Analytical form for a Bayesian wavelet estimator of images using the Bessel K form densities

Abstract

A novel Bayesian nonparametric estimator in the Wavelet domain is presented. In this approach, a prior model is imposed on the wavelet coefficients designed to capture the sparseness of the wavelet expansion. Seeking probability models for the marginal densities of the wavelet coefficients, the new family of Bessel K forms (BKF) densities are shown to fit very well to the observed histograms. Exploiting this prior, we designed a Bayesian nonlinear denoiser and we derived a closed form for its expression. We then compared it to other priors that have been introduced in the literature, such as the generalized Gaussian density (GGD) or the alpha-stable models, where no analytical form is available for the corresponding Bayesian denoisers. Specifically, the BKF model turns out to be a good compromise between these two extreme cases (hyperbolic tails for the alpha-stable and exponential tails for the GGD). Moreover, we demonstrate a high degree of match between observed and estimated prior densities using the BKF model. Finally, a comparative study is carried out to show the effectiveness of our denoiser which clearly outperforms the classical shrinkage or thresholding wavelet-based techniques.

Sparse Signal Estimation by Maximally Sparse Convex Optimization

“…The firm threshold function [30], and the smoothly clipped absolute deviation (SCAD) threshold function [23], [71] also provide a compromise between hard and soft thresholding. Both the firm and SCAD threshold functions are continuous and equal to the identity function for large |y| (the corresponding 0 (x) is equal to zero for x above some value).…”

“…Sparsity-based nonlinear estimation algorithms can also be developed by formulating suitable non-Gaussian probability models that reflect sparse behavior, and by applying Bayesian estimation techniques [1], [17], [23], [39], [40], [48], [56]. We note that, the approach we take below is essentially a deterministic one; we do not explore its formulation from a Bayesian perspective.…”

Wavelet-Based Analysis and Estimation of Colored Noise

“…Because of scaling ambiguity between √ z andũ, the hidden multiplier is often assumed to be normalized such that E [z] = 1. Prior distributions for z include Jeffrey's non-informative 3 prior Portilla et al (2003), the log-normal prior Portilla & Simoncelli (2001), the exponential distribution Selesnick (2006) and the Gamma distribution Fadili & Boubchir (2005); Srivastava et al (2002).…”

Image Denoising Using Mixtures of Projected Gaussian Scale Mixtures

“…Prior distributions f z (z) for the hidden variable z include Jeffrey's noninformative prior [17], the exponential distribution [34] and the Gamma distribution (see e.g. [8], [19], [35]). To ease the comparison with the results of Portilla et al [17], we will adapt Jeffrey's non-informative prior (i.e.…”

“…The subsequent denoising (see Section V) comes down to filtering with 1×3 horizontal filter masks. The covariance matrix C t is obtained from C y using C t = V T C y V (see (8)), which results in simply extracting elements of C y . Analogously, the diagonal elements of Ψ are computed as Ψ ii = V T C yV ii .…”