1. Introduction
Ocular motion sensing, also referred to as eye tracking, is utilized in both medical and engineering applications [
1,
2,
3,
4,
5]. These systems leverage various types of eye movements, encompassing saccades, smooth pursuit, vergences, and vestibular–ocular reflexes [
6,
7]. Among the array of technologies available for sensing eye movement, electrooculography (EOG) and videooculography (VOG) stand as the predominant techniques [
8]. EOG relies on the deployment of electrodes to measure the relative changes in the cornea-retinal potential that manifest during the simultaneous rotation of the eye and the cornea-retinal potential’s field vector [
9]. On the other hand, VOG harnesses the infrared signals’ reflection on the corneal–retinal surface to gauge the eye’s angular displacement [
8]. VOG systems are equipped with high-speed video cameras, surpassing EOG systems in accuracy. Notably, commercially available VOG systems, such as the EyeLink 1000 eye-tracker (EL), possess the capability to capture eye movements at a remarkable sampling rate of 2000
, with an accuracy reaching an impressive 0.001 degrees of visual angle (DVA) [
10]. However, it is essential to recognize that VOG systems do entail several drawbacks. They tend to be bulky, come with substantial costs, necessitate stringent or controlled lighting conditions, and fall short in their capacity to monitor closed-eye movements. These inherent limitations render EOG systems an indispensable choice for tracking eye movements across diverse scenarios where the use of VOG systems may be unfeasible. Notably, in contexts such as sleep disorder monitoring, EOG becomes indispensable, enabling the measurement of eye movements even when the eyes are closed [
2,
11].
Among the different types of eye movements, saccades hold paramount significance in medical and engineering applications. A saccade is defined as a swift and ballistic eye motion that redirects the point of fixation from one relatively stable position to another [
5]. Both EOG and VOG systems possess the capability to capture this eye movement. Nevertheless, EOG systems suffer from a notable drawback—the presence of a substantial volume of noise and artifacts that detrimentally impact the fidelity of the recorded EOG eye movements. A typical EOG signal exhibits the intrusion of diverse artifacts, including but not limited to electroencephalography (EEG) artifacts, electromyography (EMG) artifacts, blink artifacts, and an assortment of noise sources [
12,
13]. These artifacts coexist within similar frequency spectra, complicating the denoising procedure [
14,
15]. Consequently, the extraction of saccades and the associated attributes, such as saccade amplitude, velocity, and latency [
5,
10], from an EOG recording becomes an onerous and time-intensive endeavor [
16,
17]. This onerous process exerts a detrimental influence on the accurate identification and classification of saccades [
14].
Contemporary traditional saccade detection and classification methods, which rely on current threshold-based techniques, necessitate EOG signals characterized by minimal noise levels to effectively discern abrupt changes in amplitude and velocity profiles [
18]. To attain this objective, conventional signal processing techniques such as bandpass filters [
18,
19], wavelet transforms [
20,
21,
22], and smoothing filters [
14,
15], as well as specialized filters like morphological filters [
23] and dynamic time-warping filters [
24], have been employed to denoise EOG signals. Nonetheless, these traditional filtering methods often introduce distortions by inadvertently diminishing peak velocities and extending saccade duration during the denoising process [
14,
15,
18]. Furthermore, they tend to compromise the preservation of EOG saccades and result in a significant deviation from the ground truth [
10]. In response to these limitations, adaptive filtering approaches have been developed to effectively filter EOG signals while retaining the fidelity of EOG saccades [
25]. Notably, Kalman filters (KFs) [
26] have been introduced as a means to fuse the measured EOG signals with values estimated using a predefined mathematical model [
26,
27,
28]. This approach has shown substantial improvements in maintaining the shape and integrity of EOG saccades [
10]. Comparative studies have corroborated the superiority of KFs over traditional filtering methods in denoising EOG signals [
25]. Nevertheless, it is worth noting that the accuracy of a KF is contingent on the precision of the underlying state estimator [
10].
