# Black Hole shadows of $\alpha^{\prime}$-corrected black holes

F. Agurto-Sepúlveda1,2 J. Oliva1 M. Oyarzo1,3 D.R.G Schleicher2 1Departamento
de Física, Universidad de Concepción, Casilla, 160-C, Concepción, Chile
2Departamento de Astronomía, Universidad de Concepción, Casilla, 160-C,
Concepción, Chile.  3Department of Applied Science and Technology, Politecnico
di Torino, C.so Duca degli Abruzzi, 24, I-10129 Torino, Italy

###### Abstract

In this paper we study the qualitative features induces by corrections to GR
coming from String Theory, on the shadows of rotating black holes. We deal
with the slowly rotating black hole solutions up to order
$\mathcal{O}(a^{3})$, to first order in $\alpha^{\prime}$, including also the
dilaton. We provide a detailed characterization of the geometry, as well as
the ISCO and photon ring, and then we proceed to obtain the black hole images
within the relativistic thin-disk model. We characterize the images by
computing the diameter, displacement and asymmetry. A comparison with the Kerr
case, indicates that all these quantities grow due to the $\alpha^{\prime}$
correction, and that the departure from GR for different observable is
enhanced depending on the angle of view, namely for the diameter the maximum
departure is obtained when the system is face-on, while for the displacement
and asymmetry the departure from GR is maximized for edge-on point of view.

##  I Introduction

Over the last decades, the detection of compact objects such as black holes
has increased significantly, especially with new generations of
instrumentation for the observation and detection of black holes. These go
from interferometers that detect gravitational waves from black hole binaries
such as the LIGO-VIRGO collaboration [1], to the direct observation of these
objects with the Event Horizon Telescope (EHT). The latter has obtained the
first images of supermassive black holes (SMBH) at the center of the M87
galaxy [2], as well as at the center of our own galaxy with the image of the
Sgr A* SMBH [3]. These results define historical and crucial achievements in
testing the theory of General Relativity (GR) and serve as laboratories for
testing it in the strong field regime. More recent results include the study
of the relativistic jets of these systems, as well as the observation of
polarization emission due to the magnetized environment [4, 5, 6].

While GR has been able to describe the current observations to a high
precision, it is well-known that there are both theoretical and experimental
reasons that indicate the need of an extension of such framework, including
the failure to predict the accelerating expansion of the Universe, to
incorporate dark matter, as well as the incompatibility with quantum field
theory.

In this context, the inclusion of direct observation of SMBHs creates a great
opportunity to test and constrain the effects of modified gravity theories in
environments close them, which will even further improve with future
facilities like the ngEHT (Next Generation Event Horizon Telescope) 111The
ngEHT web: https://www.ngeht.org/ project, which aims at increasing the
baseline by incorporating satellites, thereby improving on the resolution
within the images, allowing for instance for very precise measurements of the
black hole spin [8].

It is currently accepted that GR has to be interpreted as an effective field
theory (EFT), emerging from a UV-complete framework, in the low energy limit.
Within such setup, corrections to GR are expected, which come suppressed by a
perturbative parameter. In the context of String Theory, such corrections come
multiplied by $\alpha^{\prime}$, namely by one over the string tension, and
are in general hard to compute. Even more, since String Theory is consistently
formulated in dimension ten, a compactification scheme is necessary to connect
the theory with four-dimensional physics. Generically, the four-dimensional
Einstein-Hilbert term, is supplemented by higher dimensional operators, which
perturbativelly modify the dynamics and the structure of the solutions.

The aim of this work is to establish the qualitative features that such terms
may imprint on the shadows of rotating black holes.

It is important to mention that the order of magnitude of $\alpha^{\prime}$ is
expected to be much lower than the values here considered, notwithstanding
this paper is devoted to study in a qualitative manner the signals induced by
the $\alpha^{\prime}$-corrections on the black hole images. This paper is
organized as follows: Section II introduces the theory containing both the
metric and the dilaton field, on the string frame, together with the
correction up to first order in $\alpha^{\prime}$. It also contains the field
equations that a perturbative solution may fulfill, by expanding around a
constant dilaton background and an Einstein geometry. Section III reviews the
construction of the $\alpha^{\prime}$-corrected, slowly rotating black hole up
to $\mathcal{O}(a^{3},a^{2}\alpha,\alpha^{2})$, presenting the expressions for
the temperature, entropy, angular velocity of the horizon and the first law of
black hole thermodynamics. Section IV contains the computation of the Petrov
type, the location of the event horizon and ergosphere, and an upper bound on
$\alpha$ that ensures the spacetime interpretation of the perturbative,
rotating solution. It also provides a detailed characterization of the
geodesics with particular emphasis on the expression for the Innermost Stable
Circular Orbits (ISCO) and the photon ring, that are of uttermost importance
for the computation of the shadows. Section V, introduces the numerical model
where the emission of the relativistic accretion disk is treated along the
thin-disk, Novikov-Thorne model. The section also contains a review of the
Ray-tracing code. Section VI presents the results of the simulations for the
black hole shadows, for different values of $\alpha^{\prime}$, focusing on the
signals induced on the diameter, displacement and asymmetry of the images,
induced by the stringy corrections. Section VII contains conclusion and
further comments.

##  II The theory

The theory we will consider emerges from the low-energy effective action of
String Theory including $\alpha^{\prime}$ corrections to the dilaton-graviton
sector. Neglecting order $\mathcal{O}(\alpha^{\prime 2})$ terms, we are able
to perform a suitable field redefinition such that the field equations are of
second order for both the metric and the dilaton, leading to the action
principle [9], [10] (see also [11] and [12])

$\displaystyle I[g_{\mu\nu},\phi]$
$\displaystyle=\int\mathrm{d}^{4}x\sqrt{-g}e^{-2\phi}[R+4(\partial\phi)^{2}$
$\displaystyle+\alpha(\mathcal{X}_{4}-16(\partial_{\mu}\phi\partial^{\mu}\phi)^{2}+\mathcal{O}(\alpha^{2})]\
,$ (1)

where the Euler density is defined as

$\displaystyle\mathcal{X}_{4}$
$\displaystyle=R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}-4R_{\mu\nu}R^{\mu\nu}+R^{2}$
(2)

providing second order equations in an explicit manner. Here the constant
$\alpha=\alpha^{\prime}/8$. See [13] for the precise dimensional reduction and
consistent truncation of String Theory leading to the action principle (1) and
[12] for the field redefinition that allows connecting both scenarios. The
field equations derived from (1) are

$\displaystyle R_{\mu\nu}+2\phi_{;\nu;\mu}+\alpha\mathcal{H}_{\mu\nu}$
$\displaystyle=0\,,$ (3) $\displaystyle
R-4(\partial\phi)^{2}+\alpha\mathcal{P}$ $\displaystyle=0\,,$ (4)

where semi-colon denotes covariant derivative and

$\displaystyle\mathcal{H}_{\mu\nu}$
$\displaystyle=-4\phi_{;\nu;\mu}(\partial\phi)^{2}+8\phi^{;\sigma}{}_{;\mu}\phi_{;\nu;\sigma}$
(5)
$\displaystyle-16\phi_{(;\mu}\phi_{;\nu);\sigma}\phi^{;\sigma}+2(\phi^{;\sigma;\lambda}-2\phi^{;\sigma}\phi^{;\lambda})R_{\mu\sigma\lambda\nu}$
$\displaystyle+g_{\mu\nu}[-56(\partial\phi)^{4}-12\phi_{;\lambda;\sigma}\phi^{;\lambda;\sigma}$
$\displaystyle-24\phi_{;\rho;\sigma}\phi^{;\rho}\phi^{;\sigma}+\frac{1}{2}R_{\lambda\delta\rho\sigma}R^{\lambda\delta\rho\sigma}]\,,$
$\displaystyle\mathcal{P}$
$\displaystyle=R_{\lambda\delta\rho\sigma}R^{\lambda\delta\rho\sigma}-16\phi_{;\mu;\nu}\phi^{;\mu;\nu}+$
(6)
$\displaystyle-64\phi^{;\rho;\sigma}\phi_{;\rho}\phi_{;\sigma}-96(\partial\phi)^{4}\,.$

Provided the fact that the action (1) is consistent neglecting
$\mathcal{O}(\alpha^{2})$, we are interested in studying consistent
$\alpha^{\prime}$–corrections of configurations $\overline{g}_{\mu\nu}$ and
$\overline{\phi}$ satisfying the equations (3) and (4) with $\alpha=0$.
Therefore, the general correction should be of the form

$\displaystyle g_{\mu\nu}$
$\displaystyle=\overline{g}_{\mu\nu}+\alpha\gamma_{\mu\nu}+\mathcal{O}(\alpha^{2})\,,$
(7) $\displaystyle\phi$
$\displaystyle=\overline{\phi}+\alpha\tau+\mathcal{O}(\alpha^{2})\,.$ (8)

where $\gamma_{\mu\nu}$ is a symmetric perturbative tensor and $\tau$ is a
scalar function of the spacetime coordinates. The situation in which
$\overline{\phi}=\phi_{0}$ is a constant is particularly interesting since the
equations for $\overline{g}_{\mu\nu}$ consistently reduce to the vacuum
Einstein equations, and we will focus in such case through out this work. It
is interesting to notice that due to the fact that the dilaton appears
linearly in the equation (3), then a consistent correction at order $\alpha$
to the metric is allowed. Consider the leading term of the dilaton being a
constant and the leading term of the metric a solution of the vacuum Einstein
equations, then a huge simplification of the equations (3) and (4) takes place
leading to

$\displaystyle\alpha\Pi_{\mu\nu}+2\alpha\overline{\nabla}_{\mu}\partial_{\nu}\tau+\frac{1}{2}\alpha\overline{g}_{\mu\nu}\overline{\mathcal{K}}$
$\displaystyle=0\,,$ (9)
$\displaystyle\alpha\overline{\Pi}_{\mu}{}^{\mu}+4\alpha\overline{\nabla}^{\mu}\partial_{\mu}\tau+\alpha\overline{\mathcal{K}}$
$\displaystyle=0\,.$ (10)

where

$\displaystyle\Pi_{\mu\nu}$
$\displaystyle=\overline{\nabla}^{\lambda}\overline{\nabla}_{(\mu}\gamma_{\nu)\lambda}-\frac{1}{2}\overline{\nabla}^{\lambda}\overline{\nabla}_{\lambda}\gamma_{\mu\nu}-\frac{1}{2}\overline{\nabla}_{\mu}\partial_{\nu}\gamma_{\lambda}{}^{\lambda}\,,$
$\displaystyle\overline{\mathcal{K}}$
$\displaystyle=R_{\lambda\delta\rho\sigma}R^{\lambda\delta\rho\sigma}|_{\overline{g}_{\mu\nu}}\,.$

The barred covariant derivatives are computed with the Levi-Civita connection
associated to the metric $\overline{g}_{\mu\nu}$. This equation can be
integrated in a variety of situations as was shown in [12] for accelerating
black holes and slowly rotating black holes. We will analyze in detail the
later situation in which the seed metric is the Kerr spacetime expanded up to
second order on the rotation parameter $a=GJ/M$.

