††thanks: 1<EMAIL_ADDRESS>
2<EMAIL_ADDRESS>
# Geodynamics in spacetime crystal under slow perturbation and deformation

Anzhuoer Li1, Liang Dong2, and Qian Niu3
leean@liangdong<EMAIL_ADDRESS>13University of Science and
Technology of China, Hefei, Anhui, China
2University of Washington, Seattle, USA

###### Abstract

We present a theory of geodynamics in a spacetime crystal based on an event
wavepacket constructed from the Floquet-Bloch waves, which not only involve a
scalar dispersion function but also a Berry curvature tensor in the phase
space manifold of spacetime and the reciprocal quasi energy-momentum. In the
presence of structural deformation, this theory is naturally extended into a
covariant form with the introduction of a lattice connection constructed out
of the gradients of the local lattice vectors. The geodesic equation for a
free particle involves not only the lattice connection but also higher-order
corrections from spacetime inhomogeneity of Berry curvatures and quasi energy-
momentum dispersion.

††preprint: APS/123-QED

_Introduction_.—Newtonian mechanics fails in the atomic scale, but a particle
view re-emerges from the Bloch bands in a crystal on length scales much larger
than the lattice constants and energy scales much smaller than the band
structure. However, besides the periodic topology of the energy dispersions,
Berry curvatures of the Bloch states are found to dramatically modify the
canonical structure of the Bloch particle dynamics, introducing a host of
transport as well as equilibrium properties of the crystalline solid [1]. What
kind of particle dynamics can emerge in a spacetime crystal?

A spacetime crystal may be defined as a medium with microscopic translational
symmetries in both space and time, which may be formed spontaneously or
imposed externally [2, 3]. We are typically concerned with the propagation of
a matter wave or excitation described by a linear partial differential
equation with periodic position and time dependence. A normal mode has the
form of Flouquet-Bloch wave, a plane wave with an amplitude varying
periodically in space and time. There has been much interest in creating such
systems and studying their novel topological properties [4, 5].

In this Letter, we seek a particle description by constructing a wave packet
out of a narrow range of Flouquet-Bloch waves of a single band, which is broad
in real space compared with the spacetime unit cell but still localized over
macroscopic scales. This is the analog of a classical event point in the meta-
spacetime, and a set of geodynamic equations of motion for the corresponding
worldline in phase space (spacetime and the reciprocal quasi energy-momentum)
can be obtained in the presence of weak external fields and various
inhomogeneity.

We are also interested in the case with the presence of slowly varying
deformation of the spacetime crystal and wish to look into how such
deformation affects particle dynamics. With the introduction of a lattice
connection from the spacetime lattice vectors and their reciprocals [6], a
technique first introduced by two of us for the study of electron
semiclassical dynamics in 3D crystals under deformation, we are able to obtain
covariant geodynamics akin to Plebanski’s formulation of general relativity
[7].

The geodesic line is of the standard form involving the lattice connection,
but both dispersion structure and Berry curvatures enter in higher order
gradients indicating the two ways of modification in effective spacetime
structure. There are also Lorentz force terms akin to gauge field effects and
correcting the lattice connection.

_Geometrowave equation and event wave packet_.—To treat space and time at
equal footing, we change the equation of different quasi-particles in crystal
to a standard form as

$\hat{\mathcal{L}}(\frac{1}{i}\partial_{x},x)\ket{\phi}=0,$ (1)

where $\ket{\phi}$ is the eigenfunction of the dispersion operator
$\hat{\mathcal{L}}$ and $x=(t,\vec{x})$ is 4-vector. Several examples are
presented here, for Bloch electron the operator is
$\hat{\mathcal{L}}=i\partial_{t}-\hat{H}(\frac{1}{i}\partial_{\vec{r}};\vec{x},t)$,
for bosons like phonons in crystal the operator is
$\hat{\mathcal{L}}=\partial^{2}_{t}-\partial_{\vec{r}}^{2}+m(\vec{x},t)$, and
for Dirac fermion is
$\hat{\mathcal{L}}=iv_{F}^{\mu}\partial_{\mu}-m(\vec{x},t)$. All equations
that can be written in this form are linear.

The explicit form of eigenfunction can be determined by symmetry analysis. The
number of spacetime groups for spacetime crystals is discrete and finite [8],
so the translation symmetry can be written in a general form as
$\hat{\mathcal{L}}(\vec{x},t)=\hat{\mathcal{L}}(\vec{x}+\vec{R}_{\alpha},t+T_{\alpha})$,
from which we can define 4-dimensional lattice vectors as
$c_{\alpha}=(T_{\alpha},\vec{R}_{\alpha})$. Reciprocal lattice vectors are
defined by orthogonal condition
$b^{\alpha}_{\mu}c_{\beta}^{\nu}=\eta^{\nu}_{\mu}\delta_{\beta}^{\alpha}$,
where $\eta_{\mu}^{\nu}$ is the metric in Minkowski spacetime manifold.
According to the Floquet-Bloch theorem

$\ket{\phi(k,x)}=e^{ik_{\mu}x^{\mu}}\ket{u(k,x)},$ (2)

where $\ket{u(k,x)}$ is the periodic part. The $k_{\mu}=(\omega,-\vec{k})$ is
the quasi-energy and quasi-momentum of the quasi-particles. It follows the
transformation rule as $k_{\mu}=\Lambda^{\nu^{\prime}}_{\mu}k^{\prime}_{\nu}$
when $\Lambda^{\nu^{\prime}}_{\mu}=\frac{\partial x^{\prime\nu}}{\partial
x^{\mu}}$ is the coordinate transformation. The Berry connections are defined
as

$A_{k_{\mu}}=i\braket{u(k,x)}{\partial_{k_{\mu}}}{u(k,x)},$ (3)

the inner product is integrated in a unit cell and a single period.

$\braket{u_{1}}{u_{2}}=\frac{(2\pi)^{4}}{TV}\int_{V,T}d^{4}ru_{1}^{*}(x,k)u_{2}(x,k).$
(4)

The dispersion relation can be directly solved by changing equation (1) to the
form

$\hat{\mathcal{L}}(\frac{1}{i}\partial_{x}+k,x)\ket{u(k,x)}=0,$ (5)

and we can have $\omega=\omega(\vec{k})$. To make the wave packet analogous to
a classical event, the wave packet needs to localize at a certain spacetime
point, which demands a separation in $k$ space by uncertainty relation. The
separation in the 4-momentum space is in contrast to the dispersion relation
$\omega=\omega(\vec{k})$, which is a 3-d surface in the 4-momentum space.

