# SDEMG: Score-Based Diffusion Model for Surface Electromyographic Signal
Denoising

###### Abstract

Surface electromyography (sEMG) recordings can be influenced by
electrocardiogram (ECG) signals when the muscle being monitored is close to
the heart. Several existing methods use signal-processing-based approaches,
such as high-pass filter and template subtraction, while some derive mapping
functions to restore clean sEMG signals from noisy sEMG (sEMG with ECG
interference). Recently, the score-based diffusion model, a renowned
generative model, has been introduced to generate high-quality and accurate
samples with noisy input data. In this study, we proposed a novel approach,
termed SDEMG, as a score-based diffusion model for sEMG signal denoising. To
evaluate the proposed SDEMG approach, we conduct experiments to reduce noise
in sEMG signals, employing data from an openly accessible source, the Non-
Invasive Adaptive Prosthetics database, along with ECG signals from the MIT-
BIH Normal Sinus Rhythm Database. The experiment result indicates that SDEMG
outperformed comparative methods and produced high-quality sEMG samples. The
source code of SDEMG the framework is available at:
https://github.com/tonyliu0910/SDEMG

Index Terms—  Surface electromyography, Score-based diffusion model, ECG
interference removal, Deep neural network

##  1 Introduction

Surface electromyography (sEMG) records the biopotential generated by motor
units during muscle contractions. This technique can noninvasively provide
valuable insights into muscle anatomy and physiology, and thus sEMG finds
applications in various clinical areas, including neuromuscular system
investigation [1], rehabilitation [2], stress monitoring [3], assessment of
neuromuscular or respiratory disorders [4, 5], and prosthesis control [6]. In
these applications, sEMG recordings may be subject to electrocardiogram (ECG)
interference if the measurement is taken near the heart [7, 8]. The ECG
contamination can distort the amplitude and frequency aspects of sEMG signals,
posing a challenge in extracting meaningful information. Hence, developing
effective ECG removal methods is crucial to enhance signal quality for various
clinical and human-computer interaction applications.

sEMG and ECG signals have frequency bands between 10 to 500 Hz and 0 to 100
Hz, respectively [9]. The overlapping frequency bands pose difficulties in
segregating the two signals. To address this issue, several single-channel ECG
removal methods have been developed, such as high-pass filters (HP) and
template subtraction (TS) [7, 10]. However, HP causes distortion by removing
the low-frequency part of sEMG signals, and TS relies on the assumption that
ECG is quasi-periodic and sEMG follows a zero-mean Gaussian distribution,
which may not hold in real-world scenarios. These limitations make these ECG
removal methods struggle under demanding conditions, such as low signal-to-
noise (SNR) ratios. Recently, neural networks (NNs) have been widely adopted
for their powerful nonlinear mapping capabilities in signal enhancement [11,
12]. Some studies have applied NNs for sEMG denoising [13, 14]. In [14], Wang
et al. developed an NN-based ECG removal method employing a fully
convolutional network (FCN) as a denoising autoencoder (DAE). The experimental
results in [14] show that FCN outperformed traditional methods in ECG artifact
removal, while it also introduced some signal distortion, which can be
problematic in clinical applications. Therefore, it is desirable to develop
more effective ECG removal methods.

The score-based diffusion model, a generative model, has excelled in producing
high-quality data in various tasks, including image and acoustic generation
[15, 16, 17]; various studies have also incorporated the model into signal
enhancement [18, 19, 20]. Compared to discriminative NNs, the score-based
diffusion model could observe and model the data distribution and reconstruct
the desired sample through sampling steps. Also, the diffusion process and the
reverse process are relatively more tractable and more flexible compared to
other generative models, such as generative adversarial networks (GANs) and
variational autoencoders (VAEs), i.e., we can determine the sample quality by
adjusting the hyperparameters empirically. The features of the score-based
diffusion model mentioned above allow us to model the distribution of the
dataset as a whole and generate samples of higher quality based on the given
conditions.

To refine sEMG denoising performance, this study proposes SDEMG, a conditional
score-based diffusion model for sEMG denoising. The proposed method
progressively adds isometric Gaussian noise to the clean sEMG during the
diffusion process. In the reverse process, we leverage the sEMG waveform
contaminated by ECG and the noise scale variable as conditions for the NN, and
the Gaussian noise is reverted to the clean sEMG segment. Experimental results
show that SDEMG outperforms the previous FCN-based denoising method in signal
quality, providing a refined ECG removal approach for clinical sEMG
applications.

