Numerous objective function archetypes are utilised in the basic formulation of the X-ray Transmission Tomography reconstruction problem. These objective functions serve as quantitative encodings of reconstruction quality, and a tomographic reconstruction which minimises (maximises) that quantity is considered a `solution' to the tomographic problem, insofar as an exact solution does not exist.
The basic formulation of the XCT reconstruction problem amounts to regularised least squares. The de facto measure of reconstruction quality is the $\ell^2$ deviation, i.e. the eponymous `least squares'. $\ell^2$ is unsuitable for use in experiments with appreciable measurement noise, resulting in divergent tomograms which are nearly unusable for quantitative analysis. Alternative measures of reconstruction exist, but (semi-)inner product measures remain attractive because they are (theoretically) invertible and globally convex. In principle these qualities allow greater convergence speed in iterative algorithms, resulting in better reconstructions when computing resources are finite.
We investigate alternative (semi-)inner products (SIPs) in their full generality, deriving some results pertaining to analytic inversion. We demonstrate how SIPs may be derived to quadratically approximate any penalty function, and apply this technique to produce maximum-likelihood reconstructions under novel (and physically relevant) noise models such as convolved Poisson-Gaussian and blurred Poisson.