To obtain an independent 4-momentum, we introduce some new parameters to
dismiss the dispersion constraint. We first interpret the equation (1) as the
zero "energy" solution of the following eigenproblem

$\hat{\mathcal{L}}(\frac{1}{i}\partial_{x},x)\ket{\phi(k,x)}=\lambda\ket{\phi(k,x)},$
(6)

another conjugate parameter, "proper time", $\tau$ is also introduced to
indicate how the state evolves by substituting
$\ket{\Phi(k,x)}=e^{-i\lambda\tau}\ket{\phi(k,x)}$ and obtain

$\hat{\mathcal{L}}(\frac{1}{i}\partial_{x},x)\ket{\Phi(k,x)}=i\frac{d}{d\tau}\ket{\Phi(k,x)},$
(7)

and $\lambda$ is conserved to $\tau$. From this point of view, the physical
world lies in a shell $\lambda=0$. We need a four-dimensional spread momentum,
so we have to go slightly off-shell with respect to the dispersion and allow
components of the event wave packet to lie on a shell with small enough
$\lambda$, which causes no problem in the subsequent evolution of the wave
packet because of the conservation law of $\lambda$. With the independent
4-momentum, the 4-dimensional wave packet can be constructed as

$\ket{W}=\int d^{4}k\ a(k,\tau)e^{ik_{\mu}x^{\mu}}\ket{u(k,x)},$ (8)

where $\ket{W}$ is the wave-packet state function and $a(k,\tau)$ is the
distribution localized in the center of the wave packet with normalization to
make $\braket{W}{W}=1$. Position and momentum of the wave-packet center can be
calculated by the expectation value of operators $\hat{x}$ and $\hat{k}$ as
$\braket{W}{\hat{x}}{W}=x_{c}$ and $\braket{W}{\hat{k}}{W}=k_{c}$. The wave
packet is localized in $x_{c}$ and $k_{c}$.

The $\lambda(k)$ is a function of energy and momentum. In the geometrowave
equation, $\lambda$ is the eigenvalue, which gives the dispersion relation
between $\omega$ and $\vec{k}$. So we call $\lambda$ the dispersion function
or dispersion for short. Also, we can obtain the dispersion relation
$\omega=\omega(\vec{k})$ by demanding that $\lambda(k)=0$. We are strongly
concerned about physics near the shell $\lambda(k)=0$, so we can take the
Taylor series near the zero points as
$\lambda=\lambda_{0}+\lambda^{\mu}k_{\mu}+\frac{1}{2}\lambda^{\mu\nu}k_{\mu}k_{\nu}$,
where the $\lambda^{\mu}$ is analogous to Fermi velocity and
$\lambda^{\mu\nu}$ is a metric term.

Figure 1: a single band spectrum of a Hamiltonian of neighboring interaction
of Floquet Bloch crystal, with parameter
$h=0.4,t=0.5,A=1.5,\Omega=1,\kappa=10$. The dispersion relation
$\omega=\omega(\vec{k})$ can be solved by taking $\lambda=0$. The oblique
Brillouin zone is predicted by the symmetry analysis before. The figure below
presents the dispersion of the band containing points A, B, and C.

To indicate the dispersion spectrum, we can select and evaluate some special
examples. We focus our discussion on a single dispersion band for simplicity.
The first is a 1-D oblique Floquet crystal. In Wannier representation, the
operator is $\hat{\mathcal{L}}=i\partial_{t}-\hat{H}$ where the Hamiltonian is

$\hat{H}=\sum_{R}[(V+A\cos(\kappa R-\Omega
t))\ket{R}\bra{R}+h(\ket{R+a}\bra{R}+h.c)].$ (9)

$R$ is the index for the lattice point with lattice constant $a$. The
parameter $\kappa$ and $\Omega$ are the outer field’s wave vector and
frequency that changes the on-site energy $t$ .$h$ is the hopping interaction.
The time translation symmetry is $H(t+2\pi/\Omega)=H(t)$. Through the exact
diagonalization method [9] we can derive the spectrum of dispersion. For the 3
points in FIG. 1, the metric term is positive for point B and minus for point
C. As for point A, the metric term is zero, and the quasiparticle of C acts
like a massless Fermion.

For bosons that exist in low-energy excitation of systems like ferromagnetism
and antiferromagnetism [10], the operator for 1-D boson with time translation
symmetry is

$\hat{\mathcal{L}}=\partial_{t}^{2}-\partial_{r}^{2}+A\cos(\Omega t-\kappa
r)+M,$ (10)

the external field $A\cos(\Omega t-\kappa r)$ is dertermined by the external
field with wave vector $\kappa$ and frequency $\Omega$. The constraint for $M$
and $A$ is $M-A\textgreater 0$ to make the wave packet have a positive mass.
We can still use the method in the supplement material of [11] and calculate
its spectrum below. For the 2 points in FIG. 2, the metric $\lambda$ is
spacelike.

Figure 2: dispersion of boson with time translation symmetry. We select only
one band to present. The parameters are $\Omega=2,\kappa=1,A=1,M=1.5$. The
dispersion relation can be solved by taking $\lambda=0$. The figure below
presents the dispersion of the band containing points D and E.

_Geodynamics under slowly varying perturbation and deformation_ —The
dispersion depends on the position of wave-packet center $x_{c}$ when there is
a slowly varying potential, and so are the Floquet-Bloch states. According to
[12], We modify equation (7) with parameter of $x_{c}$

$\hat{\mathcal{L}}(\frac{1}{i}\partial_{x},x;x_{c})\ket{\phi(k,x;x_{c})}=i\frac{d}{d\tau}\ket{\phi(k,x;x_{c})},$
(11)

and the corresponding energy-momentum relation also carries the parameter
$x_{c}$

$\hat{\mathcal{L}}(\frac{1}{i}\partial_{x},x;x_{c})\ket{\phi(k,x;x_{c})}=\lambda(k,x;x_{c})\ket{\phi(k,x;x_{c})}.$
(12)

The gradient correction is so small that the translation symmetry is still
valid under the first expansion [13].

