| This thesis presents a novel optimal methodology for dealing with linear estimation problems in spatial deterministic fields, using discrete and regularly gridded data. More specifically, a unified study of various important issues that affect the theoretical analysis and practical computations associated with signal approximation problems (namely, stability, convergence, error analysis and choice of estimation model restrictions) is performed with respect to the data resolution parameter. A combination of different mathematical tools is employed for our theoretical developments, with the underlying ideas originating from the areas of deterministic collocation in Hilbert spaces, frame signal expansions, spatio-statistical collocation and multiresolution signal analysis theory. The spatio-statistical collocation principle is used to develop a new generalized multiresolution signal analysis scheme, which offers increased flexibility (in terms of scale level restrictions) and it is more powerful (in terms of approximation performance) than the classic dyadic multiresolution analyses that are associated with standard wavelet theory. Additional investigations are conducted on interpolation error analysis with respect to the data resolution level and the used estimation kernel, as well as on aliasing error propagation in convolution integral formulas using discrete gridded input data. Most of the theoretical developments are made with practical applications in mind, which means that an extensive (and original) treatment of the optimal noise filtering problem is also included, considering the most general case with non-stationary additive noise in the gridded input data. |
