# Performance Evaluation of a Full-Duplex
UWA System in Lake Experiments

Lu Shen, , Benjamin Henson, ,
Long Shi, , Yuriy Zakharov L. Shen, B. Henson and Y. Zakharov are with School
of Physics, Engineering, and Technology, University of York, U.K. (e-mail:
<EMAIL_ADDRESS><EMAIL_ADDRESS>yury.zakharov@york.ac.uk). L.
Shi is with School of Economic Information Engineering, Southwestern
University of Finance and Economics. The work of L. Shen, B. Henson and Y.
Zakharov was supported in part by the U.K. EPSRC through Grants EP/P017975/1
and EP/R003297/1. Part of the materials in this paper was presented at the
Sixth Underwater Communications and Networking Conference (Ucomms) in Lerici,
Italy.

###### Abstract

In this work we present a full-duplex (FD) underwater acoustic (UWA)
communication system simultaneously transmitting and receiving acoustic
signals in the same frequency bandwidth. To simplify the FD hardware, the
system exploits a recently designed transducer capable of simultaneously
transmitting and receiving signals. The key challenge of implementing an FD
system is to cancel at the near-end receiver the strong self-interference (SI)
from the near-end transmitter. By using advanced adaptive filtering algorithms
providing high accuracy channel estimates, a high level of SI cancellation can
be achieved when the far-end signal is absent. However, the SI channel
estimation performance is limited in FD scenarios since the far-end signal
acts as an interference when estimating the near-end SI channel. In this
paper, we propose an FD UWA communication system which alternates between the
SI cancellation and far-end data demodulation. An adaptive Rake combiner with
multipath interference cancellation is implemented to improve the demodulation
performance in time-varying multipath channels. The performance of the FD UWA
system is evaluated in lake experiments. It is shown that the proposed
adaptive Rake combiner with multipath interference cancellation significantly
outperforms the conventional Rake combiner in all the experiments. The
experimental results demonstrate that, with the new Rake combiner, the
detection performance of the proposed FD UWA system is comparable with that of
the half-duplex system.

###### Index Terms:

Adaptive filter, Channel estimation, Full-duplex, Rake combiner, Self-
interference cancellation, Underwater acoustics.

##  I Introduction

Underwater acoustic (UWA) communication suffers from limited frequency
bandwidth for data transmission [1]. To increase the spectral efficiency
within the available bandwidth, we consider the full-duplex (FD) operation,
which allows a simultaneous transmission and reception of acoustic signals in
the same frequency bandwidth [2, 3, 4, 5]. Apart from the far-end signal, the
near-end receiver also receives a strong self-interference (SI) signal from
the near-end transmitter. Therefore, a high level of SI cancellation (SIC) is
required to detect the weak desired signal arriving from the far-end
transmitter.

In FD UWA systems, the spatial-domain SIC can be used for the SI suppression.
In [6], a spatial SIC scheme is proposed, which uses a phased array of
transducers and acoustic vector sensors. The SIC is achieved by adjusting the
beampattern of the phased array based on the direction of arrivals estimated
by the acoustic vector sensors. Sea experimental results demonstrate a spatial
SIC of 40 to 60 dB depending on the steering angle of the near-end transducer.
In [7], an FD UWA system with a combination of spatial SIC and digital SIC is
proposed and investigated in an anechoic pool; this system requires the
knowledge of the direction of the far-end transmitter, which is not always
available in practice. UWA communication systems with antenna arrays are
difficult for practical implementation. However, even without using an antenna
array, in an FD system, typically the transmitting and receiving transducers
are two separate devices. In [8], we proposed an FD transducer that can be
used for simultaneous transmission and reception of acoustic signals. We use
this transducer in our UWA FD experiments.

For FD terrestrial radio communications, a combination of analogue and digital
cancellation is normally used. Analogue cancellation is applied before digital
cancellation to avoid the analogue-to-digital converter (ADC) saturation [2,
3, 4]. In FD UWA systems, compared to terrestrial radio systems, low-frequency
signals are transmitted. In such a case, high resolution ADCs up to 24 bits
can be used, which makes it feasible to achieve a high level of cancellation
in the digital domain alone. Therefore, digital cancellation is considered as
the main SIC approach in FD UWA systems [9, 10, 11, 12]. In this work, we
implement and investigate the performance of an FD UWA system with digital
cancellation.

Two factors limiting the digital SIC performance in FD UWA systems are
nonlinearities introduced by the equipment [9, 13, 14] and fast variation of
the SI channel [11, 12]. The dominant source of the nonlinearities is the
power amplifier (PA). It is found that the SIC performance can be
significantly improved by using the PA output as the reference signal
(regressor) for SI channel estimation compared to the case of using the
original digital data [15, 13]. After addressing the nonlinearities caused by
the PA, a high level of digital SIC can be achieved in time-invariant
scenarios with a classical recursive least-squares (RLS) adaptive filter [16].
In practice, the SI channel can be fast time-varying, e.g., due to reflections
from the moving sea surface [17]. To achieve a high level of digital SIC in
time-varying scenarios, advanced adaptive filtering algorithms with good
tracking performance have been recently proposed [11, 12, 18, 19]. The SIC
performance provided by these adaptive algorithms has been evaluated in
shallow lake experiments. However, in those experiments, the far-end
transmission is not considered and the SIC performance is only limited by the
ambient noise level.

With the FD operation, the received signal includes the near-end SI,
background noise, and also the far-end signal. When estimating the near-end SI
channel, the far-end signal acts as an additional interference (noise), which
reduces the overall SI to noise ratio. To achieve a high level of SIC, the
influence of the far-end signal, in addition to the ambient noise, on the SIC
performance must be taken into account.

To avoid the impact of the far-end signal on the near-end SI channel
estimation performance, a scheme is proposed in [9] with estimation of the
near-end SI channel before the far-end signal arrives. An over-parametrization
based RLS algorithm with a sparsity constraint is proposed for the SI channel
estimation and evaluated in a water tank with a static multipath channel. This
approach works well in time-invariant scenarios. However, the SI channel needs
to be frequently re-estimated to ensure a good tracking performance in fast
time-varying scenarios. In such a case, it is difficult to guarantee
sufficiently long time intervals for the SI channel estimation, non-
overlapping with the far-end transmission.

To reduce the impact of the far-end signal on the SIC performance, we propose
an FD UWA system which alternates the near-end SIC and far-end data
demodulation in turbo iterations. The proposed system does not require a non-
overlapping period between the near-end and far-end transmissions. In the
first iteration, the SI channel is estimated, treating the far-end signal as
an extra noise. At the receiver, the residual signal after SIC is used for
tentative far-end data demodulation. In the second iteration, the far-end
signal is reconstructed using the far-end data and channel estimates and then
removed from the received signal. After that, a similar procedure is performed
as in the first iteration. As the estimation performance for both the near-end
SI channel and far-end channel is significantly improved in the second
iteration, a better demodulation performance can be achieved. Further
iterations can be useful to improve the communication performance.

In [20], the conventional Rake combiner [21] is used for demodulation of far-
end data in multipath channels. The Rake combiner provides the maximal ratio
combining for signal components propagated over various channel paths.
However, such a receiver ignores the multipath interference, i.e., the
interference from one multipath signal component to another, which
significantly degrades the receiver performance. In [22], a receiver with
linear equalization, multipath interference reconstruction and interference
cancellation is proposed. Simulation results indicate that the receiver
performance is significantly improved due to the multipath interference
cancellation. In [23], a Rake-IC combiner with multipath interference
cancellation is implemented in a UWA modem with half-duplex data packet
transmission. This Rake-IC combiner shows good demodulation performance in
static/slow time-varying scenarios. In this paper, we propose an adaptive
version of the Rake-IC combiner, which is capable of providing good
demodulation performance in FD time-varying scenarios.

To evaluate the performance of the FD UWA system, both half-duplex and FD
communication experiments are conducted using recently developed FD
transducers [8] in lake experiments. The proposed Rake-IC combiner outperforms
the conventional Rake combiner in all the experiments. Furthermore, results
indicate that the proposed FD system with the Rake-IC combiner can provide the
demodulation performance close to that of the half-duplex system. The
contributions of this paper are as follow.

  * •

An FD UWA system with multiple iterations is proposed for cancellation of the
interference from the far-end transmission when estimating the SI channel and
cancellation of the SI when demodulating the far-end data.

  * •

An adaptive version of the Rake-IC combiner is proposed to improve the
detection performance in time-varying multipath channels. Its performance is
investigated in the lake experiments.

  * •

The performance of the proposed FD UWA system is investigated and compared
with that of the half-duplex system in lake experiments.

