and SSIM +0.01 $\sim$ +0.03 over their
baselines, successively. When compared with USRGAN and BSRGAN, our models
still have overwhelming advantages.

Figure 7: Visual comparison to state-of-the-art GAN-SR methods. LR images are
from RealSR [9], NTIRE20 [26], RealSet [55], respectively. All methods are
trained on DF2K+OST. The bold +DifAugGAN represents our method, which employs
our DifAugGAN to its baseline in front of it. More visual comparisons can be
found in the supplementary materials. ($\times$4 scale. Zoomed in for a better
view.)

###  5.4 Validation on Real-world datasets

To demonstrate the generalization of the proposed DifAugGAN, we apply it to
conduct experiments on real-world images. We take DF2K+OST as the training
set, RealSR [9], NTIRE20 track2 [26], RealSet [55], DIVKRK [5], and RealSR
[19]. We take Real-ESRGAN [44], LDL [23] as our baseline. In addition, we also
adopt BSRGAN [55] as a comparative method. Since there is no ground truth for
real-world images, we give qualitative results to compare the proposed
DifAugGAN with the above methods.

Qualitative comparison. As shown in Fig.7, It is obvious that our DifAugGAN
can strongly remove the artifacts generated from Real-ESRGAN and LDL for the
patterned building structures in the first row, where BSRGAN produces over-
smooth results. For the areas with regular patterns in the second row, our
DifAugGAN encourages Real-ESRGAN and LDL to generate sharper details. For the
beard with dense textures in the third row, our DifAugGAN pushes its baselines
to generate more realistic details.

###  5.5 Ablation study

In order to better understand our method, we explore the effect of adding
noise. We show the result when we do not add the low-resolution image into the
noise. As shown in Tab. 3, we can see that adding the noise into Gaussian can
help the discriminator learn better. Both the fidelity-oriented metric (PSNR)
and perceptual quality metric (LPIPS) boost the performance after adding the
low-resolution image. Thus when generating the noise for our method, we add an
upsampled low-resolution image as the mean value.

Table 3: Results when adding up-sampled low-resolution image into the Gaussian
noise.

Dataset | Noise | PSNR $\uparrow$ | LPIPS $\downarrow$
---|---|---|---
Set14 | Gaussian | 27.30 | 0.1302
| +LR | 27.35 | 0.1096
DIV2K | Gaussian | 28.70 | 0.1135
| +LR | 28.78 | 0.1011
General100 | Gaussian | 29.97 | 0.0875
| +LR | 30.02 | 0.0790

###  5.6 Limitations

While DifAugGAN outperforms existing methods, the specific mechanisms through
which the diffusion-style augmentation improves discriminator calibration
might require further investigation and validation. More in-depth analysis
could provide better insights into the underlying reasons for its success and
potential limitations. Also, the effectiveness of the diffusion-style data
augmentation in DifAugGAN could heavily rely on the proper calibration of
diffusion parameters. Finding optimal parameter values might be challenging,
and the performance of DifAugGAN could be sensitive to these choices.
Addressing these limitations could provide a clearer understanding of the
scope and applicability of the proposed DifAugGAN approach in GAN-based image
super-resolution methods.

##  6 Conclusion

In this paper, we propose a diffusion-style data augmentation scheme to solve
the issue of the calibration of discriminators in GAN-based SR models, known
as DifAugGAN, to alleviate the large distortion result from GAN-based SR
models. It involves introducing the diffusion process in diffusion models for
calibration of the discriminator during training motivated by the successes of
data augmentation schemes in the field to achieve good calibration. Our
DifAugGAN can be a Plug-and-Play strategy for current GAN-based SISR methods
to improve the calibration of the discriminator and thus improve SR
performance. Extensive experiments on synthetic and real-world SISR datasets
demonstrate that our DifAugGAN outperforms the existing GAN-SR methods both
quantitatively and qualitatively.

