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In this paper, we introduce the homomorphic Γ-WMAP (wavelet maximum a posteriori) filter, a wavelet-based statistical speckle filter equivalent to the well known Γ-MAP filter. We perform a logarithmic transformation in order to make the speckle contribution additive and statistically independent of the radar cross section. Further, we propose to use the normal inverse Gaussian (NIG) distribution as a statistical model for the wavelet coefficients of both the reflectance image and the noise image. We show that the NIG distribution is an excellent statistical model for the wavelet coefficients of synthetic aperture radar images, and we present a method for estimating the parameters. We compare the homomorphic Γ-WMAP filter with the Γ-MAP filter and and the recently introduced Γ-WMAP filter, which are both based on the same statistical assumptions. The homomorphic Γ-WMAP filter is shown to have better performance with regard to smoothing homogeneous regions. It may in some cases introduce a small bias, but in our studies it is always less than that introduced by the Γ-MAP filter. Further, the speckle removed by the homomorphic Γ-WMAP filter has statistics closer to the theoretical model than the speckle contribution removed with the other filters.