Theory and application of optimal linear resolution to MRI truncation artifacts, multiexponential decays and in vivo multiple sclerosis pathology

It is widely believed that one of the best way to proceed when analysing data is to generate estimates which fit the data. However, when the relationship between the unknown model and data is linear for highly underdetermined systems, is it common practice to find estimates with good linear resolution with no regard for fitting the data. For example, windowed Fourier transforms produces estimates that have good linear resolution but do not fit the data. Surprisingly, many researchers do not seem to be explicitly aware of this fact. This thesis presents a theoretical basis for the linear resolution which demonstrates that, for a wide range of problems, algorithms which produce estimates with good linear resolution can be a more powerful and convenient way of presenting the information in the data, than models that fit the data.; Linear resolution was also applied to two outstanding problems in linear inverse theory. The first was the problem of truncation artifacts in magnetic resonance imaging (MRI). Truncation artifacts were heavily suppressed or eliminated by the choice of one of two novel Fourier transform windows. Complete elimination of truncation artifacts generally led to unexpectedly blurry images. Heavy suppression seemed to be the best compromise between truncation artifacts and blurriness.; The second problem was estimating the relaxation distribution of a multiexponential system from its decay curve. This is an example where hundreds of papers have been written on the subject, yet almost no one has made a substantial effort to apply linear resolution. I found the application to be very successful. As an example, the algorithm was applied to the decay of MRI data from multiple sclerosis patients in an attempt to differentiate between various pathologies.