Since there is inhomogeneity in spacetime, the Berry connection in spacetime
is not zero,

$A_{x^{\mu}}=i\braket{u(k,x)}{\nabla_{x^{\mu}}}{u(k,x)},$ (13)

so the total Berry curvatures are

$\displaystyle\Omega_{k_{\mu}k_{\nu}}=i(\braket{\partial_{k_{\mu}}u}{\partial_{k_{\nu}}u}-\braket{\partial_{k_{\nu}}u}{\partial_{k_{\mu}}u}),$
(14)
$\displaystyle\Omega_{k_{\mu}x^{\nu}}=i(\braket{\partial_{k_{\mu}}u}{\nabla_{x^{\nu}}u}-\braket{\nabla_{x^{\mu}}u}{\partial_{k_{\nu}}u}),$
(15)
$\displaystyle\Omega_{x^{\mu}x^{\nu}}=i(\braket{\nabla_{x^{\mu}}u}{\nabla_{x^{\nu}}u}-\braket{\nabla_{x^{\nu}}u}{\nabla_{x^{\mu}}u}).$
(16)

The Lagrangian of the quasi-particle is
$L=\braket{W}{i\frac{d}{d\tau}-\hat{\mathcal{L}}}{W}$, the explicit form of it
is

$L=-\lambda+k_{c\mu}\dot{x}_{c}^{\mu}+\dot{k}_{c\mu}A_{k_{c\mu}}+\dot{x}_{c}^{\mu}A_{{x_{c}}^{\mu}},$
(17)

where
$\lambda=\braket{W}{\hat{\mathcal{L}}_{c}}{W}+\braket{W}{\Delta\hat{\mathcal{L}}(k,x;x_{c})}{W}$,
the first term is simply $\lambda_{c}$ which is calculated at $x_{c}$ and
$k_{c}$, i.e $\lambda(k_{c},x_{c})$. The second part $\Delta\lambda$ is the
expectation value for gradient correction of the dispersion. When there is no
deformation, the gradient correction is
$\Delta\lambda=-Im[\braket{\nabla_{x_{c}}u}{\lambda_{c}-\hat{\mathcal{L}}_{c}}{\frac{\partial
u}{\partial_{k}}}]|_{k=k_{c}}$. [12]

The equation of motion can be derived by the variation method

$\displaystyle\dot{x}^{\mu}=\frac{\partial\lambda}{\partial
k_{\mu}}|_{\lambda=0}-\Omega_{k_{\mu}x^{\nu}}\dot{x^{\nu}}-\Omega_{k_{\mu}k_{\nu}}\dot{k}_{\nu},$
(18)
$\displaystyle\dot{k}_{\mu}=-\nabla_{x^{\mu}}\lambda|_{\lambda=0}+\Omega_{x^{\mu}x^{\nu}}\dot{x^{\nu}}+\Omega_{x^{\mu}k_{\nu}}\dot{k}_{\nu},$
(19)

where $\dot{x}^{\mu}=\frac{dx^{\mu}}{d\tau}$ and
$\dot{k}_{\mu}=\frac{dk_{\mu}}{d\tau}$. Since all the physical results are on
the shell $\lambda=0$, in the following letter, we do not write $\lambda=0$ in
the equations of motion and assume it as default.

lattice connections demonstrate the inhomogeneity of a local lattice[13] and
can be generated to the 4-d situation here. If there is deformation in
spacetime crystal, the lattice vectors become vector fields depending on the
localization where the lattice vectors are, which means
$c_{\alpha}=c_{\alpha}(x)$, where $x$ is 4-coordinate, and so are the
reciprocal lattice vector fields. By considering the lattice vectors change
$dc_{\alpha}^{\mu}=\partial_{\nu}c_{\alpha}^{\mu}dx^{\nu}$, we define the
lattice connection as

$dc_{\alpha}^{\mu}=\Gamma^{\mu}_{\sigma m}c_{\alpha}^{\sigma}dx^{m},$ (20)

where the $\Gamma^{\mu}_{\sigma m}$ is the connection in deformed spacetime
lattice. By substituting reciprocal lattice vectors and using the orthogonal
condition

$\Gamma^{\mu}_{\sigma m}=b_{\sigma}^{\alpha}\partial_{m}c_{\alpha}^{\mu},$
(21)

for an elastic spacetime crystal, the connections defined above satisfy
torsion-free condition[13], which is analogous to the Levi-Civita symbol in
general relativity[14]

$\Gamma^{\mu}_{\sigma m}=\Gamma^{\mu}_{m\sigma}.$ (22)

To get the pure dynamical aspect of particle motion, we introduce the lattice
covariant derivative to waive the influence of deformation. The first
spacetime Brillouin zone is proportionate to the range of reciprocal lattice
vectors so that it will change due to the inhomogeneity in space.
Consequently, geometrical deformation in crystals influences the
transportation of the quasi-momentum of particles. For a covariant derivative
on a function

$\nabla_{x^{\mu}}f|_{(k+2\pi b,x)}=\nabla_{x^{\mu}}f|_{(k,x)},$ (23)

where $f$ is an arbitrary periodic function in Hilbert space satisfying
$f|_{(k+2\pi b,x)}=f|_{(k,x)}$, $\nabla_{x}$ is the covariant derivative
satisfying

$\nabla_{x^{\mu}}f=\partial_{x^{\mu}}f(k,x)-k_{l}\Gamma^{l}_{\mu\nu}\partial_{k_{\nu}}f(k,x).$
(24)

It is in the similar form in [15]. When the covariant derivative acts on a
vector or a tensor, $\partial_{x^{\mu}}$ should change to a derivative
consistent with the metric tensor
$g_{\alpha\beta}=\eta_{ij}c^{i}_{\alpha}c^{j}_{\beta}$[16]. We can take the
function $f=\lambda^{\mu}k_{\mu}$ and $f=\lambda^{\mu\nu}k_{\mu}k_{\nu}$ to
obtain

$\begin{split}\nabla_{x^{\sigma}}\lambda^{\mu}=\partial_{x^{\sigma}}&\lambda^{\mu}-k_{l}\Gamma_{m\sigma}^{l}\partial_{k_{m}}\lambda^{\mu}-\Gamma^{\mu}_{\alpha\sigma}\lambda^{\alpha}\\\
\nabla_{x^{\sigma}}\lambda^{\mu\nu}=\partial_{x^{\sigma}}\lambda^{\mu\nu}-&k_{l}\Gamma_{m\sigma}^{l}\partial_{k_{m}}\lambda^{\mu\nu}-\Gamma^{\mu}_{\sigma
m}\lambda^{\nu m}-\Gamma^{\nu}_{\sigma m}\lambda^{\mu m}\end{split}.$ (25)

What’s more, we demand that the total derivative of a function can be written
as

$df(k,x)=\nabla_{x^{\mu}}fdx^{\mu}+\partial_{k_{\mu}}fDk_{\mu},$ (26)

so the total derivative of quasi-momentum without the deformation effect
writes

$Dk_{\mu}=dk_{\mu}+k_{l}\Gamma^{l}_{\nu\mu}dx^{\nu},$ (27)

$Dk_{\mu}$ tells how the momentum is changed by only dynamical properties.

