In my last post, I have discussed the difficulties when using compressed sensing in wireless telemonitoring of non-sparse physiological signals. Essentially, these difficulties come from the requirements of practical use: the recovered signals are generally further processed by other signal processing algorithms (e.g. independent component analysis) and/or pattern recognition algorithms (e.g. classification); some small characteristics in signals, although not salient, are very important for diagnosis (e.g. the QRS complexes of fetal ECG in raw abdominal recordings, the P-wave and T-wave in abnormal ECG recordings, the spike duration in EMG, and the fluctuation in some EEG signals). In the following, I will give you examples and show the results of the BSBL-BO algorithm.
The first example is the recovery of raw fetal ECG recordings (recorded from the mother's abdomen).
First, I use the fetal ECG recording in my last post to show the use of BSBL-BO. The recording contains one large peak (the QRS complex of the mother's ECG, which is not of interest) and two small peaks (the QRS complexes of the fetus' ECG, which is of interest), and large noise fluctuation. The recording can be viewed as a signal with block structure (the three peaks are blocks) but contaminated by strong noise (it's 'source noise', not the 'sensor noise' as in standard noisy compressed sensing model). Denote the recording by x. And it is compressed to y by: y = Ax, where A is a 125 x 250 binary sparse matrix (see  for details). .
Now I use BSBL-BO to recover x from y using A. Although BSBL-BO requires to know the block partition, we have pointed out that the block partition is in fact a regularization helping learn the covariance matrix of the signal in high dimensional space, and experiments also support that the user-defined block partition does not need to be consistent with the true block partition of the signal. Therefore, I randomly set the block partition like this: [1, 26, 51, 76, 101, ...], where each number indicates the starting location of each block.
If you are familiar with SBL algorithms, you must know SBL algorithms generally have a pruning-mechanism, namely they use a threshold to prune out some irrelevant coefficients. BSBL-BO also has such pruning-mechanism. But note that in this example, the recording x is a non-sparse signal; its every entry is non-zero. Therefore, I need to turn off the pruning-mechanism. This can be done by setting the input argument 'prune_gamma' to any non-positive constant (e.g. -1, 0).
Clearly, there is strong intra-block correlation in x. So we need to explore and exploit the intra-block correlation for better performance. This can be done by setting the input argument 'learntype' to 1 in the BSBL-BO code.
Then, running the algorithm, I get the results as shown below:
I encourage you to repeat the above experiment using the demo code 'DEMO_nonSparse' in the package: dsp.ucsd.edu/~zhilin/BSBL_public.zip(Sometimes the link does not work. But you can download it from the bottom of the page: https://sites.google.com/site/researchbyzhang/bsbl)
I should emphasize: in the task BSBL-BO is directly recovering a non-sparse signal, namely, using y and A to directly recover the non-sparse x from the underdetermined inverse problem: y=Ax. I didn't see any other algorithms can successfully do this (In  I compared many representative algorithms).
Of course, we can assume that x is represented in other domain (i.e x = Bz), and first recover the representation coefficients z by solving the inverse problem y=(AB) z, and then recover the original signal x by x = Bz. Using this method, BSBL-BO can get better result (using the same input parameters). And I believe, using this method, some existing algorithms may recover x as well. But note that, when other algorithms adopt this method to recover a non-sparse signal, they heavily rely on how sparse the representation coefficients z are. In fact, for more practical scenarios in telemonitoring, z is not sparse enough. In the next post, I will show you another example (should surprise you!).
 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