Diverse KF state estimators have been developed, drawing upon the mechanical [
10,
20,
29], electrical [
30,
31], and parametric [
15,
32] attributes of the eye and its ocular movements. Notably, lumped-element-based dynamic models have served as foundational state estimators in this context [
10,
20,
29]. Within these state estimators, the intricate agonist–antagonist dynamics of the extraocular muscles are characterized by their electromechanical properties to accurately estimate saccades, and these estimations are effectively integrated with EOG signals, leading to substantial enhancements in EOG saccade fidelity [
10].
While most of the model-based techniques have demonstrated proficiency in addressing issues such as the elimination of eye blinks, offset correction, and signal denoising, they have typically necessitated real-time operation facilitated by a brain or neural controller [
10,
29]. Similarly, electrical models rooted in Coulomb’s law have been developed to rectify the baseline drift in EOG signals [
25,
30], but they encounter comparable challenges in preserving the integrity of EOG saccades during the denoising process. In VOG systems, KF fusion-based methodologies have been employed to denoise ocular motion signals derived from pupil reflections [
33]. Nevertheless, these model-based denoising approaches [
34,
35], which utilize acceleration-based fusion algorithms [
25], exhibit a notable disparity in accuracy when compared to ground truth measurements.
The present study explores a novel model-based technique aimed at enhancing the denoising of EOG signals while concurrently preserving the fidelity of EOG saccades. Specifically, we introduce a constant velocity-based model that considers the relationship between the peak velocity and saccade amplitude in the human eye, serving as a state estimator for the KF. We evaluate the effectiveness of this model in retaining EOG saccades throughout the denoising process and compare its performance to several traditional and adaptive denoising methods. This paper is structured as follows:
Section 2 provides the mathematical foundation of the model-based fusion algorithms, experimental procedures employed for data acquisition and the algorithmic techniques employed for saccade identification and measurement.
Section 3 and
Section 4 present the outcomes of our analyses and the ensuing conclusions, respectively.
3. Results and Discussion
The proper calibration of the KF is of utmost importance for optimizing the performance of each of these filters. The determination of the
R values was carried out based on the results of the individual calibration trials, while the
Q values remained consistent across various methods, with
Q being determined using a trial-and-error approach as elaborated in
Section 2.2.5. The values of both
R and
Q employed in each trial are documented in
Table 2. Each experimental session consisted of two distinct types of trials, wherein the participants first engaged in a series of calibration trials followed by a sequence of experimental trials. The data recorded during the calibration trials were employed in conjunction with linear regression techniques to calibrate the subsequent experimental trials. The calibration factors generated through this process are presented in
Table 3. It is worth noting that both the EOG and EL recordings underwent the same calibration process. However, it is important to emphasize that the recorded EL data, due to its inherent high quality, did not require denoising and were considered as the ground truth for this experiment. To process the recorded signals, a polynomial piece-wise detrend function in MATLAB was employed. This function was utilized to rectify any drift between the trial markers, specifically the “start” and “end” events, and to normalize the isolated EOG saccades to their baseline value, effectively removing the systematic baseline drift. Outliers within the data were identified and removed using the interquartile range method, which entails removing data points that fall outside a range of 1.5 times the interquartile range from the first and third quartiles, thereby ensuring a dataset devoid of bias. The application of different denoising methods yielded varying results in terms of signal denoising and the reduction in outliers in the EOG saccades.
Table 2 provides a summary of the percentage of outliers removed after each denoising method was applied (namely, 13.48% for BP, 10.90% for BM, 12.05% for CVM, 12.02% for CAM, and 12.2% for LR). On average, all the model-based filters exhibited a 12.5% improvement in comparison to the bandpass filter method.
Figure 5 illustrates an example of the recorded data extracted between the “start” and “end” event markers for participant P023 during the execution of a C trial (i.e., a saccade of 11
). Subfigure (a) displays the raw recorded signal (comprising both the EOG and EL data), while subfigures (b) to (e) depict the outcomes following the application of various denoising methods. A thorough examination of these figures reveals that CVM has the capability to preserve EOG saccades more effectively when compared to other filters, as the overall signal shape closely resembles the EL records. In order to further assess and quantify the performance of the CVM, a comprehensive analysis was conducted, encompassing a correlation study, a numerical feature analysis of the EOG parameters, and an error analysis, utilizing data from all the participants.
