As I had suspected, the matrix formed by this system of equations is not invertible. For measurement spacing s and incremental shifts of t, the n x n matrix would look like this:
[ 0 s 2s ... ns ] [ t t+s t+2s ... t+ns ]A = [2t 2t+s 2t+2s ... 2t+ns ] [... ... ] [nt ... nt+ns ]Written out it is clear that the columns are all linearly dependent and that the matrix is not invertible. Ridge regression type methods could be used to make the matrix invertible. These methods work by introducing a small amount of error in the form of a scalar multiple (lambda) of the identity matrix.
A_new = A + lambda*IHowever these methods would still produce a linear mass distribution, where in reality the mass distribution is nonlinear.
I was hoping this would turn into a fun way to explore and demonstrate the applications of the radon transform to image reconstruction from filtered back projections, but it seems the fundamental concept does not work. Luckily I have had some other project ideas simmering on the back burner; I’m hoping those will yield better results than this one!