1. Introduction
Recently, wireless body area networks (WBANs), as an emerging wireless sensor networks technology that is applied to the human body [
1,
2,
3], has been more and more popular in gait telemonitoring applications [
4,
5,
6]. In such WBANs, each sensor node usually takes the form of a small wearable device equipped with a microcontroller, accelerometer, gyroscope, wireless communication module, and battery. These wearable sensors are co-located on the body for acquiring gait data of people who freely walk in a home or outdoor setting. Moreover, all acquired data can be transmitted, via internet, to a remote terminal such as a hospital for gait monitoring or assessment by further data processing such as gait classification. This greatly contributes to the telemonitoring of gait pattern change, and people do not often visit a hospital for gait function assessment [
4,
5,
6,
7]. However, in clinical applications, continuous acquisition of the multi-sensor gait data over the long term is usually required for further data processing, in order to accurately evaluate the gait pattern change [
8,
9,
10]. In such a case, there have been several challenging existing issues such as improving the energy efficiency of the sensor, getting the best classification performance, and getting a lower computation complexity. The above existing issues are strangling the WBANs-based gait telemonitoring application, and some advanced techniques for multi-sensor gait data analysis are urgently needed to tackle these existing issues [
9,
10,
11,
12].
As we know, due to the limited life of the battery, the wearable sensors are usually required to consume the least possible amount of energy for the continuous telemonitoring of gait [
10,
11,
12,
13]. How to improve the energy efficiency of the sensors has been one challenging endeavor in the WBANs-based telemonitoring applications [
10,
11,
12]. Many previous studies found that most of the energy of the sensors is consumed by wireless communication, and their works mainly focused on developing the energy-aware MAC protocols for the low energy consumption of sensors. However, these protocols don’t provide reliable data transfer because heavy collision is usually generated by some adjacent sensors [
10,
12]. In recent ten years, compressed sensing (CS)—an emerging data compression methodology that data compression and reconstruction can be perfectly performed based on data sparsity—has attracted wide attention in the study of energy-efficient gait telemonitoring applications [
14]. The basic idea is that a larger amount of the acquired gait data is firstly compressed on sensors, in order to significantly reduce the energy consumption during data transmission. Then, all compressed data received in the remote terminal are perfectly reconstructed to perform further data processing [
15,
16]. For example, Wu et al. investigated the application of CS for the energy-efficient telemonitoring of gait using acceleration data [
16]. Although significant research efforts are made, many existing works have only showed that the CS technique for energy-efficient gait telemonitoring is feasible only using single-sensor data. Theoretically, the CS technique has no ability to simultaneously compress multi-sensor data for the energy efficiency of sensors. This leads to the possible loss of the highly-correlated information regarding the intrinsic dynamics of human movement. It is essential to search for an advanced technique for jointly processing multi-sensor gait data.
Further data processing such as gait classification is also very important for gait monitoring. How to develop the best gait-classification model has been another challenging issue in gait telemonitoring applications [
16,
17,
18]. Some studies have showed that the original gait data are directly used to develop the gait-classification models based on machine learning algorithms. In these studies, some commonly used traditional machine learning algorithms include artificial neural networks (ANN) [
18], decision tree,
k-nearest neighbor (KNN) [
17], support vector machine (SVM) [
16,
19], and Hidden Markov Model (HMM) [
20,
21,
22]. Comparing these methods, the HMM technique has shown higher performance in gait pattern classification using on-body wearable sensors. For instance, Mannini et al. investigated the feasibility of human physical activity classification by using on-body wearable sensors. Especially, Taborri et al. proposed a novel HMM-based distributed classifier for the detection of gait phases by using a wearable inertial sensor network. In their proposed algorithm, a distributed stochastic model and a hierarchical-weighted classification are implemented to detect gait phases by simultaneously processing the data of multi-sensors placed on different body segments of lower limbs. Recently, an emerging sparse representation classification (SRC) algorithm has widely attracted the focus of gait classification. Its basic idea is that all gait training samples are directly used to construct an over-complete dictionary, and a test gait can be sparsely represented as a linear combination of just those training samples with same gait class. The residual error criterion is defined by solving sparse representation coefficients, in order to exactly determine the class of the test gait [
23,
24,
25]. Zhang et al. investigated the feasibility of a sparse representation-based gait-classification model using wearable sensor gait data [
24]. Also, Allen et al. studied the practicality of the sparse representation-based gait-classification model using wearable multi-sensor gait data [
25]. Although the SRC algorithm can produce a better accuracy, it has a higher computational time cost. More importantly, all above studies don’t take into account the use of the gait data received at a remote terminal to develop a gait-classification model [
26,
27].