The paper is organized as follows. In Section II, the FD UWA system structure
is presented. Section III describes the Rake-IC combiner. Section IV
investigates the performance of the FD UWA system and the proposed receiver in
lake experiments. Section V concludes the paper.

##  II System model

The block diagram of the FD UWA system is shown in Fig. 1.

Figure 1: Block diagram of the FD UWA system. The sample index with the symbol
rate $f_{d}$ is denoted by $i$; $a_{n}(i)$ and $a_{f}(i)$ represent the near-
end and far-end data symbols, respectively; $s(t)$ and $s_{f}(t)$ denote the
analogue PA outputs of the near-end and far-end transmitters; $\tilde{s}(t)$
and $\tilde{s}_{f}(t)$ represent the near-end and far-end projector outputs;
$r(t)$ denotes the received signal.

As shown in Fig. 1, the signal at the near-end receiver is given by:

$\displaystyle\begin{split}r(t)&=\tilde{s}(t)*h_{\mathrm{SI}}(t)+\tilde{s}_{f}(t)*h_{\mathrm{f}}(t)+n(t)\\\
&=r_{\mathrm{SI}}(t)+f(t)+n(t),\end{split}$ (1)

where $*$ denotes the convolution,
$r_{\mathrm{SI}}(t)=\tilde{s}(t)*h_{\mathrm{SI}}(t)$ is the SI, $\tilde{s}(t)$
is the signal emitted by the near-end projector, $h_{\mathrm{SI}}(t)$
represents the SI channel impulse response,
$f(t)=\tilde{s}_{f}(t)*h_{\mathrm{f}}(t)$ is the far-end signal,
$\tilde{s}_{f}(t)$ is the signal emitted by the far-end projector,
$h_{\mathrm{f}}(t)$ represents the far-end channel impulse response and $n(t)$
is the background noise. The near-end and far-end channel responses,
$h_{\mathrm{SI}}(t)$ and $h_{\mathrm{f}}(t)$, and the far-end signal $f(t)$
are unknown at the receiver and should be estimated.

In subsection II-A, description of signals used for near-end and far-end
transmission is given. In subsection II-B, the receiver structure is
introduced.

###  II-A Transmitted signals

The structure of the near-end transmitter is shown in Fig. 2.

Figure 2: Near-end transmitter. The sample index with the symbol rate $f_{d}$
and sampling rate $f_{s}$ are denoted by $i$ and $n$, respectively; $a_{n}(i)$
represents the near-end data symbols; $s(n)$ denotes the passband digital
signal; $s(t)$ denotes the analogue PA output.

For the near-end transmission, a pseudo-random sequence of Binary Phase Shift
Keying (BPSK) data symbols $a_{n}(i)$ is used. The data symbols are up-sampled
and pulse-shaped by a root-raised cosine (RRC) filter. The RRC filter output
is then up-converted to the carrier frequency $f_{c}$. Afterwards, the signal
is digital-to-analogue converted, amplified by a PA, and the near-end PA
output $s(t)$ is transmitted by a transducer.

For the far-end transmission, Quadrature Phase Shift Keying (QPSK) symbols
consisting of superimposed binary pilot and data symbols [24, 25] are used:

$a_{f}(i)=p(i)+jd(i),$ (2)

where $p(i)$ is a binary pseudo-random pilot sequence and $d(i)$ is the
information data symbols obtained by encoding transmitted data using a
convolutional code and interleaving the encoded symbols. The far-end symbols
$a_{f}(i)$ go through the same modulation process as shown in Fig. 2.

###  II-B Receiver structure

The receiver performs two tasks, the first task is the near-end SIC, and the
second task is the far-end data demodulation.

Figure 3: Complex demodulation. $r(n)$ and $r(i)$ represent the passband and
baseband received signals, respectively.

The passband received signal $r(t)$ is first analogue-to-digital converted by
an ADC to produce the passband samples $r(n)$, which then go through the
complex demodulation process to produce the baseband samples $r(i)$ as shown
in Fig. 3, where an RRC filter is used for the low-pass filtering.

The general structure of the receiver is shown in Fig. 4.

Figure 4: Receiver structure. $n_{it}$ is the iteration number, $r(i)$ is the
baseband received signal, $e(i)$ is the residual signal after SIC and
$\hat{f}(i)$ is the estimate of the far-end signal. [to replace
$\hat{\mathbf{h}}_{far}$ with $\hat{\mathbf{h}}_{f}$]

In the first iteration ($n_{\text{it}}=1$), since the far-end signal estimate
$\hat{f}(i)$ is not available yet, the received signal $r(i)$ is the desired
signal of an adaptive filter used for the SI channel estimation and near-end
SIC. To prevent the PA nonlinearity affecting the SI channel estimation, the
baseband digitalized PA output $\tilde{s}(i)$ is used as the regressor in the
adaptive filter, instead of utilising the symbols $a_{n}(i)$ [13]. The
baseband signal $\tilde{s}(i)$ is obtained through the same complex
demodulation process as shown in Fig. 3.

The baseband PA output $\tilde{s}(i)$ is fed to a finite impulse response
(FIR) filter whose impulse response is a delayed copy [11, 12] of the impulse
response of the adaptive filter, i.e., the SI channel estimate
$\hat{\mathbf{h}}_{\mathrm{SI}}(i)$. For performing the SIC, the FIR filter
output $\hat{r}_{\mathrm{SI}}(i)$ is subtracted from the received signal
$r(i)$. The residual signal $e(i)$ is an estimate of the far-end signal (plus
noise) which is used for the far-end channel estimation and far-end data
demodulation.

(a) Conventional Rake combiner.

(b) Rake-IC combiner.

Figure 5: Block diagrams of the Rake combiners; $e(i)$ is the residual signal
after SIC; $\hat{a}_{f}(i)$ is the far-end data estimate; $e_{l}(i)$ is the
recovered $l$th multipath signal component; $\tau_{l}$ represents a delay of
$l$ symbol intervals; $y_{\text{R}}(i)$ and $y_{\text{RIC}}(i)$ are the output
signals of the conventional Rake combiner and the Rake-IC combiner,
respectively.

The far-end channel estimate $\hat{\mathbf{h}}_{\mathrm{f}}(i)$ is obtained by
another adaptive filter. An estimate $\hat{a}_{f}(i)$ of the far-end data
$a_{f}(i)$, obtained at the previous iteration, and the residual signal $e(i)$
are used as the regressor and the desired signal of the adaptive filter,
respectively. In the first iteration ($n_{\text{it}}~{}=~{}1$), there is no
estimate of the far-end data $d(i)$, but the pilot symbols $p(i)$ are known
and therefore we use $\hat{a}_{f}(i)=p(i)$. In further iterations
($n_{\text{it}}~{}>~{}1$), there is a tentative estimate $\hat{d}(i)$ of the
far-end data, and therefore we use an improved estimate
$\hat{a}_{f}(i)=p(i)+j\hat{d}(i)$.

In this paper, two types of Rake combiner are considered for the far-end data
demodulation, the conventional Rake combiner and Rake-IC combiner with
multipath interference cancellation; they will be described in Section III.
The imaginary part of the Rake combiner output is interleaved and decoded
using the Viterbi decoder [21]. The decoded message is encoded and interleaved
in the same way as it is done in the transmitter and thus obtained sequence of
symbols $\hat{d}(i)$ is used as an estimate of the far-end symbols $d(i)$.

With $\hat{a}_{f}(i)$ and $\hat{\mathbf{h}}_{\mathrm{f}}(i)$ now available, an
estimate $\hat{f}(i)$ of the far-end signal $f(i)$ is reconstructed by
filtering $\hat{a}_{f}(i)$ in a FIR filter with the impulse response
$\hat{\mathbf{h}}_{\mathrm{f}}(i)$; this estimate will be used in the next
iteration.

##  III Rake combiner with interference cancellation

The conventional Rake combiner shown in Fig. 5(a) is implemented as an FIR
filter with its $k$th filter coefficient at the $i$th time instant given by:

$\hat{h}_{\mathrm{R}}(i,k)={\hat{h}}^{*}_{\mathrm{f}}(i,L_{\mathrm{f}}-k),$
(3)

where $\hat{\mathbf{h}}_{\mathrm{f}}(i)$ is a far-end channel estimate at the
$i$th time instant (a vector of size $L_{\text{f}}\times 1$ with channel tap
estimates) and $L_{\mathrm{f}}$ is the length of the far-end channel impulse
response. The conventional Rake combiner performs the maximal-ratio combining
but it ignores the multipath interference, while the Rake-IC combiner performs
the multipath interference cancellation before the maximal-ratio combining.