## References

  * Agustsson and Timofte [2017] Eirikur Agustsson and Radu Timofte.  Ntire 2017 challenge on single image super-resolution: Dataset and study.  In _CVPRW_ , 2017.
  * Alemi et al. [2017] Alexander A Alemi, Ian Fischer, Joshua V Dillon, and Kevin Murphy.  Deep variational information bottleneck.  In _ICLR_ , 2017.
  * Ash and Adams [2020] Jordan Ash and Ryan P Adams.  On warm-starting neural network training.  _NeurIPS_ , 2020.
  * Bai et al. [2021] Yu Bai, Song Mei, Huan Wang, and Caiming Xiong.  Don’t just blame over-parametrization for over-confidence: Theoretical analysis of calibration in binary classification.  In _ICML_ , 2021.
  * Bell-Kligler et al. [2019] Sefi Bell-Kligler, Assaf Shocher, and Michal Irani.  Blind super-resolution kernel estimation using an internal-gan.  _NeurIPS_ , 2019.
  * Bevilacqua et al. [2012] Marco Bevilacqua, Aline Roumy, Christine Guillemot, and Marie Line Alberi-Morel.  Low-complexity single-image super-resolution based on nonnegative neighbor embedding.  2012\.
  * Blau and Michaeli [2018] Yochai Blau and Tomer Michaeli.  The perception-distortion tradeoff.  In _CVPR_ , 2018.
  * Blundell et al. [2015] Charles Blundell, Julien Cornebise, Koray Kavukcuoglu, and Daan Wierstra.  Weight uncertainty in neural network.  In _ICML_ , 2015.
  * Cai et al. [2019] Jianrui Cai, Hui Zeng, Hongwei Yong, Zisheng Cao, and Lei Zhang.  Toward real-world single image super-resolution: A new benchmark and a new model.  In _ICCV_ , 2019.
  * Camuto et al. [2020] Alexander Camuto, Matthew Willetts, Umut Simsekli, Stephen J Roberts, and Chris C Holmes.  Explicit regularisation in gaussian noise injections.  _NeurIPS_ , 2020.
  * Dong et al. [2015] Chao Dong, Chen Change Loy, Kaiming He, and Xiaoou Tang.  Image super-resolution using deep convolutional networks.  _TPAMI_ , 2015.
  * Dong et al. [2016] Chao Dong, Chen Change Loy, and Xiaoou Tang.  Accelerating the super-resolution convolutional neural network.  In _ECCV_ , 2016.
  * Ferianc et al. [2023] Martin Ferianc, Ondrej Bohdal, Timothy Hospedales, and Miguel Rodrigues.  Impact of noise on calibration and generalisation of neural networks.  _ICMLW_ , 2023.
  * Freirich et al. [2021] Dror Freirich, Tomer Michaeli, and Ron Meir.  A theory of the distortion-perception tradeoff in wasserstein space.  _NeurIPS_ , 2021.
  * Guo et al. [2017] Chuan Guo, Geoff Pleiss, Yu Sun, and Kilian Q Weinberger.  On calibration of modern neural networks.  In _ICML_ , 2017.
  * He et al. [2016] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun.  Deep residual learning for image recognition.  In _CVPR_ , 2016.
  * Ho et al. [2020] Jonathan Ho, Ajay Jain, and Pieter Abbeel.  Denoising diffusion probabilistic models.  _NeurIPS_ , 2020.
  * Huang et al. [2015] Jia-Bin Huang, Abhishek Singh, and Narendra Ahuja.  Single image super-resolution from transformed self-exemplars.  In _CVPR_ , 2015.
  * Ji et al. [2020] Xiaozhong Ji, Yun Cao, Ying Tai, Chengjie Wang, Jilin Li, and Feiyue Huang.  Real-world super-resolution via kernel estimation and noise injection.  In _CVPRW_ , 2020.
  * Ledig et al. [2017] Christian Ledig, Lucas Theis, Ferenc Huszár, Jose Caballero, Andrew Cunningham, Alejandro Acosta, Andrew Aitken, Alykhan Tejani, Johannes Totz, Zehan Wang, et al.  Photo-realistic single image super-resolution using a generative adversarial network.  In _CVPR_ , 2017.
