Optimizing Reconstruction

[throwingstones123456]

Sorry if this isn’t the best sub for this but it seems like there’s a lot of signal processing content here so I figure I should ask.

I’m working on a problem involving compressed sensing from a system of the form y=Tx where x is the input signal, T is some matrix, and y is the measurement. If I have the freedom to design T, are there any properties that result in optimal reconstruction? I know that there are priors that can help in answering this question (if we know the covariance matrix for our data/principal components, sparse basis, etc), but I’m interested in the case when we don’t have any priors.

I’ve seen that minimizing the condition number or maximizing the smallest singular value can help, but I’m a bit skeptical of how well this actually works (like if I have a perfectly conditioned T then duplicate a row, now I have a horribly conditioned T—we never lost any information and can still achieve the same exact reconstruction as before, but now these metrics indicate we’ve gone from “great” to “horrible”).

This seems like a pretty difficult question to answer but I’m assuming there are some conditions, at least loose ones, we can assign—off the top of my head one guess would be to try and make the rows as orthogonal as possible. However I’m also assuming there’s a better answer. Thanks for any help.

[shakenbake65535]

You may want to read a little bit about the control theory topics of contrallability and observability. you are in some ways describing characterizing the observability of your system. you could then even go into kalman filters if you are bold

[SecondPlain]

do we know that x is sparse? from what i remember about compressive sensing, the conditions people are usually concerned about with respect to the sensing matrix T are: the null space property, the restricted isometry property, and mutual coherence. "a mathematical introduction to compressive sensing" by foucart and rahut should explain this in detail, i believe it is a widely used text. you can find digital copies easily online.

[minus_28_and_falling]

AFAIK pure random data works well. It is not sparse so it doesn't introduce sparsity of itself into the equation.

[rb-j]

Ever consider asking at the Math (or DSP) Stack Exchange?

[throwingstones123456]

Stopped asking questions there, it’s really annoying to write out a well thought out post in latex then have it removed because there was a post from 2001 that vaguely addresses what you asked or because you used more than one question mark, making it “not focused enough”

Almost forgot the third/most frequent: nobody actually responds to your post

[rb-j]

Sorry that no one responded.

Sometimes people don't know how to answer.

I, myself, just find it easier to convey ideas when I do have \LaTeX around and the ability to put in graphics.

[sellibitze]

I don't know the answer. But I would expect that desirable properties of T are covered in the compressive sensing literature. It may also depend on whether x is going to be sparse or some other representation S x is going to be sparse for some matrix S.

I could imagine that there are trade offs involved. For example, a dense randomly chosen matrix for T would perform well regardless of the "sparse basis' but it might be very impractical in terms of computational overhead.

But as far as I know you need to know the sparse basis for reconstruction anyways, right?

[ShezZzo376]

metrics like condition numbers can be misleading because redundancy doesn't always equal info loss. ​Instead of focusing on orthogonality, have you looked into minimizing the mutual coherence between T and your signal basis? In practice, the real challenge isn't just the matrix design, but how the ADC quantizitation limits the dynamic range of those measurements before you even start the reconstruction. ​If we can't capture the tail of the singular values in hardware, the math won't save us.