In this section, we introduce an adaptive version of the Rake-IC combiner
proposed in [23], which is implemented assuming that the multipath channel is
static. The adaptive Rake-IC combiner can deal with time-varying multipath
interference cancellation. The block diagram of the proposed Rake-IC combiner
is shown in Fig. 5(b).

In each branch in Fig. 5(b), only one multipath signal component is kept for
processing. The other multipath components are removed from the residual
signal $e(i)$ based on the far-end channel estimate
$\hat{\mathbf{h}}_{\mathrm{f}}(i)$. This process can either be individually
performed for each branch, or it can be performed in a more computationally-
efficient way as shown in Fig. 5(b). First, all of the signal components are
removed from $e(i)$, then every signal component is added to the difference
$e(i)-\hat{f}(i)$ individually on each branch with a corresponding delay. In
such a case, we obtain the $l$th multipath signal $e_{l}(i)$ free from the
multipath interference. Then, the maximal-ratio combining results in the
signal

$y_{\text{RIC}}(i)=\sum_{l=1}^{L_{\mathrm{f}}}e_{l}(i-L_{\mathrm{f}}+l)\hat{h}^{*}_{\mathrm{f}}(i,l).$
(4)

Fig. 5(b) shows a general version of the proposed combiner with all
$L_{\mathrm{f}}$ multipath components participating in the maximal ratio
combining.

In Section IV, we show that the demodulation performance can be significantly
improved when using the proposed Rake combiner instead of the conventional
Rake combiner.

##  IV Lake experiments

In this section, we investigate the performance of the FD UWA system in two
experimental sites. For comparison, half-duplex (HD) experiments were also
conducted, i.e., experiments with the far-end transmission only.

The lake experiments were conducted using the recently developed two-sensor FD
transducer (see Fig. 6) which contains two piezo-ceramic cylinders, one of
them is used as a projector and the other one is used as a hydrophone [8].

For both the near-end and far-end transmission, the transmitted signal is
sampled at $f_{s}=192$ kHz. The RRC filter with a roll-off factor of $0.2$ is
used for pulse-shaping and low-pass filtering. In every experiment, signals of
60 s length are transmitted. A silence period of 5 s is added at the beginning
of the transmitted signal to measure the background noise. Far-end data are
encoded with convolutional codes of different code rates as detailed in Table
I.

Figure 6:  A two-sensor transducer used in the lake experiments. Figure 7:  Setup of the FD and HD experiments in the University of York. TABLE I: List of convolutional codes [21] used in the experiments Code Rate | Constraint Length | Polynomials in Octal
---|---|---
1/4 | 8 | [235 275 313 357]
1/3 | 8 | [225 331 367]
1/2 | 6 | [53 75]
2/3 | [5 4] | [23 35 0; 0 5 13]

###  IV-A Experimental site 1

####  IV-A1 Experimental setup

The first experimental site is the shallow lake at the University of York. The
maximum depth of the experimental site is around 1 m. The experimental setup
is shown in Fig. 7. Both transducers were placed at approximately 0.5 m depth
and the distance between them was around 18 m.

Both near-end and far-end signals are transmitted at $f_{c}=36$ kHz carrier
frequency with $4$ kHz bandwidth. The far-end SNR was around $16$ dB. For the
FD experiment, the received SI to noise ratio was around $67$ dB, i.e., the
far-end signal level at the receiver was about -51 dB with respect to the SI
level.

####  IV-A2 SI and far-end channel estimation

The SI channel estimation is significantly more demanding than the far-end
channel estimation, since the level of the required SI cancellation is much
higher than that of the far-end signal cancellation. The SI and far-end
channels in this experiment were fast-varying due to the transducers
positioned close to the lake surface. Therefore, for the SI channel
estimation, we use the homotopy SRLS-L-DCD (HSRLS-L-DCD) adaptive filter [12].
This adaptive filter uses the basis function expansion of the channel
variations to improve its tracking performance and exploits the channel
sparsity to improve the estimation accuracy. For the far-end channel
estimation, the delayed sliding-window RLS (SRLSd) adaptive filter [11] is
used. The HSRLS-L-DCD algorithm could also be used for the far-end channel
estimation, but the SRLSd algorithm is less complicated.

The HSRLS-L-DCD adaptive filter length is $L_{\text{SI}}=100$ taps, which
corresponds to a delay spread of $25$ ms (the baseband sampling rate is
$f_{d}$ = 4 kHz). The sliding window length in the first iteration is set to
$M=1401$ and, in further iterations, $M=1001$. A high value of $M$ allows
better performance in noisy environment, which is the case in the first
iteration when the interference from the far-end signal has not been cancelled
yet. A smaller $M$ allows better tracking of the channel variations in less
noisy environment, which is the case in the second and further iterations
after cancelling the far-end interference. The adaptive filter uses the
parabolic approximation of the time variations of the complex multipath
amplitudes [12].

The SRLSd adaptive filter length is $L_{\mathrm{f}}=20$ taps (5 ms) and the
sliding window length is $M=1401$ in the first iteration and $M=1001$ in the
second and further iterations. The reasons for changing $M$ in the first and
further iterations are similar to that for the HSRLS-L-DCD adaptive filter.

Figure 8:  Magnitude of the impulse response estimates of the SI channel and
far-end channel in the FD experiment.

(a) Conventional Rake combiner.

(b) Rake-IC combiner.

(c) Rake and Rake-IC combiners.

Figure 9:  Experimental BER performance of HD and FD UWA systems with the
conventional Rake and Rake-IC combiners with rate-1/4 convolutional code.

An example of the magnitude of the near-end and far-end channel impulse
responses at the first and second iterations are shown in Fig. 8. In Fig. 8a,
it can be seen that the multipath structure of the SI channel estimate is much
clearer after the second iteration. The same is true for the far-end channel
estimate shown in Fig. 8b. This demonstrates that both the SI and far-end
channel estimation performance are improved at the second iteration.

(a) Conventional Rake combiner.

(b) Rake-IC combiner.

(c) Rake and Rake-IC combiners.

Figure 10: Experimental BER performance of HD and FD UWA systems with the
conventional Rake and Rake-IC combiners with rate-1/3 convolutional code.

####  IV-A3 Demodulation performance

To obtain the demodulation performance at various far-end SNRs, complex-valued
Gaussian random noise with variance $\sigma_{n}^{2}$ is added to the baseband
received signal in the HD and FD experiments, where $\sigma_{n}^{2}$ is
computed as: $\sigma_{n}^{2}=(P_{s}-P_{n})/\mathrm{SNR}-P_{n}$, and $P_{s}$
and $P_{n}$ are the average power of the baseband received signal and
background noise, respectively, in the HD experiment.

We first consider results of experiments with a rate 1/4 convolutional code,
presented in Fig. 9. The BER performance of the conventional Rake combiner in
HD and FD lake experiments is shown in Fig. 9(a). The demodulation performance
is improved in the second iteration for both HD and FD experiments, with a
performance gap of about 2 dB between these two transmission modes. In further
iterations, the performance (not shown here) is the same.

The BER performance of the Rake-IC combiner in HD and FD experiments is shown
in Fig. 9(b). Fig. 9(b) further demonstrates that using multiple iterations in
the receiver is beneficial for the demodulation performance. In this case, the
third iteration brings some extra improvement in the performance, and the
performance gap between the FD and HD modes is somewhat smaller than 2 dB.

For better comparison of the Rake combiners, Fig. 9(c) shows several BER
curves selected from Fig. 9(a) and Fig. 9(b). Here, we also show the BER
performance obtained with the same signals via numerical simulation for the
single-path static HD channel; this is a benchmark indicating the best
performance that can be achieved in the ideal conditions. It can be seen that
the Rake-IC combiner outperforms the conventional Rake combiner by 2-3 dB,
with higher improvement at lower BERs. With the Rake-IC combiner, the BER
performance in the HD experiment approaches the single-path benchmark after
two iterations. This shows that the Rake-IC combiner has achieved almost ideal
(maximal-ratio) combining of the multipath components with complete
cancellation of the multipath interference.

The demodulation performance of the Rake combiners with a rate 1/3
convolutional code are shown in Fig. 10. The BER curves in Fig. 10(a) and Fig.
10(b) further demonstrate the improvement in the demodulation performance
achieved with turbo iterations; the performance improvement is especially high
in the FD experiments. Selected BER curves for the Rake combiners in HD and FD
experiments are shown in Fig. 10(c). By comparing Fig. 9(c) and Fig. 10(c), it
can be seen that the SNR loss of the conventional Rake combiner to the single-
path benchmark has significantly increased with the higher code rate. On the
other hand, the performance of the Rake-IC combiner remains equally good for
rate-1/3 and rate-1/4 convolutional codes. At a BER of $10^{-3}$, for the
Rake-IC combiner, the SNR loss in the FD experiment is still about 2 dB
compared to the HD experiment.

