CS: Top 16 Thoughts/Ramblings on Compressive Sensing

To the question: What is Compressive Sensing ? one could give a series of answers, here is one of them:

Compressive Sensing is the multiplexed acquisition of a sparse signal.

But sometimes a definition is not enough, so here are my top 16 random and naive thoughts (it's the week-end) on the matter which may or may not bring some insights in the matter:

• Most signals are sparse or compressible since power laws are pretty universal. Hence Compresssive Sensing applies to almost all signals.
• Most current sensing instrumentation is based on single signal sensing. The challenge of Compressive Sensing is to open the door of multiplexing to the sensing world.
• Multiplexing used to be fashionable in transmission, Compressive Sensing brings it to bear to the acquisition process.
• Compressive Sensing and Group Testing are very close subject areas.
• Compressive Sensing acquisition can be performed through different types of multiplexing including linear combination of random samples (which I call random multiplexing).
• Random multiplexing seems to have a large number of additional properties including the one of not having to care about the reconstruction step.
• Dictionary building is important in the sensing of a signal only when the acquisition is non-random mutiplexing. The converse of this statement is: One could care less about building dictionaries when acquiring a signal in a random multiplex fashion.
• Decoding, demultiplexing or reconstructing a signal can be performed using a variety of techniques starting historically with Linear Programming but this technique has since been supplanted in speed with other faster techniques.
• Different analyses do make a connection between the multiplexing process and the sparsity of the signal being sampled and attendant constraints.
• The number of multiplexed measurements is less than the number of the naively sampled meassurements which makes people say it is Sub-Nyquist.
• When acquired randomly, the number of measurements are smaller yet these multiplexed measurements keep the information of interest. This is why some people want to do manifold signal processing without going through the recontruction process.
• Depending on the field you are working in: multiplexing can also mean encoding or sketching or being the embodiement of the measurement matrix
• What is now important in Compressive Sensing is finding or inventing hardware or sensing mechanisms that can acquire signals in a multiplexed fashion ? it is also important for older multiplexed sensing techniques to see how compressive sensing and its mathematical machinery (reconstruction,..) can bring new insights or better results.
• Trade studies are being performed to see how noise affects the multiplexed acquisition and eventual reconstruction.
• If you have already acquired all your data why should you care about compressive sensing ? Maybe that data's dimensionality is too large and you want to reduce it and eventually play with it in a lower dimensional manifold
• Compressive Sensing is already a breakthrough when hardware can be reprogrammed to handle multiplexing. Compressive Sensing will be a disruptive technology in areas where either single sample sensors do not or cannot exist or in areas where random materials can be rapidily evaluated.

More insights can be gained from watching any of the online talks on the subject.

Credit: NASA/JPL/Space Science Institute, Iapetus on September 14, 2009.