The equation of motion in a flat spacetime crystal can be generalized in a
deformed crystal by changing all components to covariant forms, so the
derivatives in spacetime become the covariant form. Following the rule above,
the total derivative of $k$ is changed to $Dk$ to indicate the pure dynamic
effect. Also, there is an extra term in the gradient correction of
$\Delta\lambda$

$\Delta\lambda=-Im[\braket{\nabla_{x_{c}}u}{\lambda_{c}-\hat{\mathcal{L}}_{c}}{\frac{\partial
u}{\partial
k}}]|_{k=k_{c}}+\frac{1}{2}\Gamma_{\mu\sigma}^{m}\braket{\hat{O}_{m}^{\mu\sigma}}|_{k=k_{c}},$
(28)

the detailed derivation of this term is in the Appendix.

The Lagrangian has a simple form for a quantum system controlled by the
Schrodinger equation. The dispersion induced by the operator
$\hat{\mathcal{L}}=i\partial_{t}-\hat{H}$ is always
$\lambda=-\omega+\epsilon(\vec{k},x,t)$, where $\epsilon(\vec{k},x,t)$ is the
eigenvalue of $\hat{H}$ under gradient correction. The linear dependence of
$\omega$ in dispersion makes the equation of motion for time induced to simply
$t=\tau$. Also, the state function is independent of $\omega$ so the Berry
connection in $\omega$ is zero. The Lagrangian is

$L=-\epsilon(\vec{x},\vec{k})+\vec{k}\cdot\dot{\vec{x}}+(A_{t}+\dot{\vec{x}}\cdot
A_{\vec{x}})+\dot{\vec{k}}\cdot A_{\vec{k}},$ (29)

which is consistent with the Lagrangian with deformation in [4].

_Geodesic line in deformed crystal and artificial gravity_ —Geodesic line
equation can be derived from the equation of motion, and it reflects how the
manifold is warped. We will first ignore Berry curvatures, which are small
enough terms[17], and then treat them as perturbations in the equation of
motion. After leaving Berry curvature, the equations of motion are

$\displaystyle\dot{x}^{\mu}=\frac{\partial\lambda}{\partial k_{\mu}}$ (30)
$\displaystyle\dot{k}_{\mu}=-\nabla_{x^{\mu}}\lambda,$ (31)

where $x$ and $k$ are the wave packet center position and momentum, we neglect
lower index $c$ for simplicity.

Geodesic line is an equation of second order derivative of $x_{c}$, so we take
an extra derivative of equation (30) and

$\ddot{x}^{\mu}-\Gamma^{\mu}_{\rho\sigma}\dot{x}^{\rho}\dot{x}^{\sigma}=\dot{k}_{\nu}\partial_{k_{\mu}}\dot{x}^{\nu}-\dot{x}^{\nu}\partial_{k_{\mu}}\dot{k}_{\nu},$
(32)

where $\dot{k}_{\mu}=\frac{Dk_{\mu}}{d\tau}$ and the equation is in the
covariant form. The left side of equation (32) is a geodesic line equation,
where the lattice connection plays a similar role as the Levi-Civita symbol in
general relativity, and the right side of equation (32) is a rotation in phase
space which tells how covariant scalar fields influence the wave-packet.

The dispersion of free particles, which are affected by no conservative force
without Berry curvature, should be covariant. So we demand that

$\nabla_{x_{c}^{\mu}}\lambda(k_{c},x_{c})=0,$ (33)

which is valid in a certain neighborhood of $x_{c}$ and $k_{c}$, i.e.
$\partial_{k_{c\nu}}\nabla_{x_{c}^{\mu}}\lambda(k_{c},x_{c})=0$, so there is
no local force induced by the energy-momentum relation. With this constraint,
the equation (32) is modified to

$\ddot{x}^{\rho}-\Gamma^{\rho}_{\nu\sigma}\dot{x}^{\sigma}\dot{x}^{\nu}=0,$
(34)

which is a standard geodesic line equation. The whole system is on a flat lab
spacetime manifold with only a curved lattice structure, but all the quasi-
particles in it move as in a curved universe. Unlike the Levi-Civita symbol,
the lattice connection can be detected in the lab and is physical.

Without Berry curvatures, the geodesic line equation is modified by the
higher-order corrections from dispersion directly. Concentrating on the
neighbor of zero points, the dispersion like point A in Fig (1) can be written
as linear form $\lambda=\lambda^{\mu}k_{\mu}+\lambda_{0}$, with fermi velocity
$\lambda^{\mu}$ and deformation potential $\lambda_{0}$ dependent only on $x$.
The geodesic line is

$\ddot{x}^{\mu}-\Gamma^{\mu}_{\rho\sigma}\dot{x}^{\rho}\dot{x}^{\sigma}=-\dot{x}^{\nu}\nabla_{x^{\nu}}\lambda^{\mu},$
(35)

which tells how the acceleration depends on velocity and space. For a common
dispersion, we take a momentum shift to make it quadratic for simplicity, and
the dispersion is
$\lambda=\lambda_{0}+\frac{1}{2}\lambda^{\mu\nu}k_{\mu}k_{\nu}$ where
$\lambda^{\mu\nu}$ is also dependent only on $x$. After the substitution of
the parameter, the geodesic line is

$\displaystyle\ddot{x}^{\mu}-\left(\Gamma^{\mu}_{\rho\sigma}-\tilde{\Gamma}^{\mu}_{\rho\sigma}\right)$
$\displaystyle\dot{x}^{\rho}\dot{x}^{\sigma}=\lambda^{\mu\nu}\partial_{\nu}\lambda_{0},$
(36)
$\displaystyle\tilde{\Gamma}^{\mu}_{\rho\sigma}=\frac{1}{2}\lambda^{\mu\beta}(\nabla_{\beta}\lambda_{\rho\sigma}-$
$\displaystyle\nabla_{\rho}\lambda_{\beta\sigma}-\nabla_{\sigma}\lambda_{\beta\rho}).$
(37)

The new connection, which contains a lattice connection and a higher order
correction induced by another effective metric
$\tilde{g}_{\mu\nu}=\lambda_{\mu\nu}$, links the spacetime structure and the
dispersion relation.

