# Controlling effective field contributions to laser-induced magnetization
precession by heterostructure design

J. Jarecki  Institut für Physik & Astronomie, Universität Potsdam, 14476
Potsdam, Germany  Max-Born-Institut für Nichtlineare Optik und
Kurzzeitspektroskopie, 12489 Berlin, Germany  M. Mattern  mamattern@uni-
potsdam.de Institut für Physik & Astronomie, Universität Potsdam, 14476
Potsdam, Germany  F.-C. Weber  Institut für Physik & Astronomie, Universität
Potsdam, 14476 Potsdam, Germany  J.-E. Pudell  Institut für Physik &
Astronomie, Universität Potsdam, 14476 Potsdam, Germany  European X-Ray Free-
Electron Laser Facility, Schenefeld, Germany  Helmholtz-Zentrum Berlin für
Materialien und Energie, BESSY II, 12489 Berlin, Germany  X.-G. Wang  School
of Physics and Electronics, Central South University, Changsha 410083, China
J.-C. Rojas Sánchez  Institut Jean Lamour (UMR CNRS 7198), Université
Lorraine, 54000 Nancy, France  M. Hehn  Institut Jean Lamour (UMR CNRS 7198),
Université Lorraine, 54000 Nancy, France  A. von Reppert  Institut für Physik
& Astronomie, Universität Potsdam, 14476 Potsdam, Germany  M. Bargheer
Institut für Physik & Astronomie, Universität Potsdam, 14476 Potsdam, Germany
Helmholtz-Zentrum Berlin für Materialien und Energie, BESSY II, 12489 Berlin,
Germany

###### Abstract

We experimentally demonstrate how nanoscale heterostructure design controls
the effects of laser-induced heat and strain on the magnetization precession.
By introducing insulating MgO layers into metallic Pt-Cu-Ni heterostructures
we block the propagation of hot electrons to control the magnetization
dynamics by tailoring the spatio-temporal shape of the transient heat and
strain, which we quantify by ultrafast x-ray diffraction experiments. We
access the corresponding magnetization precession in these heterostructures
using the time-resolved magneto-optical Kerr effect in polar geometry for
varying magnetic field orientations. Our analysis and modeling highlights the
importance of quasi-static strain as a driver of precession, which counteracts
the demagnetization-induced precession in situations where the magnetic
material is rapidly heated via electrons. Tailored strain pulses shaped via
nanoscale heterostructure design paves the way for opto-magneto-acoustic
devices prepared by sputtering.

##  1 Introduction

Ultrafast excitation of ferromagnets triggers phenomena such as ultrafast
demagnetization[[1, 2, 3, 4, 5]], coherent magnetization precession[[6, 7, 8,
9, 6, 10, 11, 12, 13, 14, 15, 16, 17, 18]], spin-transport[[19, 20, 21]] and
all-optical magnetization switching[[22, 23, 24]]. An optimized excitation of
coherent magnetization precession enables precessional switching[[25, 26, 27,
28]]. Most investigations have focused on mechanisms using direct excitation
of the magnetically ordered material, to drive precession via magnetic
anisotropy changes or via magneto-elastic coupling: magnetocrystalline
anisotropy is controlled by temperature[[6, 7, 8, 9]], shape anisotropy by
demagnetization[[6, 10, 12]], and strain effects are distinguished as quasi-
static expansion[[11, 12]] and propagating strain pulses[[13, 14, 15, 16, 17,
18, 27, 29]].

In a single ferromagnetic layer, the optically deposited energy leads to all
at once: Ultrafast demagnetization, temperature-induced anisotropy changes and
both quasi-static and propagating strain components, which are all a
consequence of the rapid rise of electron-, spin- and phonon-temperatures.
Each of these mechanisms can be described as pump-induced changes of effective
field contributions that drive coherent magnetization precession. Analysis and
control of the induced precession has been mainly limited to the external
magnetic field[[15, 6]], the layer thickness determining the round trip time
of acoustic strain pulses[[14, 18, 29]] or multipulse excitation schemes
tuning the strain pulse pattern[[18, 30]].

Integrating the functional magnetic layer into nanoscale heterostructures adds
multiple control scenarios: A precise adjustment of its excitation by
tailoring the heat transport and an independent tuning of the propagating
picosecond strain pulses. This includes transparent capping layers[[31, 32]],
metal-insulator superlattices[[33, 34]] and functional transducers with
tuneable stress generation mechanisms via temperature and fluence[[35, 36,
37]] or magnetic fields[[38]]. In addition, the combination of metals with
different electron-phonon coupling strength and inserting insulating
interlayers modifies the heat transport within the heterostructure[[39, 40,
41, 42, 43, 44, 45]] that enables the control of both the excitation of the
functional magnetic layer and the launched propagating strain pulses.

Recently, metallic heterostructures built of an optically excited transducer,
a propagation layer with weak electron-phonon coupling and a buried functional
magnetic layer have become intensively investigated[[43, 46, 47, 48, 49, 20,
44, 45, 50, 51, 19]]. Laser-excited hot electrons travelling through the
propagation layer induce ultrafast demagnetization in buried metallic magnetic
layers[[50, 51, 44, 45]], drive magnetization switching[[49, 20]] and excite
precession[[46]] and higher order standing spin waves in buried dielectric
ferromagnetic layers[[47, 48]]. Previous experiments have reported a
controllable attenuation and elongation of this hot electron pulse via
additional metallic interlayers exhibiting strong electron-phonon
coupling[[44, 45]].

Here, we tailor the rapid distribution of the optically deposited energy
within a Pt-Cu-Ni heterostructure by inserting an insulating MgO interlayer
that stops the heat transport via electrons. We quantify the resulting strain
response of the metallic layers via ultrafast X-ray diffraction (UXRD), to
deduce the tailored heat transport and the shape of the strain pulses. The
transient response for the buried functional Ni layer strongly depends on the
position of the MgO interlayer either before or after the Cu layer. Utilizing
the polar time-resolved magneto-optical Kerr effect (trMOKE), we probe the
laser-induced change of the out-of-plane magnetization and identify the
mechanisms (demagnetization, quasi-static strain and propagating strain),
which drive coherent precession as function of the angle of the external
magnetic field provided by a rotatable permanent magnet. We observe a
pronounced dependence of the angle-dependent precession amplitude on the
heterostructure design, which we capture by a model of the effective field as
input for the Landau-Lifshitz Gilbert (LLG) equation. We discuss the concepts
of how heterostructure design can control magnetization precession: The
thickness of the propagation layer – and hence the total multilayer stack –
sets the round trip time, which can be made resonant with the precession
frequency that is tuned via the external magnetic field angle. The position of
the MgO interlayer controls the thickness of the metal transducer and hence
tailors the shape of the strain pulses, which are most efficient in driving
the magnetization precession by phase-matching the timing of both expansive
and compressive strain to the phase of the precession. Generally the MgO
interlayer disables the ultrafast demagnetization and quasi-static expansion
of the Ni layer as driving mechanisms that otherwise drive precession
efficiently for out-of-plane external fields.