###  IV-B Experimental site 2

Figure 11:  Setup of the FD and HD experiments in the Kelk lake.

####  IV-B1 Experimental setup

Experiments were conducted in the Kelk lake in East Yorkshire. The
experimental setup is shown in Fig. 11. During the experiment, the near-end
transducer was placed at 4 m depth and the far-end transducer was placed at
approximately 1 m depth. The maximum depth of the site is around 7 m. The
distance between the transducers was around 25 m.

Both the near-end and far-end signals of 60 s length are transmitted at
$f_{c}=12$ kHz carrier frequency with $1$ kHz bandwidth. The far-end SNR was
around $19$ dB. For the FD experiment, the SI signal to noise ratio was around
$60$ dB.

####  IV-B2 SI and far-end channel estimation

In this scenario, the time-variation of the SI channel is not significant,
therefore the SRLSd adaptive filter is capable of providing good near-end SI
estimation performance in all turbo iterations; therefore, the SRLSd algorithm
is used for estimation of both the SI and far-end channels. For the SI channel
estimation, the filter parameters are: $L_{\text{SI}}=50$ taps, which
corresponds to a delay spread of $50$ ms (at the baseband sampling rate
$f_{d}$ = 1 kHz), the sliding window length in the first iteration is $M=1201$
and, in the further iterations, $M=1001$. In this scenario, the delay spread
of the far-end channel is much longer (about 30 ms) compared to that in the
first lake experiment (5 ms). For the far-end channel estimation, the filter
parameters are: the filter length $L_{\mathrm{f}}=30$ taps (30 ms) and the
sliding window length in the first iteration is $M=1001$ and, in the further
iterations, $M=801$.

Figure 12:  Far-end channel estimates in the Kelk lake experiments with a rate
1/3 convolutional code at an SNR of 3 dB.

Fig. 12 shows the far-end channel estimates in the HD and FD experiments. The
channel impulse responses are normalized with respect to the tap with the
highest magnitude and plotted in log scale. Results are obtained for rate 1/3
convolutional code when the far-end SNR is about 3 dB. It can be seen that the
accuracy of the channel estimates is significantly improved at the second
iteration in both the HD and FD experiments. Another conclusion is that the SI
signal is cancelled quite well in the FD experiment as the channel estimates
obtained in HD and FD experiments are very similar.

####  IV-B3 Demodulation performance

Complex-valued Gaussian random noise with variance $\sigma_{n}^{2}$ is added
to the baseband received signal in the HD and FD experiments as described in
subsection IV-A to obtain the demodulation performance at various far-end
SNRs.

(a) Performance of the Rake combiners in HD experiments.

(b) Performance of the Rake-IC combiner in HD and FD experiments.

Figure 13: BER performance of the Rake combiners in the Kelk lake experiments
with rate-1/3 convolutional code.

The demodulation performance of the two Rake combiners in HD experiments with
rate-1/3 convolutional code are shown in Fig. 13(a). It can be seen that the
performance of the Rake-IC combiner is very close to the single-path
benchmark; the difference is within 1 dB. Even with single iteration, the SNR
loss compared to the benchmark is only about 2 dB. On the other hand, in this
scenario, the performance of the conventional Rake combiner significantly
degrades due to the multipath interference in the far-end channel. An SNR loss
of about 10 dB can be observed when comparing with the Rake-IC combiner.

As the performance of the conventional Rake combiner is already limited in the
HD experiments, only the Rake-IC combiner is considered in the FD experiments.
The demodulation performance of the Rake-IC combiner with rate-1/3
convolutional code is shown in Fig. 13(b). It can be seen that the SNR loss in
the FD transmission mode compared to the HD mode is less than 0.5 dB.

Figure 14:  BER performance of the Rake-IC combiner in the Kelk lake
experiments with rate-1/2 convolutional code.

We then investigate the demodulation performance of the Rake-IC combiner with
rate-1/2 convolutional code. As the code rate increases, more iterations are
required to achieve the best BER performance. As can be seen in Fig. 14, the
performance in the HD experiment approaches the single-path benchmark in four
iterations. In the FD experiments, the performance is still comparable with
that in the HD experiments and with the single-path HD benchmark; the loss to
the benchmark is about 1 dB at low BERs.

Finally, the demodulation performance of the Rake-IC combiner with rate-2/3
convolutional code is shown in Fig 15. Now, five iterations are required to
approach the single-path benchmark in the HD experiments. The performance gap
between the FD and HD transmission, after five iterations, is about 1 dB at
low BERs.

Figure 15:  BER performance of the Rake-IC combiner in the Kelk lake
experiments with rate-2/3 convolutional code.

Based on the results shown in Fig. 13, 14 and Fig. 15, it can be concluded
that the proposed FD UWA system with the Rake-IC combiner is capable of
providing similar demodulation performance as that of the HD system and the
number of turbo iterations should be increased for higher data rates.

##  V Conclusions

In this paper, an FD UWA communication system with turbo iterations and the
adaptive Rake-IC combiner has been proposed. By performing the near-end SIC
and far-end data demodulation in an iterative way, the influence of the far-
end signal on the SIC performance has been reduced. The use of the adaptive
Rake-IC combiner improves the demodulation performance in time-varying
multipath channels compared to the conventional Rake combiner. The performance
of the FD UWA system has been investigated and compared with the HD
counterpart in lake experiments with different code rates. As demonstrated by
the experimental results, the proposed FD UWA system with the Rake-IC combiner
achieves good demodulation performance in all the experiments.

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# Refining Myocardial Infarction Detection: A Novel Multi-Modal Composite
Kernel Strategy in One-Class Classification

###### Abstract

Early detection of myocardial infarction (MI), a critical condition arising
from coronary artery disease (CAD), is vital to prevent further myocardial
damage. This study introduces a novel method for early MI detection using a
one-class classification (OCC) algorithm in echocardiography. Our study
overcomes the challenge of limited echocardiography data availability by
adopting a novel approach based on Multi-modal Subspace Support Vector Data
Description. The proposed technique involves a specialized MI detection
framework employing multi-view echocardiography incorporating a composite
kernel in the non-linear projection trick, fusing Gaussian and Laplacian
sigmoid functions. Additionally, we enhance the update strategy of the
projection matrices by adapting maximization for both or one of the modalities
in the optimization process. Our method boosts MI detection capability by
efficiently transforming features extracted from echocardiography data into an
optimized lower-dimensional subspace. The OCC model trained specifically on
target class instances from the comprehensive HMC-QU dataset that includes
multiple echocardiography views indicates a marked improvement in MI detection
accuracy. Our findings reveal that our proposed multi-view approach achieves a
geometric mean of 71.24%, signifying a substantial advancement in
echocardiography-based MI diagnosis and offering more precise and efficient
diagnostic tools.

Index Terms—  Echocardiography, One-class Classiﬁcation, Machine Learning,
Myocardial Infarction

##  1 Introduction

World Health Organization (WHO) recently emphasized the significant global
health impact of Coronary Artery Disease (CAD), accounting for 16% of
worldwide deaths [1]. Among the most severe outcomes of CAD is myocardial
infarction (MI), characterized by irreversible damage of the myocardium,
leading to a critical need for prompt detection [2]. MI diagnosis primarily
involves symptom identification, biochemical markers, electrocardiography
(ECG) readings, and imaging assessments. However, the initial symptoms, such
as shortness of breath and upper body pain, may not always be immediately
evident [3]. Biochemical indicators like high sensitivity cardiac troponin
(hs-cTn) also take time to manifest at levels indicative of MI [4, 5]. While
ECG provides valuable insights, its diagnostic capabilities can be limited and
often lag behind imaging methods [5]. Echocardiography, as a non-invasive
imaging technique, is crucial in early MI detection by identifying Regional
Wall Motion Abnormalities (RWMA) in the affected myocardium [6]. Its
accessibility and cost-effectiveness make it an advantageous option for early
MI detection [7].

While echocardiography is an important tool for diagnosing MI, its
effectiveness faces several limitations. The subjective nature of evaluating
RWMAs and the common occurrence of low-quality, noisy echocardiography
recordings pose significant hurdles [6, 8]. These challenges have spurred the
increased reliance on computer-aided diagnosis algorithms. Previous studies
have tested these algorithms with limited or non-diverse echocardiography
datasets, raising concerns about their reliability and robustness, especially
those based on deep learning techniques [9, 10, 11, 12]. In response, one-
class classification (OCC) models, which are trained only with samples from
the positive class, have emerged as a promising alternative [13, 14]. However,
the application of OCC models in echocardiography data analysis is still
relatively unexplored, with few studies venturing into this domain [15, 16].