To see the corrections derived by Berry curvatures, which are $U(1)$
topological fields inducing topological Hall current [18], we add back the
Berry curvatures. With our assumption about "free particle", equation(19) will
be simplified to the form below

$(\delta_{\mu}^{\nu}-\Omega_{x^{\mu}k_{\nu}})\dot{k}_{\nu}=\Omega_{x^{\mu}x^{\nu}}\dot{x^{\nu}},$
(38)

the equation (38) shows that the quasi-particles are moving under the
artificial topological gauge field in space. Another equation of motion for
the derivative of $x$ is

$\dot{x}^{\mu}=\frac{\partial\lambda}{\partial
k_{\mu}}-\Omega_{k_{\mu}x^{\nu}}\dot{x^{\nu}}-\Omega_{k_{\mu}k_{\nu}}\dot{k}_{\nu}.$
(39)

Still, we take an extra derivative to get a geodesic line equation of a wave
packet moving under the gauge fields. All terms with spacial derivatives
should be viewed as higher-order small terms and kept only up to second-order.
Considering commutation relation
$[\nabla_{x^{\nu}},\partial_{k_{\mu}}]f(k,x)=\Gamma_{\nu
m}^{\mu}\partial_{k_{m}}f(k,x)$, the equation is

$\ddot{x}^{\rho}+(\nabla_{x^{\sigma}}\Omega_{k_{\rho}x^{\nu}}-\Gamma^{\rho}_{\nu\sigma})\dot{x}^{\sigma}\dot{x}^{\nu}=(\Omega_{x^{l}x^{\alpha}}\partial_{{k_{\alpha}}}\partial_{k_{\rho}}\lambda+\Omega_{k_{\rho}k_{\nu}}\nabla_{x^{l}}\nabla_{x^{\nu}}\lambda)\dot{x}^{l}.$
(40)

Equation (40) is the equation (34) with second-order correction induced by the
connection and the gauge fields. The terms
$\partial_{k_{\alpha}}\partial_{k_{\rho}}\lambda,\nabla_{x^{l}}\nabla_{x^{\nu}}\lambda$
behave as the effective metrics in phase space coupling to the dual Berry
curvatures. The Berry curvatures are gauge fields like electromagnetic fields
in General relativity[14]. As for the correction of connection, the Berry
curvature $\Omega_{k_{\rho}x^{\nu}}$ results in a change of density states in
phase space when there is no applied fields [17]. Its covariant derivative in
space contributes to the change in the routine of wave-packet, which implies
the influence from phase space can show in real space. The free wave packet
will follow a geodesic line equation where the connection is only a gradient
of $\Omega_{kx}$ if there is no deformation.

_Summary and outlook_ —We have introduced a method for constructing a
spacetime wavepacket to represent an event in a spacetime crystal, and
obtained the equations of motion for the corresponding particle world line. We
have also brought a geometric method to describe deformation in the spacetime
crystal and obtained a covariant equation of motion in the phase space. We
have also analyzed the geodesic line equation of a free particle in real
space, with higher-order corrections induced by inhomogeneity in both the
Berry curvatures and the quasi energy-momentum dispersion.

It may be interesting to generalize our method to non-Abelian cases. When
there is spin or other degeneracy, it may couple to the torsion part of the
connection. Also, the field equation of the background is left for future
work.

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Supplemental Materials

##  I gradient expansion to dispersion

Viewed in the lab frame, the total dispersion depends on spacetime lattice
with deformation, which is hard to deal with. Since the variation is small, we
can expand the dispersion to the first order of deformation, and the zeroth
order of dispersion is locally periodic in spatial and time axes. To obtain a
periodic frame called a lattice frame, we introduce a smooth field in terms f
lattice frame coordinates. Given the 4-positions of lattice points n lab frame
$R_{l}$, where $l$ is a 4-component array to locate the lattice point, we
define a smooth lattice field $R(x^{\prime})$, which satisfies

$R_{l}=R(l),$ (41)

and lab frame coordinates are related to lattice frame coordinates as

$x=R(x^{\prime}).$ (42)

From the definition above, any deforming crystal is mapped to a periodic unit
cell lattice in a lattice frame. The physical property of lattice vector
fields comes naturally from the definition of lattice frame

$c^{\mu}_{\alpha}(\frac{l+(l+1_{\alpha})}{2})=R_{l+1_{\alpha}}-R_{l}.$ (43)

Turn back to the lab frame. We can find it is consistent with the definition
in the main part of the letter.

The local dispersion operator $\hat{\mathcal{L}}(k,r,x_{c})$ depends on the
deformation of 4-coordinates in lattice and 4-coordinates of quasi-particle.
Generally, in the lab frame, we can find the spatially related part has the
form like $\hat{\mathcal{L}}(\tilde{R}_{l}-x)$, where $x$ is the center
position of wave-packet. With the lattice field $R_{l}=R(l)$ and
$\tilde{R}_{l}=l^{\alpha}c_{\alpha}(x)$, we expand $l$ respect to the electron
position in lattice frame $x^{\prime}$, we have

$R_{l}-x=R(l)-R(x^{\prime})\approx(l-x^{\prime})^{\alpha}c_{\alpha}+\frac{1}{2}(l-x^{\prime})^{\alpha}(l-x^{\prime})^{\beta}(c_{\beta}\cdot\partial)c_{\alpha},$
(44)

and the gradient correction of effective due to deformation is

$\hat{\mathcal{L}}(\tilde{R}_{l}-x)\approx\hat{\mathcal{L}}((l^{\alpha}-x^{\prime\alpha})c_{\alpha}(x))+\frac{1}{2}\Gamma_{\mu\sigma}^{m}\hat{O}_{m}^{\mu\sigma},$
(45)

where
$\hat{O}_{m}^{\mu\sigma}=\sum_{l}\frac{\partial\hat{\mathcal{L}}\left(\left\\{\tilde{R}_{l}-x^{\prime\alpha}c_{\alpha}\right\\}\right)}{\partial\tilde{R}_{l}^{m}}\left(\tilde{R}_{l}-x^{\prime\alpha}c_{\alpha}\right)^{\sigma}\left(\tilde{R}_{l}-x^{\prime\alpha}c_{\alpha}\right)^{\mu}$,
if we turn back to lab frame we can find the correction of deformation is
$\hat{O}_{m}^{\mu\sigma}=\sum_{l}\frac{\partial\hat{\mathcal{L}}\left(\left\\{\tilde{R}_{l}-x\right\\}\right)}{\partial\tilde{R}_{l}^{m}}\left(\tilde{R}_{l}-x\right)^{\sigma}\left(\tilde{R}_{l}-x\right)^{\mu}$,
where $x$ is 4-coordinate of wave-packet center in the lab frame.