##  III Slowly rotating $\alpha^{\prime}-$ corrected BH

The slowly rotating black hole solution found in [12] corresponds to a
consistent solution of the equations (9) and (10) for $\overline{g}_{\mu\nu}$
being the slowly rotating Kerr metric with rotation parameter $a$, expanded up
to $a^{2}$ neglecting terms $\mathcal{O}(a^{3})$. We find that it is possible
to integrate the equation even considering the logarithmic branch, controlled
by an integration constant $b$, in the $\alpha$-corrected static
configuration. An analytic solution of (9)-(10) at non-linear level in the
rotation parameter is still not known in the literature, nevertheless higher
order correction in $a$ could be obtained along the lines of [13]. The slowly
rotating spacetime, neglecting orders
$\mathcal{O}(a^{2}\alpha,a^{3},\alpha^{2})$, including the logarithmic branch
is given by

$\displaystyle g_{tt}$
$\displaystyle=-1+\frac{2M}{r}+\frac{2Ma^{2}\cos^{2}\theta}{r^{3}}-\alpha
f_{1}(r)\,,$ (11) $\displaystyle g_{t\phi}$
$\displaystyle=-\frac{2Ma\sin^{2}\theta}{r}+a\alpha h_{1}(r,\theta)\,,$ (12)
$\displaystyle g_{rr}$ $\displaystyle=\frac{1}{1-\frac{2M}{r}+\alpha
g_{1}(r)}-a^{2}\frac{(2M-r)\cos^{2}\theta+r}{(2M-r)^{2}r}\,,$ (13)
$\displaystyle g_{\theta\theta}$ $\displaystyle=r^{2}+a^{2}\cos\theta\,,$ (14)
$\displaystyle g_{\phi\phi}$
$\displaystyle=(r^{2}+a^{2}+\frac{2Ma^{2}\sin^{2}\theta}{r})\sin^{2}\theta\,,$
(15)

where

$\displaystyle f_{1}(r)$
$\displaystyle=\frac{8M^{2}}{r^{4}}+\frac{10M}{3r^{3}}+\frac{4}{r^{2}}-\frac{4}{Mr}-\frac{4bM}{r}+$
(16)
$\displaystyle+\frac{b(r-3M)}{r}\log\left[\left(1-\frac{2M}{r}\right)^{2}\right]$
$\displaystyle g_{1}(r)$
$\displaystyle=-\frac{40M^{2}}{3r^{4}}+\frac{2M}{r^{3}}+\frac{2}{r^{2}}+$ (17)
$\displaystyle-\frac{bM}{r}\log\left[\left(1-\frac{2M}{r}\right)^{2}\right]\,,$
$\displaystyle h_{1}(r,\theta)$
$\displaystyle=\left(\frac{8M^{2}}{r^{4}}+\frac{6M}{r^{3}}+\frac{6}{r^{2}}+\right.$
(18)
$\displaystyle\left.-\frac{3bM}{r}\log\left[\left(1-\frac{2M}{r}\right)^{2}\right]\right)\sin^{2}\theta\,.$
(19)

While the dilaton field is given by

$\displaystyle\phi$
$\displaystyle=\phi_{0}+\alpha\left(-\frac{4M}{3r^{3}}-\frac{1}{r^{2}}-\frac{1}{Mr}+\frac{b}{2}\log\left[\left(1-\frac{2M}{r}\right)^{2}\right]\right)$

In spite of the generalization including the logarithmic branch to the slowly
rotating solution, we will concentrate through out this article in the $b=0$
case. Using the Iyer-Wald method to compute conserved charges for the theory
(1) as was explained in [12] the mass and the angular momentum neglecting
$\mathcal{O}(a^{3},a^{2}\alpha)$ are respectively

$\displaystyle\mathcal{M}$ $\displaystyle=\frac{M}{G}+\frac{\alpha}{GM}\,,$
(20) $\displaystyle\mathcal{J}$ $\displaystyle=\frac{aM}{G}\,.$ (21)

The deviation tensor $\gamma_{\mu\nu}$ is present in the components
$g_{tt},g_{t\phi}$ and $g_{rr}$. Note also that the $\gamma_{\mu\nu}$ goes to
zero fast enough as $r\to\infty$ thus the spacetime is asymptotically flat,
and the Iyer-Wald charges converge. The Hawking temperature and entropy,
computed from the Wald formula [14], of the slowly rotating black hole (11)
with $b=0$ do not receive an $a\alpha$ correction and are given by

$\displaystyle T$ $\displaystyle=\frac{1}{8\pi M}-\frac{\alpha}{8\pi
M^{3}}-\frac{a^{2}}{32\pi M^{3}}\,,$ (22) $\displaystyle\mathcal{S}$
$\displaystyle=\frac{4M^{2}\pi}{G}+\frac{12\pi\alpha}{G}-\frac{\pi
a^{2}}{G}\,,$ (23)

where consistently with the scenario we are dealing with, we have neglected
terms $\mathcal{O}(a^{3},a^{2}\alpha)$. The angular velocity of the horizon of
the slowly rotating black hole is

$\displaystyle\Omega_{\mathrm{bh}}$
$\displaystyle=\frac{a}{4M^{2}}-\frac{3a\alpha}{4M^{4}}+\mathcal{O}(a^{3},a\alpha)\,.$
(24)

The first law of thermodynamics holds by considering the variation with
respect to the integration constants $M$ and $a$ and considering the variation
of the perturbative parameter $\delta a$ being of order $a$

$\displaystyle\delta\mathcal{M}$
$\displaystyle=T\delta\mathcal{S}+\Omega_{\mathrm{bh}}\delta\mathcal{J}\,,$
(25)

again neglecting $\mathcal{O}(a^{3},a^{2}\alpha,\alpha^{2})$.

##  IV Spacetime properties

In this section we will describe further properties of the slowly rotating
$\alpha^{\prime}$-corrected geometry, comparing them with the Kerr slowly
rotating case. These properties will be important for the simulations of the
black hole shadows.

###  IV.1 Petrov clasification

To determine the Petrov type of the slowly rotating spacetime we compute the
Weyl tensor in null tetrad basis. For that, we define the following
orthonormal basis in terms of the metric component (11)

$\displaystyle e^{0}$
$\displaystyle=\sqrt{-g_{tt}}\mathrm{d}t-\frac{g_{t\phi}}{\sqrt{-g_{tt}}}\mathrm{d}\phi\,,$
(26) $\displaystyle e^{1}$ $\displaystyle=\sqrt{g_{rr}}\mathrm{d}r\,,\quad
e^{2}=\sqrt{g_{\theta\theta}}\mathrm{d}\theta\,,$ $\displaystyle e^{3}$
$\displaystyle=\sqrt{g_{\phi\phi}-\frac{g_{t\phi}^{2}}{g_{tt}}}\mathrm{d}\phi\,,$

which must be expanded in $a$ and $\alpha$ neglecting
$\mathcal{O}(a^{3},\alpha^{2},a^{2}\alpha)$. In such frame the null tetrad is
defined by

$\displaystyle k_{\mu}$
$\displaystyle=\frac{1}{\sqrt{2}}(e^{0}{}_{\mu}+e^{3}{}_{\mu})\,,\,l_{\mu}=\frac{1}{\sqrt{2}}(e^{0}{}_{\mu}-e^{3}{}_{\mu})\,,$
(27) $\displaystyle m_{\mu}$
$\displaystyle=\frac{1}{\sqrt{2}}(e^{1}{}_{\mu}-ie^{2}{}_{\mu})\,,\,\overline{m}_{\mu}=\frac{1}{\sqrt{2}}(e^{1}{}_{\mu}+ie^{2}{}_{\mu})\,.$

The components of the Weyl tensor in the null basis are given by the
projections

$\displaystyle\Psi_{0}=C_{\mu\nu\rho\sigma}k^{\mu}m^{\nu}k^{\rho}m^{\sigma}\,,\,\Psi_{1}=C_{\mu\nu\rho\sigma}k^{\mu}l^{\nu}k^{\rho}m^{\sigma}\,,$
$\displaystyle\Psi_{2}=C_{\mu\nu\rho\sigma}k^{\mu}m^{\nu}\overline{m}^{\rho}l^{\sigma}\,,\,\Psi_{3}=C_{\mu\nu\rho\sigma}l^{\mu}k^{\nu}l^{\rho}\overline{m}^{\sigma}\,,$
$\displaystyle\Psi_{4}=C_{\mu\nu\rho\sigma}l^{\mu}\overline{m}^{\nu}l^{\rho}\overline{m}^{\sigma}\,.$
(28)

For the spacetime (11) the following expressions hold

$\displaystyle\Psi_{0}$
$\displaystyle=\mathcal{O}(a^{3},\alpha^{2},a^{2}\alpha)\,,$
$\displaystyle\Psi_{3}$
$\displaystyle=\mathcal{O}(a^{3},\alpha^{2},a^{2}\alpha)\,,$
$\displaystyle\Psi_{0}\Psi_{4}-9(\Psi_{2})^{2}$
$\displaystyle=\mathcal{O}(a^{3},\alpha^{2},a^{2}\alpha)\,.$

Indicating that the corrected spacetime is Petrov type D, as the seed
spacetime.

###  IV.2 Event horizon and ergosphere

In [12] the location of the event horizon for a non-rotating black hole was
calculated from the conditions $g^{rr}(r_{+})=g_{tt}(r_{+})=0$, assuming that
$r_{+}=2M+\alpha\mu_{1}$. Solving for $\mu_{1}$, the location of the horizon
was found at

$r_{+}=2M+\dfrac{\alpha}{6M}.$ (29)

Furthermore, the rotating case possesses a curvature singularity at $r=0$,
this can be inferred by analyzing the Kretschmann scalar

$\displaystyle R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}$ $\displaystyle=$
$\displaystyle\frac{48M^{2}}{r^{6}}-\frac{1008\cos^{2}\theta
M^{2}a^{2}}{r^{8}}$ $\displaystyle+$
$\displaystyle\frac{\left(-128M^{3}-96M^{2}r-96M\,r^{2}+96r^{3}\right)\alpha}{r^{9}},$

which diverges as $r\rightarrow 0$. The rotating spacetime also possesses an
event horizon, located at $r=r_{+}$ such that $g^{rr}\left(r_{+}\right)=0$ ,
which turns out to be equivalent to the condition

$g_{tt}g_{\phi\phi}-g_{t\phi}^{2}=0,$ (31)

Solving this equation using the location of the horizon in the Kerr metric and
with the ansatz $r_{+}^{\alpha^{\prime}}=r_{+}^{SK}+\alpha\mu_{1}$, we found
that the location of the event horizon for this spacetime is

$r_{+}=2M-\frac{a^{2}}{2M}+\frac{\alpha}{6M}+\mathcal{O}(a^{2}\alpha,\alpha^{2}).$
(32)

Using the condition $g_{tt}(r)=0$ to find the location of the ergosphere, one
finds

$r_{erg}^{\alpha^{\prime}}=2M-\frac{a^{2}}{2M}\cos^{2}\theta-\dfrac{\alpha}{12M^{2}}+\mathcal{O}(a^{2}\alpha,\alpha^{2}).$
(33)

Note that the $\alpha^{\prime}$-corrected ergosphere is smaller than the
ergosphere of Kerr, namely
$r^{\alpha^{\prime}}_{\mathrm{erg}}<r_{\mathrm{erg}}^{SK}$, while the location
of the event horizon of the $\alpha^{\prime}$-corrected geometry is larger
than the location of the Kerr event horizon.

