I'm trying to use l1 magic to reconstruct an image from a single pixel camera I've developed. The test functions used are random binary patterns projected onto the object scene, so each pattern is represented as a row vector of 0's and 1's and form the rows of the test function matrix phi. ie.
b = phi.x = phi.psi.alpha
where b is a column vector of measurements (from a photodiode), phi is a matrix with rows representing the projected binary patterns (of 0's and 1's), and x represents a column vector of the image. x is deconstructed into psi (a transform matrix) and column vector alpha (representing the image).
The equation is to be solved with the condition that the l1 norm of alpha is minimised (i.e. it's sparse)
However, the matrix phi is not positive definite, so the basis persuit function throws out an error. This is confusing me as most of the literature promotes using random test functions. Is there something wrong here? Do you have to ensure your "random" test functions eventually form a positive definite matrix?
Thanks for your help