HKF Hierarchical Kalman Filtering with Online Learned Evolution Priors for Adaptive ECG Denoising

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HKF: Hierarchical Kalman Filtering

with Online Learned Evolution Priors

for Adaptive ECG Denoising

Guy Revach, Timur Locher, Nir Shlezinger, Ruud J. G. van Sloun, and Rik Vullings

Abstract—Electrocardiography (ECG) signals play a pivotal

role in many healthcare applications, especially in at-home

monitoring of vital signs. Wearable technologies, which these

applications often depend upon, frequently produce low-quality

ECG signals. While several methods exist for ECG denoising to

enhance signal quality and aid clinical interpretation, they often

underperform with ECG data from wearable technology due

to limited noise tolerance or inadequate ﬂexibility in capturing

ECG dynamics. This paper introduces HKF, a hierarchical and

adaptive Kalman ﬁlter, which uses a proprietary state space

model to effectively capture both intra- and inter-heartbeat

dynamics for ECG signal denoising. HKF learns a patient-speciﬁc

structured prior for the ECG signal’s intra-heartbeat dynamics

in an online manner, resulting in a ﬁlter that adapts to the speciﬁc

ECG signal characteristics of each patient. In an empirical study,

HKF demonstrated superior denoising performance (reduced

mean-squared error) while preserving the unique properties of

the waveform. In a comparative analysis, HKF outperformed

previously proposed methods for ECG denoising, such as the

model-based Kalman ﬁlter and data-driven autoencoders. This

makes it a suitable candidate for applications in extramural

healthcare settings.

I. INTRODUCTION

In the pursuit to halt the increase in healthcare costs and

create a sustainable healthcare system, progressively more

patients should be monitored outside the hospital environment.

In recent years, various technologies have emerged for remote

and ambulatory monitoring of vital signs, including wearable

electrocardiography (ECG) monitoring devices [2], [3].

The ECG reﬂects the electrical activity of the heart and is

considered one of the most important and informative monitor-

ing modalities, which reveals information about cardiac func-

tion and possible pathologies. More speciﬁcally, an ECG can

play a large part in the clinical detection of diseases, including

coronary heart diseases, heart attacks, and arrhythmia that can

lead to even more severe conditions, such as stroke [4], [5].

ECG signal analysis can also play a crucial part in detecting

the asphyxia of a fetus during labor [6].

Parts of this work were presented at the IEEE International Conference

on Acoustics, Speech, and Signal Processing (ICASSP) 2023 [1]. G. Revach

and T. Locher and are with the Institute for Signal and Information Process-

ing (ISI), D-ITET, ETH Z¨

urich, Switzerland (e-mail: grevach@ethz.ch). N.

Shlezinger is with the School of ECE, Ben-Gurion University of the Negev,

Beer Sheva, Israel (e-mail: nirshl@bgu.ac.il). R. J. G. van Sloun is with

the EE Dpt., Eindhoven University of Technology, The Netherlands, (e-mail:

r.j.g.v.sloun@tue.nl). R. Vullings is with the EE Dpt., Eindhoven University of

Technology, and with Nemo Healthcare, Veldhoven, The Netherlands, (e-mail:

r.vullings@tue.nl). We thank Hans-Andrea Loeliger for helpful discussions,

and Mehdi Bakka for helping with the empirical evaluation.

The speciﬁc shape of the ECG waveform is used by medical

professionals and cardiologists to diagnose the speciﬁc heart

condition and, therefore, it is essential that the ECG recording

is as clean as possible. To diagnose the speciﬁc condition,

or deterioration thereof, of a patient, a medical professional

primarily focuses on speciﬁc characteristics in the ECG. These

characteristics can differ between applications. For instance, in

case of myocardial infarction or hypoxia in the fetus, the ST-

segment often provides vital information [6], [7]. Monitoring

the ST-interval or the occurrence of a negative T-wave ampli-

tude can also be beneﬁcial since these indicate compromised

cardiac performance [8], [9].

Compared with in-hospital monitoring, at-home ECG mon-

itoring comes at the expense of signal quality, e.g., electrodes

incorporated in garments that are used for recording the ECG

generally provide noisier signals, with more artifacts than the

adhesive electrodes that are typically used in the hospital [10].