Given the utilization of two distinct recording devices in this study, an examination was conducted to assess the feature correlation between different features and the applied denoising methods, thereby establishing a relationship between each filter and the features derived from the two devices. Specifically, the
,
,
, and
of EOG and EL were computed for each denoising method and are comprehensively presented in
Table 4. Notably, all these methods exhibited a robust correlation, with the KF-based filter techniques demonstrating an enhanced correlation of approximately 0.8% in amplitude. It is pertinent to note that
, representing the average peak values of the saccade at fixation, is a key feature influencing the shape of the signal. Furthermore, the SNRs were computed and their results are displayed in
Table 5 to affirm the efficacy of all the employed denoising methods in generating a potent output signal. As anticipated, the average
(also showcased in
Table 5) was higher for the model-based KF in comparison to the BP method, aligning with expectations.
To conduct a more comprehensive examination of the capacity to retain EOG saccades within the proposed denoising procedure, four key parameters were subjected to analysis.
Figure 6 provides a visualization of the average values of these primary parameters, considering data from all the participants (encompassing both EOG and EL saccades) throughout the study, subsequent to the application of various denoising methods. In
Figure 6a,b, the behavior of
following the application of each denoising method is depicted. As summarized in
Table 6, the utilization of the CVM has demonstrated a notable enhancement of the EOG signal by 28.7% in comparison to the BP method, resulting in a reduction in the
. Furthermore, when compared to the EL signal, the overall EOG signal has experienced a 22.3% improvement (
).
Figure 6c illustrates the
of the saccades, revealing that the application of these denoising methods does not have a significant impact on the EOG saccade
. In
Figure 6d, the relationship between the peak velocity (
) and magnitude after the application of various denoising methods is elucidated. The filtered saccades were fitted to Equation (
5) and compared with the data from reference [
32]. The resulting fitted curves produced values of 62.13 for CVM, 110.63 for BP, 83.86 for CAM, 18.78 for BM, and 16.11 for LR, all fitted with
. Although BM and LR exhibited low RMSE values, their
was diminished post-filtering. In contrast, the CVM, BP, and CAM yielded far more realistic
values compared to the peak velocity versus amplitude data published in reference [
32]. Among these methods, CVM displayed the lowest RMSE, indicating the closest fit to the
formula. This analysis underscores that the CVM filter has not only effectively denoised the signal but has also preserved the EOG saccades during the denoising process. Representative examples of randomly selected trials are provided in
Appendix A for further illustration.
To conduct a more in-depth performance analysis of the CVM in comparison to other denoising methods, we examined the kernel density of the probability distribution of the percentage of the normalized error, as depicted in
Figure 7. This was computed by evaluating the (measured value − true value)/true value after the removal of outliers using 1.5 times the IQR criterion. The calculated mean values (
: EL = 0.18, CVM = 0.28, CAM = 0.3, BP = 0.38, and BM = 0.55) of this distribution serve as an indicator of the accuracy of each technique concerning EL. It is evident from the analysis that the CVM exhibits an improvement in accuracy of 2%, 10%, and 27% when compared to the CAM, BP, and Brownian techniques relative to EL. Furthermore, two-sample
t-tests were conducted against EL, and all the filtering techniques demonstrated the rejection of the null hypothesis, thereby establishing the significance of the difference between the recorded percentages of the normalized error values (
p values: CVM—
, CAM—
, BP—
, and BM—
. Consequently, the CVM showcases superior accuracy in comparison to other techniques concerning EL. The variance (
: EL = 0.03, CVM = 0.06, CAM = 0.07, BP = 0.08, and Brownian = 0.08) of the percentages of the normalized error values also reveals that the CVM demonstrates a 1%, 2%, and 2% enhancement in precision relative to the CAM, BP, and Brownian techniques concerning EL.
