Normalized and Differential Convolution Methods for Interpolation and Filtering of Incomplete and Uncertain Data

In this paper it is shown how false operator responses due to missing or uncertain data can be significantly reduced or eliminated. Perhaps the most well-known of such effects are the various ‘edge effects’ which invariably occur at the edges of the input data set. Further, it is shown how operators having a higher degree of selectivity and higher tolerance against noise can be constructed using simple combinations of appropriately chosen convolutions. The theory is based on linear operations and is general in that it allows for both data and operators to be scalars, vectors or tensors of higher order. Three new methods are presented: Normalized convolution, Differential convolution and Normalized Differential convolution. All three methods are examples of the power of the signal/certainty philosophy, i.e. the separation of both data and operator into a signal part and a certainty part. Missing data is simply handled by setting the certainty to zero. In the case of uncertain data, an estimate of the certainty must accompany the data. Localization or ‘windowing’ of operators is done using an applicability function, the operator equivalent to certainty, not by changing the actual operator coefficients. Spatially or temporally limited operators are handled by setting the applicability function to zero outside the window. Consistent with the philosophy of this paper all algorithms produce a certainty estimate to be used if further processing is needed. Spectrum analysis is discussed and examples of the performance of gradient, divergence and curl operators are given.