The remainder of this paper is organized as follows. Section 2 reviews related
works. Section 3 introduces the proposed approach. Section 4 presents the
experimental setup and results. Finally, Section 5 concludes the paper and
discusses future works.

##  2 Related Work

###  2.1 ECG interference removal methods

Several single-channel ECG removal methods have been developed in previous
studies, including HP and TS [7, 10]. HP removes the frequency band of the
ECG, inevitably leading to the loss of the low-frequency part of the sEMG
signal. In contrast, TS removes ECG artifacts in the time domain. It extracts
ECG templates for subtraction by either filtering or waveform averaging [7,
21], and the ECG artifacts are subtracted from the contaminated sEMG waveform.
However, the effectiveness of TS relies on the assumption that sEMG signals
are zero-mean Gaussian distributions, which may not be satisfied in real-world
scenarios. This study applies both HP and TS to serve as comparative sEMG
denoising methods. HP is implemented with a cutoff frequency of 40 Hz, and the
TS method was followed using the HP method for optimal results.

Beyond conventional techniques, NN-based methods have demonstrated exceptional
performance in sEMG denoising [13, 14]. In [14], Wang et al. proposed an FCN
as a DAE to eliminate ECG interference from sEMG. The proposed FCN consists of
two parts: an encoder and a decoder. Both parts consist of multiple
convolutional layers with different filter sizes and strides. The encoder
receives contaminated sEMG waveform as input and performs downsampling to
generate high-level feature maps. Conversely, the decoder attempts to
reconstruct the clean sEMG waveform from the encoder output. According to the
study, the FCN demonstrates more effective denoising performance than
conventional methods.

###  2.2 Score-based diffusion models

Score-based diffusion models are a category of deep generative models
initially devised for image generation, and they have further been adapted to
signal enhancement, including speech enhancement and ECG denoising [19, 20].
It is preferable to other generative methods, such as GANs and VAEs. To
elaborate, GANs are notoriously unstable during the training process, and VAEs
lack sample quality. Score-based diffusion models overcome these limitations
and achieve remarkable performance in numerous applications [15, 22, 23].

The process of the score-based diffusion model starts with gradually adding
Gaussian noise to input data at different scales. The NN within the diffusion
model aims to estimate the noise segment at each particular time step.
Subsequently, desired samples are generated during the sampling process.

$s(y)=\nabla_{x}\log p(x).$ (1)

To further introduce the score-based diffusion model, we start with modeling
the underlying data distribution with the Stein score function as shown in Eq.
(1), which is the gradient of data log-density $\log p(x)$ concerning data
$x$:

$\mathbb{E}_{p(\mathbf{x})}[\|\nabla_{\mathbf{x}}\log
p(\mathbf{x})-\mathbf{s}_{\theta}(\mathbf{x})\|_{2}^{2}].$ (2)

We can train an NN, a score-based model $s_{\theta}$, to minimize the Fisher
divergence between the model and the actual data distribution as shown in Eq.
(2). As presented, we calculate the L2 distance between the target data score
and the score-based model with score-matching methods [24, 25]:

$\mathbf{x}_{i+1}\leftarrow\mathbf{x}_{i}+\epsilon\nabla_{\mathbf{x}}\log
p(\mathbf{x})+\sqrt{2\epsilon}~{}\mathbf{z}_{i},\quad i=0,1,\cdots,K.$ (3)

After training the score-based model $s_{\theta}$, we can generate samples
from the specific density, $\tilde{x}\sim p(x)$, via Langevin dynamics
presented in Eq. (3):

Fig. 1: The diffusion process and the sampling in SDEMG.

##  3 The Proposed SDEMG Method

###  3.1 Training process

In this study, we trained a score-based model to estimate the noise
distribution and generate high-quality and high-fidelity sEMG samples based on
the diffusion probabilistic model proposed by Ho et al. [15]. The original
model was devised for unconditional image generation, i.e., the model
generates samples from Gaussian noise without any inputs. In our case, we aim
to retrieve clean sEMG samples from noisy sEMG waveforms contaminated with
ECG. Therefore, we require the noisy signal $\tilde{x}$ as a condition. The
score-based denoising framework takes sEMG mixed with actual noise as input
and attempts to reconstruct the clean waveform.

