Imposing sparse structure in Gabor space I have a 2-D signal collected in the frequency vs time domain. There is a feature of interest in this signal (a chirp) appearing at some arbitrary times which I have successfully extracted using a properly orientated 2-D Gabor filter.

What I would like to do is exploit the detection of this feature to represent the data in a sparse domain, so that Compressed Sensing can be used to capture the data regions which contain the chirp trait without sampling the entire data set a-la Nyquist.

I have searched for a way to do this online but haven't found very much. The literature I've read says that the basis must be orthogonal for perfect reconstruction, and that Gabor domain is most times not, but given that I'm only interested in a faster detection (yes contains/no, does not contain chirp) and not really a perfect reconstruction, is such a thing possible? If the approach above is not the best way to go about doing this, recommendations would be welcome as well.
In my experience, you do not necessarily need an orthogonal dictionary, see for example Compressed Sensing with Coherent and Redundant Dictionaries (DOI: 10.1016/j.acha.2010.10.002).

The paper Quantifying the performance of compressive
sensing on scalp EEG signals (DOI: 10.1109/ISABEL.2010.5702814) seems to be an example of the use of Gabor dictionaries (in EEG) in compressed sensing. I would just go ahead and try it.

If the chirp you are looking for can have a fairly fixed location in the Gabor domain, it might be possible to use compressive detection or classification instead of actual reconstruction. This is explained in Signal Processing With Compressive Measurements (10.1109/JSTSP.2009.2039178. You can see an example of how we have used it for symbol detection in IEEE 802.15.4 communication.