Although simple ﬁltering can suppress certain noises and

artifacts [11], its effectiveness is limited for additive Gaussian

noise (AGN), due to a partial overlap between the signal and

noise bandwidth [12]. The challenge of denoising ECG signals

corrupted by AGN is the primary focus of this paper.

In the ﬁeld of ECG denoising, recent literature presents a

variety of approaches, ranging from classical signal processing

techniques to cutting-edge deep learning methods. Model-

based techniques are built upon predeﬁned statistical models

that capture the intrinsic characteristics of the ECG waveform.

In contrast, non-parametric methods, such as the wavelet

transform and EMD, steer clear of rigid model assumptions.

They decompose the ECG signal into different frequency com-

ponents or intrinsic mode functions, facilitating the separation

of noise. On the other end of the spectrum, deep neural

network architectures harness vast amounts of data to learn an

empirically optimal process for denoising an ECG signal [13]–

[16].

Deep learning-based approaches, speciﬁcally training deep

neural networks end-to-end to minimize a loss function

with vast datasets, have become powerful tools for various

tasks [17], including ECG denoising [18]. Examples include

using a recurrent neural network, as in [19], and employing a

fully convolutional denoising autoencoder, as in [20]. In [21],

a multi-channel fetal ECG denoising was considered based

on deep convolutional neural networks. In [22], a generative

adversarial network architecture was considered, while in [23],

a deep learning framework based on stacked cardiac cycle

tensors was introduced.

arXiv:2210.12807v3 [eess.SP] 20 Nov 2023

A primary advantage of deep learning is its data-driven

nature. When trained on diverse and comprehensive datasets,

it can yield superior accuracy and robustness. The efﬁcacy

of these models is heavily inﬂuenced by the quality and

diversity of the training data, highlighting the importance of

meticulously curated datasets. However, these models also

pose challenges. A signiﬁcant constraint is their reliance on ag-

gregated patient datasets. When trained with an mean-squared

error (MSE) criterion, deep networks often exhibit a strong

bias towards the mean. Consequently, in situations with noise

where multiple plausible waveforms might have produced the

observed data, these MSE-trained networks often output the

posterior mean. This bias complicates the task of tailoring the

model to an individual patient’s unique ECG characteristics.

Furthermore, the complexity of these architectures requires

substantial computational resources. Additionally, their ”black

box” nature introduces challenges in model interpretability.

Among the arsenal of effective ECG denoising techniques,

non-parametric methods have attracted considerable atten-

tion [24]. empirical mode decomposition (EMD) methods [25],

such as [26]–[28], stand out for their ability to decompose a

signal into intrinsic mode functions (IMFs), with each IMF

representing a speciﬁc oscillatory mode within the original

ECG. While EMD’s decomposition provides a clear repre-

sentation of the ECG’s frequency components and facili-

tates effective noise separation, it can sometimes be sensitive

to ﬂuctuations, leading to mode mixing or the emergence

of spurious modes. Additionally, wavelet transforms offer

a robust multi-resolution analysis, enabling a precise time-

frequency representation of signals [29]–[31]. Methods, such

as [32]–[36], distinguish essential ECG components from

high-frequency noise and facilitate selective noise removal,

preserving the integrity of the original ECG waveform. While

their computational efﬁciency makes them suitable for real-

time clinical applications, their success heavily relies on the

correct choice of wavelet and its parameters, and incorrect

tuning can lead to ECG distortion. Moreover, in the face of

heavy noise, wavelets might not be fully effective.

Model-based techniques have emerged as a notable al-

ternative for ECG denoising. Central to this approach is

the utilization of state space (SS) models, complemented

by various variations of the Kalman ﬁlter (KF) [37], as

in [38]–[42]. Local approximations of the ECG waveforms

have been explored through windowed SS models in [43]–

[45], and via autoregressive models [46], [47]. Nevertheless,

these approximations can sometimes fall short of capturing

the intricate intra-heartbeat dynamics and the inherent quasi-

periodicity between consecutive heartbeats, posing challenges

for optimal denoising.

