As discussed above, the parameters of non-Gaussian noise are not preserved by convolution filtering. From the various types of non-Gaussian noise that we generated, uniform noise was selected because of the perfect fit of the wavelet-based GLM. We consider image $x$ contaminated by additive noise as defined in Eq. (13). Hence, the additive model in the wavelet domain is given as $Y=X+N$, where $Y=UWT{y}$ denotes the noisy observation (i.e., the acquired data), $N=UWT{n}$ presents noise, and $X=UWT{x}$ is the ideal noise-free image. The conditional mean of the posterior PDF $pX|Y(x|y)$ provides the least square estimation of $X$. The formula for the MMSE^{16} estimator runs as Display Formula
$X^(Y)=\u222b\u2212\u221e+\u221epX|Y(x|y)xdx=\u222b\u2212\u221e+\u221epY|X(y|x)pX(x)xdx\u222b\u2212\u221e+\u221epY|X(y|x)pX(x)dx,=\u222b\u2212\u221e+\u221epN(y\u2212x)pX(x)xdx\u222b\u2212\u221e+\u221epN(y\u2212x)pX(x)dx,$(32)
where $pY|X(y|x)$ denotes the likelihood function, $pX(x)$ represents the a priori model, and $pN(x)$ stands for the noise model. Both these random variables are modeled by the wavelet-based GLM. Similarly to ^{30}, we define the theoretical central moments of $Y$. The second moment is given by Display Formula$m2(Y)=\tau 2\Gamma (3\lambda )\Gamma (1\lambda )+s2\Gamma (3\nu )\Gamma (1\nu )=m2(X)+m2(N),$(33)
where $\tau $ and $\lambda $ are the GLM parameters of the signal. The fourth moment of $Y$ runs as Display Formula$m4(Y)=\tau 4\Gamma (5\lambda )\Gamma (1\lambda )+6s2\tau 2\Gamma (3\nu )\Gamma (3\lambda )\Gamma (1\nu )\u2009\Gamma (1\lambda )+s4\Gamma (5\nu )\Gamma (1\nu )=m4(X)+6m2(N)m2(X)+m4(N).$(34)
The GLM parameters of the noise-free signal are estimated from second and fourth moment of the observed signal and noise using the kurtosis formula Display Formula$\kappa X=\Gamma (5\lambda )\Gamma (1\lambda )\Gamma 2(3\lambda )=m4(Y)\u2212m4(N)\u22126m2(N)[m2(Y)\u2212m2(N)](m2(Y)\u2212m2(N))2.$(35)
From Eq. (33), we may derive Display Formula$\tau =[m2(Y)\u2212m2(N)]\Gamma (1\lambda )\Gamma (3\lambda ).$(36)
The values of the moments used in the above equations are estimated from the data using the sample moments. The $k$’th central sample moment of $X$ is given by $Mk(X)=1I\u2211i=1I[Xi\u2212E(X)]k$.