The score-based diffusion framework comprises two steps: a diffusion process
(algorithm 1) and a sampling process (algorithm 2). The diffusion process
(algorithm 1), otherwise forward process, is defined as a Markov Chain as
shown in Eq. (4):

$q(x_{1:T}|x_{0})=\prod_{t=1}^{T}q(x_{t}|x_{t-1}).$ (4)

During this process, Gaussian noise is gradually added (Eq. (5)) with respect
to different noise schedules $\beta_{1},\dots,\beta_{N}$. The original signals
will be perturbed more significantly as $t$ becomes larger.

$q(x_{t}|x_{t-1})=\mathcal{N}(x_{t};\sqrt{1-\beta_{t}}x_{t-1},\beta_{t}I).$
(5)

Ho et al. [15] observed that: Let $\alpha_{t}=1-\beta_{t}$ and
$\bar{\alpha}_{t}=\prod_{i=1}^{t}\alpha_{i}$, the resultant distribution of
the diffusion process can be presented in Eq. (6) with variance
$1-\bar{\alpha_{t}}$:

$q(\mathbf{x}_{t}|\mathbf{x}_{0})=\mathcal{N}(\mathbf{x}_{t};\sqrt{\bar{\alpha}_{t}}\mathbf{x}_{0},(1-\bar{\alpha}_{t})\mathbf{I}).$
(6)

As mentioned earlier, to avoid signal distortion and generate high-quality
samples, during the training process, we introduce noisy sEMG as the condition
to assist the model in retrieving more features regarding the data
distribution. Furthermore, Song & Ermon [26] discovered that the selection of
noise schedule is critical to sample quality, and Chen et al. [17] proposed
the continuous condition $\bar{\alpha}$ instead of discrete time step variable
$t$ will provide more auxiliary information to the model. In the methods
proposed by Li et al. [20], the reparameterized condition was also adopted and
generated high-quality samples. Consequently, we utilized the cosine beta
scheduler as it was more helpful for the model to generate more robust
predictions [27] and adapted similar condition methods to our denoising
framework and selected the corresponding noise scale $\bar{\alpha}$ from the
noise scale schedule
$\gamma=\\{1,\sqrt{\bar{\alpha_{0}}},\dots,\sqrt{\bar{\alpha_{T}}}\\}$ as one
of the inputs. As shown in Algorithm 1, the model will receive clean sEMG
$x_{0}$, noisy sEMG $\tilde{x}$, and noise scale parameter $\bar{\alpha}$ as
inputs and aims to predict the isotropic Gaussian noise $\epsilon$. We will
minimize the mean squared error of the predictions during the training
process.

1:repeat

2: $x_{0},\tilde{x}\sim q(x_{0},\tilde{x})$

3: $t\sim Uniform(\\{1,\dots,T\\})$

4: ${\bar{\alpha}}\sim Uniform(\gamma_{t-1},\gamma)$

5: $\epsilon\sim{\mathcal{N}}(0,I)$

6: Take gradient descent step on
$\nabla_{\theta}||\epsilon-\epsilon_{\theta}(\sqrt{{\bar{\alpha}}}x_{0}+\sqrt{1-{\bar{\alpha}}}\epsilon,\tilde{x},{\bar{\alpha}})||^{2}$

7:until Converged

Algorithm 1 Training

###  3.2 Sampling process

In the reverse sampling process, as shown in Algorithm 2, the initial sample
is random Gaussian noise $x_{T}$. The denoising framework will reconstruct the
desired sEMG signals through $T$ sampling steps. The sampling process is
defined as a reverse Markov chain in Eq. (7):

$p_{\theta}(\mathbf{x}_{0:T})=p(\mathbf{x}_{T})\prod^{T}_{t=1}p_{\theta}(\mathbf{x}_{t-1}|\mathbf{x}_{t},\tilde{x}).$
(7)

During each iteration, SDEMG will generate a sample $x_{t-1}$ from the
previous sample $x_{t}$ and the input noisy data $\tilde{x}$:

$p_{\theta}(\mathbf{x}_{t-1}|\mathbf{x}_{t},\tilde{x})=\mathcal{N}(\mathbf{x}_{t-1};\boldsymbol{\mu}_{\theta}(\mathbf{x}_{t},t,\tilde{x}),\boldsymbol{\Sigma}_{\theta}(\mathbf{x}_{t},t,\tilde{x})).$
(8)