At the same time, other recent studies have focused on using the received gait data to develop gait-classification models. In view of the energy efficiency of sensors, recent relevant studies mainly concentrated on the combination of the CS technique with machine learning algorithms for gait classification. Their basic idea is that all compressed gait data received at a remote terminal are reconstructed by the CS technique, and the reconstructed gait data are then employed to develop gait-classification models based on machine learning algorithms [
14,
16]. In such works, one important step is to reconstruct gait data with a higher quality. However, the traditional CS technique has no ability to perfectly reconstruct the gait data; this is because gait data such as acceleration data are poorly sparse [
14,
16]. This possibly deteriorates the further gait-classification performance. Moreover, these hybrid techniques possibly produce a higher computational complexity. This motivates us to search for advanced approaches to the energy-efficient telemonitoring of multi-sensor gait [
28,
29].
It is well known that multi-sensor gait data generally interact in a complex fashion, an observation attributable to the intrinsic dynamics of human gait. Theoretically, we can assume multi-sensor gait data as a data ensemble with joint sparsity, in order to jointly process multi-sensors and thus capture the higher-correlation information associated with gait. This can potentially allow the best classification performance and energy efficiency of sensors. Recently, there have emerged some advanced techniques that have a superior ability to jointly process multi-signals based on data joint sparsity. Successful examples include distributed compressed sensing (DCS) [
30,
31] and the joint sparse representation classification (JSRC) model [
32,
33,
34]. These two techniques have been widely applied in video coding [
35,
36,
37], image fusion [
38], and multichannel physiological monitoring [
39]. So far, no studies have been reported where DCS and JSRC have been employed for jointly processing multi-sensor gait data.
In this study, a novel advanced hybrid technique of DCS and JSRC for the telemonitoring of multi-sensor gait is proposed based on data joint sparsity. Unlike the recent relevant studies, the advantage of our proposed technique is that we directly take advantage of the multi-sensor gait data compressed by DCS to develop the novel neighboring JSRC gait-classification models with better performance. This can not only avoid the complicated step of joint reconstruction of multi-sensor gait data, but can also produce the best classification performance as well as a lower computational time. In our proposed technique, the DCS technique is firstly utilized to simultaneously compress multi-sensor gait data, in order to potentially gain higher-correlation information as well as improved energy efficiency of the sensors. Then, all jointly compressed gait data are directly used to develop a novel neighboring sample-based JSRC model by defining the sparse representation coefficients-inducing criterion (SRCC), not to perform the reconstruction task. The test gait can be sparsely represented by constructing a new over-complete dictionary that includes a few local neighboring training samples containing more valuable information. This possibly produces the best classification performance as well as a lower computational time. Our proposed technique has great potential to enable continuous collection of people’s gait information and accurate human gait measurement anywhere and at any time. This possibly provides benefits in telemedicine applications such as the early identification of at-risk gait of elderly in the community, as well as the monitoring of gait rehabilitation progress of people at home. The multi-sensor gait data are selected from an open wearable action recognition database (WARD), in order to validate the feasibility of our proposed method. The results show the superior classification performance of our proposed techniques to the traditional techniques such as SRC and JSRC. Moreover, a lower computational time cost is spent in our proposed technique.
The rest of the paper is organized as follows. In
Section 2, we describe the DCS technique for multi-sensor gait data. A novel neighboring JSRC model for multi-sensor gait classification is presented in
Section 3. In
Section 4, we describe the materials and methods for evaluating our proposed technique. The evaluation results obtained from multi-sensor gait data are given in
Section 5. Discussion and conclusions are presented in
Section 6 and
Section 7, respectively.
