Statistics of 2-D DT-CWT Coefficients for a Gaussian Distributed Signal
This paper deals with the statistical properties of the two-dimensional dual-tree complex wavelet transform (DT-CWT) coefficients of a Gaussian distributed signal both in the Cartesian and polar forms. The first level of decomposition of the DT-CWT uses the wavelet filters that form only an approximate Hilbert-pair, while those at the higher levels form almost an exact Hilbert-pair. Hence, a significant correlation exists between the quadrature-filtered coefficients of the two trees in the first level of decomposition as compared to the other levels. As a consequence, in the Cartesian representation, the real and imaginary components of the complex coefficients are modeled as independent zero-mean Gaussian having unequal variances for the first level of decomposition and equal variances for the higher levels. In the polar representation, the magnitude components are modeled by a generalized Gamma probability density function (PDF) for the first-level decomposition and a Rayleigh PDF for the higher levels. The corresponding phase components are modeled by an analytic PDF. The Monte Carlo simulations show that the proposed PDFs of the transform coefficients match very well with the empirical ones. It is shown that the moments of the corresponding PDFs closely approximate the estimated sample moments. Finally, two techniques, namely, maximum a posteriori-based estimation and phase-based ridge detection are developed using the proposed PDFs. Simulation studies are carried out showing that the use of the proposed techniques provides improved estimation and detection performance of images in a noisy environment.