Tomographic and spectral analysis using noise statistics

A unified theoretical treatment of constrained optimisation methods for tomographic and spectral estimation from discrete data is given. The solution is shown to be equivalent to the unconstrained optimisation of a dual functional in which the image or spectrum is modelled in terms of a Lagrange multiplier vector and the kernel of the constraint integrals. In order to obtain the best possible solution it is important to consider the effects of noise in the constraints. The problem is reformulated using the above models and the exact data is replaced with the noise statistics as constraints; this is solved using a penalty method. A very fast direct algorithm is also introduced which matches the noise variance provided the signal to noise ratio is approximately known.