###  IV.3 Lorentz signature

We will be exploring the geodesic orbits on this geometry, therefore, we
impose the preservation of the Lorentzian signature as a consistency
constraint on the perturbative construction. Since, the determinant of the
Slowly rotating Kerr metric is given by

$g^{SK}=-r^{2}\sin^{2}\theta\left(2a^{2}\cos^{2}\theta+r^{2}\right)\ ,$ (34)

the determinant of the spacetime defined by the metric functions (11) is

$\displaystyle
g^{\alpha^{\prime}}=g^{SK}+\frac{\alpha\left(32M^{2}r+18Mr^{2}+12r^{3}\right)\sin^{2}\theta}{3M}.$
(35)

As we remark, the $\alpha^{\prime}$-correction in the metric tensor goes to
zero at spatial infinity, thus the signature of the metric can change only in
the interior of the spacetime. In particular we are interested in the
signature of the metric in the near horizon region. Computing the quotient
between the slowly rotating Kerr determinant and the corrected metric
determinant we can find some restrictions for the possible values of the
$\alpha^{\prime}$ parameter

$\displaystyle\frac{g^{\alpha^{\prime}}}{g^{SK}}=1-\alpha\left(\frac{32M}{3r^{3}}+\frac{6}{r^{2}}+\frac{4}{Mr}\right)\,.$
(36)

To preserve the signature of the spacetime outside the horizon, namely $r\geq
r_{+}$, it is a necessary condition that the second term of (36) to be smaller
than 1. In particular, close to the horizon $r=r_{+}^{\alpha^{\prime}}$, the
signature should be preserved as well

$\frac{g^{\alpha^{\prime}}}{g^{SK}}(r_{+})=1-\frac{29\alpha}{6M^{2}}+\mathcal{O}(a^{2}\alpha,\alpha^{2}),$
(37)

where we have expanded around the rotation parameter and the coupling term.
This equation leads to the bound

$\alpha<\frac{6M^{2}}{29}+\mathcal{O}(a^{2}\alpha,\alpha^{2}),$ (38)

which is a consistent upper limit for the $\alpha$ coupling.

###  IV.4 Circular geodesic motion

The circular geodesic motion around compact objects for a stationary
axisymmetric space-time can be described with the equatorial approximation,
$|\theta-\frac{\pi}{2}|\ll 1$. Therefore, under this condition, the components
of the metric $g_{\mu\nu}$ depends only on the radial coordinate $r$. The
geodesic equation reads

$\frac{d^{2}x^{\alpha}}{d\tau^{2}}=-\Gamma^{\alpha}_{\beta}{}_{\gamma}\frac{dx^{\beta}}{d\tau}\frac{dx^{\gamma}}{d\tau},$
(39)

where $\Gamma^{\alpha}_{\beta}{}_{\gamma}$ are the Christoffel symbols and
$\tau$ is the affine parameter. Since the metric presents axi- and reflection
symmetry, for particles on circular orbits, the geodesic equation in the
radial direction, $\alpha=r$ in (39), reduces to

$\frac{1}{2}\partial_{r}g_{tt}\left(\frac{dt}{d\tau}\right)^{2}+\partial_{r}g_{t\phi}\frac{dt}{d\tau}\frac{d\phi}{d\tau}+\frac{1}{2}\partial_{r}g_{\phi\phi}\left(\frac{d\phi}{d\tau}\right)^{2}=0,$
(40)

where $\partial_{r}=\partial/\partial r$. From this equation, the Keplerian
frequency in the equatorial plane can be expressed as

$\displaystyle\Omega_{\phi}$ $\displaystyle=$
$\displaystyle\frac{p^{\phi}}{p^{t}}=\frac{-\partial_{r}g_{t\phi}\pm\sqrt{(\partial_{r}g_{t\phi})^{2}-\partial_{r}g_{tt}\partial_{r}g_{\phi\phi}}}{\partial_{r}g_{\phi\phi}},$
(41) $\displaystyle p^{\phi}$ $\displaystyle\equiv$
$\displaystyle\dot{\phi}\,,\quad p^{t}\equiv\dot{t}\,,$ (42)

where the upper sign refers to prograde orbits and the lower sign to
retrograde orbits, depending on the value of the spin parameter $a$.

Since the metric $g_{\mu\nu}$ is stationary and axisymmetric (i.e independent
of the $t$ and $\phi$ coordinates), it has a timelike and an azimuthal Killing
vector, leading to the existence of two conseved quantities, the specific
energy $\tilde{E}$ and the specific angular momentum $\tilde{L}$. Those
correspond to the $p_{t}=-\tilde{E}$ and $p_{\phi}=\tilde{L}$ components of
the four-momentum, from which the following geodesic equations can be derived,

$\dot{t}=\frac{\tilde{E}g_{\phi\phi}+\tilde{L}g_{t\phi}}{g_{t\phi}^{2}-g_{tt}g_{\phi\phi}},\\\
$ (43)
$\dot{\phi}=-\frac{\tilde{E}g_{t\phi}+\tilde{L}g_{tt}}{g_{t\phi}^{2}-g_{tt}g_{\phi\phi}},$
(44)

where $\dot{t}=dt/d\tau$ and $\dot{\phi}=d\phi/d\tau$.

The four momentum is normalized as

$g_{\mu\nu}p^{\mu}p^{\nu}=p^{\mu}p_{\mu}=-\epsilon^{2},$ (45)

where $\epsilon^{2}=+1,-1,$ for time-like and space-like geodesics,
respectively. As usual, it is possible to obtain the expression for the
effective potential in the equatorial plane as follows

$g_{rr}\dot{r}^{2}=V_{eff}(r)\ ,$ (46)

where the effective potential is given by

$V_{eff}(r)=-\epsilon^{2}+\frac{\tilde{E}^{2}g_{\phi\phi}+2\tilde{E}\tilde{L}g_{t\phi}+\tilde{L}^{2}g_{tt}}{g_{t\phi}^{2}-g_{tt}g_{\phi\phi}}\
.$ (47)

Our primary focus is on circular orbits confined to the equatorial plane for
massive, timelike particles ($\epsilon^{2}=1$), for which the orbits are
governed by

$V_{eff}(r)=0,$ (48) $\frac{\partial}{\partial r}V_{eff}(r)=0.$ (49)

Using these conditions it is possible to obtain the expression for the
specific energy $\tilde{E}$ and angular momentum $\tilde{L}$ in a circular
orbit

$\tilde{E}=\frac{-g_{tt}-g_{t\phi}\Omega_{\phi}}{\sqrt{-g_{tt}-2g_{t\phi}\Omega-
g_{\phi\phi}\Omega^{2}}},$ (50)
$\tilde{L}=\frac{g_{t\phi}+g_{\phi\phi}\Omega_{\phi}}{\sqrt{-g_{tt}-2g_{t\phi}\Omega-
g_{\phi\phi}\Omega^{2}}}.$ (51)

On the other hand, since one of our goals is to obtain simulations of a thin
accretion disk around a black hole described by this solution, we must
determine the location of ISCO. It is possible to find it from the condition
$\partial^{2}V_{eff}(r)/\partial r^{2}=0$, which leaves us with the following
expression

$\tilde{E}^{2}\partial_{r}^{2}g_{\phi\phi}+2\tilde{E}\tilde{L}\partial_{r}^{2}g_{t\phi}+\tilde{L}^{2}\partial_{r}^{2}g_{tt}-\partial_{r}^{2}(g_{t\phi}^{2}-g_{tt}g_{\phi\phi})=0.$
(52)

Since the metric given by Eq. (11) satisfies all the characteristics
previously mentioned, it is possible to obtain the Keplerian frequency from
Eq. (53) in $\mathcal{O}(\alpha)$, leading to

$\displaystyle\Omega_{\phi}$ $\displaystyle=$
$\displaystyle-\frac{Ma}{r^{3}}\pm\sqrt{\frac{M}{r}}\frac{\left(M^{3}a^{2}+M^{2}r^{3}\right)}{r^{4}M^{2}}$
(53) $\displaystyle\pm$
$\displaystyle\frac{\alpha\left(-8M^{3}-\frac{5}{2}r\,M^{2}-2M\,r^{2}+r^{3}\right)}{r^{\frac{9}{2}}M^{\frac{3}{2}}}$
$\displaystyle+$
$\displaystyle\frac{a\alpha\left(32M^{2}+18Mr+12r^{2}\right)}{2r^{6}}\ .$

The expressions for the specific energy and angular momentum, from Eqs.
(50-51) at $\mathcal{O}(\alpha)$ are

$\displaystyle\tilde{E}=$
$\displaystyle\frac{r-2M}{r^{1/2}(r-3M)^{1/2}}+\frac{a^{2}M^{2}}{2\sqrt{r}(r-3M)^{5/2}}$
(54) $\displaystyle\mp\frac{aM^{3/2}}{r(r-3M)^{3/2}}$
$\displaystyle-\alpha\frac{(34M^{3}+29M^{2}r-36Mr^{2}+6r^{3})}{6Mr^{5/2}(r-3M)^{3/2}}$
$\displaystyle\mp\frac{\alpha
a}{6\sqrt{M}r^{4}(r-3M)^{5/2}}(120M^{4}-16M^{3}r+$
$\displaystyle+15M^{2}r^{2}-12Mr^{3}+6r^{4})$

and

$\displaystyle\tilde{L}=$
$\displaystyle\pm\frac{rM^{1/2}}{(r-3M)^{1/2}}-a\frac{3M(r-2M)}{\sqrt{r}(r-3M)^{3/2}}+$
(55) $\displaystyle\pm\frac{a^{2}M^{1/2}}{2r(r-3M)^{5/2}}(6M^{2}-5Mr+2r^{2})+$
$\displaystyle+\frac{\alpha a}{2r^{5/2}(r-3M)^{5/2}}(18M^{3}+25M^{2}r$
$\displaystyle-6Mr^{2}-6r^{3})+$
$\displaystyle\pm\frac{\alpha}{6r^{2}M^{3/2}(r-3M)^{3/2}}(72M^{4}-28M^{3}r$
$\displaystyle-3M^{2}r^{2}-12Mr^{3}+6r^{4})$

where the upper and lower sign correspond to prograde and retrograde orbits,
respectively.

And the effective potential, according to Eq. (47) is

$\displaystyle V_{eff}(r)=$
$\displaystyle\left(r^{3}+\left(2M+r\right)a^{2}\right)E^{2}$ (56)
$\displaystyle+$
$\displaystyle\left(\frac{\left(32M^{2}r^{3}+18r^{4}M+12r^{5}\right)\alpha}{3M\,r^{3}}\right)E^{2}$
$\displaystyle+$
$\displaystyle\left(-\frac{4\left(20M^{2}+9Mr+3r^{2}\right)a\alpha}{3r^{3}}-4Ma\right)LE$
$\displaystyle+$
$\displaystyle\left(-r+2M+\frac{\left(40M^{3}-6r\,M^{2}-6r^{2}M\right)\alpha}{3M\,r^{3}}\right)L^{2}$
$\displaystyle-$ $\displaystyle
r^{3}+2r^{2}M-a^{2}r+\frac{\left(40M^{3}r^{2}-6M^{2}r^{3}-6r^{4}M\right)\alpha}{3M\,r^{3}}.$

Consequently, from Eq. (52) the expression for the ISCO is

$\displaystyle
r_{isco}^{\alpha^{\prime}}=r_{isco}^{SK}+\alpha\,\frac{163}{54M}-a\alpha\frac{362\,\sqrt{6}}{243M^{2}}+\mathcal{O}(a^{2}\alpha,\alpha^{2})\,,$
(57)

where

$r_{isco}^{SK}=M\left(6-\frac{4\sqrt{6}a}{3M}-\frac{7a^{2}}{18M^{2}}\right)+\mathcal{O}(a^{3}).$
(58)

We use the Kretschmann scalar to define an upper limit for the rotation
parameter $a$ to study the motion of test particles and the accretion disks
around slowly rotating space-times. Figure 1 shows the ratio between the
Kretschmann scalar for the slowly rotating approximation,

$\mathcal{K}_{SK}=\frac{48M^{2}}{r^{6}}-\frac{1008\cos^{2}\theta
M^{2}a^{2}}{r^{8}}+\mathcal{O}(a^{3}),$ (59)

and the Kretschmann scalar for the regular Kerr solution,

$\displaystyle\mathcal{K}$ $\displaystyle=$
$\displaystyle\frac{-48M^{2}}{\left(r^{2}+a^{2}\cos^{2}\theta\right)^{6}}\Big{(}a^{6}\cos^{6}\theta-15\cos^{4}\theta
a^{4}r^{2}$ (60) $\displaystyle+$ $\displaystyle 15\cos^{2}\theta
a^{2}r^{4}-r^{6}\Big{)}.$

In Figure 1, the ratio of Kretchmann scalars near the ISCO radius starts
departing significantly from one for values around $a\approx 0.3M$, in
consequence we restrict our attention to values of the spin parameter
$a\lesssim 0.3M$.

