Bound-Optimization based Block Space Bayesian Learning - implementation -

Zhilin just wrote the provocative Is compressed sensing really useful for wireless telemonitoring of non-sparse physiological signals (Part 1)? where he makes the case that in most physiological signals the small stuff is sometimes as important as the large ones and that in effect in most conditions these signals are not sparse or barely compressible. This is exactly what Yonina Eldar was saying at the MIA 2012 meeting when she was talking about compressed beamformining [1] additonal work into sub-Nyquist sampling can be found in [2] and [3]. Zhilin  however takes a different route it seems, i.e. more in line with structured sparsity. Going back to Zhilin's argument:

.....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)....

The paper introducing the Bound-Optimization based Block Space Bayesian Learning is: Low Energy Wireless Body-Area Networks for Fetal ECG Telemonitoring via the Framework of Block Sparse Bayesian Learning by Zhilin Zhang, Tzyy-Ping Jung. , Scott Makeig , Bhaskar D. Rao . The abstract reads:

Fetal ECG (FECG) telemonitoring is an important branch in telemedicine. The design of a telemonitoring system via a low-power wireless body-area network for ambulatory use is highly desirable. As an emerging technique, compressed sensing (CS) shows great promise in compressing data with low power consumption. However, due to some specific characteristics of FECG recordings such as non-sparsity and strong noise contamination, current CS algorithms generally fail in this application. In this work we utilize the block sparse Bayesian learning (bSBL) framework, a recently developed framework solving the CS problems. To illustrate the ability of the bSBL methods, we apply it to two representative FECG datasets. In one dataset the fetal heartbeat signals are visible, while in the other dataset are barely visible. The experiment results show that the bSBL framework is capable of compressing FECG raw recordings and successfully reconstructing them. These successes rely on two unique features of the bSBL framework; one is the ability to reconstruct less-sparse but structured signals, and the other is the ability to learn and exploit correlation structure of signals to improve performance. These two abilities of the framework greatly enhance the potential use of bSBL in telemonitoring of other physiological signals.

The BSBL-BO algorithm to break through the bottleneck in wireless telemonitoring using compressed sensing is here. Stay tuned to part 2 of this argument: 

[1] Compressed Beamforming in Ultrasound Imaging by Noam Wagner, Yonina C. Eldar, Arie Feuer, Zvi Friedman.

[2]  Innovation Rate Sampling of Pulse Streams with Application to Ultrasound Imaging by Ronen Tur,  Yonina C. Eldar, Zvi Friedman. 

[3] Sub-Nyquist Radar, Omer Bar-Ilan and Yonina C. Eldar





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