Compressed Sensing for Voxel Data Compression

Compressed sensing is not meant to be a compression technique: The whole point is using compression techniques to interpolate from sparse data, but you need to start with a compression technique.

I get what you're going for: You throw a huge set of basis functions at your data; you find a representation in terms of a small number of them; and you send the coefficients.

From one angle, this is the same reason all the Fourier image compression techniques work (JPEG and its descendents): It happens that, with this choice of basis, most of your image energy gets concentrated in a small number of coefficients (in this case, even without doing anything to explicitly encourage sparsity). So why not try other bases and actually reward sparsity with L1 regularization? It sounds plausible, right?

Unfortunately, other people have had the same idea, and as far as I know, none of them have actually been able to achieve competitive compression of e.g. images in this manner.

So if this is for your job and you need results soon, I might encourage you to look elsewhere. But if this is for research or a hobby project -- then who knows? Go grab a convex programming solver and see if you get anywhere.

Compressed sensing is not meant to be a compression technique: The whole point is using compression techniques to interpolate from sparse data, but you need to start with a compression technique.

I think a lot of people conflate "compressed sensing" with L1 regularization...'

Taking a step back from L1 regularization, and thinking just about a choice of basis: I wonder how far OP'd get with splitting the grid up into blocks and doing a Hadamard transform of each. It's more-or-less the direct generalization of JPEG to the "3d binary pixels" case...

To answer my own question: When you throw out the small coefficients of the Hadamard transform in 1d, what you get looks awful (see attached image).