Space-Frequency Localization as Bivariate Mother Wavelets Selecting Criterion for Hyperanalytic Bayesian Image Denoising

Abstract

Communicated by Stefania ResidoriThroughout recent years, many wavelet transforms (WTs) were used in digital image processing: the discrete WT (DWT), the stationary WT (SWT) or the hyperanalytic WT (HWT). All these transforms have in common a feature, the mother wavelets (MW). A great number of MWs was already proposed in literature. The purpose of this paper is the selection of MW for hyperanalytic Bayesian image denoising on the basis of its space-frequency localization. The MW with the same space-frequency localization as the elements of the input image gives the better results. Some procedures for the evaluation of the space-frequency localization of MWs and input images are proposed and applied to optimize the results obtained by the simulations of denoising, indicating the most appropriate MW. Keywords: Mother wavelets; space-frequency; denoising. 1250009-1 Fluct. Noise Lett. 2012.11. Downloaded from www.worldscientific.com by MCMASTER UNIVERSITY on 02/06/15. For personal use only. A. IsariH{ψ} and ψ a = ψ + iH{ψ} are also MWs. H denotes the Hilbert transform operator. This wavelets pair (ψ, iH{ψ}) defines a complex discrete wavelet transform (CDWT). A complex wavelet coefficient is obtained by interpreting the wavelet coefficient from one DWT tree as being its real part, whereas the corresponding coefficient from the other tree is considered its imaginary part. In [4], the DTCWT, which is a quadrature pair of DWT trees similar to the CDWT, is developed. The DTCWT coefficients may be interpreted as arising from the DWT associated with a quasi-analytic wavelet. Both DTCWT and CDWT are invertible and quasi shiftinvariant. The implementation of analytic wavelet transform (AWT) is presented in [6]. First, a Hilbert transform is applied to the data. The real wavelet transform is then applied to the analytical signal associated to the input data, obtaining complex coefficients. The DTCWT and the AWT are equivalent because: d DTCWT [m, n] = x(t), ψ m,n (t) + iH{ψ m,n (t)} = x(t), ψ m,n (t) − i x(t), H{ψ m,n (t)} = x(t), ψ m,n (t) + i H{x(t)}, ψ m,n (t) = x(t) + iH{x(t)}, ψ m,n (t) = d AWT [m, n]