Fig. 1: The proposed framework for early detection of MI with multi-modal one-
class classiﬁcation over multi-view echocardiography. The workflow begins with
image pairs representing the apical four-chamber (A4C) and apical two-chamber
(A2C) views. Feature sets $\mathbf{F}_{A}$ and $\mathbf{F}_{B}$ extracted from
the left ventricle motion characteristics undergo a transformation through
NPT, followed by a projection into an optimized lower-dimensional shared space
through the projection matrix $\mathbf{Q}$. The process concludes with a
multi-modal one-class classification that determines the presence or absence
of MI.

In this study, we have developed a novel framework using OCC techniques for
the early detection of MI using multi-view echocardiography. This framework
focuses on extracting key features from apical 4-chamber (A4C) and apical
2-chamber (A2C) views by utilizing Active Polynomials (APs) to monitor left
ventricle (LV) movement [17]. A crucial aspect of the proposed method is the
implementation of a composite kernel that combines Gaussian and Laplacian
sigmoid kernels, designed to capture a more comprehensive representation of
echocardiography data, thereby increasing the model’s sensitivity to MI
indicators. Moreover, we introduce an innovative strategy for optimizing the
projection matrices, a central element in our OCC method. This optimization
involves the combinations of gradient ascend and descent variations in the
optimization process of data description and subspace learning to refine the
model’s performance. By fine-tuning the projection matrices, our model adapts
more precisely to the distinctive MI features present in multi-view
echocardiography. These advancements in composite kernel design and projection
matrix optimization are extensively tested using the HMC-QU dataset. In
addition to leveraging the advantages of multi-modal OCC, the proposed
approach also explores the efficacy of both uni-modal and multi-modal
algorithms in the early detection of MI through multi-view echocardiography.

The paper is organized in the following manner: Section 2 presents the
proposed framework for early MI detection with the details of the methodology
and innovations introduced in the Multi-modal Subspace Support Vector Data
Description (MS-SVDD) approach. Section 3 is dedicated to the experimental
results and discussion including the analysis of the framework’s performance
on the HMC-QU111The benchmark HMC-QU is publicly shared at the repository
https://www.kaggle.com/datasets/aysendegerli/hmcqu-dataset. dataset. Finally,
Section 4 concludes the paper.

##  2 Methodology

As illustrated in Figure 1, the proposed OCC method is structured around the
integration of three critical components: Multi-modal Subspace Support Vector
Data Description, the composite kernel in non-linear projection trick (NPT),
and adapting maximizing strategy for the projection matrix update. These
elements work collectively to enhance the detection of MI through OCC using
multi-view echocardiography. By combining these advanced techniques, our
approach aims to improve the accuracy and efficiency of MI diagnosis.

###  2.1 Feature Engineering and Data Preparation

As in [17], the endocardial boundary of the left ventricle (LV) from A4C and
A2C views of echocardiography recordings is extracted using Active Polynomials
(APs). From the APs formed on each frame of the echocardiography recording,
the extracted LV wall is tracked for myocardial motion analysis, where
myocardial motion curves are obtained for each of the 12 myocardial segments.
Then, the feature vectors are extracted from each myocardial segment motion
curve in both A4C and A2C view echocardiography recordings over one cardiac
cycle. Thus, A4C and A2C myocardial motion features are obtained for each
myocardial segment in each echocardiography view, as depicted in Figure 1.
Lastly, the extracted feature vectors are utilized for the detection of
myocardial infarction with OCC techniques.

###  2.2 Multi-modal Subspace Support Vector Data Description

In the proposed framework, we used OCC for the detection of MI. Contrary to
class-speciﬁc algorithms, the OCC models do not require information from
negative samples during training. For the OCC model, we used the MS-SVDD as in
[18] due to its feasibility with multi-view echocardiography data. MS-SVDD
plays a pivotal role in our methodology, particularly in the context of multi-
view echocardiography analysis. This approach focuses on mapping multi-view
feature vectors into a lower-dimensional optimized feature space tailored for
OCC.

The mathematical foundation of MS-SVDD is built around the concept of
iteratively transforming the multi-modal (view) data from the original feature
space to a new joint feature space and finding a joint compact description of
data. Let the instances in each view $v$, $v=1,\dots,V$, are represented by
$\mathbf{F}_{v}=[\mathbf{f}_{v,1},\mathbf{f}_{v,2},\dots\mathbf{f}_{v,N}]$,
$\mathbf{f}_{v,i}\in\mathbb{R}^{D_{v}}$, where $D_{v}$ denotes the
dimensionality of the feature space of view $v$. A projection matrix
$\mathbf{Q}_{v}\in\mathbb{R}^{d\times D_{v}}$ is learned to transform the
feature vectors from the corresponding original $D_{v}$-dimensional space
associated with view $v$ into a lower $d$-dimensional shared subspace
optimized for OCC. The total number of target class instances ($N$) is
projected as follows:

$\mathbf{y}_{v,i}=\mathbf{Q}_{v}\mathbf{f}_{v,i},\forall
v\in\\{1,\dots,V\\}\>\>,\forall i\in\\{1,\dots,N\\}.$ (1)

The objective of MS-SVDD is to find a compact hypersphere that encloses the
target class data; the optimization function is defined as:

$\displaystyle\min F(R,a)=R^{2}+C\sum_{v=1}^{V}\sum_{i=1}^{N}\xi_{v,i}$
$\displaystyle\text{s.t.}\quad||\mathbf{Q}_{v}\mathbf{f}_{v,i}-a||^{2}\leq
R^{2}+\xi_{v,i},$ $\displaystyle\xi_{v,i}\geq 0,\quad\forall
v\in\\{1,\ldots,V\\},\forall i\in\\{1,\ldots,N\\},$ (2)

where $R$ is the radius of the hypersphere, $a$ its center, $\xi_{v,i}$ are
slack variables, and the term $C$ controls the penalty for outliers in the
training set. The Lagrangian function of MS-SVDD is as follows:

$\displaystyle
L=\sum_{v=1}^{V}\sum_{i=1}^{N}\alpha_{v,i}\mathbf{y}_{v,i}^{\intercal}\mathbf{y}_{v,i}-\sum_{v=1}^{V}\sum_{i=1}^{N}\sum_{n=1}^{V}\sum_{j=1}^{N}\alpha_{v,i}\mathbf{y}_{v,i}^{\intercal}\mathbf{y}_{n,j}\alpha_{n,j}.$
(3)

where $\alpha$ is the Lagrange multiplier, and maximizing (3) will give us
$\alpha$ values for all training instances defining their position in the data
description. It should be noted that optimizing (3) for $\alpha$ corresponds
to the traditional SVDD [19] applied in the subspace [19]. We update the
projection matrix $\mathbf{Q}_{v}$ iteratively as
$\mathbf{Q}_{v}\leftarrow\mathbf{Q}_{v}-\eta\Delta L$, where $\Delta L$ is the
gradient of the Lagrangian for the corresponding modality $v$, and $\eta$ is
the learning rate. The gradient with respect to the projection matrix is
defined as:

$\displaystyle\Delta
L=2\sum_{i=1}^{N}\alpha_{v,i}\mathbf{Q}_{m}\mathbf{f}_{v,i}\mathbf{f}_{v,i}^{\intercal}-2\sum_{i=1}^{N}\sum_{j=1}^{N}\sum_{n=1}^{V}\mathbf{Q}_{n}\mathbf{f}_{n,j}\mathbf{f}_{v,i}^{\intercal}\alpha_{v,i}\alpha_{n,j}+\beta\Delta\omega,$
(4)

where $\omega$ is the regularization term incorporated into MS-SVDD, taking
into account the covariance of data from different modalities in the shared
subspace, the regularization term $\omega$ expresses the covariance and can
take different forms depending on the instances taken into account during the
regularization strategy. Its contribution is weighted by a hyper-parameter
$\beta$. For a more in-depth understanding of various regularization
techniques, readers can refer to [18], where comprehensive details are
provided. We maintain consistency in terminology and denote the regularization
strategies using the same symbols outlined in the aforementioned reference.

This mathematical framework of MS-SVDD underpins our approach to optimizing
the feature space for effective OCC in echocardiography. The incorporation of
regularization techniques and the projection of features into a shared
subspace are crucial aspects that enhance the robustness and accuracy of MI
detection.