Our report on the combined study of trMOKE and UXRD experiments is structured
as follows: Section 2 presents the magnetization dynamics triggered in three
Pt-Cu-Ni heterostructures that only differ in the position of an insulating
MgO layer. Section 3 shows how UXRD quantifies the transient strain and heat
in these heterostructures, before Section 4 discusses a model based on the LLG
equation, where the extracted strain profile is used as an input parameter for
the transient effective magnetic field.

Figure 1:  Magnetization precession depending on the orientation of the
external magnetic field: a) Sketch of the trMOKE experiment including a
rotatable permanent magnet providing a magnetic field of
$\mu_{\mathrm{0}}H_{\mathrm{ext}}=390\,\text{mT}$ at an angle $\xi$ with
respect to the sample normal. b) Schematics of the three heterostructures with
the Pt cap layer as the main light absorber and Cu as propagation layer for
hot electrons. Sample 1 is a purely metallic structure and samples 2 and 3
contain an MgO layer to stop electron propagation. c) Polar trMOKE measured
for sample 1 for a full rotation of the external magnetic field by
$360^{\circ}$. d) Fourier-transformation of the trMOKE signal shows an
increase of the precession frequency with in-plane orientation of the external
field and a maximum FFT amplitude $A$ for $0^{\circ}$ and $47^{\circ}$. e,f)
Same for samples 2 and 3, where only the maximum at $47^{\circ}$ is observed.

##  2  Observation of the magnetization precession by trMOKE

In the all-optical trMOKE experiment sketched in Figure 1a, the Pt-Cu-Ni
heterostructures are excited from the Pt side by an $100\,\text{fs}$ pump-
pulse with a central wavelength of $800\,\text{nm}$ at normal incidence with a
repetition rate of $0.5\,\text{kHz}$, a beam footprint of $580\times
480\,\text{\textmu m}^{2}$ and a pump energy density of
$7.5\,\text{mJ}/\text{cm}^{2}$. We probe the transient out-of-plane
magnetization of the Ni film through the glass substrate by detecting the
polarization rotation of a reflected $400\,\text{nm}$ probe pulse. A
commercial rotatable Neodymium permanent magnet (dimension: $60\times 30\times
15\,\text{mm}^{3}$) provides an external magnetic field of
$\mu_{\mathrm{0}}H_{\mathrm{ext}}=390\,\text{mT}$ at the probed sample region
under an angle $\xi$ with respect to the surface normal. All three
investigated heterostructures shown in Figure 1b have the same structure of a
$7$ to $10\,\text{nm}$ thick Pt cap layer that absorbs the pump pulse, a thick
Cu transport layer and a $22\,\text{nm}$ buried Ni layer. The samples mainly
differ by the ultrafast energy transfer between the Pt and Ni layer, which is
supported by the electrons in the $95\,\text{nm}$ Cu film (sample 1). Samples
2 and 3 suppress the electronic heat transport by an insulating MgO
interlayer, either before or after the Cu layer. The $8\,\text{nm}$ amorphous
Ta seed layer ensures textured crystalline growth of Ni, Cu and Pt with
(111)-orientation via sputtering on the glass substrate (Corning 1737 AMLCD).

Figure 1c shows the polar trMOKE signal, i.e. the transient change of the out-
of-plane magnetization $\Delta M_{z}$ in $\Delta\xi=5^{\circ}$ steps for a
full rotation of the magnet for sample 1. The variation from out-of-plane
($0^{\circ}$ and $180^{\circ}$) to in-plane ($90^{\circ}$ and $270^{\circ}$)
external magnetic fields shows a fourfold symmetry of the precession.
Therefore, we only discuss the range from $0$ to $90^{\circ}$ in the
following. The Fourier transform of these data in Figure 1d reveals an
increasing precession frequency as the out-of-plane component of the external
field decreases and the magnetization tilts towards the sample plane, which
represents the magnetic easy axis. Since the external magnetic field is below
the saturation field $\mu_{\mathrm{0}}H_{\mathrm{sat}}=620\,\text{mT}$, the
magnetization is always oriented along an angle between the external field and
the sample plane before laser excitation. Figure 1e,f show the same dependence
of the precession frequency on $\xi$ for samples 2 and 3. The color code for
the precession amplitude shows that the presence of the MgO layer in these
samples suppresses the excitation of precession for an out-of plane
orientation of the external field ($\xi=0^{\circ}$) which is pronounced for
the purely metallic sample 1. All three samples, however, exhibit a maximum
amplitude around $\xi=47^{\circ}$, with considerably smaller amplitude for
sample 3.

Figure 2 directly compares the driven magnetization precession in the three
magnetic heterostructures: The extracted frequency in panel b confirms that
the angle-dependence of the precession frequency is the same for all three Ni
layers. Panel a provides a comparison of the precession amplitudes extracted
from the FFT. Indeed all samples show a maximum at the external magnetic field
angle $\xi=47^{\circ}$ for which the precession frequency matches the inverse
round-trip time $T=L/v_{\text{s}}$ of longitudinal strain pulses through the
multilayer stack (dashed horizontal line), which is essentially given by the
total thickness $L$ and the thickness-weighted average sound velocity
$v_{\text{s}}$. While in the presence of an MgO interlayer (samples 2 and 3)
the precession amplitude rapidly decreases for larger and smaller $\xi$ values
until it completely vanishes, we observe an additional increase of the
precession amplitude towards $\xi=0^{\circ}$ in the absence of an MgO
interlayer. These systematics already hint at the fact that the maximum at
intermediate angles is a magneto-acoustic resonance that links the precession
to the propagating strain pulses. In contrast, the low frequency precession
close to $\xi=0^{\circ}$ is only enabled by the rapid heat transfer into Ni
via hot electrons that leads to ultrafast demagnetization and expansion.

The time-domain out-of-plane magnetization response (symbols) shwon in Figure
2c,d provides further insights into the laser-induced driving of the
magnetization precession. In the absence of an MgO interlayer, we clearly
observe an ultrafast demagnetization originating from the rapid excitation of
the Ni layer by hot electrons. This rapid excitation drives a precession at
$\xi=5^{\circ}$, which is absent in samples 2 and 3, where electrons are
stopped by the MgO interlayer and we only observe a slow demagnetization on
hundreds of picoseconds (yellow and blue transients in Figure 2). At
$\xi=47^{\circ}$ we observe pronounced oscillations of varying amplitude for
all three heterostructures. The precession amplitude even slightly increases
from the first to the second maximum indicating the periodically acting
driving force by propagating strain pulses. In addition, we note an immediate
start of the precession upon laser-excitation in sample 1 and 2, while sample
3 clearly responds not earlier than at about $20\,\text{ps}$, which is the
time it takes acoustic strain pulse to travel from the excited Pt cap layer
through the Cu layer.

