Quadratic Form Minimisation in X-ray Computed Tomography
Creators: name: Grewar, Murdock
Publisher: name: The Australian National University
Year: 2022

description: The simplest physical model of X-ray attenuation through matter (i.e. Beer's Law) gives rise to a linear equality between a vector of measured attenuations, m, and the projection of an attenuating volume vector x. In equation form, m = Ax where A is determined by the tomographic experiment parameters. Due to measurement noise in the vector m, this equality can never be satisfied. In lieu of an exact solution, a quantitative reconstruction chooses and minimises some objective function (explicit or implicit) between m, A and x. In this thesis we develop a systematised approach to constructing and optimising quadratic approximations to objective penalty functions f(Ax) on projected attenuations. We describe how the inversion problem for quadratic approximations can be expressed precisely as x = (A^T S A)^{-1} A^T S c where c and S are the expansion point and Hessian respectively of the quadratic approximation to f(Ax) as a function of Ax. This conceptual framework encapsulates Generalised Least Squares and Weighted Least Squares, is more general, and offers an immediate proof that the objective function is a best possible quadratic-form approximation to f(Ax) in the neighbourhood of a solution. The primary challenge to minimising the quadratic approximations to penalty functions f(Ax) lies in actually applying the operator (A^T S A)^{-1} computationally. We derive some results pertaining to the exact expression (A^T S A)^{-1}, including a computationally-practicable version for 2D tomography for arbitrary choices of f(Ax)-- unfortunately we did not find a non-trivial example stable enough for use. The theoretical result is still of interest, if for no other reason than there will likely exist some choices of f(Ax) for which it is in fact stable. An expression for the exact inverse also serves as a foundation for further study into suitably stable approximations. Approximations to (A^T S A)^{-1} (also known as preconditioners of the gradient descent of f(Ax)) can be more computationally tractable and also more stable than the exact formula; in reconstruction there is a spectrum of tradeoff between exactness (speed) and stability (robustness) of the preconditioner. To this end, we propose a stable preconditioner (A^T S A)^{-1} ~= diag(A^T S A 1)^{-1} (where 1 is a vector of ones) which is the (unique) generalisation of the one used in the seminal SIRT algorithm by Peter Gilbert. We illustrative reconstructions based on this preconditioner (via the Conjugate Gradient Method) in the simulations section of this thesis, as well the potential gain in reconstruction quality (objective and subjective) from this generalisation. The resulting reconstructions are minimisers of f(Ax), and so they also demonstrate what could be achieved with direct methods if stable inverses to (A^T S A)^{-1} exist and could be found. Thus the problem of exact inversion of (A^T S A)^{-1} is of great interest for application to tomographic experiments with significant noise and/or minor nonlinearities, both of which may be incorporated into a quadratic penalty function f(Ax). This thesis is a foray into the theory underlying the inversion of $\bsaproj$, and a brief exhibition of a new series of practical iterative algorithms which may be applied to arbitrary quadratic forms.