Figure 1: Ratio between the Kretschmann scalar in the slowly rotating Kerr
metric and the standard Kerr metric, for different values of the spin
parameter $a$, where vertical lines are the location of the ISCO for each $a$.

In Figure 2 we show the specific energy, angular momentum, Keplerian frequency
and the effective potential from equations (53)-(56) for different values of
the coupling $\alpha$. It can be noticed that as the value of $\alpha$
increases and approaches its upper limit, clearer differences appear in the
Keplerian frequency, energy and specific angular momentum. As a test particle
approaches the event horizon, when looking at the plots of the energy and
specific angular momentum for values of $\alpha$ larger than $0.05$ it seems
that an unstable circular orbit emerges near the horizon within the ISCO, as
well as a more intricate behaviour in the Keplerian frequency.

Figure 2: The upper-left panel shows the specific energy $\tilde{E}(r)$, the
upper-right panel the specific angular momentum, the bottom-left panel the
Keplerian frequency and the bottom-right panel shows the effective potential
for the values of $E=1$ and $L=4.5$, for the $\alpha^{\prime}$-corrected BH
with a mass of $M=6.2\times 10^{9}M_{\odot}$ and different values of $\alpha$
from $0$ to the upper limit $\sim 2$ in equation (38).

###  IV.5 Photon rings of slowly rotating $\alpha^{\prime}$-corrected black
hole

We address now the problem of the separability of the Hamilton-Jacobi equation
for null geodesics, namely

$\displaystyle g^{\mu\nu}\frac{\partial S}{\partial x^{\mu}}\frac{\partial
S}{\partial x^{\nu}}$ $\displaystyle=0\,.$ (61)

We assume a separable ansatz for the Jacobi characteristic function

$\displaystyle S$ $\displaystyle=-Et+L\phi+W(r)+P(\theta)\ .$ (62)

Interestingly, equation (61) is compatible with the separable ansatz, leading
to the following two equations

$\displaystyle\left(\frac{\mathrm{d}P}{\mathrm{d}\theta}\right)^{2}$
$\displaystyle+\frac{\xi^{2}}{\sin^{2}\theta}-a^{2}\cos^{2}\theta-\xi^{2}-\eta=0\,,$
(63)

and

$\displaystyle-\tilde{\Delta}\left(\frac{\mathrm{d}W(r)}{\mathrm{d}r}\right)^{2}+\frac{(\xi^{2}-4M^{2})r-2M\xi^{2}}{r(2M-r)^{2}}a^{2}+$
$\displaystyle+\left(\frac{4\xi(4M^{2}+3Mr-3r^{2})}{3r^{2}(2M-r)^{2}}\alpha-\frac{4M\xi}{2M-r}\right)a+$
$\displaystyle-\frac{2(12M^{3}+5M^{2}r+6r^{2}M-6r^{3})}{3(2M-r)^{2}M}\alpha+$
$\displaystyle+\frac{(r-2M)\xi^{2}-r^{3}}{2M-r}-\eta$ $\displaystyle=0$ (64)

where $\eta$ is the separation constant and we have defined
$\xi\equiv\tilde{E}/L$ normalizing all the quantities respect to $\tilde{E}$.
Without loss of generality, we fix $\tilde{E}=1$, and define

$\displaystyle\tilde{\Delta}$ $\displaystyle\equiv
r^{2}-2Mr+a^{2}+\frac{-40M^{2}+6Mr+6r^{2}}{3r^{2}}\alpha\,.$ (65)

From here we define the functions

$\displaystyle R(r)$
$\displaystyle\equiv\frac{\tilde{\Delta}^{2}}{\tilde{E}^{2}}\left(\frac{\mathrm{d}W(r)}{\mathrm{d}r}\right)^{2}\,,$
(66) $\displaystyle\Theta(\theta)$
$\displaystyle\equiv\frac{1}{\tilde{E}^{2}}\left(\frac{\mathrm{d}P(\theta)}{\mathrm{d}\theta}\right)^{2}\,.$
(67)

Following these definitions the Jacobi characteristic functions is written as

$\displaystyle S$
$\displaystyle=-t+\xi\phi+\int_{r_{0}}^{r}\mathrm{d}r\frac{\sqrt{R(r)}}{\tilde{\Delta}}+\int_{\theta_{0}}^{\theta}\mathrm{d}\theta\sqrt{\Theta(\theta)}$
(68)

and the first order equations for the non-trivial coordinates reduce to

$\displaystyle g_{rr}^{2}\dot{r}^{2}$
$\displaystyle=\frac{R(r)}{\tilde{\Delta}^{2}}\,,$ (69) $\displaystyle
g_{\theta\theta}^{2}\dot{\theta}^{2}$ $\displaystyle=\Theta(\theta)$ (70)

Then it is possible to find conditions for the existence of circular orbits
without restricting to the equatorial plane, by imposing the following two
conditions for all $r$

$\displaystyle R(r)$ $\displaystyle=0\,,$ (71)
$\displaystyle\frac{\mathrm{d}R(r)}{\mathrm{d}r}$ $\displaystyle=0\,.$ (72)

These equations can be solved in terms of the constants $\xi$ and $\eta$
giving two families of solutions. Considering the branch that connects with
the Kerr solution perturbatively, we have

$\displaystyle\xi$
$\displaystyle=a\frac{M+r}{M-r}+\frac{r^{3}-3r^{2}}{a(M-r)}+$ (73)
$\displaystyle+\frac{a\alpha}{6r^{4}(2M-r)^{2}(M-r)M}(-240M^{7}+592M^{6}r+$
$\displaystyle+62M^{5}r^{2}-347M^{4}r^{3}+44M^{3}r^{4}+25M^{2}r^{5}+24Mr^{6}-12r^{7})+$
$\displaystyle+\frac{\alpha}{a}\frac{1}{3r(M-r)^{2}(2M-r)M}(-336M^{5}+327M^{4}r+$
$\displaystyle-66M^{3}r^{2}+16M^{2}r^{3}-18r^{4}M+3r^{5})$

and

$\displaystyle\eta$
$\displaystyle=\frac{4Mr^{3}}{(M-r)^{2}}-\frac{(3M-r)^{2}r^{4}}{(M-r)^{2}a^{2}}+\frac{\alpha}{3r^{2}(M-r)^{3}(2M-r)^{2}}\times$
$\displaystyle\times(-720M^{7}+2016M^{6}r-118M^{5}r^{2}-1619M^{4}r^{3}$
$\displaystyle+781M^{3}r^{4}-47M^{2}r^{5}+21Mr^{6}-18r^{7})+$
$\displaystyle-\frac{\alpha}{a^{2}}\frac{2r(r-3M)}{(r-2M)(r-M)^{3}}(r^{5}-6r^{4}M+\frac{16}{3}M^{2}r^{3}+$
$\displaystyle-22M^{3}r^{2}+109M^{4}r-112M^{5}).$ (74)

The impact parameters $x^{\prime}$ and $y^{\prime}$ defined as in Bardeen
[15], are the usual Cartesian coordinates of the image plane for a observer
located at $r=r_{0}$ coordinate distance and with inclination angle of view
$\theta=i$. In the case of $r_{0}\rightarrow\infty$, the expression for the
impact parameters are

$\displaystyle x^{\prime}$ $\displaystyle=$ $\displaystyle-\frac{\xi}{\sin
i},$ (75) $\displaystyle y^{\prime}$ $\displaystyle=$
$\displaystyle\pm\sqrt{\Theta(i)},$ (76)

where $\Theta(i)=a^{2}\cos^{2}i-\xi^{2}\cot i+\eta$.

The shape of the photon rings in the usual Kerr metric has been already
studied previously by several authors Bardeen [15], Luminet [16], Takahashi
[17], Beckwith and Done [18], Amarilla  _et al._ [19], Johannsen [20], Giribet
_et al._ [21]. Using the previous equations it is possible to obtain images of
the photon rings in the slowly rotating $\alpha^{\prime}$-corrected metric
(see Figure 3) in order to observe the effects of $\alpha$ in the shape of the
photon ring and to compare it with the Kerr case.

Figure 3: Photon rings around the $\alpha^{\prime}$-corrected black holes at
an inclination angle of $\theta=i=0^{o}$ (left panel) and $i=90^{o}$ (right
panel), where the (red line) represents a black hole with spin parameter
$a=0.2M$ in the Kerr approximation. The (orange dashed line to blue dashed
line) correspond to black holes with the coupling
$\alpha=[0.04,0.08,0.12,0.16,0.2]$, respectively, in units of the
gravitational radius squared $r_{g}^{2}$.

##  V Numerical model

In this section, we describe our numerical model, including the assumptions
for the structure of the thin disk as well as the ray-tracing code RAPTOR I
[22].

###  V.1 Thin-disk model

To analyze and obtain the shadow of the black hole, we assume a thin accretion
disk, using the standard relativistic thin-disk model from Novikov and Thorne
[23]. This model describes a geometrically thin disk with steady and axially
symmetric states of the matter around the black hole. The distribution of the
matter starts at the location of the ISCO radius and extends to some very
large radius $r_{out}$, forcing the particles to move in nearly circular
geodesics in the equatorial plane along the disk.

From the equations of conservation of energy, rest mass, and angular momentum
in the disk paticles [23], the radiation flux emitted by the surface of the
disk follows as

$F(r)=\frac{\dot{M}}{4\pi M^{2}}f_{disk}(r),$ (77)

where $\dot{M}$ is the mass accretion rate and

$f_{disk}(r)=-\dfrac{d\Omega}{dr}\dfrac{M^{2}}{\sqrt{-g}(\tilde{E}-\Omega\tilde{L})^{2}}\int_{r_{in}}^{r}(\tilde{E}-\Omega\tilde{L})\dfrac{d\tilde{L}}{d\overline{r}}d\overline{r}$
(78)

is a dimensionless function of the radius, $\Omega$ is the Keplerian
frequency, $\tilde{E}$ and $\tilde{L}$ are the specific energy and angular
momentum, $r_{in}$ is the location of the ISCO radius and $g$ is the
determinant of the metric in cylindrical coordinates $(t,r,z,\phi)$. In the
case of SK we have $\sqrt{-g}=r$ and for the metric defined in Eq. (11), one
has

$\sqrt{-g}=\sqrt{r^{2}-\alpha\left(\frac{32M^{2}+18Mr+12r^{2}}{3Mr}\right)}.$
(79)

Figure 4 shows the flux for this space-time compared to the slowly rotating
Kerr case, where it can be seen that as $\alpha$ grows the flux moves slightly
outwards from the black hole.

Figure 4: The energy flux of a BH with a rotation parameter $a=0.2M$, mass of
$M_{BH}=6.2\times 10^{9}M_{\odot}$ and mass accretion rate
$\dot{M}=1.172\times 10^{23}[g/s]$ [24] for different values of $\alpha$,
where vertical lines represents the ISCO value for each $\alpha$.

