Abstract:
The Maximum Entropy Method (MaxEnt) is a Bayesian technique for the reconstruction of images and spectra from imperfect data. The method is founded on the principles of maximum entropy where entropy is considered to be a measure of uncertainty. In the situation where a number of given theories fit the data equally well, entropy is maximised by choosing the most uniform theory. MaxEnt has been used with great success in a wide range of fields since the late 1970's. These include radio astronomy image reconstruction, molecular biology, nuclear physics, medical tomography and spectral analysis.
Two major applications of the MaxEnt method are covered in this thesis. The first concerns the development of a tomographic approach to the estimation of fish densities in the New Zealand hoki (Macruronus novazelandiae) fishery. The individual trawls carried out during the fishing season are used as tomographic lines which provide a cross section of the image - the density of fish in a particular area of the sea. MaxEnt image reconstruction techniques are then used to provide an estimate of the image. A detailed analysis of a two week fishing period as well as an analysis of four seasons of data on a weekly basis are presented. Results provide information on the distribution and movement of fish on a scale never before available. Comparison with other measures on the fishery indicates that the densities produced correspond to real phenomena. The success of the method opens up a wide range of possibilities for further investigations in this field.
The second application covered is that of free form spectral estimation where the aim is to determine the frequencies present in a data series composed of harmonically varying functions. Along with the MaxEnt method, the traditional Fourier analysts methods and other Bayesian techniques are also discussed. The MaxEnt technique is applied to three separate problems: variable star data, wind speed data and telephone tones. Results illustrate the benefits and the versatility of the method. In cases of high noise and incomplete data, noise and artifacts are suppressed while the prominent features of the data are highlighted.