Figure 2:  Heterostructure-dependent magnetization precession: a) Precession
amplitude $A$ as function of the orientation of the external magnetic field.
The extracted experimental data (symbols) are well reproduced via the LLG
model described in Section 4 (solid lines) for all three heterostructures. The
black solid line denotes the hypothetical precession amplitude for sample 1
without any contribution from propagating strain pulses. b) Measured
precession frequency $f$ (symbols) that is independent of the heterostructure,
since all structures share the same Ni layer. c,d) Laser-induced change of the
out-of-plane magnetization $\Delta M_{z}$ that displays the heterostructure-
dependent demagnetization and magnetization precession at $\xi=5^{\circ}$ and
$\xi=47^{\circ}$, respectively. The solid coloured lines represent the
modeling described in Section 4, while the grey solid lines represent the
magnetization precession if we neglect the ultrafast demagnetization as driver
of the precession in sample 1. The results are offset for clarity.

To rationalize the origin of the heterostructure-dependent driving of the
precession, we quantify the laser-induced strain response of the Ni layer and
its transient temperature increase by UXRD. Using the measured transient
temperature rise and the laser-induced strain as input for our LLG model
described in Section 4 it nicely captures the heterostructure-dependent
precession (solid lines in Figure 2) and unambiguously identifies the role of
the different driving mechanisms.

##  3  Characterization of strain and heat via UXRD

In the UXRD experiment, the Pt-Cu-Ni heterostructures are excited by
femtosecond p-polarized laser pulses with a central wavelength of
$800\,\text{nm}$ as in the trMOKE experiment. We probe the transient layer-
specific expansion by the shift of the layer-specific Bragg peaks with the
reciprocal space slicing method[[52]] at a laser-driven table-top plasma X-ray
source[[53]]. The incident angle $\alpha_{\text{in}}$ of the
$200\,\text{fs}$-long hard X-ray pulses with a photon energy of $\approx
8\,\text{keV}$ is kept fixed at $19.5^{\circ}$ to probe the strain response of
the Pt layer and at $22.1^{\circ}$ for the Cu and Ni layers. At an
intermediate angle we are able to record all three Bragg peaks
simultaneously[[42]] on a pixelated area detector as displayed in the detector
inset of Figure 3a that sketches the experimental geometry with the pump beam
incident at $50^{\circ}$ with respect to the sample normal.

Figure 3:  UXRD quantifies heat transport and layer-specific strain: a) Sketch
of the UXRD geometry: The detector records layer-specific Bragg peaks
separated along the reciprocal space coordinate $q_{z}$. b) Integration of the
detector signal along $q_{y}$ yields a reciprocal space slice, which encodes
the mean out-of-plane lattice constant of the Pt, Cu and Ni layer within the
heterostructures. c) Laser-induced strain response of the Pt layer (symbols)
determined from the transient shift of the Bragg peak. The excitation fluence
of $13.2\,\text{mJ}/\text{cm}^{2}$ for sample 1 and 2 was reduced to
$7.5\,\text{mJ}/\text{cm}^{2}$ for sample 3 to avoid damage to the MgO
interlayer. The solid lines represent our model of the transient strain
response utilizing the modular Python library udkm1Dsim[[54]] and a single set
of thermo-elastic parameters collected in Table 1. The strain responses of the
Pt capping layer are offset for clarity. d,e) Same for the Cu and Ni layers.
Note the clear difference of the initial response of the Ni layer, which is
expansive for sample 1, compressive for sample 2 and zero for sample 3.

Figure 3b displays the Bragg peaks of the $(111)$-oriented Pt, Cu and Ni
layers of sample 1. Their positions along the reciprocal space coordinate
$q_{z}$ encodes the mean out-of-plane lattice constant $d$ of the respective
layer via $q_{z}=2\pi/d$. The separated Bragg peaks enable a layer-specific
probing of the laser-induced expansion via the shift of the respective Bragg
peak along $q_{z}$. This peak shift yields the layer-specific transient strain
$\eta=\Delta d/d_{0}$ as relative change of the mean out-of-plane lattice
constant $\Delta d$ with respect to its value $d_{0}$ before excitation.

Figure 3c–e display the layer-specific strain responses (symbols) to an
excitation of $13.2\,\text{mJ}/\text{cm}^{2}$ for sample 1 and 2 and
$7.5\,\text{mJ}/\text{cm}^{2}$ for sample 3 choosen to avoid damage to the MgO
interlayer. The solid lines represent the modeled transient strain response of
the heterostructures utilizing the modular Python library udkm1Dsim[[54]] and
a single set of thermo-physical parameters stated in Table 1. To model the
strain response, we solve a diffusive two-temperature model[[43, 42]] to
determine the spatio-temporal distribution of the laser-deposited energy
within the heterostructures, which linearly relates to a spatio-temporal
laser-induced stress via subsystem-specific Grüneisen parameters. The solution
of the elastic wave equation for these stress contributions yields the spatio-
temporal strain response, which is translated into Bragg peak shifts via
dynamical X-ray diffraction theory that yield the transient layer-specific
strain as in the UXRD experiment. Details of our model approach were reported
previously[[42]].