2. Distributed Compressed Sensing for Multi-Sensor Gait Data
Theoretically, the DCS technique combines distributed source coding theory with compressed sensing theory [
14,
30,
31]. It can take advantage of data joint sparsity to capture both inter- and intra-signal correlation among multi-sensor signals. In the DCS technique, three different joint sparsity models (JSM), such as JSM-1, JSM-2, and JSM-3 [
30,
31], are usually used to perform the simultaneous compression and the joint reconstruction task in the multi-signal case. In this study, since each sensor can acquire the same gait data, such as acceleration data, we assume that each sensor’s data has the same sparsity pattern but with different coefficients. Here, JSM-2 is adopted to jointly process the multi-sensor gait data [
30,
31]. That is, all
sensors’ gait data in WBANs are assumed as a data ensemble
, where
denotes the length of each sensor data
. Here, we assume that each sensor data
is sparsely represented as
where
denotes dictionary matrix vectors and
is the corresponding sparse vector that satisfies the same support set
. Then, the multi-sensor gait data ensemble
can be represented as
where
is the joint sparse coefficients vector and
is the corresponding dictionary matrix. Then, the multi-sensor gait data ensemble
can be rewritten as
Next, we define a measurement matrix
as
Thus, the multi-sensor gait data ensemble
can be simultaneously compressed as
where
is the jointly compressed multi-sensor data. Here, it is noted that each
is performed independently of the others.
In the DCS technique, joint reconstruction is another important task. Its basic idea is to obtain an accurate solution of the joint sparse representation coefficients
by solving the following
-norm minimization:
Then, the joint reconstruction of multi-sensor data can be estimated as
. The commonly used joint reconstruction algorithms include the One-Step Greedy Algorithm (OSGA) [
32] and the Simultaneous Orthogonal Matching Pursuit (SOMP) [
31]. However, since multi-sensor gait data such as accelerometer and gyroscope data are poorly sparse, it is very difficult to guarantee the best solution of joint representation coefficients
by Equation (6). In fact, the jointly compressed multi-sensors data
by DCS possibly contain enough highly-correlated discriminative information associated with gait. Theoretically, the compressed multi-sensor data
can be directly used to develop a JSRC-based classification model, but not to perform a joint reconstruction task. This helps to improve multi-sensor gait-classification performance.
3. A Novel Neighboring JSRC Model for Gait Classification
Theoretically, the JSRC model is used to generalize the traditional SRC model to the multi-signal case [
33,
40,
41,
42,
43]. Its basic idea is to explore the joint sparse representation for the multiple input signals in the classification task, and the joint sparsity can be enforced by imposing a joint sparsity-inducing norm(such as
-norm,
-norm) penalty on representation coefficients [
23,
24,
25]. However, due to the large-scale gait training samples in this study, more computational time and higher memory cost are needed for the matrix inverse operation in the JSRC algorithm. This possibly yields a high computational complexity as well as a poor classification performance. It is essential to search for a few training samples to develop the best JSRC-based gait-classification models. In this study, a novel neighboring JSRC model is developed based on a few neighboring training samples selected by the sparse representation coefficients-inducing criterion (SRCC). As we know, the same gait class can be theoretically spanned by the training samples (i.e., over-complete dictionary atoms) of the same class in a low-dimensional subspace [
33,
41,
42,
43,
44,
45,
46,
47,
48]. The spanned training sample can be sparsely represented in the high-dimension space. Those samples with high values of sparse representation coefficients are selected as the nearest neighbor samples because they possibly contain the more valuable information associated with gait pattern change. So, these selected neighboring samples are used to reconstruct a new over-complete dictionary for multi-sensor gait classification. The block diagram of the proposed novel neighboring JSRC model is shown in
Figure 1. In order to clearly describe the novel neighboring JSRC model, we firstly describe the construction of an over-complete dictionary using all original training samples.