A fundamental principle in numerous ECG denoising

methodologies is to model the signal’s complex evolution

by capitalizing on its quasi-periodicity. This principle is

subsequently utilized as a prior belief in Bayesian ﬁltering

techniques [48], such as the KF [38]. For instance, [40]

omits intra-heartbeat variations and chooses instead to rep-

resent the evolution of consecutive heartbeats with an identity

function. This approach is grounded in the premise that, in

the absence of arrhythmia, two successive heartbeats, when

centered around the R-peak, closely resemble each other. In

essence, this method is equivalent to a weighted average of

multiple heartbeats. While this strategy enhances the Signal

to Noise Ratio (SNR), it also runs the risk of obscuring

vital physiological dynamics. The research presented in [41]

incorporates the expectation-maximization (EM) algorithm to

determine the evolution function. When combined with a bank

of KFs, this method efﬁciently targets both high and low-

frequency noise. However, its application is limited by its

reliance on a linear prior function, its constraint to ﬁlter signals

of a ﬁxed length, and its omission of the ECG’s periodic

information. In contrast, [49] introduces non-linear priors us-

ing partial differential equation models [38]. While innovative,

these models frequently face challenges in capturing patient-

speciﬁc variations. To address this issue, [42] attempts to

automatically ﬁt model parameters using a least squares (LS)

optimizer, based on several pre-recorded heartbeats.

In this work, we propose a hierarchical Kalman ﬁlter

(HKF) for ECG denoising, which enhances signal quality

without obscuring dynamic changes potentially linked to

pertinent pathophysiology. Our HKF is designed based on

our innovative hierarchical SS model, which describes the

ECG signal dynamics both within individual heartbeats and

across consecutive heartbeats. Speciﬁcally, HKF consists of an

online learned structured evolution prior for a single heartbeat;

a Rauch-Tung-Striebel (RTS) intra-heartbeat smoother [50]

that harnesses this prior; and an inter-heartbeat KF [37] for

denoising spanning multiple heartbeats. The online warm-up

phase is meticulously designed to tackle challenges such as the

highly patient-speciﬁc heartbeat shape and substantial noise

variation resulting from myriad factors, ranging from equip-

ment intricacies to room temperature variations. While one

might conceptualize a typical ECG signal shape, the reality is

that there’s considerable inter-variability among patients. Not

only can the signal shape vary signiﬁcantly between patients,

but there’s also intra-patient variability; the placement and ori-

entation of electrodes can introduce alterations in the observed

waveform. This renders the task of crafting a universal prior

quite challenging. Crucially, HKF doesn’t require supervised

pre-training and is inherently patient-adaptive, due to its online

covariance estimation and its learned structured evolution

prior. Yet, it preserves the transparent and interpretable nature

of the KF.

Our experimental study shows that the proposed HKF effec-

tively denoises ECG signals, even in challenging setups, while

retaining the subtle, clinically valuable structures within the

signals. These attributes make it especially suited for medical

and healthcare applications where a high degree of conﬁdence

and reliability is essential.

The remainder of this paper is organized as follows: Sec-

tion II formulates the task and introduces the hierarchical SS

model. Section III delves into the details of the proposed

HKF. Section IV elaborates on parameter estimation. Finally,

Section Vpresents our empirical study, demonstrating that the

HKF surpasses both model-based and data-driven benchmarks.

(a) Heart Musculature [51]

Time

Amplitude

80 ms 100 ms 160 ms

(b) An ECG Waveform Illustration

Fig. 1: Illustration of a Heart and an ECG Waveform

II. SYSTEM MODEL AND TASK FORMULATION

In this section, we lay the groundwork for the subsequent

derivation of the HKF. We begin with a review of essential

information on the ECG signal. Following that, we delve into

the task of multi-channel ECG signal denoising. We conclude

by presenting our unique hierarchical SS model, which serves

as the foundation for the HKF design.