| Pt | Cu | Ni | Ta | MgO | glass
---|---|---|---|---|---|---
$\gamma\textsuperscript{S}$ (mJ cm-3 K-2) | 0.73[[55]] | 0.10[[56]] | 1.06[[55]] | 0.38[[57]] | - | -
$C\textsubscript{ph}$ (J cm-3 K-1) | 2.85[[58]] | 3.44 | 3.94[[59]] | 2.33[[57]] | 3.32 [[60]] | 1.80 [[61]]
$\kappa_{\text{el}}^{0}$ (W m-1 K-1) | 66[[62]] | 396[[55]] | 81.4[[55]] | 52.0 | - | -
$\kappa\textsubscript{ph}$ (W m-1 K-1) | 5.0[[62]] | 5.0 | 9.6[[55]] | 5.0 | 50 [[63]](2.5) | 1.0 [[61]]
$g$ (PW m-3 K-1) | 375[[64]] | 95[[56]] | 360[[56]] | 100 | - | -
$\rho$ (g cm-3) | 21.45 | 8.96 | 8.91 | 16.68 | 3.58 | 2.54 [[61]]
$v\textsubscript{S}$ (nm ps-1) | 4.2[[65]] | 5.2[[66]] | 6.3[[67]] | 4.2[[68]] | 9.1 [[69]] | 5.7 [[61]]
$\Gamma$el | 2.4[[70]](1.2) | 0.9[[70]] | 1.4[[71]] | 1.3[[70]] | - | -
$\Gamma$ph | 2.6[[72]] | 2.0[[73]] | 1.8[[71]] | 1.6[[70]] | 1.7 [[74]] | 0.3 [[61]]
Table 1: Thermo-elastic parameters of the strain model taken from the
literature: The Sommerfeld constant $\gamma\textsuperscript{S}$, the specific
heat of the phonons $C\textsubscript{ph}$, the electron-phonon coupling
constant $g$, and the electron $\kappa_{e}^{0}$ and phonon
$\kappa\textsubscript{ph}$ heat conductivity determine the spatio-temporal
energy distribution upon laser-excitation in the framework of a diffusive two-
temperature model[[43, 42]]. The subsystem-specific Grüneisen parameters
$\Gamma$el and $\Gamma$ph linearly relate the spatio-temporal energy density
to an elastic stress on the lattice driving a quasi-static expansion and
strain pulses propagating with sound velocity $v\textsubscript{S}$ according
to the elastic wave equation[[42]]. Values in brackets are optimized values
for the strain modeling. The significant reduction of the thermal conductivity
of the MgO interlayer is related to a Kapitza interface resistance.

In all three heterostructures, the optical energy is mainly deposited to the
Pt layer[43]. The resulting expansion of the Pt layer (Figure 3c) peaks at
$1.5\,\text{ps}$, when the expansive strain pulses launched at the surface and
the Pt-Cu interface have propagated through the Pt layer at the speed of sound
$v_{\text{Pt}}=4.2\,\text{nm/ps}$[[65]]. The following strain oscillation can
be interpreted as a breathing of the Pt layer with a
$T_{\text{Pt}}=2L_{\text{Pt}}/v_{\text{Pt}}\approx 3\,\text{ps}$ period
determined by the layer thickness $L_{\text{Pt}}$. The partial transmission of
strain pulses through the impedance mismatched Pt-Cu or Pt-MgO interfaces
leads to a decay of the oscillation of the Pt layer thickness within tens of
picoseconds. This propagating strain pulse is superimposed on the quasi-static
expansion of Pt due to heating. While the oscillatory Pt response is very
similar for all three heterostructures, the quasi-static expansion is much
increased for sample 3 where the MgO layer keeps the thermal energy within Pt,
because it prohibits electronic heat transport from Pt into the thick Cu film.

This suppression of the rapid excitation of Cu as well as the buried Ni layer
in sample 3 results in a delayed expansion on hundreds of picoseconds due to
slow energy transfer into these layers via phonons. Therefore, the strain
response within the first tens of picoseconds is dominated by the strain pulse
with leading compression driven by the rapid expansion of the Pt layer. This
strain pulse immediately compresses the adjacent Cu layer and subsequently
also compresses the Ni layer when it enters at $20\,\text{ps}$ after having
propagated through Cu. In sample 2, moving the MgO interlayer in front of Ni
enables a rapid distribution of the optically deposited energy among Pt and Cu
via hot electrons[[43]]. This rapid excitation of Cu results in a rapid
expansion (Figure 3d) that shortens the initial compression by the expanding
Pt layer. Additionally, the expansion of Cu drives a bipolar strain pulse that
compresses the buried Ni layer already within the first picoseconds and
superimposes with the strain pulse launched by Pt entering Ni at
$20\,\text{ps}$.

Removing the MgO interlayer enables an additional fast diffusive transport of
hot electrons from the optically excited Pt layer to the buried Ni through Cu
as observed previously[[43]]. Equilibration of the electronic system is very
fast, and the large electronic specific heat capacity and the strong electron-
phonon coupling in Ni originating from the large density of states of flat
$3\text{d}$-bands close to the Fermi level rapidly localizes a large fraction
of the deposited energy in Ni. The large amount of energy deposited to Ni
drives an expansion, launching an additional compression into the adjacent Cu
layer that extends its compression to $6\,\text{ps}$. In total, the strain
response of Ni is the superposition of the quasi-static expansion due to
heating and the strain pulses launched by Cu and Pt, the entering of the
latter at $20\,\text{ps}$ is clearly observable (Figure 3e).

Our model of the layer-specific transient strain response captures these
experimental observations and yields the transient average electron
$T^{\text{el}}_{\text{Ni}}(t)$ and phonon $T^{\text{ph}}_{\text{Ni}}(t)$
temperature as well as the average transient strain $\eta_{\text{Ni(t)}}$ of
the buried Ni layer in all three heterostructures. In the following, these
quantities validated by the good agreement of our model with the UXRD
experiment serve as input for our LLG model of the heterostructure-dependent
magnetization precession.

##  4  Modeling of the magnetization precession and Discussion

In this section, we relate the heterostructure-dependent strain response and
heating of the Ni layer quantified by UXRD in Section 3 to the precession
observed by polar trMOKE in Section 2 and discuss how heterostructure design
can control the driving mechanisms of magnetization precession.

We model the precessional motion of the macrospin vector
$\vec{M}(t)=M_{\text{sat}}(m_{x}(t)\vec{e}_{x}+m_{y}(t)\vec{e}_{y}+m_{z}(t)\vec{e}_{z})$
that represents the average magnetization of the Ni layer in the
heterostructures utilizing the LLG equation:

$\frac{\partial\vec{m}}{\partial
t}=-\gamma\vec{m}\times\vec{H}_{\text{eff}}+\alpha\vec{m}\times\frac{\partial\vec{m}}{\partial
t}\,,$ (1)

where $\gamma$ represents the gyromagnetic ratio and $\alpha$ the
phenomenological Gilbert damping. The effective field
$\vec{H}_{\text{eff}}(t)$ drives the magnetization precession and is
determined by the material-specific free energy of the macroscopic
magnetization $F\textsubscript{M}$ via
$\mu_{0}\vec{H}_{\text{eff}}=-\vec{\nabla}_{M}F\textsubscript{M}$. In our
model of the macrospin in Ni, we consider a Zeeman energy originating from the
external magnetic field $\vec{H}_{\text{ext}}$, a shape anisotropy resulting
from the thin film geometry and a magneto-acoustic contribution induced by the
average out-of-plane strain within the Ni layer $\eta_{\text{Ni}}(t)$:

$\begin{split}\vec{H}_{\text{eff}}(\vec{m},\eta)&=\vec{H}\textsubscript{ext}-\vec{H}\textsubscript{d}-\vec{H}\textsubscript{me}\\\
&=\vec{H}\textsubscript{ext}-M\textsubscript{sat}m_{z}(t)\vec{e}_{z}-\frac{2b_{1}\eta_{\text{Ni}}(t)}{\mu_{0}M\textsubscript{sat}}\,m_{z}(t)\vec{e}_{z}\;.\end{split}$
(2)

In agreement with the literature[[11, 12]] we neglect any magnetocrystalline
anisotropie, which is supported by a negligible in-plane saturation field and
a quantitive agreement of the out-of-plane saturation field with
$\vec{H}\textsubscript{d}$ for the saturation magnetization
$M_{\text{sat}}=4.9\cdot 10^{5}\,\text{A/m}$ characterized via vibrating
sample magnetometry (VSM). The parameter $b_{1}$ determines the strength of
the coupling of the laser-induced strain to the out-of-plane magnetization
$M_{z}$. In addition to the magneto-elastic coupling, our model considers the
demagnetization of Ni as driving mechanism of magnetization precession. The
reduced length of the magnetization vector
$\vec{m}(T^{\text{ph}}_{\text{Ni}}(t))$ as function of the phonon temperature
$T^{\text{ph}}_{\text{Ni}}$ introduces a change of the effective field via the
changing demagnetization field. The transient strain $\eta_{\text{Ni}}(t)$ and
the transient temperature $T^{\text{ph}}_{\text{Ni}}(t)$of the Ni layer are
not adjustable parameters in our model: We use the time dependence that is
quantified from modeling the laser-induced strain response of all three layers
quantified by UXRD. For all three heterostructures, we use the same parameter
set of the gyromagnetic ratio $\gamma/2\pi\mu_{0}=30.8\,\text{GHz/T}$, the
damping $\alpha=0.06$, the magneto-acoustic coupling $b_{1}=-8.2\cdot 10^{6}$
J/m3 and saturation magnetization of $M_{\text{sat}}=4.9\cdot
10^{5}\,\text{A/m}$ characterized via VSM.

Figure 4:  Driving mechanisms of the precession: a) Sketch of the change of
the effective field induced by demagnetization (left) and quasi-static
expansion (right). The dotted lines indicate the maximal excursions of the
magnetization vector. Note that the demagnetization yields a very rapid
decrease on $m_{z}$, and additionally a change $\Delta H_{\text{eff}}$ whereas
the quasi-static expansion only induces a $\Delta H_{\text{eff}}$ with
opposite sign. Note that strain pulses, i.e. time-dependent strain $\eta(t)$
yields similar time-dependent changes of the effective field with both
polarities. b) The measured average strain in the Ni layer of sample 1 is
separated into a strong step-like quasi-static expansion (red), a long bipolar
strain pulse launched by the thick Cu layer (blue) and a sharper strain pulse
originating from Pt (yellow). c) Same for sample 2, where the quasi-static
expansion is missing because MgO prevents energy flow into Ni. d) Same for
sample 3, where only the strain pulse from Pt persists, as the MgO layer
confines the energy there.

With an excellent match to the experimental data, Figure 2a,b displays the
modeled precession frequency and amplitude for all heterostructures as
function of the angle $\xi$ of the external magnetic field. According to
Equation 2 the precession frequency increases with $\xi$ due to an increasing
$H_{\text{eff}}$ as the demagnetization field $\vec{H}_{\text{d}}$ only
compensates the out-of-plane component of the external field while its in-
plane component prevails. A comparison of the model to the time-domain out-of-
plane magnetization reponse in Figure 2c,d confirms also an excellent match
regarding the phase of the precession and the temperature-induced
demagnetization.

Based on our model of the effective field $H_{\text{eff}}$ in Equation 2, we
distinguish two main contributions to the laser-induced change of the
effective field: (i) the demagnetization of Ni originating from the laser-
induced temperature-increase in Ni and (ii) the transient strain in Ni that is
the superposition of a quasi-static expansion and propagating strain pulses.
In the absence of an MgO interlayer, the rapid heating of Ni induces both
demagnetization and quasi-static expansion. Figure 4a illustrates their
counteracting influence on the effective field in Ni. The ultrafast
demagnetization shortens the magnetization vector $\vec{m}$. This reduces the
demagnetization field $\vec{H}_{\text{d}}$ via the reduced out-of-plane
component $m_{z}$. As a result, the effective field change
$\Delta\vec{H}_{\text{eff}}=\Delta\vec{H}_{\text{d}}$ points along the
positive $z$-direction and the shorter magnetization vector precesses around
the new effective field $H_{\text{eff,0}}+\Delta\vec{H}_{\text{eff}}$. In
contrast, the magneto-elastic field $\vec{H}\textsubscript{me}$ induced by the
expansion of Ni causes an additional contribution to the effective field
pointing along the negative $z$-direction. This opposite change of the
effective field results in opposite starting directions of the precession
corresponding to a phase shift of $\pi$. If we neglect the demagnetization as
driving mechanism, the agreement of the model (grey solid line) with the
experimentally observed precession in Figure 2c,d is much worse. Especially,
the ratio of the precession amplitude at $\xi=5^{\circ}$ and $\xi=47^{\circ}$
can not be reproduced. This highlights the important role of demagnetization
to partially counterbalance the effect of quasi-static expansion. The fact
that the precession for sample 1 and $\xi=5^{\circ}$ starts with an additional
decrease of $m_{z}$ after the demagnetization indicates the dominance of the
quasi-static strain over the demagnetization contribution. In Figure 2a, the
black solid line represents the modeled precession amplitude considering only
the quasi-static expansion and demagnetization of Ni in sample 1. The
pronounced deviation from the measurement indicates the crucial role of
propagating strain pulses providing an additional driving force of the
magnetization precession.

To rationalize the heterostructure-dependent precession amplitude, we
reconsider the transient strain response of Ni
$\eta_{\text{Ni}}(t)=\eta_{\text{qs}}(t)+\eta_{\text{p}}(t)$ quantified by
UXRD (Section 3) that is the superposition of a quasi-static strain
$\eta_{\text{qs}}$ and propagating strain pulses $\eta_{\text{p}}$ launched by
rapidly expanding layers. Figure 4b–d dissects the heterostructure-dependent
$\eta_{\text{Ni}}(t)$ into contributions highlighted in different colors.
While for sample 3 (panel d) only a sharp bipolar strain pulse launched by the
expansion of the Pt layer is present, sample 2 (panel c) exhibits a
superposition of this strain pulse from Pt and a pulse with longer period that
is attributed to the expansion of the thicker Cu layer. For the all-metallic
sample 1 (panel b), the ultrafast energy transfer to the Ni layer itself
additionally yields the quasi-static expansion of Ni highlighted in red.

