Positive solutions to linear inverse problems

We discuss two methods for incorporating the prior knowledge that the solution is positive into the truncated singular value decomposition method for solving linear inverse problems. The methods are based on mathematical programming techniques. One method can be viewed as a primal method and the other as its dual. Provided the singular functions are analytic these methods both deliver the same solution - namely the positive solution of minimum 2-norm which agrees with the truncated singular function expansion in its known terms - and this solution also appears to possess higher resolution. In the presence of noise both methods can sometimes fail to converge and in these situations we give simple remedies which yield approximate solutions. While there may be no reason to suppose the unknown object should be that of minimum 2-norm, our method has the advantage over other nonlinear methods that for a noiseless band-limited object the exact solution is delivered.