A. The ECG Signal

The heart is composed of two main types of muscle: the

atrial muscle and the ventricular muscle, as illustrated in

Figure 1a. During a heartbeat (HB), these different muscles

contract and relax in response to electrical impulses, which

depolarize and repolarize the heart. These impulses propagate

as an electrical ﬁeld through the body and can be detected by

electrodes on the skin. The voltage variation recorded over

time is what we refer to as the ECG signal. The typical

ECG signal comprises three segments: the P-wave, the QRS

complex, and the T-wave, as shown in Figure 1b. Each of these

segments represents a different stage of heart contraction. The

P-wave corresponds to the contraction of the atrial muscle; the

QRS complex indicates ventricular depolarization; and the T-

wave reﬂects ventricular repolarization. As its name suggests,

the QRS complex consists of three smaller waves (Q-wave, R-

wave, and S-wave) associated with the depolarization of the

ventricular muscle. The entire QRS complex cycle takes about

100 [msec][52]. While a typical ECG signal depicts positive

peaks for the P-wave, R-wave, and T-wave, the direction of

these peaks can vary depending on the placement of the

electrodes. The bandwidth of an ECG signal usually falls

within the 0.05-100 Hz range [53].

B. Multi-Channel ECG Denoising Task Formulation

The electrical activity of the heart is observed and moni-

tored by placing multiple electrodes on the human body and

recording the noisy amplitudes as a vector time series. Here,

each electrode is referred to as a channel. The vector yt∈Rm

denotes the observed noisy amplitudes across mchannels in

a discrete-time index t∈Z. The speciﬁc features of this

recording, such as ECG shape, amplitude, noise, and other

artifacts, depend on the electrode and its placement on the

body. We assume that yt, the noisy recordings, originated

Single

Heartbeat

Channels

Consecutive

Heartbeats

1 2 ... ... t... ...

Intra-Heartbeat:

Inter-Heartbeat:

Fig. 2: Hierarchical System Model as a Tensor

from xt∈Rm, multi-channel noiseless signals, that were then

corrupted by AGN vt. The ECG denoising task is hereby

deﬁned as the reconstruction of xt, the mclean channels,

from {yi}t

i=1, their corresponding past and current noisy

observations, namely

Ψ : {yi}t

i=1 7→ ˆ

xt,Ψ∗= arg min E∥ˆ

xt−xt∥2.(1)

The denoiser, formulated as a mapping Ψ, is designed to

minimize a cost function. A natural choice for this function is

the MSE.

The underlying ground truth heart activity is commonly

described as a cardiac vector, a stochastic process in R3, that

captures the direction and magnitude of the electrical impulses

as they propagate through the heart. Therefore we assume

that xt, describing the mchannels, is intra-correlated, and

originates from the same hidden ground truth activity signal

This assumption will be used in our hierarchical system model

as shown next.

C. Hierarchical System Model

The relationship between the observed ECG signal ytto its

corresponding noiseless instance xt, can be modeled as a dy-

namical system. A canonical way to model dynamical systems

in discrete-time is by using SS models [54]. SS models blend

well with numerous ﬁltering and smoothing techniques, and

can accommodate the integration of both physical and data-

driven learned models. To exploit the periodicity, which plays

a key role in the ECG signal, we model the signal dynamics

using a hierarchical SS model.

In the following, we make a reasonable assumption that

individual HBs can be accurately segmented. Therefore, we

divide the signal into periodic segments of length T, where

each such segment represents a single HB. Our model draws

inspiration from the decimated components decomposition

method, which is used to represent periodic systems and

cyclostationary signals [55] as multivariate (tensor) stationary

ones. The dynamics within a single HB are described using

an intra-HB (internal) SS model, while the dynamics between

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HKF:HierarchicalKalmanFilteringwithOnlineLearnedEvolutionPriorsforAdaptiveECGDenoisingGuyRevach,TimurLocher,NirShlezinger,RuudJ.G.vanSloun,andRikVullingsAbstract—Electrocardiography(ECG)signalsplayapivotalroleinmanyhealthcareapplications,especiallyinat-homemonitoringofvitalsigns.Wearabletechnologies...

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HKF Hierarchical Kalman Filtering with Online Learned Evolution Priors for Adaptive ECG Denoising

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