Difference between compressive sensing and DCT-based compression?

Question

I am working on transmitting EEG signals over wireless body area network. I have applied two different compression techniques: DCT-based and compressive sensing (CS-based) approach. I noticed that the reconstructed signal is better using the DCT-based approach. Is this reasonable? I know that compression using DCT is lossy. The metrics used for evaluating the quality of the reconstructed signal is the normalized mean square error (NMSE) and the structural similarity index metric (SSIM).

Thanks in advance

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In case of not fulfilling the CS theoretical conditions, is it recommended to use DCT based compression for compressing biomedical signals to transmit them wirelessly over a body area network? I am asking this question because most papers recommend the use of CS compression, due to the energy efficient consumption property

Answer 1

Compressed sensing is not a particularly efficient type of compression, see e.g. Goyal, Fletcher, Rangan 2008 DOI: 10.1109/MSP.2007.915001. That being said, compressed sensing per se is not even compression in the information-theoretical sense - it is a dimensionality reduction (reduce large vectors of continuous values to smaller vectors of continuous values). This does not take into account that the measurements must be discretised (quantised) in order to be able to talk about compression. Doing this (discretisation), you have a trade-off for a given bit-rate between few measurements at high resolution vs many measurements at low resolution which can affect the reconstruction quality a lot. This trade-off is not particularly well characterised in the literature (but see e.g. Laska & Baraniuk 2011 DOI: 10.1109/TSP.2012.2194710) and even then, reconstruction quality will depend a lot on the reconstruction algorithm and how it possibly takes the discretisation into account.

In other words, you might be able to get better results by trying more measurements at a coarser quantisation or fewer measurements at a finer quantisation.

Compressed sensing (with quantisation implied) has also been proposed as a "cheap" digital compression technique for low-energy wireless sensors (e.g. Liu, Zhang, Xu, Fan, Fu 2013 DOI: 10.1016/j.bspc.2014.02.010). That is, you first sample "normally" in the Nyquist sense and then apply the measurement matrix digitally. When considering compressed sensing like this (which I am guessing is what you did), the argument is that applying this compression (depending on how dense the measurement matrix is and whether it can be applied via "fast" operators such as FFT) is cheaper than traditional compression and the "hard work" of reconstruction in the case of compressed sensing is off-loaded to the receiver side.

Particularly if you can use a Fourier-like transform (DFT, DCT, DST...) as your sparsifying dictionary, you can get away with sampling at lower rate in the time domain in a simple way where you take fewer samples than the Nyquist rate dictates at discrete (preferably random) time points.

I general, I find that compressed sensing mostly makes sense in applications where the possibly analog "compression" represented by the compressed sensing measurement matrix enables sensing that was not otherwise possible - for example by enabling cheaper sensors or lowering the sampling rate or reducing the total sampling time from an unrealistic to a feasible sampling level. First sampling a signal traditionally according to Nyquist and then applying compressed sensing in the digital domain does not make much sense - with low-energy wireless sensors as a possible exception in some cases.

Answer 2

Comparing compressive sensing and traditional compression is a complicated task. Even comparing two compression algorithms.

One the one hand you have compression ratios, making sure the bits sent can be decompressed without other hidden bits. On the other hand, error measurements, rarely perfectly matched to the actual meaningful distorsions. [EDIT] Standard SSIM or NMSE are not well adapted to sparse signal.

Then, knowing the range over which one algorithm is better than the other is important, because one can be better at low-rates, the other at high-rates. There is rarely a free lunch in compression.

From a very mundane point of view, if you are in a compression range where the signal is not "sparse" enough (with respect to some basis) and the CS quantization is not really taken into account, yet sufficiently harmonic, it is well understandable that a DCT-like engineered algorithm can do better than a CS method if the CS theoretical conditions are not fulfilled.

Answer 3

Obviously the DCT compression, would have better results. From your question I feel there are two misassumptions you have on CS.

I know that compression using DCT is lossy.

So is CS. I mean not only DCT compression is lossy but also signal reconstruction in CS theory is. The CS reconstruction is not perfect. There are error bound in signal reconstruction in theory, where in practice, errors get even larger. The algorithms perform recovery until getting a certain error point while even theory doesn't guarantee perfect or even near perfect reconstruction. It Only says, if you increase your measurements, you'll (with high probability) end up with less error in reconstruction.

...is it recommended to use DCT based compression for compressing biomedical signals...

Yes, CS is not an method for signal compression. It tries to offer a solution for under-determined systems of equations, assuming the solution has a sparse representation in some domain. From there, it provides a framework to sample(sense) sparse signals under the Nyquist rate and recovery. CS based sampling performs two jobs at the same time, sampling&compression. It might be recommended, because sampling through Analog to Digital Conversion is costly while CS requires minimum number of samples, hence less power is consumed in ADC portion and of course in wireless transmission circuitry.

And lastly,

The metrics used for evaluating the quality of the reconstructed signal is the normalized mean square error (NMSE) and the structural similarity index metric (SSIM).

There is nothing wrong with your metrics.

I would suggest this paper by Fred Chen et.al. which deals with exactly this subject (although through even talking to the author I am still pessimist toward using CS for EEG signals!)