Besides the deformation of lattice points, the dispersion also depends on the
position of the center location of the wave packet, so use covariant
derivative and equation(10) to get the gradient correction caused by the
center position of the wave-packet as

$\Delta\hat{\mathcal{L}}^{\prime}=\frac{1}{2}[(\hat{x}-x_{c})\cdot\nabla_{x_{c}}\hat{\mathcal{L}}_{c}+\nabla_{x_{c}}\hat{\mathcal{L}}_{c}\cdot(\hat{x}-x_{c})],$
(46)

then the total correction is

$\Delta\hat{\mathcal{L}}_{c}=\frac{1}{2}[(\hat{x}-x_{c})\cdot\nabla_{x_{c}}\hat{\mathcal{L}}_{c}+\nabla_{x_{c}}\hat{\mathcal{L}}_{c}\cdot(\hat{x}-x_{c})]+\frac{1}{2}\Gamma_{\mu\sigma}^{m}\hat{O}_{m}^{\mu\sigma},$
(47)

with the total correction of effective, the approximation of eigenvalue
$\lambda$ is the expectation value of correction of dispersion with a wave-
packet state function,

$\Delta\lambda=\braket{W}{\Delta\hat{\mathcal{L}}_{c}}{W},$ (48)

Still according to [12],

$\Delta\lambda=-Im[\braket{\nabla_{x_{c}}u}{\lambda_{c}-\hat{\mathcal{L}}_{c}}{\frac{\partial
u}{\partial_{k}}}]|_{k=k_{c}}+\frac{1}{2}\Gamma_{\mu\sigma
c}^{m}\braket{\hat{O}_{m}^{\mu\sigma}}|_{k=k_{c}}.$ (49)

##  II lattice connection in one-dimensional spacetime crystal

A more direct model is presented here to help illustrate our theory. Consider
a 1+1D spacetime crystal with arbitrary dispersion, we are going to write a
geodesic line equation with parameter $t$ instead of $\tau$. The two geodesic
line equations are

$\displaystyle\ddot{t}+\Gamma^{0}_{00}\dot{t}^{2}+(\Gamma^{0}_{01}+\Gamma^{0}_{10})\dot{x}\dot{t}+\Gamma^{0}_{11}\dot{x}^{2}=0$
(50)
$\displaystyle\ddot{x}+\Gamma^{1}_{00}\dot{t}^{2}+(\Gamma^{1}_{01}+\Gamma^{1}_{10})\dot{x}\dot{t}+\Gamma^{1}_{11}\dot{x}^{2}=0.$
(51)

Through the relation $\dot{x}=\dot{t}x^{\prime}$ where
$x^{\prime}=\frac{dx}{dt}$, we have the equation of $x$

$x^{\prime\prime}=\Gamma^{1}_{11}+(\Gamma^{1}_{01}+\Gamma^{1}_{10}-\Gamma^{0}_{00})x^{\prime}+(\Gamma^{1}_{11}-\Gamma^{1}_{01}-\Gamma^{1}_{10})x^{\prime
2}-\Gamma^{0}_{11}x^{\prime 3},$ (52)

which indicates the acceleration law in the deformed spacetime crystal.

Another simple model to indicate the true spacetime structure in geodesic line
equation is the one-dimensional spacetime crystal with Hamiltonian

$\hat{H}=\sum_{R}[(V+A\cos(\kappa R-\Omega
t))\ket{R}\bra{R}+h(\ket{R+a}\bra{R}+h.c)],$ (53)

where the lattice $a$, frequency $\Omega$ and $\kappa$ are all dependent in
space and time. The lattice vectors are $(\frac{2\pi}{\Omega},0)$ and
$(\frac{\kappa\cdot a}{\Omega},a)$. The reciprocal lattice vectors are
$(\frac{\Omega}{2\pi},\frac{\kappa}{2\pi})$ and $(0,-\frac{2\pi}{a})$. The
lattice connection of this lattice is

$\displaystyle\Gamma^{0}_{00}=-\frac{\pi^{2}\partial_{t}\Omega}{\Omega},\Gamma^{0}_{01}=-\frac{\pi^{2}\partial_{x}\Omega}{\Omega},\Gamma^{0}_{10}=\frac{-\pi^{2}\kappa\partial_{t}\Omega}{\Omega^{2}},\Gamma^{0}_{11}=\frac{-\pi^{2}\kappa\partial_{x}\Omega}{\Omega^{2}};$
(54)
$\displaystyle\Gamma^{1}_{00}=\frac{2\pi(a\Omega\partial_{t}\kappa+\kappa(\Omega\partial_{t}a-a\partial_{t}\Omega))}{a\Omega^{2}}$
(55)
$\displaystyle\Gamma^{1}_{01}=\frac{2\pi(a\Omega\partial_{x}\kappa+\kappa(\Omega\partial_{x}a-a\partial_{x}\Omega))}{a\Omega^{2}}$
(56)
$\displaystyle\Gamma^{1}_{10}=\frac{2\pi\partial_{t}a}{a},\Gamma^{1}_{11}=\frac{2\pi\partial_{x}a}{a}.$
(57)

The parameters are constrained by both the elastic condition which guarantees
the symmetry of the lattice connection and the gradient of dispersion is zero.
By substituting all the connections, we can obtain the first-order geodesic
line without Berry curvature. For the space part equation, it is

$x^{\prime\prime}-\frac{2\pi\Omega\partial_{t}a}{a\kappa}-\frac{2\pi(a\Omega\partial_{t}\kappa+\kappa(\Omega\partial_{t}a-a\partial_{t}\Omega))}{a\Omega^{2}}x^{\prime}-\frac{2\pi(a\Omega\partial_{x}\kappa+\kappa(\Omega\partial_{x}a-a\partial_{x}\Omega))}{a\Omega^{2}}x^{\prime}-\frac{2\pi\partial_{x}a}{a}x^{\prime
2}=0.$ (58)

This equation is the acceleration law in the 1+1D oblique spacetime crystal.


