Strongly Consistent Estimation of the Sample Distribution of Noisy Continuous-Parameter Fields

The general problem of defining and determining the sample distribution in the case of continuous-parameter random fields is addressed. Defining a distribution in the case of deterministic functions is straightforward, based on measures of sublevel sets. However, the fields we consider are the sum of a deterministic component (nonrandom multidimensional function) and an i.i.d. random field; an attempt to extend the same notion to the stochastic case immediately raises some fundamental difficulties. We show that by “uniformly sampling” such random fields the difficulties may be avoided and a sample distribution may be compatibly defined and determined. Not surprisingly, the obtained result resembles the known fact that the probability distribution of the sum of two independent random variables is the convolution of their distributions. Finally, we apply the results to derive a solution to the problem of deformation estimation of one- and multidimensional signals in the presence of measurement noise.