  * Li et al. [2022] Zheyuan Li, Yingqi Liu, Xiangyu Chen, Haoming Cai, Jinjin Gu, Yu Qiao, and Chao Dong.  Blueprint separable residual network for efficient image super-resolution.  In _CVPR_ , 2022.
  * Liang et al. [2021] Jingyun Liang, Jiezhang Cao, Guolei Sun, Kai Zhang, Luc Van Gool, and Radu Timofte.  Swinir: Image restoration using swin transformer.  In _ICCV_ , 2021.
  * Liang et al. [2022] Jie Liang, Hui Zeng, and Lei Zhang.  Details or artifacts: A locally discriminative learning approach to realistic image super-resolution.  In _CVPR_ , 2022.
  * Lim et al. [2017] Bee Lim, Sanghyun Son, Heewon Kim, Seungjun Nah, and Kyoung Mu Lee.  Enhanced deep residual networks for single image super-resolution.  In _CVPRW_ , 2017.
  * Lu et al. [2022] Zhisheng Lu, Juncheng Li, Hong Liu, Chaoyan Huang, Linlin Zhang, and Tieyong Zeng.  Transformer for single image super-resolution.  In _CVPRW_ , 2022.
  * Lugmayr et al. [2020] Andreas Lugmayr, Martin Danelljan, and Radu Timofte.  Ntire 2020 challenge on real-world image super-resolution: Methods and results.  In _CVPRW_ , 2020.
  * Lugmayr et al. [2021] Andreas Lugmayr, Martin Danelljan, and Radu Timofte.  Ntire 2021 learning the super-resolution space challenge.  In _CVPR_ , 2021.
  * Lyn [2020] Jiawen Lyn.  Multi-level feature fusion mechanism for single image super-resolution.  _arXiv preprint arXiv:2002.05962_ , 2020.
  * Ma et al. [2020] Cheng Ma, Yongming Rao, Yean Cheng, Ce Chen, Jiwen Lu, and Jie Zhou.  Structure-preserving super resolution with gradient guidance.  In _CVPR_ , 2020.
  * Ma et al. [2019] Hailong Ma, Xiangxiang Chu, Bo Zhang, and Shaohua Wan.  A matrix-in-matrix neural network for image super resolution.  _arXiv preprint arXiv:1903.07949_ , 2019.
  * Mei et al. [2021] Yiqun Mei, Yuchen Fan, and Yuqian Zhou.  Image super-resolution with non-local sparse attention.  In _CVPR_ , 2021.
  * Naeini et al. [2015] Mahdi Pakdaman Naeini, Gregory Cooper, and Milos Hauskrecht.  Obtaining well calibrated probabilities using bayesian binning.  In _AAAI_ , 2015.
  * Neelakantan et al. [2015] Arvind Neelakantan, Luke Vilnis, Quoc V Le, Ilya Sutskever, Lukasz Kaiser, Karol Kurach, and James Martens.  Adding gradient noise improves learning for very deep networks.  _arXiv preprint arXiv:1511.06807_ , 2015.
  * Niu et al. [2022] Axi Niu, Yu Zhu, Chaoning Zhang, Jinqiu Sun, Pei Wang, In So Kweon, and Yanning Zhang.  Ms2net: Multi-scale and multi-stage feature fusion for blurred image super-resolution.  _IEEE TCSVT_ , 2022.
  * Niu et al. [2023a] Axi Niu, Pei Wang, Yu Zhu, Jinqiu Sun, Qingsen Yan, and Yanning Zhang.  Gran: Ghost residual attention network for single image super resolution.  _Multimedia Tools and Applications_ , 2023a.
  * Niu et al. [2023b] Axi Niu, Kang Zhang, Trung X Pham, Pei Wang, Jinqiu Sun, In So Kweon, and Yanning Zhang.  Learning from multi-perception features for real-word image super-resolution.  _arXiv preprint arXiv:2305.18547_ , 2023b.
  * Park et al. [2023] JoonKyu Park, Sanghyun Son, and Kyoung Mu Lee.  Content-aware local gan for photo-realistic super-resolution.  In _ICCV_ , 2023.
  * Saharia et al. [2022] Chitwan Saharia, Jonathan Ho, William Chan, Tim Salimans, David J Fleet, and Mohammad Norouzi.  Image super-resolution via iterative refinement.  _TPAMI_ , 2022.
  * Soh et al. [2019] Jae Woong Soh, Gu Yong Park, Junho Jo, and Nam Ik Cho.  Natural and realistic single image super-resolution with explicit natural manifold discrimination.  In _CVPR_ , 2019.