# detection and identification accuracy of PCA-accelerated real-time
processing of hyperspectral imagery

###### Abstract

Real-time or near real-time hyperspectral detection and identification are
extremely useful and needed in many fields. These data sets can be quite
large, and the algorithms can require numerous computations that slow the
process down. A common way of speeding up the process is to use principal
component analysis (PCA) for dimension reduction. In the reduced dimensional
space, provided by a subset of the principal components, fewer computations
are needed to process the data resulting in a faster run time. In this paper,
we propose a way to further decrease the time required to use PCA by
investigating how many principal components may be omitted with minimal impact
on the detection rate. Using ACE to perform the detection, and then
probability, and spectral fit for identification, we find that the number of
principal components can be reduced by a substantial amount before seeing a
noticeable change in detection rates.

Index Terms—  Hyperspectral data, Principal component analysis, Target
detection and identification

##  1 Introduction

Military, industry, and humanitarian organizations use hyperspectral imagery
to detect and identify targets of interest [1]. Ideally, the identification
occurs in real-time; however, the necessary data cleaning and detection
algorithms can be both time and resource costly making the real-time aspect
slower than desired. While there are different ways to adjust the algorithms
and data to speed up calculations, this usually is accomplished by reducing
either the number of computations or the size of the data. Dimension reduction
is a method of projecting a data set into a lower dimensional space without
losing valuable information. One common way of doing dimension reduction is
principal component analysis (PCA)[2].

To help speed up computations we want to use a minimal number of principal
components (PCs) without sacrificing information. Standard PCA algorithms
naturally ordered the PCs by importance and thus each additional PC used has
diminishing returns. We propose that the latter PCs may not be worth the time
to process them. The question then becomes, when to stop calculating PCs. [3].
In this paper, we will look at the target detection rates and identification
accuracy when we change only the number of PCs.

##  2 Data Summary

For this research, the hyperspectral data collected from Aberdeen Proving
Ground’s H-Field from August 18th to 31st 1995 was used. This data set has a
total of 77 different types of targets, which are a mix of vehicles and target
panels. The data was collected with the Hyperspectral Digital Imagery
Collection Experiment (HYDICE). It collected 210 channels from 0.4 $\mu m$ to
2.5 $\mu m$ [4]. After removing bands that were hindered by the atmosphere
there were 145 bands, each of which creates a principal component.

##  3 Detection Summary

ACE, a well-established algorithm for target detection, stands for adaptive
coherence estimator and is defined by Formula 1 where $\sum$ is the covariance
matrix, $t$ is the target of interest to detect, and $x$ is the pixel you are
looking at. [5]

$ACE(x)=\frac{x\sum{t}^{-1}}{\sqrt{x\sum{x}^{-1}}\sqrt{t\sum{t}^{-1}}}$ (1)

This formula returns a score of how likely the pixel contains the target
material. If the ACE score passes our threshold, set to 0.5, then we labeled
that position a region of interest (ROI). Once a ROI has been detected as a
possible target it is passed to an identification step where spectral fit and
probability techniques are used[6] [5]. In this identification step, the pixel
is then compared against the background and confusers as well as the target
spectra. The background is found by looking at pixels around the ROI with low
ACE scores and confusers are a set of objects known to look similar to the
target. This step concludes with the pixel classified as either a real target
or a non-target.

It is important to remember that this sample is strictly looking at man-made
materials therefore it may not transfer well to other types of targets such as
organic materials. However, with 77 different types of targets, we believe
that this data set provides a good sample of varying object materials and is
representative of most hyperspectral datasets.

##  4 PCA summary

Principal component analysis determines which ratio of variables creates the
most variance in the data set and therefore the most important direction of
spread, which is designated as the first PC. Then the algorithm looks for the
next largest variance by looking at the directions perpendicular to all
previous PCs. Once the next PC is determined, the process continues.

The PCA, shown in Formulas 2 and 3, is used to find the column vector which is
called the principal component of the image. There is a column for each PC
that is used.

$W=P\sqrt{D^{-1}}$ (2)

The diagonal matrix, $D$, has the eigenvalues of the covariance matrix on the
diagonal. The $P$ matrix is built such that its column vectors are the
eigenvectors of the covariance matrix. We can use the $W$ matrix for the
whitening transform shown in Formula 3. This also requires the average of the
image’s spectra, $\mu$, and $a$ the particular spectra of interest. [2]

$\hat{a}=W^{T}(\textbf{a}-\mu)$ (3)

Only considering directions perpendicular to the previous PCs, the algorithm
ensures any new PCs do not include variance in the previously determined
directions. This implies the current PC focuses on variance in unaccounted for
directions by the PCs which preceded it.

##  5 Primary Results

The PCA already orders components by importance and the principal components
which are calculated near the end of the process may be omitted to provide a
more efficient run time with the goal of providing real-time identification.
We ran the ACE algorithm to perform the detection on the data while varying
the PC numbers from 5 to 145 in increments of 5 PCs. In the Figures 1 and 2,
the ACE scores are shown for each number of PCs used. In Figure 1 the green
line represents the mean of the ACE score of a real target found, with the
shaded region being one standard deviation away from the mean. In this case,
the ROIs were confirmed by the identification step. The blue line is the mean
non-target ACE score, plotted with a shaded region of one standard deviation
in either direction. With the blue line, the ROIs were determined not to be
real targets and were false detections.

There is an inconstancy seen with the usage of 30 PCs. In this case, the ACE
and identification methods only found a single ROI. Our initial analysis
showed this was a result of how the algorithms worked with this data set and
the drop with subsequent jump should be ignored for the purposes of this
paper.

Fig. 1: A envelope plot of ACE scores for targets and non-targets across PC
counts ranging from 5 to 145 increasing by increments of 5.

To look at this analysis in more detail, the exact ACE score for a single
object is shown in Figure 2 while the number of PCs used varies. Each green
line represents a different target, and each blue line represents a different
non-target.

Fig. 2: The ACE scores for targets and non-targets across PC counts ranging
from 5 to 145 by increments of 5 for each individual target.