###  3.3 Model Architecture

The network within SDEMG is adapted from the model proposed by Li et al. [20].
The model comprises two data streams, one for the clean signal and the other
for the noisy data, and the bridges to aggregate the features extracted from
different layers. The data streams are composed of multiple Half Normalized
Filters (HNF), which are computation blocks adopted from Multi-Kernel Filter
design [28] and HiNet [29]. The HNF blocks act as a feed-forward module that
combines features extracted by different convolutional layers with various
receptive fields with channel-wise concatenation. Moreover, half of the
features will be normalized as this technique has demonstrated more stability
during the training process [29]. These blocks utilize residual connections to
produce outputs from input data and processed features. The Bridge blocks are
modified from the feature-wise linear modulation (FiLM) block [30] and
condition on the input noise scale $\sqrt{\bar{\alpha}}$. Notably, we
increased the input dimension of the model from 64 to 128 to better
accommodate the complexity and dynamic nature of sEMG data, since sEMG is
inherently more intricate and variable than ECG data.

##  4 Experiments

###  4.1 Datasets

The sEMG signals used in this study are from the DB2 subset of the Non-
Invasive Adaptive Prosthetics (NINAPro) database [31], which contains 12
channels of sEMG recordings of hand movement made by 40 intact subjects.
Notably, the sEMG recordings were acquired from the upper limb. The DB2 subset
includes 3 sessions, Exercise 1, 2, and 3, which involve 17, 22, and 10
movements, respectively. Each movement was repeated six times for five
seconds, followed by a three-second rest interval. Previous studies [14, 32]
have applied filters to obtain clean sEMG data from this database.

As for ECG interference, this study adopts MIT-BIH Normal Sinus Rhythm
Database (NSRD) from the PhysioNet database [33]. In this data set, 2-channel
ECG recordings were collected from 18 healthy subjects, with a sampling rate
of 128 Hz. Previous studies have adopted the ECG signals from this data set as
the interference in sEMG signals [14, 32].

1:$x_{T}\sim{\mathcal{N}}(0,I)$

2:for $t=T,\dots,1$ do

3: $z\sim{\mathcal{N}}(0,I)$ if $z>1$ else $z=0$

4:
$x_{t-1}=\frac{1}{\alpha_{t}}\left(x_{t}-\frac{1-\alpha_{t}}{\sqrt{1-{\bar{\alpha}}_{t}}}\epsilon_{\theta}(x_{t},\tilde{x},\sqrt{\bar{\alpha_{t}}})\right)+\sigma_{t}z$

5:end for

6:return $x_{0}$

Algorithm 2 Sampling

###  4.2 Data preprocessing and preparation

The sEMG data were processed by a 4th-order Butterworth bandpass filter with
cutoff frequencies of 20 and 500 Hz and downsampled to 1kHz. Subsequently, all
sEMG signals were normalized and divided into 10-second segments. ECG data in
Channel 1 from the MIT-BIH NSRD were filtered by a 3rd-order Butterworth high-
pass and low-pass filter with cutoff frequencies of 10 and 200 Hz to discard
possible noise in ECG signals [7].

sEMG segments of Channel 2, Exercise 1 and Channel 2, Exercise 3 from 30
subjects were selected as the training and validation set, respectively. For
each segment in the training set, 10 randomly selected ECG signals from 12
subjects in the MIT-BIH NSRD were considered ECG artifacts and were
superimposed onto the clean sEMG segments at 6 SNRs (-5, -7, -9, -11, -13, and
-15 dB). For the validation set, three other subjects in the MIT-BIH NSRD were
considered ECG artifacts, and the SNRs are identical to the training sets.

The mismatch conditions between the training and testing sets were considered
to evaluate the generalizability of the proposed framework. sEMG segments of
Channel 9, 10, 11, and 12, Exercise 2, from the remaining 10 subjects were
selected as the testing set. The ECG data from the remaining 3 subjects
(16420, 16539, and 16786) were selected for interference, and the SNRs are
-14-0 dB with an increment of 2 dB. Please note that sEMG subjects, sEMG
movements, sEMG channel, ECG subjects, and SNRs are entirely different from
the setting of the training set.