Since the radiative description of this model is defined by thermal
equilibrium black-body emission due to its simplicity and effectiveness, the
flux is related to the disk temperature according to the standard Stefan-
Boltzmann law as

$F(r)=\sigma T_{eff}(r)^{4},$ (80)

where $\sigma$ is the Stefan-Blotzmann constant and
$T_{eff}(r)=f_{col}^{-1}T_{col}(r)$. Here the specific intensity can be
defined as

$I_{\nu}=\frac{1}{f_{col}^{4}}B_{\nu}(\nu,T_{col}),$ (81)

where $f_{col}$ is the spectral hardening factor and

$B_{\nu}(\nu,T_{col})=\dfrac{2h\nu^{3}}{c^{2}}\dfrac{1}{\exp(h\nu/k_{B}T_{col})-1}$
(82)

is the Planck distribution for a blackbody spectrum. Here $\nu$ is the
frequency of emission and $h$ and $k_{B}$ are the Planck and Boltzmann
constants, respectively.

###  V.2 Ray-tracing code RAPTOR I

To obtain images of the BH shadow in the slowly rotating $\alpha^{\prime}$
space-time, we employ a modified version of the open-source ray tracing code
RAPTOR I [22], which is capable to work in arbitrary space-times. The code
calculates the null geodesic around the BH and solves the relativistic
radiative transfer equation along the geodesics. To use it for obtaining
images of BHs in this specific space-time, we modified the metric of the code,
the connection and also included the thin accretion disk model [23] with the
specific changes of the different physical quantities such as the location of
the ISCO, the velocity of the disk and the flux emitted according to this
model as we discussed in section V.1 for this space-time.

The code starts creating a virtual camera at a distant observer position to
define the initial conditions for the light rays; this is performed following
the general approach from Bambi [25].

Therefore, RAPTOR I integrates the geodesics along the path of the rays which
interacts with the structure of the disk that extends from the ISCO radius
(57) to $r_{out}=1000r_{g}$. In a final step the code employs the calculation
of the radiative transfer equation at the observer’s location as

$I_{\nu,obs}=\overline{g}^{3}I_{\nu},$ (83)

where $I_{\nu}$ is the intensity emitted from Eq. (81) and $\overline{g}$ is
the redshift factor

$\overline{g}=\frac{\nu_{obs}}{\nu},$ (84)

where $\nu_{obs}$ is the frequency at observer frame and

$\nu=k^{\mu}u_{\mu}$ (85)

is the frequency in the plasma frame, $k^{\mu}$ is the contravariant wave
vector and $u_{\mu}$ is the covariant four-velocity.

##  VI Black hole simulations and shadow characterization

In the following, we present our simulation results with an analysis of the
diameter, displacement and asymmetry of the obtained images.

###  VI.1 Diameter, displacement and asymmetry

To characterize the shadow of the different black holes in order to compare
them, we employ the method used by [20, 26] that consists in the study of the
peaks of the intensity profile in each image of the black hole to calculate
the displacement, diameter and asymmetry through the following expressions.
The displacement is defined as

$D\equiv\frac{|x_{max}+x_{min}|}{2},$ (86)

where $x_{max}$ and $x_{min}$ are the locations of the two peaks in the
horizontal intensity profile. Analogously, it is possible to define the
displacement in the vertical direction. Then the average radius is defined by

$\langle\overline{R}\rangle\equiv\dfrac{1}{2\pi}\int_{0}^{2\pi}\overline{R}d\alpha,$
(87)

where
$\overline{R}\equiv\sqrt{(x^{\prime}-D_{x})^{2}+(y^{\prime}-D_{y})^{2}}$. Thus
the diameter is $L=2\langle\overline{R}\rangle$.

Consequently, we get the following expression for the asymmetry:

$A\equiv
2\sqrt{\dfrac{\int_{0}^{2\pi}\left(\overline{R}-\langle\overline{R}\rangle\right)^{2}d\alpha}{2\pi}}.$
(88)

###  VI.2 Simulations

We performed simulations with RAPTOR I of an $\alpha^{\prime}$-corrected BH
that are shown in Figure 5, with mass similar to M87*, for a thin accretion
disk model. The settings of the simulations are shown in Table 1.

Table 1: Setup for the simulation, values extracted from Akiyama  _et al._ [24] Variable |  Value
---|---
Mass |  $6.2\times 10^{9}$ $\mathrm{M_{\odot}}$
Distance |  $16.9$ $\mathrm{Mpc}$
$a$ | $0.2\ M$
$r_{\text{camera}}$ | $10^{4}\ \mathrm{r_{g}}$
Range for $(x^{\prime}$, $y^{\prime})$ | $[-20,20]\ \mathrm{r_{g}}$
Resolution $(x,y)$ |  $200$ $\mathrm{px}$
Frequency |  $230$ $\mathrm{GHz}$
Inclination (°) | $[0,30,60,90]$
$\dot{M}$ | $1.71\times 10^{23}\mathrm{g/s}$
$\alpha^{\prime}$ | $[0,0.1,0.15,0.2]$

To compare the shadow and disk structure of this BH with the usual Kerr BH in
the same physical conditions, we performed a similar analysis as Agurto-
Sepulveda  _et al._ [26], calculating the intensity profile of each image, for
different viewing angles from face-on to edge-on and then we obtained the
values of the diameter, displacement and asymmetry of the shadow from
equations (86-88) for each simulation.

A comparison is shown in Figure 6, where one can see that an increasing value
of $\alpha^{\prime}$ leads to an increase of the diameter of the image
compared to Kerr while it decreases when going from a face-on to an edge-on
viewing angle. Here, it is also important to notice that the Kerr case and the
slowly rotating Kerr case also differ in the diameter for a few values of
inclination. In the case of the displacement, we can see a similar situation,
where the displacement of the shadow increases with the value of
$\alpha^{\prime}$, and decreases abruptly for an edge-on viewing angle. Here,
the difference between the Kerr case and the $\alpha^{\prime}$-corrected case
is also clearly visible. Finally, the asymmetry values do not differ as much
as their predecessors and are quite close to the Kerr case, yet for increasing
$\alpha^{\prime}$ differences can be noticed for the first lower half of the
viewing angles and stay close to Kerr’s case for the top half.

Figure 5: Simulation performed by Raptor I code of the Novikov-Thorne thin
accretion disk model for a slowly rotating $\alpha^{\prime}$-corrected SMBH
with a mass of $M=6.2\times 10^{9}M_{\odot}$, a mass accretion rate of
$\dot{M}=1.172\times 10^{23}[g/s]$, a spin parameter $a=0.2M$ and couplings
$\alpha=[0.1,0.15,0.2]$, in units of the $r_{g}^{2}$. Figure 6: Comparison of
the diameter, displacement and asymmetry values, for simulations of Kerr
space-time (red dots), slowly rotating Kerr approximation (orange triangles)
and $\alpha^{\prime}$-corrected BH with different values of the coupling term
$\alpha=[0.1,0.15,0.2]\ r_{g}^{2}$ (cyan stars, blue diamonds and darkcyan
plus sign, respectively).

###  VI.3 Effects of $\alpha^{\prime}$-corrected terms in the shadow

One way to observe possible signals of the deviations from the Kerr shadow due
to the $\alpha^{\prime}$-corrected terms, is setting the structure of the disk
from a given value of ISCO radius $r_{isco}$ to a big outer radius $r_{out}$
for the two different space-times in the simulations.

As we have seen before, increasing the value of $\alpha^{\prime}$ correspond
to an increase in the diameter and displacement of the BH shadow. Since
$\alpha^{\prime}$ stands for the inverse of the tension of the string, it must
be positive. Therefore, to have a same value for the ISCO radius with
different values of $\alpha^{\prime}$ and spin parameter $a$ we proceed as
follows: we fix the ISCO radius as $r_{isco}=7.31r_{g}$, which in Kerr metric
yields a value of spin parameter $a=-0.42M$, while in the slowly rotating
$\alpha^{\prime}$ metric yields a spin parameter $a=-0.3M$ and the coupling
value of $\alpha=0.189\ r_{g}^{2}$.

Figure 7 shows the difference between the two images of a Kerr black hole
(left panel) and the slowly rotating $\alpha^{\prime}$-corrected black hole
(right panel) with the same location of the ISCO corresponding to the
previously described setting. A careful look at the images shows a slight
difference in the shadow size, being larger in the case of
$\alpha^{\prime}-$corrected space-time, as well as a slight difference in the
intensity of the images.

In Figure 8 we show the differences in the diameter, displacement and
asymmetry values for different angles of view for the same configuration,
including also a comparison with the slowly rotating Kerr spacetime. There, it
is possible to notice a greater difference between the images. The
$\alpha^{\prime}$-corrected BH shows a bigger diameter compared to the Kerr
BH, for an angle of view near face-on, while the diameters get closer as one
approaches the edge angle. In the case of the displacement, it is clearly seen
that for increasing angle of view the displacement also increase for the
$\alpha^{\prime}$-corrected BH, when compared to the Kerr BH. In the case of
asymmetry, the difference between the two images seems to be less clear, a
slight difference may be visible for an angle of view closer to the edge-on,
but still the values seem to be close to each other.

Figure 7: Comparison between two simulations with the similar configuration of
Table 1 for a BH in a Kerr space-time (left panel) with a spin parameter
$a=-0.42M$ and inclination angle of view $\theta=[0,80]$ degrees and a slowly
rotating $\alpha^{\prime}$-corrected BH (right panel) with different values of
the coupling term $\alpha=0.189\ r_{g}^{2}$ and spin parameter $a=-0.3M$.
Figure 8: Comparison of the diameter, displacement and asymmetry values, for
simulations of BHs with the same ISCO radius in the Kerr space-time (red dots)
and slowly rotating Kerr approximation (orange triangles) with a retrograde
spin parameter $a=-0.42M$ and $\alpha^{\prime}$-corrected BH with a spin
parameter $a=-0.3M$ and $\alpha^{\prime}=0.189$.

Table 2 shows several simulations with the same approach, fixing the ISCO
radius, which were performed to observe the effect of $\alpha^{\prime}$ for
different values of spin $a$ and the same angle of view of $75^{o}$.

