Bayesian Field Theory Applied to Scattered Data Interpolation and Inverse Problems

Problems of scattered data interpolation are investigated as problems in Bayesian statistics. When data are sparse and only available on length scales greater than the correlation length, a statistical approach is preferable to one of numerical analysis. However, when data are sparse, but available on length scales below the correlation length it should be possible to recover techniques motivated by more numerical considerations. A statistical framework, using functional integration methods from statistical physics, is constructed for the problem of scattered data interpolation. The theory is applicable to (i) the problem of scattered data interpolation (ii) the regularisation of inverse problems and (iii) the simulation of natural textures. The approaches of Kriging, Radial Basis Functions and least curvature interpolation are related to a method of 'maximum probability interpolation'. The method of radial basis functions is known to be adjoint to the Universal Kriging method. The correlation functions corresponding to various forms of Tikhonov regularisation are derived and methods for computing some samples from the corresponding probability density functionals are discussed.