###  4.3 Evaluation metrics

To compare SDEMG with prior methods, this study follows the earlier work [14,
12, 34] and evaluates the performance with two aspects of criteria: signal
reconstruction quality and feature extraction errors. SNR improvement (SNRimp)
and root-mean-square error (RMSE), respectively, reflect the quality of the
reconstructed signal by presenting the difference of SNR between the inputs
and the outputs and the variance between the outputs and the ground truth.
Furthermore, the RMSE of the average rectified value (ARV) and mean frequency
(MF) feature vectors are adopted as the metrics to evaluate the features
extracted from the sEMG signals [7]. The settings for calculating these
metrics follow the previous study [14]. Better signal reconstruction quality
and higher fidelity can be represented by smaller RMSE values, larger SNRimp
values, and smaller RMSE values of the extracted feature vectors ARV and MF.

Table 1: Overall performance of HP, TS, FCN, and SDEMG.

|  SNRimp (dB) | RMSE |  RMSEARV |  RMSEMF (Hz)
---|---|---|---|---
HP | 13.885 | 1.735e-2 | 3.06e-3 | 17.688
TS | 14.279 | 1.626e-2 | 3.86e-3 | 23.149
FCN | 17.758 | 1.178e-2 | 3.86e-3 | 18.038
SDEMG(Ours) | 18.467 | 1.138e-2 | 2.81e-3 | 14.435

Fig. 2: Comparison of all methods with SNRimp under different SNR input.

###  4.4 Results and discussion

We compared the performance of SDEMG with three previous methods, namely HP,
TS, and FCN [14]. Table 1 illustrates the overall performance of all methods
evaluated by SNRimp, RMSE, and RMSE of ARV and MF. The results show that SDEMG
can achieve higher SNR improvement, lower RMSE value, and better ARV and MF.
Furthermore, Fig. 2 presents the SNRimp of ECG elimination approaches under
various SNRs. The proposed approach performs best across all SNR conditions,
demonstrating its superior effectiveness and robustness in ECG removal.

Notably, this study evaluated the performance of SDEMG under the specific
scenario in which biceps brachii sEMG [7, 10] (Channel 11 in DB2) is
introduced as simulation data. These trunk sEMG signals are prone to
contamination of ECG at around SNR -10dB according to a previous study [35].
Consequently, we investigate the performance of SDEMG in this experimental
setup along with other methods, as shown in Fig. 3. The analysis demonstrates
that the use of SDEMG retains its superiority in most of the cases. Moreover,
compared to our previous study [14], we observed performance degradation of
FCN, particularly in feature extraction error metrics. This discrepancy might
stem from the reduced length of sEMG segments (from 60s to 5s), which is
designed to make the denoising framework more efficient and practical. It is
discovered that FCN trained with a smaller segment size tends to introduce
more distortion to sEMG signals, increasing the errors in sEMG feature
extraction. Conversely, SDEMG demonstrates the capability to alleviate signal
distortion, generating higher-quality sEMG signals. This highlights the
potential of SDEMG to provide improved signal quality for clinical assessment
and evaluation.

Fig. 4 presents an example of ECG contamination removal using SDEMG. It can be
observed that the ECG artifacts in the noisy sEMG (SNR=-8 dB) are eliminated
in the denoised waveform, and the sEMG waveform exhibits minimal distortion
when compared to the clean sEMG. This underscores the capability of SDEMG to
provide high-quality sEMG signals. One challenge of SDEMG is its relatively
high computational effort for optimal performance. This issue may be addressed
by involving ODE solvers or applying parameters pruning and quantization.

(a)

(b)

(c)

(d)

Fig. 3: Performance of simulated trunk sEMG denoising results evaluated by (a)
SNRimp, (b) RMSE, (c) RMSEARV, and (d) RMSEMF Fig. 4: The waveform of (a)
clean sEMG (b) noisy sEMG (c) denoised sEMG by SDEMG.

##  5 Conclusion

In this study, we have proposed SDEMG, a score-based diffusion model, to
reconstruct high-quality and high-fidelity sEMG samples from ECG-interfered
sEMG signals. SDEMG requires no additional reference signal and can directly
process raw sEMG signal. The experimental results demonstrate that the
proposed approach outperforms the comparative methods across a broad range of
SNRs and various evaluation metrics under training-testing mismatched
conditions. To the best of our knowledge, this is the first study developing
diffusion models for sEMG denoising. In the future, we will explore applying
SDEMG on a real contaminated sEMG dataset. Moreover, we will further
investigate the ability of SDEMG to perform denoising on other kinds of sEMG
contaminants.

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