###  2.3 MS-SVDD with Composite Kernel (MS-SVDD-CK)

In our study, we have augmented the MS-SVDD by incorporating a composite
kernel in the non-linear data description. The composite kernel is a crucial
component in our methodology, combining the strengths of Gaussian and
Laplacian sigmoid kernels. This combination improves the feature extraction
capability of our model, which is crucial for effective myocardial infarction
detection. The composite kernel is expressed as:

$\mathbf{K}_{ij}=\gamma\exp\left(-\frac{||f_{i}-f_{j}||^{2}}{2\sigma^{2}}\right)+(1-\gamma)\tanh(\kappa(f_{i}^{\intercal}f_{j})+\theta),$
(5)

where $\gamma$ is the balancing parameter determining the weight given to each
kernel type within the composite kernel, $\sigma$, is the scale parameter for
the Gaussian kernel influencing the extent to which it captures global
variations in the data, and $\kappa$ and $\theta$ are unique parameters of the
Laplacian sigmoid kernel. Consequently, $\kappa$ modulates the steepness of
the sigmoid function, while $\theta$ shifts the function along the x-axis.
These parameters allow the Laplacian component to effectively capture local
variations in the data. In this study, we set $\gamma=0.5$ to indicate equal
importance to both the Gaussian and Laplacian sigmoid components. After
obtaining the non-centered composite kernel matrix, we proceed with the Non-
Linear Projection solution outlined in [20]. Subsequently, the composite
kernel is centered as,

$\displaystyle\mathbf{\hat{K}}=(\mathbf{I}-\frac{1}{N}\mathbf{1}\mathbf{1}^{\intercal})\mathbf{K}(\mathbf{I}-\frac{1}{N}\mathbf{1}\mathbf{1}^{\intercal}),$
(6)

where $\mathbf{1}\in\mathbb{R}^{N}$ represents a vector with all elements set
to one, and $\mathbf{I}\in\mathbb{R}^{N\times N}$ is an identity matrix. The
centered kernel matrix $\mathbf{\hat{K}}$ is decomposed by using
eigendecomposition as follows:

$\displaystyle\mathbf{\hat{K}}=\mathbf{U}\mathbf{A}\mathbf{U}^{\intercal},$
(7)

where $\mathbf{A}$ is a diagonal matrix containing the non-negative
eigenvalues of the matrix $\mathbf{\hat{K}}$ in its diagonal. The
corresponding eigenvectors aligned with these eigenvalues are stored in the
matrix $\mathbf{U}$ columns. The representation of data for non-linear data
description is obtained as follows:

$\displaystyle{\Phi}=(\mathbf{A}^{\frac{1}{2}})^{+}\mathbf{U}^{+}{\mathbf{\hat{K}}},$
(8)

where $+$ in the superscript denotes the pseudo-inverse. After obtaining the
data in $\Phi$ space, we apply all the steps for linear data description
described in 1.

The proposed approach, merging both Gaussian and Laplacian sigmoid kernels,
provides our Multi-modal Subspace Support Vector Data Description with
Composite Kernel (MS-SVDD-CK) model with a robust and versatile tool for
analyzing echocardiography data. The Gaussian component captures broad trends
and general data characteristics, whereas the Laplacian sigmoid kernel focuses
on detailed and local variations. This synergistic combination ensures a
comprehensive analysis, enhancing the model’s ability to detect myocardial
infarction in echocardiography accurately.

###  2.4 Optimization Techniques and Enhanced Adaptability

We initialize the projection matrix $\mathbf{Q}$ for the corresponding v using
Principal Component Analysis (PCA). The projection matrix is orthogonalized
and normalized at every iteration so that
$\mathbf{Q}\mathbf{Q}^{\intercal}=\mathbf{I}$. Our approach enhances the MS-
SVDD model through the Symmetric Descent (SD) and Asymmetric Descent (AD)
methods. These methods are designed to update the projection matrix
$\mathbf{Q}_{v}$ more effectively, catering to the unique characteristics of
each data modality.

Symmetric Descent Method: The SD method involves a consistent application of
gradient operations across all modalities. It is denoted as SD- for gradient
descent and SD+ for gradient ascent, applied uniformly across different data
modalities.

Asymmetric Descent Method: The AD method applies divergent gradient operations
to different modalities, employing descent for one and ascent for another. It
allows for tailored updates according to the specific requirements of each
modality. This method is represented by AD-+ and AD+-, indicating the
direction of the gradient update for the first and second modalities,
respectively.

Mathematical Formulation: The projection matrix $\mathbf{Q}$ for each modality
$v$ is updated using the gradient $\Delta L(\mathbf{Q}^{(t)})$ of the loss
function at iteration $t$ and the learning rate $\eta$ as follows:

$\mathbf{Q}^{(t+1)}=\begin{cases}\mathbf{Q}^{(t)}-\eta\Delta
L(\mathbf{Q}^{(t)})&\text{for SD-}\\\ \mathbf{Q}^{(t)}+\eta\Delta
L(\mathbf{Q}^{(t)})&\text{for SD+}\\\ \mathbf{Q}^{(t)}\mp\eta\Delta
L(\mathbf{Q}^{(t)})&\text{for AD-+ or AD+-}\\\ \end{cases},$ (9)

where the $\mp$ symbol in the AD method indicates the application of negative
gradient steps for the first modality and positive for the second in AD-+, and
the reverse in AD+-. Algorithm 1 describes the overall MS-SVDD algorithm.

The SD and AD optimization strategies significantly enhance the MS-SVDD model.
By fine-tuning the unique aspects of each data modality in the multi-view
echocardiography dataset, these strategies ensure a more precise, robust, and
efficient myocardial infarction detection model. The adaptability offered by
SD and AD methods allows the MS-SVDD model to effectively address the
complexities of multi-modal datasets, leading to improved accuracy and
robustness. This approach is particularly effective for the early detection of
myocardial infarction and has substantial potential to enhance anomaly
detection and classification in real-world applications.

##  3 Experimental Evaluation

This section details our experimental setup and the results obtained over the
benchmark HMC-QU dataset [21]. We compare the results with [22], focusing on
both multi-modal linear and non-linear approaches with novel optimization
techniques and a composite kernel in the non-linear data description.

###  3.1 Experimental Setup

Inputs : ${\mathbf{F}}_{v}$ ($\Phi$ in NPT case) // Input data

$\beta$ // Regularization parameter for $\omega$

$\eta$ // Learning rate parameter

$d$, // Dimensionality of joint subspace

$C$ // Regularization parameter in SVDD

$V$ // Total number of modalities (Views)

Outputs : $\textbf{Q}_{v}$ for each $v=1,...,V$ // Projection matrices

$R$ // Radius of hypersphere

$\alpha$ // Defines the data description

for  _v=1:V_ do

Initialize $\mathbf{Q}_{v}$ via PCA;

end for

for  _$t=1:max\\_iter$_ do

For each $v$, map $\mathbf{F}_{v}$ to $\mathbf{Y}_{v}$ using Eq. (1);

Form Y by combining all $\mathbf{Y}_{v}$’s;

Solve SVDD in the subspace to obtain $\alpha$ in Eq. (3);

for  _v=1:V_ do

Calculate $\Delta L$ using Eq. (4);

Update $\mathbf{Q}_{v}$ according to (9);

Orthogonalize and normalize $\mathbf{Q}_{v}$ using QR decomposition;

end for

end for

For each $v$, compute $\mathbf{Y}_{v}$ using Eq. (1);

Form $\mathbf{Y}$ by combining all $\mathbf{Y}_{v}$’s;

Solve SVDD to obtain the final data description;

Algorithm 1 The optimization process of the MS-SVDD algorithm.

To evaluate the efficacy of our proposed framework, we conducted an extensive
set of experiments on the HMC-QU dataset. This dataset comprises 260
echocardiography recordings, capturing both A4C and A2C views from a cohort of
130 individuals. The ground truths of the dataset are divided into two groups:
88 MI patients and 42 non-MI subjects. The implementation of all OCC models is
carried out in MATLAB R2022b.