Table 2: Comparison between the diameter, displacement and asymmetry for a set of simulations with the same ISCO radius and different values of the spin parameter $a$ and the coupling $\alpha^{\prime}$. $r_{isco}\ [\mathrm{r_{g}}]$ |  $a\ [\mathrm{M}]$ | $\alpha^{\prime}[\mathrm{r_{g}^{2}}]$ |  $L\ [\mathrm{r_{g}}]$ |  $D\ [\mathrm{r_{g}}]$ |  $A\ [-]$ |  $\theta\ [{}^{o}$]
---|---|---|---|---|---|---
| $-0.05$ | $0$ | $15.389$ | $1.158$ | $7.121$ |
$6.162$ | $0.05$ | $0.102$ | $16.481$ | $1.461$ | $7.555$ | $75$
| $0.1$ | $0.145$ | $16.816$ | $1.561$ | $7.753$ |
| $0.15$ | $0.185$ | $17.105$ | $1.561$ | $7.857$ |
| $-0.1$ | $0$ | $15.642$ | $1.057$ | $7.167$ |
$6.322$ | $-0.05$ | $0.056$ | $16.375$ | $1.360$ | $7.472$ | $75$
| $0.05$ | $0.152$ | $17.146$ | $1.561$ | $7.814$ |
| $0.1$ | $0.193$ | $16.415$ | $1.662$ | $8.581$ |
| $-0.15$ | $0$ | $15.883$ | $1.158$ | $7.258$ |
$6.481$ | $-0.1$ | $0.059$ | $16.705$ | $1.360$ | $7.583$ | $75$
| $-0.05$ | $0.112$ | $17.151$ | $1.461$ | $7.774$ |
| $0.05$ | $0.201$ | $17.647$ | $5.088$ | $9.524$ |
| $-0.2$ | $0$ | $16.143$ | $1.158$ | $7.358$ |
$6.639$ | $-0.15$ | $0.063$ | $17.031$ | $1.360$ | $7.680$ | $75$
| $-0.1$ | $0.119$ | $17.419$ | $1.461$ | $7.855$ |
| $-0.05$ | $0.168$ | $17.733$ | $1.662$ | $8.035$ |
| $-0.3$ | $0$ | $16.659$ | $1.158$ | $7.502$ |
$6.949$ | $-0.25$ | $0.074$ | $17.590$ | $1.360$ | $7.849$ | $75$
| $-0.2$ | $0.136$ | $18.039$ | $1.662$ | $8.054$ |
| $-0.15$ | $0.189$ | $18.459$ | $1.662$ | $8.268$ |
| $-0.4$ | $0$ | $17.153$ | $1.158$ | $7.639$ |
$7.254$ | $-0.3$ | $0.160$ | $18.773$ | $1.561$ | $8.268$ | $75$

Our results indicate that there are particular observable quantities, for
which the differences on the BH image, for a given ISCO, are enhanced and
therefore one could look for such observables to obtain a greater difference,
when looking for qualitative signals beyond GR, in the context of the
$\alpha^{\prime}$-corrected black holes.

##  VII Conclusions and discussions

Summarizing, the solution that we presented in this paper is a extension of
our slowly rotating solution described in [12], including the logarithmic
branch from the static solution. We presented the conserved charges, the
Hawking temperature and the entropy for this space-time in the particular case
of $b=0$. Then, we studied several properties of the metric, first we assign a
Petrov D type to the solution, we found a curvature singularity inside the
event horizon, the location of the event horizon and ergosphere, as well as an
upper limit for $\alpha$, all the quantities related to the Kerr metric. We
then move on to the study of circular geodesics motion of test particles where
we find that the circular orbits around this black hole are different from the
Kerr orbit, presenting an unstable circular orbit close to the horizon. We
also find that our solution is separable under the Hamilton-Jacobi equations,
allowing us to find expressions for the photon rings where it is possible to
see that as $\alpha$ grows, the ring diameter grows as well. Finally, we move
on to the study of black hole shadowing by means of numerical ray tracing
simulations for a thin accretion disc, and then a post-analysis is carried
out. Determining that increasing values of $\alpha$ increase the diameter of
the shadow and subtly affect the shadow offset and asymmetry. This is evident
from simulations where the accretion disc has the same starting point, i.e.
the same ISCO radius.

###### Acknowledgements.

We thank B. Bandyopadhyay, A. Cisterna and G. Giribet for enlightening
comments. FAS and DRGS gratefully acknowledge support via the ANID BASAL
project FB210003 and via Fondecyt Regular (project code 1201280). DRGS thanks
for funding via ANID QUIMAL220002 and via the Alexander von Humboldt -
Foundation, Bonn, Germany. JO is partially supported by FONDECYT Regular
1221504. MO is partially funded by Beca ANID de Doctorado grant 21222264.

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# PCA Event-Based Optical Flow for Visual Odometry

Mahmoud Z. Khairallah Fabien Bonardi David Roussel Samia Bouchafa
IBISC  Univ. Evry  Université Paris-Saclay  91025  Evry  France
{mahmoud.khairallah, fabien.bonardi, david.roussel, samia.bouchafa}@univ-
evry.fr

###### Abstract

With the advent of neuromorphic vision sensors such as event-based cameras, a
paradigm shift is required for most computer vision algorithms. Among these
algorithms, optical flow estimation is a prime candidate for this process
considering that it is linked to a neuromorphic vision approach. Usage of
optical flow is widespread in robotics applications due to its richness and
accuracy. We present a Principal Component Analysis (PCA) approach to the
problem of event-based optical flow estimation. In this approach, we examine
different regularization methods which efficiently enhance the estimation of
the optical flow. We show that the best variant of our proposed method,
dedicated to the real-time context of visual odometry, is about two times
faster compared to state-of-the-art implementations while significantly
improves optical flow accuracy.

##  1 Introduction

Neuromorphic cameras are considered to be one of the revolutionary visual
sensors that have been developed over the past decade[7]. Based on the demand
for robotics research to overcome all the known problems caused by standard
cameras (i.e. image blur, low dynamic range and data redundancy), neuromorphic
sensors are still currently being developed to overcome most of the standard
cameras’ drawbacks. Instead of transmitting a sequence of whole images at a
constant rate, which includes a lot of redundant information and can cause
high latency, the pixels of these neuromorphic chips operate independently and
asynchronously.

Figure 1: The setting of our environment where the checkerboard is tracked by
ATIS camera. Right: the change detection output obtained by the camera shown
above the grayscale output

These sensors can provide two kinds of information: the differentiation of
light intensity of the environment as a raw output and the integration of this
output to recover grayscale information (see Figure 1). Due to the
independence of each pixel, the raw output of a neuromorphic sensor can be
provided with a very low temporal resolution that may be less than 1 $\mu s$.
This high-debit datastream solves the problem of motion blur and extends the
dynamic range to more than $120$ dB.

Considering events in the design of new algorithms naturally suggests fully
exploiting neuromorphic sensors for a complete and precise estimation of
optical flow. Indeed, events are due to changes in the environment and
therefore indirectly linked to optical flow information. Moreover, optical
flow is essential in many applications like robotics since it can be used as a
core stage for many tasks (visual odometry or SLAM) that require accurate and
fast optical flow algorithms. In this paper, we show improvements in event-
based optical flow estimation using a PCA (Principal Component Analysis)
scheme and explore the availability of using our algorithm in real-time
applications.

##  2 Related Work

The events generated by neuromorphic vision sensors carry intrinsically
optical flow information because they respond only to changes in luminosity.
The asynchronous quasi-continuous nature of these sensors promotes their
incorporation in critical robotics applications which require agility and
accuracy. As one of the first attempts to tackle the challenge of proposing an
event-driven optical flow method, Delbruck [5], considers it as a problem of
data association since each event is created when gradients change. Events are
correlated using the difference of timestamps between a considered event and
the events in its spatio-temporal neighborhood. Hence, the edge that created
the event is assigned one of four directions from
$\mathrm{0}\mathrm{\SIUnitSymbolDegree}$ to
$\mathrm{1}\mathrm{3}\mathrm{5}\mathrm{\SIUnitSymbolDegree}$ separated by
$\mathrm{4}\mathrm{5}\mathrm{\SIUnitSymbolDegree}$. The final direction and
magnitude of the event’s optical flow are determined using the time of flight
between the events stacked perpendicular to the edge direction. This method
can be effective to augment the amount of information induced by each event
but can not be considered as a suitable method of optical flow estimation
because it gives highly discretized directions and a noisy estimated
magnitude.

Since each event is the differential log output of light intensity, Benosman
et al [4], integrate (stack) the events in the event spatio-temporal
neighborhood and use the count of events at each pixel as an intensity-
equivalent value. The count of events is inserted in an adapted event-based
Lucas-Kanade scheme to estimate the optical flow. This method can be
considered as the first event-based optical flow algorithm. However, the
intensity-equivalent value is not the real intensity value. As a consequence,
the optical flow magnitude may not correspond accurately to the real
magnitude. Moreover, to get sufficiently accurate estimations of optical flow,
the value of $\Delta t$ that is related to the size of the spatio-temporal
neighborhood needs to be tuned each time for different dynamics of the
environment.

Figure 2: set of $\mathbb{R}^{3}$ scattered points with their original axes
“in black” and their normalized principal axes“in red”

Benefiting from the time being represented as a monotonically increasing
function, the generated events in a small temporal shift will form planar
surfaces (as the surface of extruded material). Based on this property,
Benosman et al [3], propose a local plane fitting method where a mapping of
clustered events in a spatio-temporal neighborhood can be used to fit a plane
(in $x$, $y$ and $t$ coordinates system). The $x$ and $y$ components of the
vector normal to the plane represent an estimate of the optical flow. To make
sure the estimated plane is the correct fit, an iterative scheme is used to
reject events that are very far from the plane estimate until convergence is
met. Local Plane fitting methods provide acceptable optical flow accuracy
while being intuitively more realistic to describe events’ nature, although
tuning for $\Delta t$ is also required and plane estimation could be
deteriorated due to falsely generated events. To alleviate the problem of
$\Delta t$ fine-tuning, Mueggler et al [10], use the so-called active events
surface (a surface that stores only the last generated events at each pixel)
to cluster events in the neighborhood and then adopt a RANSAC scheme for a
better plane fitting. In [10], to refine the output and get a better event’s
lifetime estimate (which is also the main purpose of our work) an optimization
scheme is proposed. This scheme is able to smooth the total output of optical
flow but did not provide higher accuracy while increasing more computation
time.

Many other contributions in the field of event-based optical flow estimation
have been proposed [1, 2, 15]. These methods differ from one another in the
use of the type of domain (spatial or frequency) or the segmentation method
(based on contrast maximization optimization or spatio-temporal
regularization). All these existing methods provide good accuracy but suffer
from extremely heavy computation time. Since it is accepted that these
algorithms will be integrated in more complex schemes to achieve the given
tasks, they are considered out of interest for real-time robotics applications
even if accuracy and robustness are guaranteed.

In this paper, we present an optical flow algorithm that can provide accurate,
yet simple to implement, estimates while maintaining relatively low
computational time. The paper is organized as follows: in section 3, we
present the specific properties of event-based cameras and the reason why they
are well suited to provide good optical flow algorithm performance. Section 4
demonstrates the concept of Principle Component Analysis (PCA) and the
benefits from its adaptation to event-based optical flow scheme. Results are
shown in section 5. Section 6 summarizes the work presented in this paper and
the proposed future work.

Figure 3: In red: points spanning the real actual plane. In green: noisy
points representing a plane consensus. A closer look of all the coordinates
systems shows up to the right where, aside from plane fitting axes, PCA and
the actual plane coordinates look almost identical. In black: the actual plane
coordinates system. In blue, the estimated coordinates system estimated using
PCA. In red: the coordinates system estimated using plane fitting least
square.

##  3 Event-Based Nature

Neuromorphic vision sensors, also called event-based cameras, feature pixels
that operate asynchronously and independently. Each pixel responds to log
intensity change in the environment and triggers events whenever a threshold
is attained. The asynchronous and independent nature of event-based cameras
privileges them thanks to their very high temporal resolution where many
events may be triggered in a microsecond. Each event is encoded as a tuple
$\langle x,y,t,p\rangle$ where $x$ and $y$ are the pixel coordinates, $t$ is
the timestamp of the triggered event and $p$ is the polarity in $\\{1,-1\\}$
that corresponds to positive or negative luminosity change respectively. An
event is triggered whenever a change exceeds a certain threshold $\delta_{t}$
which may vary between $10$ and $15\%$ of the luminosity of the last triggered
event at each pixel according to the following equation:

$\Delta L(x_{i},y_{i},t_{i})=L(x_{i},y_{i},t_{i})-L(x_{i},y_{i},t_{i}-\Delta
t)=p_{i}\delta_{l}$ (1)

where $L(x_{i},y_{i},t_{i})$ is the log intensity at the current time and
$L(x_{i},y_{i},t_{i}-\Delta t)$ is the log intensity of the event created
previously at this pixel at $t_{i}-\Delta t$. Changes in luminosity are
supposed to be only due to movements since small time intervals are
considered. Events encompass optical flow information as it can be shown by
approximating $\Delta L(x_{i},y_{i},t_{i})$ using Taylor expansion:

$\Delta L(x_{i},y_{i},t_{i})\approx\frac{\partial L}{\partial
t}(x_{i},y_{i},t_{i})\Delta t=\frac{\partial L}{\partial x}\frac{\partial
x}{\partial t}+\frac{\partial L}{\partial y}\frac{\partial y}{\partial
t}+\frac{\partial L}{\partial t}\\\ $ (2)

Considering that event-based cameras encapsulate optical flow information and
provide quasi-continuous signals, suggests pushing the limits of classical
optical flow algorithms by proposing new adaptations to event-based cameras.
Even though many improvements for event-based optical flow have been
introduced in the state-of-the-art, many restrictions like accuracy and
computational power still prevent the adoption of optical flow with
neuromorphic vision sensors. As a step to overcome these burdens, a novel
algorithm for optical flow estimation is presented in the next section.