In total, all three layers exhibit propagating strain pulses their (partial)
reflection at the Ta-glass interface and the surface results in a periodic
contribution to $\eta_{\text{Ni}}(t)$ given by the round-trip time through the
metallic heterostructure of $57\,\text{ps}$. This provides a periodic driving
force that most efficiently drives the magnetization precession when it
matches the precession frequency, which results in a maximum precession
amplitude at $\xi=47^{\circ}$ and $f=17.5\,\text{GHz}$ for all three
heterostructures. However, the efficiency of this magneto-acoustic resonance
depends on shape of the strain pulse. The reduced precession amplitude for
sample 3 in comparison to sample 2 highlights that the spatially extended
strain pulse launched by the Cu layer drives the magnetization precession more
efficiently than the short high frequency strain pulse launched by the Pt
layer. We attribute this to the $\approx 40\,\text{ps}$ period of the Cu
strain pulse that leads to an efficient phase matching of the positive and
negative excursions of the strain pulse to the magnetization precession. We
note that the amplitude of the bipolar strain pulse associated with Pt is much
larger than for the Cu layer, due to the higher energy density in Pt. The
repetition of this bipolar pulse with the same round trip time of about
$57\,\text{ps}$ through the entire heterostructure gives rise to considerable
Fourier amplitude at the precession frequency, which is enhanced by the phase
patching. Nonetheless the precession is not effectively driven in sample 3,
because within a small fraction of the precession period the large positive
and negative strain in the bipolar pulse nearly cancel.

In the absence of an MgO interlayer, the rapid energy deposition to Ni via hot
electrons gives rise to a rapidly rising quasi-static expansion and an
ultrafast demagnetization that both serve as additional driving mechanism of
the magnetization precession. Simultaneously, the reduced excitation of Cu
attenuates the launched low frequency strain pulse lowering its contribution
to the transient effective field. This results to a reduced precession
amplitude at $\xi=47^{\circ}$ with respect to sample 2 (cf. Figure 2b).

##  5  Conclusion

In summary, we conducted systematic UXRD and trMOKE experiments to identify
the mechanisms driving magnetization precession in Ni and to demonstrate their
control by tailoring the heat transport within Pt-Cu-Ni heterostructures via
insulating MgO interlayers.

The insulating MgO interlayer stops the heat transport via electrons, which
disables the quasi-static expansion and ultrafast demagnetization as driving
mechanisms by suppressing the rapid excitation of the buried Ni layer. In the
absence of an MgO interlayer, these counteracting mechanisms drive the
precession most efficiently for out-of-plane magnetic field orientations. Our
analysis of the precession phase reveals the quasi-static expansion as the
dominant driver in the explored fluence regime. This efficient magneto-
acoustic mechanism is similarly active at an external field orientation that
brings the precession into resonance with acoustic sound propagation through
the multilayer stack. By considering the actual shape of the strain pulses
quantified by UXRD, we moreover show that in addition to this resonance,
magneto-acoustic efficiency is boosted roughly by a factor $3$ by phase
matching, i.e. when positive and negative strain contributions each act on the
magnetization vector for about half of the precession period.

Our experiment demonstrates the crucial role of laser-induced strain as
driving mechanism of magnetization precession and highlights heterostructure
design as a promising approach to tune the efficiency of driving the
magnetization precession that is central to pave the way towards coherent
magnetization switching via strain. In this publication, the heterostructure
serving as effective tuneable transducer consists of simple metals and an
insulating layer blocking the heat transport via electrons. In future
applications, the controllable manipulation of buried layers by
heterostructures may be extended by utilizing materials with stress generation
mechanisms or heat transport properties tunable by external parameters such as
temperature or electric and magnetic fields.

###### Acknowledgements.

We acknowledge the DFG for financial support via No. BA 2281/11-1 and Project-
No. 328545488 – TRR 227, project A10. We thank Jamal Berakdar (project B06)
for fostering the discussions about modeling the magnetization dynamics.

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#  _Seeing the Bigger Picture:_ Improving Otsu’s Thresholding Method of Global
Binarization Using Ring Theory for Ultrasonographies of Congestive Heart
Failure

Alisa Rahim  Liberal Studies Department, New York University (NYU), 726
Broadway, 6th Floor, New York, NY 10003.  Esley Torres  Centro de
Investigación en Ciencias (CINC), Instituto de Investigación en Ciencias
Básicas y Aplicadas (IICBA), Universidad Autónoma del Estado de Morelos
(UAEM), Mexico.

###### Abstract

Ring Theory states that a ring is an algebraic structure where two binary
operations can be performed among the elements: addition and multiplication.
Binarization is a method of image processing where values within pixels are
reduced to a scale from zero to one, with zero representing most absence of
light and one representing most presence of light. Currently, sonograms —
computer images composed by movement of sound waves — are implemented in
scanning for congestive heart failure. However, the renowned “Playboy Bunny”
symbol representing the ailment becomes increasingly difficult to isolate due
to surrounding organs and lower quality image productions. This paper examines
the Otsu thresholding method and incorporates new elements to account for
different image features meant to better isolate congestive heart failure
indicators in ultrasound images.

††journal: ol

##  1 Introduction

Ring Theory is a branch of abstract algebra within the field of pure
mathematics, stating that a ring is a set of elements with two binary
operations — addition and multiplication. The term ”elements” is used rather
than ”numbers” because rings are used to generalize complex mathematical
concepts, including matrices and polynomials with real coefficients. These
operations, along with subtraction, can be performed within a ring. However,
division may not always be possible among the elements in a ring, and
multiplication is not guaranteed to be commutative among all rings. To
subtract within a ring would essentially mean to add an element to its
additive inverse. Likewise, to divide would mean to multiply an element by its
multiplicative inverse. Doing so would bring the element back to the additive
identity zero and the multiplicative identity one, respectively [1].

Binarization is a subprocess of image segmentation whereby an image becomes a
binary resolution - one that is grayscale and where the pixel values range
from zero to one in regards to presence of darkness. The process of
binarization starts with the image being converted to grayscale, then applying
an adaptive threshold to the final resolution. The adaptive threshold allows
for the image to output different iterations of the same picture. There are
two types of binarization: global (focusing on the entire image) and local
(honing in on a specific region). Binarization, like other methods of image
segmentation, has various uses — including document scanning, forensic
analysis, and medical imagery [2].

Congestive heart failure is an illness where the heart cannot control blood
flow adequately and often requires conduction of medical imagery — including
echocardiograms (EKGs), magnetic resonance imaging (MRIs), or computed
tomography (CT) scans among other methods. The heart may fail to pump blood
(systolic heart failure) or fill up with blood (diastolic heart failure) in an
efficient manner[3]. Previous research acknowledges that congestive heart
failure is detected by a “Playboy Bunny” sign: denoted by heavy demarcations
in the left, middle, and right hepatic veins, located by the liver and within
the inferior vena cava. A medical image production can be inaccurate due to
either the presence of surrounding organs or underperforming technology, which
can lead to an incorrect or missed diagnosis altogether. This research seeks
to improve the detection of congestive heart failure by revising Otsu’s
thresholding method, a means of global binarization.