  * Tashiro et al. [2020] Yusuke Tashiro, Yang Song, and Stefano Ermon.  Diversity can be transferred: Output diversification for white-and black-box attacks.  _NeurIPS_ , 2020.
  * Tian et al. [2022] Chunwei Tian, Xuanyu Zhang, Jerry Chun-Wei Lin, Wangmeng Zuo, Yanning Zhang, and Chia-Wen Lin.  Generative adversarial networks for image super-resolution: A survey.  _arXiv preprint arXiv:2204.13620_ , 2022.
  * Timofte et al. [2017] Radu Timofte, Eirikur Agustsson, Luc Van Gool, Ming-Hsuan Yang, and Lei Zhang.  Ntire 2017 challenge on single image super-resolution: Methods and results.  In _CVPRW_ , 2017.
  * Wang et al. [2018] Xintao Wang, Ke Yu, Shixiang Wu, Jinjin Gu, Yihao Liu, Chao Dong, Yu Qiao, and Chen Change Loy.  Esrgan: Enhanced super-resolution generative adversarial networks.  In _ECCVW_ , 2018.
  * Wang et al. [2021] Xintao Wang, Liangbin Xie, Chao Dong, and Ying Shan.  Realesrgan: Training real-world blind super-resolution with pure synthetic data supplementary material.  _ICCVW_ , 2021.
  * Wang et al. [2022] Yufei Wang, Renjie Wan, Wenhan Yang, Haoliang Li, Lap-Pui Chau, and Alex Kot.  Low-light image enhancement with normalizing flow.  In _AAAI_ , 2022.
  * Whang et al. [2022] Jay Whang, Mauricio Delbracio, Hossein Talebi, Chitwan Saharia, Alexandros G Dimakis, and Peyman Milanfar.  Deblurring via stochastic refinement.  In _CVPR_ , 2022.
  * Woo et al. [2018] Sanghyun Woo, Jongchan Park, Joon-Young Lee, and In So Kweon.  Cbam: Convolutional block attention module.  In _ECCV_ , 2018.
  * Wu et al. [2023] Gang Wu, Junjun Jiang, and Xianming Liu.  A practical contrastive learning framework for single-image super-resolution.  _TNNLS_ , 2023.
  * Wu et al. [2021] Haiyan Wu, Yanyun Qu, Shaohui Lin, Jian Zhou, Ruizhi Qiao, Zhizhong Zhang, Yuan Xie, and Lizhuang Ma.  Contrastive learning for compact single image dehazing.  In _CVPR_ , 2021.
  * Xie et al. [2023] Liangbin Xie, Xintao Wang, Xiangyu Chen, Gen Li, Ying Shan, Jiantao Zhou, and Chao Dong.  Desra: Detect and delete the artifacts of gan-based real-world super-resolution models.  2023\.
  * Xue et al. [2023] Danna Xue, Luis Herranz, Javier Vazquez Corral, and Yanning Zhang.  Burst perception-distortion tradeoff: Analysis and evaluation.  In _ICASSP_ , 2023.
  * Zeyde et al. [2012] Roman Zeyde, Michael Elad, and Matan Protter.  On single image scale-up using sparse-representations.  In _International conference on curves and surfaces_ , 2012.
  * Zhang et al. [2018a] Hongyi Zhang, Moustapha Cisse, Yann N Dauphin, and David Lopez-Paz.  mixup: Beyond empirical risk minimization.  In _ICLR_ , 2018a.
  * Zhang et al. [2020] Kai Zhang, Luc Van Gool, and Radu Timofte.  Deep unfolding network for image super-resolution.  In _CVPR_ , 2020.
  * Zhang et al. [2021] Kai Zhang, Jingyun Liang, Luc Van Gool, and Radu Timofte.  Designing a practical degradation model for deep blind image super-resolution.  In _ICCV_ , 2021.
  * Zhang et al. [2022] Linjun Zhang, Zhun Deng, Kenji Kawaguchi, and James Zou.  When and how mixup improves calibration.  In _ICML_ , 2022.
  * Zhang et al. [2018b] Richard Zhang, Phillip Isola, Alexei A Efros, Eli Shechtman, and Oliver Wang.  The unreasonable effectiveness of deep features as a perceptual metric.  In _CVPR_ , 2018b.
  * Zhang et al. [2018c] Yulun Zhang, Kunpeng Li, Kai Li, Lichen Wang, Bineng Zhong, and Yun Fu.  Image super-resolution using very deep residual channel attention networks.  In _ECCV_ , 2018c.