4. Conclusions
In this paper, we introduce an adaptive filtering method grounded in the assumption of constant velocity, which entails considering the rate of change in the corneal–retinal potential as constant. We then conduct a comparative assessment of this CVM against established techniques, such as BP, BM, CAM, and LR-based methods. To rigorously evaluate these methods, we employ controlled experiments wherein saccades are concurrently measured using an OpenBCI Cyton Biosensing board (for EOG signal acquisition) and an EyeLink 1000 eye-tracker. We extract essential parameters including , , , and from the signals and employ them to assess the performance of each denoising method, with particular emphasis on characterizing the efficacy of the recently introduced CVM approach. The CVM exhibited the most superior performance, manifesting in the filtered data by achieving the lowest errors ( and ). The outcomes reveal that, in comparison to the physical size of the saccade, the CVM enhanced the EOG signal by approximately 29%, and in relation to the EyeLink (EL) recordings, it improved it by over 22% when contrasted with the BP filter. Furthermore, compared to the physical size of the saccade, the CVM outperformed the BP filter by approximately 2% and surpassed it by over 21% when evaluated against the EL recordings. The kernel density distribution of the normalized error percentages indicates that the introduction of sound mathematical models can augment the filtering capabilities of Kalman filters (KFs). As demonstrated by the results, both the CVM and CAM substantially elevated the accuracy and precision of the eye movement recordings. Specifically, the CVM exhibited an average improvement of 13% in accuracy and 3% in precision when compared to other denoising methods.
In
Figure 7, we present the computed means derived from the entire dataset, characterizing the four key saccade features: amplitude (
Figure 6a), error (
Figure 6b), peak velocity (
Figure 6c), and latency (
Figure 6d). A comparative analysis is performed between the means obtained from the EyeLink and the EOG signals, utilizing various filtering methods, including the BP, Brownian, CAM, and CVM. The accompanying error bars depict the standard deviation. It is important to note that the EyeLink signal is regarded as the primary reference point for comparison, given its recognized accuracy in measurements. Upon examining
Figure 6a,b, a discernible pattern emerges, wherein saccades with smaller amplitudes exhibit superior accuracy in the EOG signal when compared to saccades with relatively larger amplitudes. Specifically, targets A and D correspond to the most peripheral goal locations, entailing saccade amplitudes of
concerning the home position, where the induced charge approximates zero. This peripheral positioning may introduce nonlinearity into the signal, potentially diminishing the accuracy of the recording. A similar trend is observed in
Figure 6b, where the saccades directed toward targets B and C display reduced errors in comparison to those aimed at targets A and D. It is noteworthy that the application of the filtering methods has a negligible impact on the latency of the signal. This outcome aligns with expectations, as these filters are not anticipated to influence the temporal resolution of the signal.
The CVM-based KF employed fixed values for
Q and
, as outlined in
Section 2.2.5. These values were established at the outset of each session and remained unaltered throughout the signal processing phase. It is plausible that this static configuration may have constrained the algorithm’s overall performance. Thus, it is posited that the implementation of a methodology to dynamically compute real-time adjustments for the
Q and
R values could enhance the efficacy of the signal filtering process. As indicated in
Figure 6d, saccades demonstrate a constant velocity behavior for only high degrees and speeds. Consequently, there may be a necessity to modify this method for effective application in very short range saccades and low-speed scenarios, such as those encountered in smooth pursuit eye movements. Furthermore, it is important to note that the method does not encapsulate any mechanical or electrical characteristics of the eyeball or eye muscles, unlike models such as LR which incorporate electromechanical properties. Therefore, in instances of extreme conditions, such as individuals with disorders affecting the eyeball or eye muscles, the method, in its current state, may encounter challenges in accurately enhancing signals. Furthermore, the amalgamation of sensor data from both EyeLink (EL) and EOG sources has the potential to yield highly accurate EOG saccades. However, such an approach may impose limitations on the practical applicability of this technology, particularly in scenarios requiring the detection of closed-eye movements. One notable limitation in the present study pertains to the collection of EOG data under conditions of consistent illumination. Given the sensitivity of the corneal–retinal potential to variations in lighting conditions, this factor assumes significance in real-world applications where illumination levels are subject to change. Future research endeavors could extend upon this foundation by exploring the filtering capabilities of this algorithm when applied to EOG records acquired under varying lighting conditions. Additionally, there exists the opportunity to investigate novel machine learning-based techniques for EOG denoising that offer comparable capabilities. The current investigation is specifically concentrated on saccadic eye movements, given their prevalence and frequent occurrence in daily human activities. Subsequent research endeavors are anticipated to extend the scope of these methods to encompass less common eye movement types, such as smooth pursuit movements and vestibular–ocular reflexes.