3.1. Constructing an Over-Complete Dictionary for JSRC
In this study, we assume that each sensor node is equipped with a triaxial accelerometer
and a biaxial gyroscope
[
24,
25], in order to capture more valuable information associated with gait. Firstly, we define a vector
to denote the gait data of each sensor
at time
, i.e.,
Then, a vector
is defined to represent multi-sensor gait data of all
sensors at time
, i.e.,
Next, based on Equation (6), a matrix
is defined to represent the multi-sensor gait data ensemble during the length of time
:
where the sub-vector
denotes the gait data of sensor
with duration
,
.
Here, assuming that there are
different gait classes, we can define a new matrix
that concatenates the training sample from all
gait classes:
where sub-matrix
represents all training sample data of gait class
, and
denotes the total of training samples of class
. In such a case, we can employ the matrix
to construct an over-complete dictionary that consists of
sub-dictionary
with respect to all
classes [
23,
25,
33,
41,
43]. Thus, based on the constructed over-complete dictionary
, a test gait
(
) with class
can be joint-sparsely represented as
where
denotes the joint sparse representation coefficients matrix whose entries are zero except those associated with gait class
, and the total number of samples
.
Usually, the joint sparse coefficients matrix
can be estimated by solving the following joint sparse optimization problem [
33,
41,
43]:
where
denotes the Frobenius norm,
refers to the mixed norm that can be phrased as performing the
-norm cross the column and then the
-norm along the row, i.e.,
.
denotes the maximum number of the nonzero coefficients in
.
In fact, because the regularization norm needs the shared gait pattern across observation, the computation complexity reaches for the optimization solution of non-zero sparse coefficients . That is, if large-scale training sample data are directly used to construct the over-complete dictionary, then a higher computational complexity is possibly yielded in estimating the optimization solution of . Therefore, it is necessary to find a few training samples to reconstruct an over-complete dictionary for developing the JSRC-based multi-sensor gait-classification model with high quality.
3.2. Reconstructing a New Over-Complete Dictionary for JSRC
Here, the sparse representation coefficients-inducing criterion (SRCC) is used to find a few nearest neighboring samples containing the most valuable information, in order to reconstruct a new over-complete dictionary for JSRC. The detailed procedure of selecting the nearest neighbor samples is presented as follows.
Step 1 The test sample
from sensor
is linearly represented as
Then, the sparse representation coefficients
are obtained by solving the following
-minimization problem:
where
is a positive regularization, and the
-norm is defined as
.
Step 2 Based on the solution of
, the
training samples with the larger sparse coefficients values are selected as the nearest neighbor training samples. For gait class
, a new training sample set is defined as
Step 3 Based on Equation (15), a new over-completed dictionary is reconstructed as
Therefore, a test gait
with class
can be joint-sparsely represented as
Here, the joint sparse coefficients matrix
in Equation (17) can be estimated by solving the following
-minimization problem:
The detailed procedure of solving joint sparse representation coefficients
is presented in
Section 3.3.
3.3. MBCS Technique for Solving Joint Sparse Representation Coefficients
In this study, the Multitask Bayesian Compressive Sensing (MBCS) technique is applied to solve the joint sparse representation coefficients
in Section 2.2.2 [
48], in order to capture more of the temporal–spatial correlation information regarding gait. Due to multi-sensor gait data that are contaminated by noise, the test gait
can be represented as
, i.e.,
where
denotes the noise. Here, the noise
satisfies a Gaussian distribution with an unknown parameter
, and the prior distribution function of
is defined as
Then, the prior distribution function of the sparse coefficients
is defined as
where
denotes the
th sparse representation coefficient of sensor
and
refers to the Gaussian distribution function whose mean is equal to zero and variance
.
After the optimal parameters
are estimated by constructing a likelihood function, we can solve the joint sparse representation coefficient
. The production of the solution for
is presented in the
Appendix A, and the detailed procedure for the solution can be found in reference [
48].
