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Wednesday, May 30, 2012

Is compressed sensing really useful for wireless telemonitoring of non-sparse physiological signals (Part 3)?

In the last post, the signal to recover is much easy, because the two fetal QRS complexes are relatively strong. Now I show you another example, in which the fetal QRS complexes are almost invisible.

Below are 8-channel abdominal recordings (download link can be found in [1]). The peaks you see are the maternal ECG's QRS complexes, which we are not interested in. What we are interested in are the QRS complexes of fetal ECG, which peak around 6, 217, 433, 643, 854, 1065, 1278, 1488, 1697, 1912, 2110, 2317 sampling points. However, you almost can't observe such peaks in each channel recording, since the fetal ECG is very weak and is buried in noise. So, any sparsifying processing can completely remove the fetal ECG QRS complexes.
The 8 recordings are compressed using the same binary sparse matrix A in my last post, and recovered by BSBL-BO using the same input parameters (using the direct recovery way). Results are shown below:
Apparently, the whole 8-channel recordings are recovered well. But note that the recovered recordings will be processed by independent component analysis (ICA) to extract a clean fetal ECG. Therefore, if there is any small distortion in the recovered recordings (the distortion may be not discernible but could destroy the mutual structure across the channel recordings), the ICA decomposition of the recovered recordings will be different from the ICA decomposition of the original recordings. To ensure the ICA decomposition is unchanged is crucial to practical clinical diagnosis.

Hence, we perform ICA decomposition on the recovered recordings using FastICA, and for comparison, we also perform the same ICA decomposition on the original recordings. The ICA decomposition of the original recordings is given below:

The ICA decomposition of the recovered recordings is given below:

Clearly, there is no significant difference between the two ICA decompositions. More importantly, the separated fetal ECG and maternal ECG from the recovered recordings are the same as those from the original recordings. For you convenience, I zoom in some of the separated components. Below are the separated maternal ECG (a)(b) and fetal ECG (c) from the original recordings:
And below are the separated maternal ECG (a)(b) and fetal ECG (c) from the recovered recordings:

Clearly, the fetal ECG separated from the recovered recordings is almost the same as the one separated from the original recordings.

Note that, in this task the BSBL-BO recovered the 8- recordings using the direct recovery method (without first recovering the representation coefficients and then reconstructing the original recordings) as stated in my last post. The results show the BSBL-BO is very powerful to recover non-sparse structured signals.

As I said early, the in-direct recovery method, i.e. first recovering the representation coefficients and then reconstructing the original recordings, is not effective in every situation, because for the applications considered here, small representation coefficients (in transformed domain) are very important and should be recovered. In [1] I have compared several well-known algorithms, which used the in-direct method. They all recovered the significant representation coefficients, and thus the maternal ECG could be obtained from the ICA decompostion. However, the ICA decompositions of their recovered recordings were still largely different from the genuine ICA decomposition, and no fetal ECG could be obtained. This is because the representation coefficients in the transformed domain are still not sparse enough, and these algorithms could not recover the majority of small representation coefficients. Interested readers can refer to Fig.13 in [1].

PS: Using the in-direct recovery method, BSBL-BO can get better performance.

We have other results when using BSBL-BO to recover EEG for brain-computer interface. We are now revising the paper. Once we have done, I will post the paper in my homepage.

[1] 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

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