Overall, there is a considerable difference between 5 and 10 PCs looking over
a 5 PC increment; however, there is a minimal difference between 100 and 105
PCs over the same sized increment. This validates our earlier claim that we do
not need to calculate all the PCs to detect and identify the ROI.
Nevertheless, there does seem to be some optimal number of PCs, in this case
around 40 PCs, which are required for accurate classification. Also, note that
the number of PCs does not affect the target ACE score as much as it impacts
the ACE score for non-targets. In cases where the hyperspectral detection is
strictly used to detect targets, a lower number of PCs may be used.

###  5.1 Visual Explanation

To further observe the differences between the PCs used and targets or non-
targets identified we plotted the ROI locations on the visible image in
Figures 3 and 4. The red pixels in the image are all the locations with an ACE
score above our threshold of 0.5 given the full number of PCs. Recall, not all
of these ROIs will pass the identification step, which is why objects like
parts of the road are detected even though there is not an actual target
there. Each $\times$ in the image represents the center of a ROI and the color
denotes the number of PCs used to find the ROIs. Ideally, the ROIs should be
on the red marks. When an ROI is found on a part of the image that is not on
top of a red pixel, the process is finding a false detection which would not
have found using the full 145 PCs. In general, when the PC count is too low,
the false detection rate rises.

To process the data in a reasonable amount of time the number of ROIs
processed for identification were limited to the first 100. This is the reason
there are targets that went undetected. The scope of this paper was to compare
how the number of PCs used affects the ROIs and not to identify every target,
using a 100 ROIs limit will not affect the outcome.

Figure 3 shows the results when using 130 to 145 PCs. In this case, the ROIs
tend to be lined up on the targets most notably in the center of the figure
where the targets are positioned in rows and columns. Also, note there is not
a large difference between the purple, green, and yellow marks, indicating
that in the 130 to 145 PC range, including the 15 additional PCs did not
improve the detection of ROIs by much.

Fig. 3: ROIs plotted on the ground truth for PC counts from 130 to 145
compared with the full 145 PCs.

We can then look at the same view, using 5 to 25 PCs in Figure 4. We can see
the ROIs are much more scattered correlating with our previous results, Figure
1. In Figure 4, the purple, green, and yellow marks are more spread out
implying, the increment from 5 to 25 PCs is important and cannot be ignored.
In addition, the false detection rate is much higher.

Fig. 4: ROIs plotted on the ground truth for PC counts from 5 to 25 compared
with the full 145 PCs.

There is a significant difference in the ACE scores for a lower number of PCs;
however, at some point, increasing the number of PCs does not affect the
output of the detection step. This is redundancy that should be removed to
minimize the computations needed for detection.

###  5.2 Individual Targets

Lastly, we looked how particular targets are affected by the number of PCs
used. We took each ROI that was identified as a target when using the full 145
PCs and called each one an object. Then we looked at every ROI found within
$\pm$ 3 pixels of the center of the object for any PC amount. We plotted an
$\times$ in the center of each of these ROIs on the following figures. Since
we are only considering a single object for each figure, if PCs do not affect
the detection and identification then there should be no change in the
outputs. Therefore, by looking at the changes for different amounts of PCs we
can see how the PC counts are affecting the algorithms, Figures 5 through 9.

In Figure 5(a), the ACE scores of the ROIs are broken into targets in blue and
false detections in orange. The spectral fit, seen in Figure 5(b), and
probability, shown in Figure 5(d), are part of our identification process.
These panels show the probability of that the ROI is the identified target and
the spectral fit of that ROI to the identified target. Figure 5(e) shows which
material the ROI had the highest probability of being identified as when
compared to our target library. The visual location of the target, Figure
5(c), and a more detailed visual, 5(f), are provided. As before, each $\times$
is the center of the ROI for a particular number of PCs used and the red
pixels are the targets found using the full set of PCs.

In Figure 5 the target can be found with most PC levels tested, but the
reliability and quality of the detection start to decrease when fewer PCs are
used. In Figure 5(e) the material identification for this ROI jumps between 3
different classifications but this ROI is correctly identified as F8, with all
levels of PCs. The designation after F8 represents different measurements of
the same material.

Fig. 5: An example of a standard target panel for a single target to
demonstrate how this target’s detection and identification change for
different PC used.

In Figure 6 the small target goes undetected when using less than 70 PCs. In
general, the analysis of the whole data set suggests using 40 PCs would be
sufficient for detection; however, such a low PC would be problematic for this
target or targets of similar size.

Fig. 6: A small object completely undetected when fewer than 70 PCs are used.

Figure 7 is an example of another a small target. Due to the small size of the
target, detection becomes unreliable when using lower levels of PCs. With a
small number of PCs the identification is inaccurate, predicting a different
type of material for the target. In this case, the material identification
improves when a large number of PCs are used.

Fig. 7: A small target with good detection and identification in higher PC
levels which becomes harder to detect and identify below 40 PCs.

With Figure 8, the target can be consistently detected with a lower number of
PCs but the false detection rate climes with fewer PCs. In this particular
case, there are two materials for identification on the target, a vehicle with
a fabric net, which may be why this ROI identification seems sensitive to
changes in the level of PCs used.

Fig. 8: A vehicle target with two materials demonstrates identification issues
below 40 PCs.

Lastly, the target in Figure 9 is another type of vehicle. In this case, the
target is always identified as the correct type of object; however, its
detection rate is not consistent until about 35 PCs. It is also notable how
much the spectral fit fluctuates with different numbers of PCs.

Fig. 9: A vehicle target with good detection though poor identification for
all levels of PCs used.

##  6 Conclusion

Using the full number of PCs is redundant and the process may be sped up
without sacrificing detection by using a fewer number of PCs. In general,
using about 40 PCs does not seem to cause the ACE algorithm to miss the ROIs.
However, reducing the number of PCs used does cause higher ACE scores for non-
targets.

Although the probabilities are consistent across the different numbers of PCs
the spectral fit tends to not work as well when using fewer PCs. This may be
due to the pixels used to detect the background since those pixels are also
based on the ACE scores. It is possible that one could account for this by
adjusting the thresholds that are used to determine ROIs by growing and
masking pixels for spectral fit. This would reinforce the spectral fit which
should improve the identification output.

Finally, it is necessary when deciding how many PCs to use to determine if
target detection is appropriate or if further identification is important.
This may drastically impact the number of PCs required and will need to be
balanced with the calculation speed, especially in a situation where real-time
detection is needed.

##  7 Acknowledgments

Thanks to Geospatial Technology Associates for access to the raw data used in
the paper.

## References

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