Nonuniqueness and instability

We cannot avoid defining $\sigma^2$,because without it, any region of zero signal would get an infinite weight. This is likely to lead to undesirable performance: in other words, although with the data of Figure 2 I found rapid convergence to a satisfactory answer, there is no reason that this had to happen. The result could also have failed to converge, or it could have converged to a nonunique answer. This unreliable performance is why academic expositions rarely mention estimating weights from the data, and certainly do not promote the nonlinear-estimation procedure. We have seen here how important these are, however.

I do not want to leave you with the misleading impression that convergence in a simple problem always goes to the desired answer. With the program that made these figures, I could easily have converged to the wrong answer merely by choosing data that contained too much crosstalk. In that case both images would have converged to $\bold s$.Such instability is not surprising, because when $\alpha$ exceeds unity, the meanings of $\bold v$ and $\bold h$ are reversed.