Covariances of Linear Filter Outputs in Computer Vision

The use of linear filters, i.e. convolutions, inevitably introduces dependencies in the uncertainties of the filter outputs. Such non-vanishing covariances appear both between different positions and between the responses from different filters (even at the same position). This report describes how these covariances between the output of linear filters can be computed. We then examine the induced covariance matrices for some typical 1D and 2D filters. Finally the total noise reduction properties are examined.