Abstract
It is shown how false operator responses due to missing or
uncertain data can be significantly reduced or eliminated. 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
represented: 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 are handled simply 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
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