Wavelet based compression can be achieved by suppressing coefficients below a threshold (or preserving a predetermined number of significant coefficients). Compressed sensing on the other hand projects the data to a lower dimension (assuming sparsity in a domain such as wavelets) and reconstructs by solving (a relaxed version of) an optimization problem. What is the advantage of one over the other? CS seems like a round about way of doing things that can be achieved straightforward by DWT. Although one needs to keep track of the index of significant coefficients in wavelet compression, I am ending up with DWT based compression outperforming CS! Is there a distinct advantage of CS for compression?
See my other answer. I recommend that you start with the most general mathematical formulation in a vector space, and only then do you specialise the reasoning with hypotheses. It's the best way to keep in control and know exactly what you are doing.
Stay close to the physics of the problem you are addressing
Wavelet compression takes advantage of Spatial Redundancy, Coding Redundancy and Psycho-visual Redundancy of the data. It also supports multi-resolution representation. These qualities usually yield higher compression ratio and better subjective quality of the reconstructed data, particularly, image and video data.
The difference - For only WC, you need to have the full information of the data beforehand that you want to compress. You will compute wavelet co-efficients and discard some of them you feel redundant. However with CS, you don't need to have all the information (part of which you will deem redundant afterwards) and directly attempt to reconstruct an image with less data. Advantage example of CS - helpful in slow data acquisition systems like MRI where had you been forced to take all the data samples, it would have taken 15 mins more inside the MRI machine. But due to CS like methods, you can speed up data acquisition.
Any transform based compression (DWT, cosine transform, etc.) transmits to the decoder a subset of the transform coefficients. One tries to select the transform so that as much of the signal energy is represented by the transmitted coefficients, and one can even do what you suggested, to dynamically select the coefficients of largest magnitude and transmit those coefficients along with their corresponding indices. However, at the end of the day the decoder has only partial information about the signal and it has to "guess" the values for the coefficients which were not transmitted. Typically, the decoder simply sets them to zero, which leads to decoding errors. The main idea of CS is that during the decoding you can apply some auxiliary information about the the signal which allows you to guess the missing coefficients more accurately. This auxiliary information may be that the signal is sparse with respect to a specific basis, or that the signal is a low rank matrix, or other more specific characterizations. You can use your DWT compression and still use CS decoding in order to significantly improve your decoding accuracy. Hence DWT and CS can, in principle, be combined.
When speaking about compression we often care about the number of bits needed to transmit the coefficients rather than the number of coefficients transmitted. Designing a module which will compress the selected coefficients and encode them as a bit sequence is a large subject, but you cannot really compare performance without taking it into account.
Other practical considerations that affect the design are the relative complexity of the encoder and the decoder, and the sensitivity to transmission errors. These too are large issues which need to be taken into account in assessing advantages and disadvantages.
CS can be considered as an enhancement tool for wavelet video coding. Moreover, it can be also applied as an error-concealment tool as well (in case of video streaming). For example, my experiements show that 3-D SPIHT compression can be improved by 1.5 dB by appliing CS at the decoder side, and in case of losses the error-concealment based on CS gives 4.9 dB improvement. For more details please have a look into my paper E.Belyaev et al., Error concealment for 3-D DWT based video codec using iterative thresholding, IEEE Communications Letters, 2017.
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