##  4 PCA Optical Flow

PCA [11] was first introduced to the scientific community by Karl Pearson as a
linear dimensionality reduction method. The principle is to map higher
dimensional spaces $\mathbb{R}^{n}$ data to lower ones by providing a
hierarchical orthogonal coordinates system, changing the basis that spans data
dimensions111PCA provides hierarchical principal axes where each axis is
scaled to span a principal dimension of the data. (see Figure 2).
Dimensionality reduction is done using PCA by maximizing the variance of the
data around the principal axes in the lower dimension space. Since variance is
maximized around the singular vectors representing the data, eigenvalue
decomposition can be used to find the principal components of the data.

Based on the hypothesis that triggered events create a plane in a small
neighborhood, PCA can be employed to find the two orthogonal vectors spanning
the created plane and the third will be the plane’s normal222plane normal
corresponds to 0 variance for a perfect plane (no thickness for flat
surfaces).. PCA is found to be able to estimate the plane’s normal and provide
better accuracy than least-square plane fitting algorithms provided in the
state of the art [3, 10, 13] (see Figure 3). Applying PCA in an optical flow
event-based scheme is carried out in three steps. First, events provided by
the camera are conditioned to put aside events that are redundant or created
by noise or strong luminosity changes. Second, the optical flow is estimated
using PCA method on conditioned events. Finally, the optical flow is
regularized to assure robustness of the estimated optical flow. These steps
are detailed extensively in the following section.

###  4.1 Events Filtering

Provided events on pixel-level are due to changes in the environment makes
event-based cameras highly prone to different noise sources. Events filtering
step is required before applying any algorithm to verify that triggered events
correspond to real changes in the environment. First, positive and negative
events are separated and processed independently where a refractory filter
filters out events that may be triggered due to sharp changes in intensity.
Events are filtered if an event has already been triggered on the same pixel
within a certain time interval. The limit is chosen to be 20 $ms$ for events
triggered of the same polarity and 1 $ms$ for opposite polarity. These values
are chosen based on practical experiments. To make sure that all events belong
to real motion, an adaptive activity filter [8] is applied. Adaptive filtering
in this context consists of storing 8-pixel neighborhood events in a buffer
and checking whenever a new event is triggered. For each event, the time
difference between it and at least 3 of the 8-pixel neighborhood events in the
buffer does not exceed the adaptive support time $T_{f}$ which is computed as
a function of events frequency333Events frequency varies according to the
environment dynamics which affects event:

$\displaystyle\alpha$ $\displaystyle=$ $\displaystyle\frac{k}{\log f_{e}}$ (3)
$\displaystyle T_{f}$ $\displaystyle=$
$\displaystyle\frac{T_{max}-T_{min}}{\alpha_{max}-\alpha_{min}}(\alpha-\alpha_{min})+T_{min}$
(4)

Data: $e\\{x,y,t,p\\}_{i=0}^{N-1}$

Result: $e\\{x,y,t,p,v_{x},v_{y}\\}_{i=0}^{N-1}$

1 Initialize: $\epsilon\;,n=\\#\;of\;pixels,\;ev=zeros(col,row,2)\;,$

2 $flow=zeros(col,row,2,3),$

3 $t_{s}=20\;ms\;,\;t_{o}=1\;ms$

4 for  _$i\leftarrow 0$ to $N-1$_ do

5 $flag=0$

6 $x,y,t,p=e_{i}$

7 $p_{o}=opposite\;polarity$

8 if  _$(t_{i}-ev(x_{i},y_{i},p_{i}) <t_{s}$) &
$(t_{i}-ev(x_{i},y_{i},p_{o})<t_{o})$_ then

9 $T_{f}$ = adaptive time (eqn(3), (4))

10 if  _in neigh $(t_{i}-t_{neigh})<T_{f}$_ then

11 $flag=1$

12 $ev(x_{i},y_{i},p_{i})=t_{i}$

13 end if

14

15 end if

16 if  _$flag=1$_ then

17 $neigh\leftarrow$ events in $ev$ in $n\times n$ neighborhood

18 if  _$size(neigh) >3$_ then

19 Estimate: $\mathbf{\mu_{x},\mu_{t},\mu_{t}}$

20 Construct: $N$ = eqn(5), $\Sigma$ = eqn(6)

21 Compute: [V,E] = eig($\Sigma$)

22 if  _$e(1,1) <\epsilon$_ then

23 [$v_{xi},v_{yi}$] = eqn(7)

24

25 else

26 $[v_{xi},v_{yi]}=[0,0]$

27 end if

28

29 else

30 $[v_{xi},v_{yi]}=[0,0]$

31 end if

32

33 end if

34

35 end for

Algorithm 1 PCA Optical Flow Estimation

Figure 4: Three levels of neighborhood are shown, $5\times 5$ neighborhood ”in
blue”, $7\times 7$ neighborhood ”in red”, $9\times 9$ neighborhood ”in green”
and the extreme points (outliers) ”in yellow”. PCA estimation of the three
levels are shown.

###  4.2 PCA Optical Flow Estimation

PCA was first adapted to event-based nature in a line segmentation scheme [6],
here we adapt it for optical flow estimation. We estimate best plane fit of
triggered events to obtain spatial components where the plane is expressed as:

$\begin{bmatrix}a\\\ b\\\ c\\\
d\end{bmatrix}\begin{bmatrix}x&y&t&1\end{bmatrix}=0$ (5)

$(a\;b\;c\;d)$ are the estimated plane parameters. Principal components of a
set of data points can be computed by finding the eigen vectors of the
covariance matrix $\mathbf{\Sigma}$ of this set of centralized points. The
first principal component (vector) will be the one corresponding to the
largest eigenvalue and the next ones correspond to lower eigenvalues
respectively. For a set of $n$ polarized events $\\{x_{i},y_{i},t_{i}\\}$
created in a neighborhood $\mathbf{N}$ in $\mathbb{R}^{3}$, they are
centralized by subtracting the mean of each dimension such that:

$\mathbf{N}=\begin{bmatrix}x_{1}-\mathbf{\mu_{x}}&y_{1}-\mathbf{\mu_{y}}&t_{1}-\mathbf{\mu_{t}}\\\
\vdots&\vdots&\vdots\\\
x_{n}-\mathbf{\mu_{x}}&y_{n}-\mathbf{\mu_{y}}&t_{n}-\mathbf{\mu_{t}}\\\
\end{bmatrix}$ (6)

The matrix $\mathbf{N}$ is used to construct the covariance matrix as:

$\mathbf{\Sigma}=\begin{bmatrix}\mathbf{\sigma_{xx}}&\mathbf{\sigma_{xy}}&\mathbf{\sigma_{xt}}\\\
\mathbf{\sigma_{yx}}&\mathbf{\sigma_{yy}}&\mathbf{\sigma_{yt}}\\\
\mathbf{\sigma_{tx}}&\mathbf{\sigma_{ty}}&\mathbf{\sigma_{tt}}\\\
\end{bmatrix}$ (7)

where $\mathbf{\sigma_{ij}}=\sum_{k=0}^{n}i_{k}j_{k}$. The eigen vectors of
$\mathbf{\Sigma}$ represent the principal components spanning the
neighborhood, two orthogonal vectors span the plane and the third one
corresponding to the smallest eigenvalue is perpendicular to the plane. The
vector $\mathbf{V_{p}}$ corresponding to the smallest to eigenvalue is
considered the plane normal. This vector can be used directly to estimate the
optical flow, however we employ a step to validate the accuracy of estimated
parameters first. We estimate the plane parameters as follows;

$\displaystyle a=V_{px}\;,\;b=V_{py}\;,\;c=V_{pt}$ (8) $\displaystyle
d=-(V_{px}x+V_{py}y+V_{pt}t)$ (9)

$(V_{px},V_{py},V_{pt})$ are the components of the vector $V_{p}$, $(x,y,t)$
are the coordinates of the event under test. The plane parameters are used to
evaluate $t_{est}$ value for each pixel used to fit the plane where only the
event spatial coordinates are used.

$t_{est}=-\frac{ax+by+d}{c}$ (10)

if the absolute difference between $t_{est}$ and the actual triggering time of
the event $t$ exceeds a threshold $\delta$ the event is considered outlier. A
consensus of inliers should be more than $(1-\epsilon)N^{2}/2$ where
$\epsilon$ is the accepted ratio of outliers, otherwise the estimated plane
will be rejected. The estimation of optical flow is obtained as follows:

$\begin{bmatrix}v_{x}\\\
v_{y}\end{bmatrix}=\frac{-V_{pt}}{V_{px}^{2}+V_{py}^{2}}\begin{bmatrix}V_{px}\\\
V_{py}\end{bmatrix}$ (11)

Hence the lifetime of each event is obtained:

$t_{lifetime}=\frac{V_{px}^{2}+V_{py}^{2}}{V_{pt}}$ (12)

Algorithm 1 demonstrates a step-by-step pseudo-algorithm of our method.

###  4.3 Spatio-Temporal Optical Flow Regularization

Due to the sparsity and noisy nature of neuromorphic sensors, it is required
to go the extra mile and refine the estimated optical flow to make sure it
depicts the true optical flow. We try two different spatio-temporal
regularization techniques to see which would work best w.r.t estimation
quality and computation time.

Figure 5: Variation of weighting functions plotted according to real data
collected. Inverse exponential (in green) gives extreme weights, inverse
logarithmic gives averaging-like effect and the inverse gives weightings to be
distributed in between.

Methods in the state-of-the-art demand solving a system of equations that
depends on the size of the chosen neighborhood. The first technique benefits
from the fact that the neighborhood’s size will not affect much the
computation time for PCA444we always compute eigen vectors of a $3\times 3$
matrix which would take most of the time. We use different levels of the
neighborhood to estimate the optical flow and take the mean as the best
estimate of optical flow to be sure that extremes are filtered out (see Figure
4).

The other technique consists of storing the optical flow in a buffer called
“active optical flow” and then weightings are given to the optical flow
estimated previously in a specific neighborhood of the event under test.
Weightings $\theta_{i}$ are chosen to be inversely proportional to the
difference of timestamps of the event under test and events triggered in the
neighborhood so that the final optical flow estimation will be:

$\begin{bmatrix}v_{x}\\\
v_{y}\end{bmatrix}=\sum_{i=1}^{n}\theta_{i}\begin{bmatrix}v_{xi}\\\
v_{yi}\end{bmatrix}$ (13)

Allegedly, choosing the inverse of exponential would be a reasonable choice
for its vanishing values for higher time difference. The choice of this
function is found to greatly suppress the effect of old optical flow on the
neighborhood (see Figure 5). Among all possible weighting functions, we chose
to use the inverse of the difference because it gives relatively correctly
distributed weightings while being the fastest to compute. It has been found
that choosing a smaller neighborhood for weightings555i.e. $5\times 5$ for a
$7\times 7$ neighborhood to estimate optical flow. than the neighborhood
chosen to estimate the optical flow gives better results. The regularization
process is shown in Algorithm 2.