Table 1: Average myocardial infarction detection performance results (%)
computed over the test sets of each $5-$fold in HMC-QU dataset. Optimization
strategies(OS): ’SD-’ is represented by ’\--’, ’SD+’ by ’++’, ’AD-+’ by ’-+’,
and ’AD+-’ by ’+-’, NA (Not Applicable) designates when the OS and/or
regularization is not applicable.

| Target: MI |  | Target: non-MI
---|---|---|---
| $OS$ | $r$ | Sen | Spe | Pre | F1 | Acc | GM |  | $OS$ | $r$ | Sen | Spe | Pre | F1 | Acc | GM
Non-linear |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
$\text{MS-SVDD-CK}_{ds_{1}}$ | $-+$ | $\omega_{4}$ | $70.45$ | $66.67$ | $81.58$ | $75.61$ | $69.23$ | $\mathbb{68.53}$ |  | $++$ | $\omega_{0}$ | $73.81$ | $60.23$ | $46.97$ | $57.41$ | $64.62$ | $66.67$
$\text{MS-SVDD-CK}_{ds_{2}}$ | $-+$ | $\omega_{4}$ | $56.82$ | $47.62$ | $69.44$ | $62.50$ | $53.85$ | $52.02$ |  | $++$ | $\omega_{2}$ | $64.29$ | $64.77$ | $46.55$ | $54.00$ | $64.62$ | $64.53$
$\text{MS-SVDD-CK}_{ds_{3}}$ | $+-$ | $\omega_{4}$ | $56.82$ | $59.52$ | $74.63$ | $64.52$ | $5769$ | $58.16$ |  | $+-$ | $\omega_{3}$ | $73.81$ | $55.68$ | $44.29$ | $55.36$ | $61.54$ | $64.11$
$\text{MS-SVDD-CK}_{ds_{4}}$ | $-+$ | $\omega_{5}$ | $84.09$ | $42.86$ | $75.51$ | $79.57$ | $70.77$ | $60.03$ |  | $++$ | $\omega_{6}$ | $66.67$ | $76.14$ | $\mathbb{57.14}$ | $\mathbb{61.54}$ | $\mathbb{73.08}$ | $\mathbb{71.24}$
$\text{MS-SVDD}_{ds_{1}}$ | $-+$ | $\omega_{1}$ | $67.05$ | $64.29$ | $79.73$ | $72.84$ | $66.15$ | $65.65$ |  | $-+$ | $\omega_{1}$ | $71.43$ | $62.50$ | $47.62$ | $57.14$ | $65.38$ | $66.82$
$\text{MS-SVDD}_{ds_{2}}$ | $+-$ | $\omega_{3}$ | $62.50$ | $42.86$ | $69.62$ | $65.87$ | $56.15$ | $51.75$ |  | $++$ | $\omega_{3}$ | $64.29$ | $63.64$ | $45.76$ | $53.47$ | $63.85$ | $63.96$
$\text{MS-SVDD}_{ds_{3}}$ | $-+$ | $\omega_{0}$ | $77.27$ | $35.71$ | $71.58$ | $74.32$ | $63.85$ | $52.53$ |  | $-+$ | $\omega_{2}$ | $69.05$ | $62.50$ | $46.77$ | $55.77$ | $64.62$ | $65.69$
$\text{MS-SVDD}_{ds_{4}}$ | $-+$ | $\omega_{1}$ | $68.18$ | $61.90$ | $78.95$ | $73.17$ | $66.15$ | $64.97$ |  | $-+$ | $\omega_{2}$ | $71.43$ | $64.77$ | $49.18$ | $58.25$ | $66.92$ | $68.02$
$\text{MS-SVDD}_{ds_{1}}$ | $--$ | $\omega_{1}$ | $70.45$ | $61.90$ | $79.49$ | $74.70$ | $67.69$ | $66.04$ |  | $--$ | $\omega_{0}$ | $71.43$ | $53.41$ | $42.25$ | $53.10$ | $59.23$ | $61.77$
$\text{MS-SVDD}_{ds_{2}}$ | $--$ | $\omega_{2}$ | $63.64$ | $42.86$ | $70.00$ | $66.67$ | $56.92$ | $52.23$ |  | $--$ | $\omega_{0}$ | $61.90$ | $73.86$ | $53.06$ | $57.14$ | $70.00$ | $67.62$
$\text{MS-SVDD}_{ds_{3}}$ | $--$ | $\omega_{2}$ | $55.68$ | $59.52$ | $74.24$ | $63.64$ | $56.92$ | $57.57$ |  | $--$ | $\omega_{0}$ | $57.14$ | $57.95$ | $39.34$ | $46.60$ | $57.69$ | $57.54$
$\text{MS-SVDD}_{ds_{4}}$ | $--$ | $\omega_{6}$ | $39.77$ | $\mathbb{76.19}$ | $77.78$ | $52.63$ | $51.54$ | $55.05$ |  | $--$ | $\omega_{2}$ | $73.81$ | $67.05$ | $51.67$ | $60.78$ | $69.23$ | $70.35$
ES-SVDD | NA | $\psi_{0}$ | $73.86$ | $38.10$ | $71.43$ | $72.63$ | $62.31$ | $53.05$ |  | NA | $\psi_{0}$ | $69.05$ | $56.82$ | $43.28$ | $53.21$ | $60.77$ | $62.64$
S-SVDD | NA | $\psi_{2}$ | $59.09$ | $54.76$ | $73.24$ | $65.41$ | $57.69$ | $56.88$ |  | NA | $\psi_{1}$ | $54.76$ | $52.27$ | $35.38$ | $42.99$ | $53.08$ | $53.50$
SVDD | NA | NA | $80.68$ | $38.10$ | $73.20$ | $76.76$ | $66.92$ | $55.44$ |  | NA | NA | $69.05$ | $71.59$ | $53.70$ | $60.42$ | $70.77$ | $70.31$
OC-SVM | NA | NA | $42.05$ | $71.43$ | $75.51$ | $54.01$ | $51.54$ | $54.81$ |  | NA | NA | $35.71$ | $\mathbb{82.95}$ | $50.00$ | $41.67$ | $67.69$ | $54.43$
Linear |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
$\text{MS-SVDD}_{ds_{1}}$ | $++$ | $\omega_{5}$ | $72.73$ | $59.52$ | $79.01$ | $75.74$ | $68.46$ | $65.80$ |  | $-+$ | $\omega_{2}$ | $71.43$ | $64.77$ | $49.18$ | $58.25$ | $66.92$ | $68.02$
$\text{MS-SVDD}_{ds_{2}}$ | $-+$ | $\omega_{5}$ | $60.23$ | $73.81$ | $82.81$ | $69.74$ | $64.62$ | $66.67$ |  | $+-$ | $\omega_{2}$ | $85.71$ | $47.73$ | $43.90$ | $58.06$ | $60.00$ | $63.96$
$\text{MS-SVDD}_{ds_{3}}$ | $+-$ | $\omega_{5}$ | $57.95$ | $76.19$ | $\mathbb{83.61}$ | $68.46$ | $63.85$ | $66.45$ |  | $++$ | $\omega_{5}$ | $66.67$ | $63.64$ | $46.67$ | $54.90$ | $64.62$ | $65.13$
$\text{MS-SVDD}_{ds_{4}}$ | $-+$ | $\omega_{5}$ | $85.23$ | $47.62$ | $77.32$ | $\mathbb{81.08}$ | $\mathbb{73.08}$ | $63.71$ |  | $++$ | $\omega_{4}$ | $73.81$ | $61.36$ | $47.69$ | $57.94$ | $65.38$ | $67.30$
$\text{MS-SVDD}_{ds_{1}}$ | $--$ | $\omega_{5}$ | $81.82$ | $47.62$ | $76.60$ | $79.12$ | $70.77$ | $62.42$ |  | $--$ | $\omega_{5}$ | $73.81$ | $62.50$ | $48.44$ | $58.49$ | $66.15$ | $67.92$
$\text{MS-SVDD}_{ds_{2}}$ | $--$ | $\omega_{2}$ | $54.55$ | $59.52$ | $73.85$ | $62.75$ | $56.15$ | $56.98$ |  | $--$ | $\omega_{2}$ | $78.57$ | $36.36$ | $37.08$ | $50.38$ | $50.00$ | $53.45$
$\text{MS-SVDD}_{ds_{3}}$ | $--$ | $\omega_{0}$ | $67.05$ | $59.52$ | $77.63$ | $71.95$ | $64.62$ | $63.17$ |  | $--$ | $\omega_{5}$ | $73.81$ | $50.00$ | $41.33$ | $52.99$ | $57.69$ | $60.75$
$\text{MS-SVDD}_{ds_{4}}$ | $--$ | $\omega_{5}$ | $85.23$ | $42.86$ | $75.76$ | $80.21$ | $71.54$ | $60.44$ |  | $--$ | $\omega_{0}$ | $\mathbb{80.95}$ | $59.09$ | $48.57$ | $60.71$ | $66.15$ | $69.16$
ES-SVDD | NA | $\psi_{3}$ | $82.95$ | $35.71$ | $73.00$ | $77.66$ | $67.69$ | $54.43$ |  | NA | $\psi_{3}$ | $45.24$ | $67.05$ | $39.58$ | $42.22$ | $60.00$ | $55.08$
S-SVDD | NA | $\psi_{3}$ | $70.45$ | $45.24$ | $72.94$ | $71.68$ | $62.31$ | $56.45$ |  | NA | $\psi_{2}$ | $50.00$ | $70.45$ | $44.68$ | $47.19$ | $63.85$ | $59.35$
SVDD | NA | NA | $\mathbb{86.36}$ | $33.33$ | $73.08$ | $79.17$ | $69.23$ | $53.65$ |  | NA | NA | $69.05$ | $69.32$ | $51.79$ | $59.18$ | $69.23$ | $69.18$
OC-SVM | NA | NA | $44.32$ | $73.81$ | $78.00$ | $56.52$ | $53.85$ | $57.19$ |  | NA | NA | $47.62$ | $81.82$ | $55.56$ | $51.28$ | $70.77$ | $62.42$