Figure 1: Examples of ultrasonographies showing the "Playboy Bunny" symbol of
congestive heart failure.

##  2 Related Literature

Image segmentation is a popular practice in the field of medical imagery. Ritu
and Ginni Garg assert the following in their publication entitled “Brain Tumor
Detection and Classification based on Hybrid Ensemble Classifier”: “The
challenging task in Brain Tumor is due to high variability and inherent MRI
data characteristics, e.g., variability in tumor sizes or shapes, tumor
detection, area calculation, segmentation, classification, and finding
uncertainty in segmented region. The most significant task in image
understanding is image segmentation because it helps in feature extraction,
area calculation, and significance in many real-life applications. It can be
used, for example, estimation of tumor volume, tissue classification, blood
cell delineation, and localization of tumors, matching of an atlas, surgical
planning, and image registration” [4]. Garg reiterates difficulties posed in
medical image segmentation, applicable to imagery any bodily organ. Current
thresholding methods are still vulnerable to inciting misdiagnoses due to
poorly produced scans.

Sensitivities in medical imagery apparatus can also lead to faulty image
reproductions. In “Long-evolution ascites in a patient with constrictive
pericarditis”, Nunes and colleagues wrote: “The electrocardiogram is not
characteristic and may reveal nonspecific repolarization changes and low-
voltage electrical activity. The presence of pericardial calcification on
chest radiograph is typical but not routinely found. Echocardiography offers
important clues that strongly suggest the diagnosis but CT scan and MRI are
the most sensitive methods to identify pericardial thickening… CP is a complex
disease, systemic in nature, and many of the diagnostic problems come from its
insidious course and lack of distinctive cardiopulmonary symptoms. A
particular subgroup of patients may represent an interesting diagnostic
challenge, presenting with ascites and signs of chronic liver disease.” (Nunes
et al., 2016) A single heart scan can bring back issues with surrounding
organs, or images that reflect them. For instance, signs of congestive heart
failure could be intertwined with symptoms being shown within the liver.
Failures in deduction can lead to an incorrect diagnosis or medicalization of
a different body system [4].

Additionally, different illnesses show similar symptoms, and an objective
heart scan will not be sufficient enough in identifying the exact ailment in
question. Yusuf Bayraktar, MD states in “Hepatic venous outflow obstruction:
Three similar syndromes” that “although patients with [hepatic venous outflow
obstruction] generally present with abdominal pain due to hepatomegaly, and
jaundice and ascites due to portal hypertension; chronic [Budd-Chiari
Syndrome] patients may first present with cirrhosis and its complications. BCS
patients with inferior vena cava obstruction may also have leg edema and
venous collaterals over the trunk. Additionally, signs and symptoms of heart
failure such as jugular venous distention, leg edema, and dyspnea may be seen
in patients with [congestive hepatopathy]. Nonetheless, the histological
findings in all three syndromes are almost identical and include sinusoidal
congestion and hepatocyte necrosis predominating in perivenular areas of
hepatic acini which eventually leads to bridging fibrosis between adjacent
central veins… The laboratory findings in [veno-occulsive disease], BCS, and
CH are also very similar.” (Bayraktar 2007) Different heart conditions can
coexist, which can lead doctors or medical technicians to improperly diagnose
a patient. Ultrasounds of a heart can show passive liver congestion, inferior
vena cava probes, or enlarged lymph nodes.

Implementing or improving on current thresholding methods is essential to the
continuation of better CT, MRI, and/or PET scans among others. In “Automatic
image thresholding using Otsu’s method and entropy weighting scheme for
surface defect detection”, Truong et al. dessert the following about the Otsu
thresholding method: “Otsu’s method, a state-of-the-art automatic thresholding
technique, is the basis for several automatic defect detection proposals. It
determines optimal threshold values that maximize the between-class variances
of the foreground and background. Studies demonstrate that Otsu’s method is
effective for thresholding a histogram with bimodal or multimodal
distribution. Otsu’s method fails if the histogram is unimodal or almost
unimodal (Ng, 2006). Therefore, it provides acceptable results for
thresholding general real-life images and fails when being applied in visual
inspection systems because of the nature of the input images” [5]. When an
image has a bi-modal histogram, its foreground and background pixels are
easily discernible. This is not always the case with medical image scans,
especially with ailments that continue developing and spreading across the
body in stages (cancers, tumors, etc). At said ailments’ final stages, images
can come back and appear unimodal. Therefore, examining and revising the Otsu
thresholding method is vital to the production of higher quality scans.

##  3 Theoretical Aspects

###  3.1 Ring Theory

A ring must satisfy the following axioms:

  * •

The ring, under addition, is an abelian group.

  * •

The multiplication operation is associative, and therefore closed.

  * •

All operations satisfy the distributive law of multiplication over addition.

An example of a ring includes the set of real polynomials. The set of real
polynomials can be denoted as:

$R[x]={a_{n}x^{n}+a_{n-1}x^{n-1}+\ldots+a_{1}x+a_{0}\;|\;a_{i}\in R}.$

Within this ring, you can freely add, subtract, and multiply one polynomial,
essentially an element within the ring, to get another polynomial - another
element. The additive identity is presented as zero. Since zero is a constant
polynomial, this value is also considered to be an element in the ring of real
polynomials. The multiplicative identity is presented as one. Since
multiplication is always commutative among all polynomials, the ring of real
polynomials is deduced as a commutative ring with an identity element.

###  3.2 Otsu Thresholding Method

The Otsu thresholding method has both mathematical and computational
representations. When calculating the appropriate threshold, users must look
for the minimum within class variance (sum of 2 variances multiplied by
associated weights):

$(\sigma_{W}^{2})=W_{b}(\sigma_{b}^{2})+W_{f}(\sigma_{f}^{2})$

and maximum between class variance (largest difference between overall
variance and within class variance):

$\sigma_{B}^{2}=\sigma^{2}-\sigma_{W}^{2}$

where:

  * •

$\sigma_{W}^{2}$ represents within class variance,

  * •

$W_{b}$ represents the weight (light presence) of the background pixels,

  * •

$\sigma_{b}^{2}$ represents the variance of the background,

  * •

$W_{f}$ represents the weight of the foreground pixels,

  * •

$\sigma_{f}^{2}$ represents the variance of the background,

  * •

$\sigma^{2}$ represents the overall image variance, and

  * •

$\sigma_{B}^{2}$ represents the between class variance.

