Deconvolution by Modified Wiener Filtering: Interpretation for an Imperfectly Known Wavelet.

Deconvolution in the presence of additive noise is a well known problem for which there exists a Wiener filter which simultaneously spectrally whitens while suppressing noise. A simple variant of this standard Wiener filter incorporates a parameter, P say, which is intended to allow further weight to be given to noise suppression. We shall call such a filter a modified Wiener filter. To design such a filter it is required to know precisely the frequency response of the spread function or wavelet, plus the spectra of the input and additive noise. In practice some response function is taken to be appropriate, and the modified Wiener filter designed from it. If the design response function is thought of as one chosen from a set of allowable response functions - a realistic practical viewpoint - then it is shown how the selection of the design response, the chosen value of the parameter P and the noiseinput power spectral ratio effectively determine the characteristics of this set of possible wavelet response functions. This is demonstrated for two different error criteria - i the minimization of the average mean-squared error, and ii the minimization of the maximum mean-squared error. It is shown how to calculate deconvolution filters which solve sub-optimal versions of i and ii, but which are robust to uncertainty in the wavelets frequency response.