In the training phase of our OCC models, we designated MI and non-MI as the
target classes, respectively. This allowed us to explore the model’s
performance in identifying both the presence and absence of MI in different
settings. To comprehensively assess the effectiveness of our models, we
employed several standard performance metrics, each offering a unique insight
into the model’s diagnostic capabilities: Sensitivity (Sen), indicating the
proportion of true positives correctly identified; Specificity (Spe),
reflecting the accurate identification of true negatives; Precision (Pre),
measuring the rate of positive class predictions; F1-Score (F1), the harmonic
mean of Sensitivity and Precision; Accuracy (Acc), the overall proportion of
correct classifications across the dataset; and Geometric Mean (GM),
representing the balance between Sensitivity and Specificity. Collectively,
these metrics provide a comprehensive evaluation of our model’s performance in
identifying and differentiating MI and non-MI cases.

In our study, OCC models are extensively evaluated through a stratified 5-fold
cross-validation (CV) scheme, allocating 80% of the dataset for training and
20% for testing. To optimize performance, we conducted an exhaustive search
for the best hyperparameters over a stratified 10-fold CV scheme, focusing on
maximizing GM during the training phase. The experimental assessment is
conducted on the HMC-QU dataset, where we explore both linear and non-linear
versions of the MS-SVDD models, along with the enhanced MS-SVDD with Composite
Kernel, MS-SVDD-CK. The aim was to assess the efficacy of these models in
myocardial infarction detection, employing various optimization strategies for
improved accuracy. To ensure a fair comparison with previous studies, all
parameters and settings were aligned with those cited in the referenced
literature. The hyperparameters were configured as follows:

  * •

Gaussian Kernel Scale ($\sigma$): The range
$\\{10^{-2},10^{-1},1,10,10^{2},10^{3}\\}$ is explored, allowing us to adjust
the kernel’s sensitivity to the spread of the data.

  * •

Laplacian Kernel Scale ($\kappa$): Set as $\kappa=\frac{1}{d}$, where $d$ is
the dimensionality of the projected subspace. This scale parameter helps to
control the kernel’s focus on the local structure of the data.

  * •

Learning Rate ($\eta$): The range $\\{10^{-4},10^{-3},10^{-2},10^{-1},1\\}$ is
set for $\eta$. This parameter controls the rate of learning in the
optimization process.

  * •

Regularization parameter ($\beta$): The values for $\beta$ were explored
within $\\{10^{-4},10^{-3},10^{-2},10^{-1},1,10,10^{2},10^{3},10^{4}\\}$,
providing a range of options to balance the model complexity and control
overfitting.

  * •

Penalty Parameter ($C$): We examined $C$ within the range
$\\{0.01,0.05,0.1,0.2,0.3,0.4,0.5,0.6\\}$. This parameter determines the
trade-off between maximizing the margin and minimizing classification errors.

  * •

Dimensionality ($d$): For multi-modal approaches, the projected dimensionality
is set in the range $[1,5]$, with increments of 1 at each step.

  * •

MS-SVDD Decision Strategies ($ds$): The MS-SVDD employs various decision
strategies, namely $ds_{1}$, $ds_{2}$, $ds_{3}$, and $ds_{4}$. The specifics
of these strategies are detailed in the reference [18].

The tuning of hyperparameters ensures that our approach is optimally
configured to accurately and reliably detect myocardial infarction in
echocardiography data; thus, enhancing the effectiveness of our diagnostic
framework.

###  3.2 Results and Discussion

A comprehensive study is conducted evaluating the performance of both MS-SVDD
and MS-SVDD-CK models in OCC using the HMC-QU dataset, focusing on both MI and
non-MI targets. The models are developed in both linear and non-linear
versions, incorporating various optimization and decision strategies. The
performance metrics are computed as the average of test sets of 5 folds in the
HMC-QU dataset, as shown in Table 1. In addition, Table 2 presents confusion
matrices for optimal models based on the GM for MI and non-MI target classes.

Moreover, we compare our results with uni-modal OCC algorithms, including the
One-class Support Vector Machine (OC-SVM) [23], the Support Vector Data
Description (SVDD) [19], the Subspace SVDD (S-SVDD) [14], and the Ellipsoidal
S-SVDD (ES-SVDD) [24]. It is crucial to note that the last eight models for
both linear and non-linear cases presented in Table 1 originate from our
previous study [22]. In referenced uni-modal subspace OCC methods, S-SVDD and
ES-SVDD), the regularization term is denoted by $\psi$ and represents the
class variance in the projected space. Different regularization strategies
($\psi_{0}$-$\psi_{3}$) control the impact of individual samples in the
regularization term. More details of the regularization strategies can be
found in [25]. Using these benchmarks, a comparative analysis is conducted
with the advancements constructed in the current study. The methodologies and
optimization strategies employed in these models, as detailed in [22], laid
the foundation for the improvements realized in our latest research. By
comparing the current study’s outcomes with these established models, we can
clearly illustrate the progress in myocardial infarction detection using
echocardiography.

There is a significant improvement in MI detection performance with the MS-
SVDD-CK models. The model $\text{MS-SVDD-CK}{ds_{1}}$, leveraging optimization
AD-+ and composite kernel with regularization $\omega_{4}$, achieved a GM of
68.53% for the MI target. This represents an approximate 3.77% improvement
over the previously best-performing model, $\text{MS-SVDD}{ds_{2}}$ with
$\omega_{5}$ and SD- optimization, which had a GM of 66.04%. For non-MI
targets, the $\text{MS-SVDD-CK}{ds_{4}}$ model exhibited an even higher GM of
71.24%, an enhancement of approximately 1.27% compared to the previous model
$\text{MS-SVDD}{ds_{4}}$ with a GM of 70.35%.

Table 2: Confusion matrices of optimal models for MI and Non-MI target
classes.

$\text{MS-SVDD-CK}_{ds_{1}}$ (Target: MI)
---
| Predicted Non-MI | Predicted MI
Ground Truth Non-MI | 28 | 14
Ground Truth MI | 26 | 62

$\text{MS-SVDD-CK}_{ds_{4}}$ (Target: non-MI)
---
| Predicted Non-MI | Predicted MI
Ground Truth Non-MI | 28 | 14
Ground Truth MI | 21 | 67

The high precision of the linear $\text{MS-SVDD}_{ds_{1}}$ (AD+-) model
suggests its potential utility in clinical settings where it is important to
minimize false positives. There is, however, a trade-off implied by the lower
sensitivity that may need to be addressed in future research. With a superior
F1-Score, the linear $\text{MS-SVDD}_{ds_{4}}$ (AD-+) model is suitable for
scenarios where both false positives and false negatives are of equal concern.

As a result of our study, it was found that non-linear models with composite
kernels are particularly effective at balancing sensitivity and specificity,
which is essential for reliable medical diagnosis. As indicated by the varied
results across different configurations, optimization strategy, and
regularization technique play a significant role in model performance.

##  4 Conclusion

This study presents a significant advancement in the detection of MI using
multi-view echocardiography. Through the integration of a novel MS-SVDD-CK, we
have enhanced the adaptability and effectiveness of OCC models. Incorporating
Gaussian and Laplacian kernels with Symmetric and Asymmetric Descent methods
has yielded notable improvements in model performance. Based on the
comprehensive experiments conducted over the HMC-QU dataset, we demonstrate a
superior performance of our multi-modal approach, particularly its ability to
adapt to different data modalities. This is evidenced by achieving the best
geometric mean of 71.24% for myocardial infarction detection. In addition to
providing a robust framework for early MI diagnosis, this study also paves the
way for future research in applying advanced machine learning techniques to
cardiac health diagnostics. In the future, we will investigate using graph-
embedded-based subspace learning methods [26] for myocardial infarction
detection from multi-view echocardiography.

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