3.4. The Definition of Minimal Residual Error Rule for Gait Classification
Based on the estimated solutions of the joint sparse representation coefficients
, we firstly define the matrix indication function
whose entries are zero except for those associated with gait class
. i.e.,
Then, the approximation of test gait sample
can be defined as
The minimal residual error between
and
can be defined as
Thus, the class label of test gait
can be identified by the following minimal residual error rule [
23,
33,
41]:
6. Discussion
The results of present studies demonstrate that the hybrid technique of DCS and JSRC can take advantage of data joint sparsity to jointly process multi-sensor gait data, which can capture the more valuable high-correlation information hidden in multi-sensor gait data. This greatly contributes to the best classification performance as well as to the energy efficiency of sensors in WBANs-based gait telemonitoring applications. Currently, the discovery of powerful techniques for the quantitative analysis of multi-sensor gait data has become a research focus in the gait telemonitoring field [
10,
11,
12]. One important issue is how to gain the higher-correlation information regarding gait for the best gait-classification performance as well as for improving the energy efficiency of sensors [
13,
26,
27,
28].
In the present studies, considering that both DCS and JSRC have the best ability to jointly process multi-signals based on data joint sparsity, we investigated the feasibility of the hybrid technique of DCS and JSRC for gait telemonitoring by jointly processing multi-sensor gait data [
32,
33,
34]. In particular, we try to take advantage of the joint sparsity of multi-sensor gait data to develop a novel neighboring JSRC model for the best classification performance as well as lower computation time. In this study, we firstly evaluate the effect of the simultaneous compression of multi-sensor data by DCS on the JSRC-based classification model. As illustrated in
Table 2, all JSRC-based gait-classification models can yield better accuracy. This suggests that the DCS technique has the superior ability to capture the higher-correlation information regarding gait by jointly compressing multi-sensor gait data, and all compressed multi-sensor gait data containing the more valuable information significantly helps to produce the better classification performance of the JSRC-based model. These results further suggest that it is feasible that DCS and JSRC can jointly process multi-sensor gait data for gait classification based on data joint sparsity.
Next, we evaluate the feasibility of our proposed method for multi-sensor gait classification based on the different compression ratios. As shown in
Figure 2, all selected JSRC models can yield a better gait-classification performance when compression ratios are properly chosen. In the comparison, our proposed model is best. This suggests that our proposed model potentially yields the best performance as well as the best energy efficiency of sensors. In addition, we also compared our proposed method to the traditional SRC, NBC, and KNN models. As shown in
Figure 3, our proposed method significantly outcompetes the selected traditional methods. A possible reason for this is that the selected neighboring training samples can contain the more distinctive and high-correlation information associated with gait, which significantly improves multi-sensor gait-classification performance. However, the traditional SRC model does not gain the valuable high-correlation information as it does not jointly processes multi-sensors gait data [
15,
16,
24,
25]. Similar results in a study on hyperspectral image classification have been found in [
45]. Besides this, from
Figure 2, we also find that all JSRC-based gait-classification models result in poor accuracy when the compression ratios are more than 70%. This is because the compressed multi-sensor gait data possibly contains more redundant information, thus destroying the classification performance. Similar results in studies on gait telemonitoring have been also reported in [
10,
15,
16,
29].
In addition, we also evaluate the computational time cost corresponding to all JSRC-based models based on the different compression ratios. As illustrated in
Figure 4, in comparison, our proposed method has the lowest computational cost. The main reasons for this are that the proposed novel neighboring JSRC algorithm has a maximum time complexity of only
(
denotes the time for searching the
nearest neighbor samples) to produce the best classification performance. However, the traditional JSRC algorithm has a computational complexity of
(
,
is the total number of training sample data points) for multi-sensor gait performance [
33,
41,
43]. In conclusion, all the above results show that our proposed technique has the potential ability to gain the best classification performance, lower computational time cost, and better energy efficiency of sensors in multi-sensor gait classification.
In this study, our proposed model is also employed to classify nine different multi-sensor gait patterns, in order to further examine the practicality of gait telemonitoring applications. As shown in
Table 5, our proposed technique shows high-quality gait-classification performance. This suggests that our proposed technique may enforce the robustness of coefficient estimation, which helps to exactly solve the joint sparse representation coefficients. In particular, this is because our hybrid technique can take advantage of the joint sparsity of multi-sensor gait data to gain the more distinctive information associated with the spatial, temporal, and dynamic correlations of gait [
13,
26,
27,
28], which greatly contribute to identifying multi-sensor gait pattern change. Similar findings have been reported in [
9,
12,
46].