Is compressed sensing really useful for wireless telemonitoring of non-sparse physiological signals (Part 1)?

Applying compressed sensing to wireless telemonitoring of physiological signals is not new. There have been a number of work published in recent years. However, the achievement is very limited.

This is because the following reasons:

1. Most published work were only successful on few types of physiological signals, such as adult ECG. This is because these signals are relatively sparse (or become sparse after some sparsifying processing). However, most physiological signals are non-sparse, such as fetal ECG, EEG, EMG, etc.

2. Current published work required that signals contain less noise and interference (otherwise the noisy signals are not sparse). However, the main goal of wireless telemonitoring is to allow people can walk freely at home or office. Thus artifacts/interference from muscle movement is unavoidable.

3. Most published work required some sparsifying pre-processing, namely setting the entries of signals with small values to zero. This sparsifying pre-processing requires some complicated techniques to set a threshold (used to set the entries to zero). But the sparsifying pre-processing has three crucial problems:
   (1) These techniques change the underlying structure in signals. For fetal ECG, which is very weak and even invisible in the recorded signals, this technique can completely remove the fetal ECG's QRS complexes.
   (2) These techniques need to occupy extra on-chip computation and thus consume extra energy, while energy consumption is a crucial problem in wireless telemonitoring.
   (3) The setting of thresholds may have problems in practical scenarios, such as when people are walking or in some abnormal situations (remember the target market of telemonitoring is for patients).

4. More importantly, practical wireless telemonitoring systems generally collect/send multi-channel signals. For example, in telemonitoring of EEG for BCI, the channel number generally ranges from 2 to 8; in telemonitoring of fetal ECG, the channel number generally ranges from 4 to 12. These multi-channel signals are sent to remote terminals and will be further processed by other advanced signal processing techniques, such as independent component analysis (ICA). These multi-channel signal processing techniques require that the underlying structure across multi-channel signals is intact. For example, ICA requires the underlying ICA mixing structure in collected multi-channel signals is intact. Therefore, a practical telemonitoring system requires that the raw physiological signals can be completely recovered even including the noise/interference contained in the recorded signals. In other words, this requires the non-sparse signals can be completely recovered, including its entries with small amplitudes. 

Clearly, we can see, if a compressed sensing algorithm can be widely used in wireless telemonitoring, it must has the ability to recover non-sparse physiological signals, which completely contradicts the sparsity assumption in all the compressed sensing algorithms.

One may ask, there is another way for compressed sensing of non-sparse signals: suppose a non-sparse signal can be represented in a transform domain (e.g. wavelet domains, DCT domains, etc), such that the representation coefficients are sparse, i.e.,
x = Bz,
where x is the signal, B is the basis of the transform domain, and z is the representation coefficients. Then, the signal x is compressed by,
y = Ax,
where y is the compressed signal, and A is the sensing matrix. In the remote terminal, the compressed data y and the matrix product AB are used to recover the sparse coefficients z  according to the relation: y = (AB) z. And then the original signal x is recovered by the relation x = Bz.

However, the success of this method relies on the sparsity of the representation coefficients z. Unfortunately, for most physiological signals, z is not sparse enough, and completely recovering z is still a challenge for most compressed sensing algorithms (especially the sensing matrix A is a binary sparse matrix, a requirement of telemonitoring with low energy consumption).

Here is an example. We transform a segment of fetal ECG signal into the DCT domain, and see the representation coefficients:

 

As we can see, the coefficients are still not sparse. There are many coefficients with small values. For any compressed sensing algorithms, it is not a problem to recover the big coefficients, but is a big problem to recover the small coefficients. Remember, recovering the small coefficients is very important to maintain the underlying structure of signals, especially for later multi-channel signal processing. For this example, recovering the small coefficients is important to recover the fetal ECG's QRS complexes (i.e. the peaks at 65 and 180 in figure(a)) and to maintain the underlying structure for later ICA decomposition (the goal is to use ICA to extract the weak fetal ECG, not just to recover the noisy recording).

In the next post, I will introduce my work:

Zhilin Zhang, Tzyy-Ping Jung, Scott Makeig, Bhaskar D. Rao, Low Energy Wireless Body-Area Networks for Fetal ECG Telemonitoring via the Framework of Block Sparse Bayesian Learning, submitted to IEEE Trans. on Biomedical Engineering

and I will show you how to use my BSBL-BO algorithm to break through the bottleneck in wireless telemonitoring using compressed sensing.