Data: $[v_{x},v_{y},t],\;flow$

Result: regularized $[v_{x},v_{y}]$

1 Initialize: $reg$, $n_{l}=\\#\;of\;levels$

2 if  _$reg=weighting$_ then

3 $t_{n}\leftarrow$ times in $n\times n$ neighborhood in $flow$
$v\leftarrow[v_{x},v_{y}]$ in $n\times n$ neighborhood in $flow$

4 $W=\frac{1}{t-t_{n}}$

5 $W=\frac{W}{norm(W)}$

6 $[v_{x},v_{y}]_{reg}$ calculated using eqn(8)

7else

8 $v\leftarrow[v_{x},v_{y}]$ for  _$i\leftarrow 1$ to $n_{l}$_ do

9 Recall: Algorithm(1) with $n=n-1$

10 $v\leftarrow v+[v_{x},v_{y}]$

11 $[v_{x},v_{y}]_{reg}=\frac{v}{n_{l}}$

12 end for

13

14 end if

15

Algorithm 2 PCA Optical Flow Regularization

We tested the availability to apply PCA optical flow algorithms with and
without those different regularization techniques and compare them with state-
of-the-art event-based optical flow algorithms and show the results in the
following section.

##  5 Results

To validate the reliability and accuracy of the algorithm we propose, we used
the data-sets provided previously [8, 10]. We used two sequences of the first
data-set which cover translational and rotational dominant motion and the data
set of moving stripes to validate the estimated lifetime. In [8], a VICON
system is used to track the motion of multiple objects and 3D/2D projection is
used to create optical flow ground truth (see Figure 6). These sequences are
recorded using an ATIS (Asynchronous Time-based Image Sensor) camera [12]
provided by PROPHESEE with a $480\times 360$ resolution which can provide
change detection, events detection and exposure measurements (see Figure 1).
The camera is handheld and tracks a checkerboard while the desired motion is
fulfilled. We test the validity of PCA optical flow with and without
regularization and compare it with the methods of event-based Lucas-Kanade and
plane fitting methods adapted with Savitzky-Golay filter as described in [14].
Usage of Savitzky-Golay filter is proven to improve accuracy and reduce
computation time. Experiments are carried out with a spatial neighborhood of
$7\times 7$, three levels for pyramidal regularization are chosen and $\Delta
t$ for Lucas-Kanade method is chosen to be $10\;ms$.

Four metrics are used to provide a quantitative and comprehensive comparison.
The Average End Point Error (AEPE) measures differences in optical flow
magnitude, The Average Angular Error (AAE) measures differences in optical
flow orientation, Lifetime error to check the difference between estimated and
real lifetime of each event and computational time shows the availability to
use these algorithms in real-time applications. Table 3 summarizes the
computation time required to estimate optical flow for each algorithm. Table 2
shows AEPE and AAE performance metrics evaluated using our recorded sequences.

Figure 6: Probability Density Function(PDF) of the estimated lifetime for only PCA (Blue), weighted PCA (Red) and leveled PCA (Yellow) of the sequence [10] stripes Algorithm | $\tt{slow\\_stripes}$ | $\tt{fast\\_stripes}$
---|---|---
max bin | % | max bin | %
(ms) | (ms)
PCA Only | 5.35 | 52.19 | 10.65 | 17.95
Weighted PCA | 5.55 | 57.72 | 11.05 | 19.44
Leveled PCA | 5.65 | 38.58 | 11.45 | 15.11
Table 1: The maximum bin value for the life time estimation of the dataset
provided in [10] and the percentage of each bin

###  5.1 Average End Point Error

The average end point error is defined as:

$AEPE=\frac{1}{N}\sum\limits_{i=1}^{N}||\textbf{u}_{i}-\hat{\textbf{u}}_{i}||$
(14)

where $\textbf{u}_{i}$ and $\hat{\textbf{u}}_{i}$ are the estimated and the
ground truth optical flow respectively. AEPE illustrates the error in the
magnitude of estimated optical flow.

Figure 7: the results obtained from our algorithm and methods in the state of
the art, up from left to right: lucas-kanade with Savitzky Golay filter, plane
fitting with Savitzky Golay filter and PCA with no regularization. down from
left to right, PCA with weighting regularization, PCA with pyramidal levels
regularization and finally the synthitically generated ground truth is
displayed on top of real events. Red arrows represent optical flow estimated
from positive events, green arrows represent optical flow estimated from
negative events. The ground truth is represented by blue arrows.

Translation Scenario: The usage of PCA, in general, shows better accuracy in
magnitude estimation. Applying PCA only improved the AEPE by more than $8\%$
w.r.t true optical flow compared to the local plane fitting method. Applying
leveled PCA regularization provides the best magnitude estimation and improves
the overall estimation by more than $11\%$. Weighted PCA shows a slight
improvement in magnitude estimation compared to PCA only.

Algorithm | translation
---|---
Magnitude (rel.) | Orientation ($\mathrm{\SIUnitSymbolDegree}$)
PCA only | $0.069\pm 0.024$ | $7.872\pm 4.665$
PCA /w weights | $0.061\pm 0.023$ | $\mathbf{5.671\pm 4.667}$
PCA /w levels | $\mathbf{0.046\pm 0.015}$ | $6.599\pm 5.108$
Local Plane | $0.158\pm 0.099$ | $13.158\pm 7.542$
Lucas-Kanade | $0.237\pm 0.085$ | $19.540\pm 8.25$
Algorithm | rotation
Magnitude (rel.) | Orientation ($\mathrm{\SIUnitSymbolDegree}$)
PCA only | $0.081\pm 0.032$ | $11.854\pm 4.513$
PCA /w weights | $0.075\pm 0.0401$ | $\mathbf{11.236\pm 3.345}$
PCA /w levels | $\mathbf{0.071\pm 0.048}$ | $12.014\pm 4.868$
Local Plane | $0.173\pm 0.054$ | $15.568\pm 6.540$
Lucas-Kanade | $0.275\pm 0.082$ | $22.634\pm 8.215$
Table 2: Relative average end point error and average angular error for rotational and translational sequences. Algorithm | Computational Time
---|---
per Event ($\mu s$)
Lucas-Kanade | $3.32\pm 1.07$
Plane Fitting | $1.18\pm 0.74$
PCA only | $\mathbf{0.29\pm 0.05}$
PCA with weights | $0.51\pm 0.06$
PCA with Levels | $0.78\pm 0.07$
Table 3: Computation times needed for calculations per event.

Rotational Scenario: Estimating optical flow in rotation motion is considered
more critical because magnitude would vary in a small neighborhood. For this
reason, AEPE is shown to be slightly higher than in the translational
scenario. Leveled PCA regularization is shown to give the best results and
relatively provide the same percentage of improvement in rotational scenarios
compared to the state-of-the-art.

###  5.2 Average Angular Error

The metric to measure the differences of orientation is the average angular
error and is defined as:

$AAE=\frac{1}{N}\sum\limits_{i=1}^{N}cos^{-1}\left(\frac{\hat{\textbf{u}}_{i}^{T}\textbf{u}_{i}}{||\hat{\textbf{u}}_{i}||||\textbf{u}_{i}||}\right)$
(15)

Figure 8: From left to right, Events stacked during $20ms$ for the sequence
$\tt{stripes}$, active events only shown according to PCA only algorithm,
weighted PCA algorithm and leveled PCA algorithm all at $136ms$.

Translation Scenario: The choice of neighborhood size affects deeply the
accuracy of estimating the orientation, for which reason, weighted PCA
regularization provided best results. Weighted PCA gives best results because
it uses a smaller level of neighborhood for regularization and also because it
uses a consensus of neighborhood optical flow which creates a smoothing
effect. Leveled PCA regularization improved the estimation of orientation and
boosted the accuracy but did not maintain maximum accuracy due to using many
levels of the neighborhood.

Rotational Scenario: As expected, rotational scenarios are harder to estimate
and accuracy may be reduced. Results provided by the three methods of PCA did
not much vary, however PCA with weightings regularization attained the best
results.

###  5.3 Lifetime Estimation

Lifetime is considered as one of the important features assigned to event-
based cameras. It helps sharpening image-like matrices used in event-based
visual odometry and SLAM algorithms [10]. With sufficiently acceptable
accuracy, all PCA algorithms provided reliable lifetime estimation compared to
the state-of-the-art [10, 9]. We used the sequence $\tt{stripes}$ where two
stripes are moving at different constant speed and hense constant lifetime for
each event generated by these stripes. Figure 6 shows the probability density
function (PDF) of the estimated lifetime. Using PCA only, $52.19\%$ of events
where assigned lifetime of $5.35\;ms$ with error of $10.83\%$ of the slow
stripes and $17.95\%$ are assigned $10.65\;ms$ with error of $11.25\%$ of the
fast stripes. Weighted PCA gave the thinnest distribution around the maximum
bins value (least variance). The nearest large bin to slow stripes has the
value of $5.55\;ms$ with $57.72\%$ of all events and $7.5\%$ error. For Fast
stripes, the largest bin’s lifetime is $11.05\;ms$ with $19.44\%$of all events
which translates to an error of $7.91\%$ of the actual lifetime. Leveled PCA
gives the best results but with the largest variance around actual lifetime
with percentage of $38.58\%$ and $15.11\%$ around the slowest and the fastest
stripes respectively and errors of $5.83\%$ and $4.58\%$. The high variance of
Leveled PCA creates many falsely estimated lifetimes (see Figure 8).

###  5.4 Computational Time

PCA method is based on computing the eigen vectors of a $3\times 3$ matrix
that is constructed by increments based on the chosen neighborhood. This
results in a much lower computation time compared to other techniques. For the
same reason, the change of the neighborhood size, on the contrary to other
methods, does not affect the computation time. PCA method only is shown to be
5 times faster than the local plane fitting method and 10 times faster than
Lucas-Kanade method. As expected, the usage of levels regularization, which is
the slowest PCA method, increased the computation time, however it is still
about 2 times faster than local plane fitting methods and about 4 times faster
than Lucas-Kanade method. Although these algorithms are implemented on a Linux
machine with Core i5 3.10 GHz processor, the provided computation time of PCA
variants is shown to be competitive to be selected for real-time applications
if implemented using low-level language.

##  6 Conclusion and Future Work

In this paper, we propose a novel method for event-based optical flow
estimation based on PCA. We have shown that the proposed method is more
adapted to the sparse nature of event-based cameras, and produces
significantly less noise than other methods. Although improvements in
performance always come at cost of computational power, the method we proposed
makes sparing use of the processor and drastically reduces computation time.
The simple and straightforward procedure of PCA event-optical flow approach,
besides providing accurate and rapidly computed estimation, is the strength
point of the method. We have shown improvements in event-based optical flow
estimation and these results incite to pursue the study on event-based optical
flow performance for more complex schemes.

Visual odometry is one of the most critical robotic applications that require
the accuracy and agility of the provided data. Based on neuromorphic vision
sensors nature and the accomplished gain, a pragmatic decision would be to
consider an optical-flow-based visual odometry scheme. To the best of our
knowledge, this area is not widely investigated and still requires due
attention. To exploit event-based optical flow in robotic applications, a
better understanding of its limitations is required. We intend to analyze the
capacity of these optical-flow-based schemes in future work.

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