Inverse problems is the scientific art of going from effect to cause. This course is the second course on computational inverse problems.
In medical tomography, the “cause” is the X-ray attenuation coefficient inside a patient. The “effect” is a collection of X-ray images of the patient recorded along several directions of view. Recovering the inner structure of the patient from a set of noisy X-ray images is an inverse problem.
Mathematically, the X-ray images can be seen as a collection of line integrals of a non-negative function. Johann Radon proved in 1917 that a (reasonably regular) planar function can be reconstructed from the knowledge of its integrals along every line. This idea was refined and implemented by Hounsfield and Cormack in the 1960’s and 1970’s, resulting in modern CT scanners used in hospitals every day. And in Nobel Prizes for Hounsfield and Cormack.
Recently there has been growing interest in designing medical X-ray tomography that deliver a smaller radiation dose to the patient than the classical CT scanners. One way to do this is to take fewer X-ray images, but then the inverse problem becomes more ill-posed. Ill-posedness means that the solution is highly sensitive to modelling errors and measurement noise. Robust solution of limited-data tomography problems is based on regularization.
The same mathematical model appears in a variety of tomographic applications, not necessarily based on X-rays. Monitoring the ozone layer can be done tomographically using satellite-borne star occultation measurements. Electron microscopy can reveal the structure of molecules once a limited-angle tomography problem is solved in a regularized way. Secret chambers inside the Giza pyramids in Egypt have been found using tomographic imaging based on cosmic muon rays. Neutron tomography can be used for studying large metallic objects.
The course starts by showing how tomographic imaging problems can be modelled using matrices. First, naive reconstruction is attempted computationally using a least-squares solution. This fails because of the ill-posedness of the inverse problem. The source of the difficulties is identified using Singular Value Decomposition (SVD).
The difficulties can be overcome by regularization. Several regularized reconstruction methods are discussed both theoretically and computationally: truncated SVD, generalised Tikhonov regularization, Total Variation regularization, and wavelet-sparsity regularization.
The goals of the course:
(1) understand X-ray tomography as a matrix model,
(2) learn least-squares solution technique and see that it is not enough for tomography,
(3) show how to use SVD to assess the ill-posedness of a tomographic problem,
(4) understand why tomography needs special regularised methods,
(5) write robust Matlab programs for tomographic image reconstruction,
(6) learn how to extend the solution algorithms to large-scale problems using matrix-free methods.