How we can encode/decode sparse signals?

Question

I have question and looking for help. Suppose we have a vector of real values (lat's say 64 length resulting from factorization 8*8 block image). We got a sparse representation of that vector (let's suppose that the sparse vector is 128 length with very few active elements, about 4 coefficients). I want to send this sparse vector through a channel, so I need bit stream of 1,0. Are there any efficient encoding/decoding technique for the sparse signals ?

Answer 1

Since you have tagged your question compressive-sensing, I would guess that you are interested in a compressive sensing solution to this. In principle, this could work by taking a number (less than 64) of random linear combinations (measurements) of your length 64 vector and transmitting these. At the receiver, the fact that this vector has a sparse representation allows you to reconstruct it again from these measurements. However, compressive sensing does not work that well on low-dimensional vectors (64 is relatively small) and so I do not expect this scheme to work very well.

A more natural solution, since you mention that you know the sparse length 128 representation of the vector, is to use run-length encoding (Wikipedia explanation) of the sparse representation and transmit that. Since the sparse vector is almost all zeros, this should be very efficient.