Adaptive Bayesian Denoising for General Gaussian Distributed Signals

Abstract

Optimum Bayes estimator for General Gaussian Distributed (GGD) data in wavelet is provided. The GGD distribution describes a wide class of signals including natural images. A wavelet thresholding method for image denoising is proposed. Interestingly, we show that the Bayes estimator for this class of signals is well estimated by a thresholding approach. This result analytically confirms the importance of thresholding for noisy GGD signals. We provide the optimum soft thresholding value that mimics the behavior of the Bayes estimator and minimizes the resulting error.The value of the threshold in BayesShrink, which is one of the most used and efficient soft thresholding methods, has been provided heuristically in the literature. Our proposed method, denoted by Rigorous BayesShrink (R-BayesShrink), explains the theory of BayesShrink threshold and proves its optimality for a subclass of GDD signals. R-BayesShrink improves and generalizes the existing BayesShrink for the class of GGD signals. While the BayesShrink threshold is independent from the wavelet coefficient distribution and is just a function of noise and noiseless signal variance, our method adapts to the distribution of wavelet coefficients of each scale. It is shown that BayesShrink is a special case of our method when shape parameter in GGD is one or signal follows Laplace distribution. Our simulation results confirm the optimality of R-BayesShrink in GGD denoising with regards to Peak Signal to Noise Ratio (PSNR) and Structural Similarity (SSIM) index. Index Terms-Bayesian estimation, wavelet shrinkage and denoising, soft thresholding

Thresholding or Bayesian LMMSE/MAP Estimator, which one Works Better for Despeckling of True SAR Images?

“…Although these methods provide better denoising performance, their computational complexity is significantly much higher and in case the speed and time are of great importance we may still prefer to use simple thresholding or Bayesian LMMSE/MAP estimators. As validated in [7], the actual Bayesian estimator for generalized Gaussian distributed data behaves similar to a piecewise linear thresholding (shrinkage) function. Therefore, it is of special interest to provide simple optimized shrinkage functions for SAR denoising purposes.…”

“…The two simple, yet efficient, methods for SAR despeckling are using (i) thresholding (or shrinkage) estimators [7], and (ii) Bayesian estimators such as linear minimum mean square error (LMMSE) estimator or maximum a posteriori (MAP) estimator [8]. The SAR image is first converted into the transform domain like the NSCT.…”

“…In addition to the shape of mapping function, finding the optimum value of the threshold is also of great importance. For multiresolution transform based analysis like NSCT, the threshold selection can be adaptive or non-adaptive per subband [7] [10].…”

Extended Shearlet HMT Model-Based Image Denoising Using BKF Distribution

“…The major novelty of the proposed approach is that the parameters of the GGD PDF are taken to be space-varying within each wavelet frame. Hashemi et al [16] studied behavior of the Bayesian estimator for noisy GGD data, and showed that this estimator can be well estimated with a simple shrinkage function. Azzari et al [17] introduced an adaptive Gaussian-Cauchy mixture modeling for the likelihood of pairwise mean/standard deviation scatter points found when estimating signal-dependent noise.…”

“…Although seemingly very different, they all share the same property: to keep the meaningful edges and remove less meaningful ones. The existing image denoising work can be roughly divided into Nonlocal Methods [3][4][5], Random Fields [6][7][8], Bilateral Filtering [9][10][11], Anisotropic Diffusion [12,13], and Statistical Model [15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32]. In addition, many authors have developed image denoising algorithms based on support vector machine (SVM) classification [14].…”