###  3.3 Gaussian Filtering

Gaussian filters are used to smooth images and remove any noise [6], denoted
by:

$G(x)=\frac{1}{\sqrt{2\pi\sigma^{2}}}\cdot\exp{\left(-\frac{x^{2}}{2\sigma^{2}}\right)},$

where $\sigma$ represents the distribution standard deviation, with the mean
assumed to be zero.

Table 1: Significant Values for Gaussian Filters when $\sigma$ = 1 x | $0$ | $1$ | $2$
---|---|---|---
$G(x)$ | 0.399 | 0.242 | 0.05
$G(x)/G(0)$ | 1 | 0.6 | 0.125

In MATLAB, we create a new variable, Iblur1, consisting of the target image I
and the smoothing level (changed to an integer).

###  3.4 K-Means Clustering Methods

The K-Means clustering method is used to group data where it is otherwise not
explicitly segmented [7]. Muthukrishnan remarks that "assuming we have input
data points $x_{1}$, $x_{2}$, $x_{3}$, $\ldots$, $x_{n}$ and value of $K$ (the
number of clusters needed), we follow the below procedure:

  1. 1.

Pick $K$ points as the initial centroids from the dataset, either randomly or
the first $K$.

  2. 2.

Find the Euclidean distance of each point in the dataset with the identified
$K$ points (cluster centroids).

  3. 3.

Assign each data point to the closest centroid using the distance found in the
previous step.

  4. 4.

Find the new centroid by taking the average of the points in each cluster
group.

  5. 5.

Repeat 2 to 4 for a fixed number of iteration or till the centroids don’t
change.

Centroids are used as the average values of each cluster, and Euclidean
distance is calculated using:

$d(p,q)=\sqrt{(q_{1}-p_{1})^{2}+(q_{2}-p_{2})^{2}}$

where p and q represent two points in space, $p_{1}$ and $q_{1}$ represent the
x-coordinates of the point, and $p_{2}$ and $q_{2}$ represent the
y-coordinates of the point.

The output result is 3 different images, each with a different number of
clusters and ultimately different reproduced figure. Figures 2 and 3
demonstrate $K$-Means Clustering using the images from Figure 1.

Figure 2: Image reproductions of the first "Playboy Bunny" segmentation.

Figure 3: Image reproductions of the second "Playboy Bunny" segmentation.

###  3.5 Ring Theory in Binarization

For the sake of the research at hand, the method of clustering being
implemented treats the pixels of an image in the same way elements of a ring
are handled. In computing different segmentations of the same image, the
method forms different "rings" and attempts to keep all elements within their
respective clusters. The focus of this research is to reproduce an image that
better indicates the renowned Playboy Bunny symbol[8], and Ring Theory can be
used to achieve this by isolating the region of interest from surrounding
organs (distinguishing foreground clusters from background clusters).

##  4 Improvements to Otsu Thresholding Method

In order to improve the quality of the image reproduction, the Otsu
thresholding method will instead incorporate both Gaussian filters and
$K$-Means Clustering. The clustering is done first to separate the symbol
pixels from the surrounding organs, and the Gaussian smoothing is encoded
afterwards to remove any noise from any other elements in the environment and
further isolate the region of interest.

Since the first produced image using K-Means for both original segmentations
provides the clearest distinction between the notable symbol and surrounding
parts, the code runs under the assumption that $k$ = 1.

##  5 Experimental Results

Using the above code, segmentations will be run on both original images.
Figures 4 and 5 show the original images alongside their reproductions using
the improved Otsu method.

Figure 4: First original image (left) and image produced by the improved Otsu
method (right). Figure 5: Second original image (left) and image produced by
the improved Otsu method (right).

##  6 Conclusions

The original method of Otsu thresholding failed to account for noise removal,
which ultimately caused reproduced images to be of lower quality. The
improvements made to the Otsu thresholding method herein account for both the
removal of bias in clustering symbol elements from image background and
smoothing the image altogether to achieve a replica of higher resolution.

K-Means was implemented as this specific method can identify more
unconventional figures, whereas other segmentation practices (i.e. Mean Shift
Iterative Algorithm) are meant to identify uniform shapes or require
additional training, similar to a convolutional neural network.

##  7 Limitations

  * •

This project works specifically with MATLAB, and therefore has not been tested
or revised in other programming languages.

  * •

Since the focus of this research was to better detect the notable “Playboy
Bunny” symbol for congestive heart failure, other ailment detection methods
have not been looked into, and the improvements made within this project have
not been tested to better pinpoint other illness indicators.

  * •

Due to the ongoing global health crisis, access to resources and mentorship
were both inevitably limited.

  * •

Image reproduction is directly correlated to the original image quality; in
images with poorer production methods, isolation of the region of the interest
becomes increasingly lower in quality.

##  8 Suggestions for Further Research

  * •

Expand into other programming languages, most likely starting with Python
and/or Javascript.

  * •

Look into other ailments and different detection methods, most notably MRI and
CT scans of brain tumors or soft tissue sarcomas.

  * •

When possible, continue networking to outside professionals and laboratories
to implement improved methods into medical technology apparatus.

  * •

Begin testing the new thresholding method on a larger set of images and
looking into different methods of original image production.

Disclosures The authors declare no conflicts of interest.

##  9 Acknowledgements

This project would not have been made possible without the multitude of
support I have received from friends, family, and fellow academic
professionals. I would like to especially thank Dr. Rishi Nath of York College
for first introducing me to Ring Theory. Additionally, I have both Professors
Yasel Garcés Suárez and Esley Torres of the Institute of Cybernetics to thank
for introducing me to binarization and assisting with the format of this
paper, respectively.

## References

  * [1] A. Rahim, National High School Journal of Science (2019).
  * [2] Y. Zhu and X. Liang, “Exploiting fully convolutional network and visualization techniques on spontaneous speech for dementia detection,” (2020).
  * [3] U.-D. Bayraktar, S. Seren, and Y. Bayraktar, World journal of gastroenterology 13, 1912 (2007). Publisher: Baishideng Publishing Group Co., Limited.
  * [4] G. Garg and R. Garg, CoRR abs/2101.00216 (2021).
  * [5] M. T. Truong and S. Kim, Soft Comput. 22, 4197–4203 (2018).
  * [6] https://www.cs.auckland.ac.nz/courses/compsci373s1c/PatricesLectures/Gaussian%20Filtering_1up.pdf (2010).
  * [7] https://muthu.co/mathematics-behind-k-mean-clustering-algorithm/ (2018).
  * [8] A. Hokama, S. Arakaki, S. D. M. Tatsuji, K. Fukunori, and J. Fujita, Western Journal of Emergency Medicine 12, 433 (2011).