Forward Problem: Numerical Solution

Let us denote the Markov PDF of the corrupted signal by $P_C ({\bf\tilde Z})$. Let us model the Markov PDF of the uncorrupted signal using Parzen-windowing as:

$$P_\mathcal{U} ({\bf z}) = \frac {1} {\vert\mathcal{U}\vert} \sum_{u \in \mathcal{U}} G ({\bf z} - {\bf z}_u, \sigma),$$ (117)

where $\{ {\bf z}_u \}_{u \in \mathcal{U}}$ denotes the means of the Gaussians and $\sigma$ their standard deviation along each dimension. This nonparametric model is a general model capable of representing arbitrary PDFs for large $\vert\mathcal{U}\vert$. The goal is to estimate the set $\{ {\bf z}_u \}_{u \in \mathcal{U}}$ and $\sigma$, i.e., the parameters of the model, based on the knowledge of the observed corrupted-signal Markov statistics and the Rician corruption process. The key idea is as follows. An estimate of the uncorrupted-signal model parameters and the Rician noise level gives us an estimate of the corrupted-signal statistics. In the inverse-methods literature, this is the process of solving the so-called forward problem. We must match this estimate of the corrupted-signal Markov PDF with the Markov PDF obtained from the corrupted data by suitably updating the prior-model parameters. We use the KL-divergence measure to quantify the goodness of the match. We now analyze the noise model in detail and present a numerical scheme for solving the forward problem.

The Rician noise model corresponds to a linear shift-variant system whose impulse response for an impulse PDF located at $x \ge 0$ is

$$P ( \tilde x \vert x ) = \frac {\tilde x} {\sigma_R^2} \exp \Bigg... ... x^2} { 2 \sigma_R^2} \Bigg) I_0 \Bigg( \frac {\tilde x x} {\sigma_R^2} \Bigg),$$ (118)

where $\sigma _R$ is the noise level and $I_0 (\cdot)$ is the zero-order modified Bessel function of the first kind. For $x \gg 3 \sigma_R$, Rician noise corrupts in a way very similar to additive independent Gaussian noise. For smaller $x$, though, the effect is more complex. For a Gaussian input PDF $G (x - \mu, \sigma)$, a general analytical formulation of the output PDF makes the denoising framework very cumbersome. To alleviate this problem, we compute the system response numerically and approximate it by a Gaussian. We construct two lookup tables $\mathcal{L}_{\mu} (\cdot)$ and $\mathcal{L}_{\sigma} (\cdot)$ that provide the means and variances of the output Gaussians $G (x' - \mu', \sigma')$, given the means $\mu$ and variances $\sigma^2$ of input Gaussians and the noise level $\sigma _R$. We discretize the input parameters at a sufficiently-high resolution and employ bilinear interpolation to read values from the table.

We must be aware of some important issues while computing the system response. The Rician PDF $P ( \tilde x \vert x )$ is defined only for nonnegative $x$. However, the Parzen-window model with Gaussian kernels extends to negative values too. This model approximates the system poorly in cases where $\sigma$ values are relatively large as compared to the magnitude of their means $\parallel {\bf z}_u \parallel$. In such cases, the Rician corruption process that applies only to the nonnegative part of the Gaussian input (a truncated Gaussian) and produces an output that may not be fitted well by a Gaussian. However, we can view the situation more positively because of the implications of the central limit theorem [167,123,78,12]. This classic theorem [167,123] states that the PDF for the sum of independent RVs asymptotically approaches a Gaussian. In the same vein, there exists a central limit theorem for arbitrary dependent RVs too [78,12] that proves their sum to approach a Gaussian RV. The theorem concerning dependent RVs applies to the Rician corruption process--the functional form of $P ( \tilde X \vert x )$ depends on $x$. In our case, while one of the RVs is a Gaussian (input PDF), the other (Rician PDF) resembles a Gaussian in general and approaches a Gaussian for specific parameter values. These facts help us obtain good fits. Figure 5.1 shows that the fitted Gaussians approximate the Rician-corrupted output PDFs reasonably well. We observe that for input Gaussians that extend significantly to the negative axis, in Figure 5.1(a)-(b), the fit is not perfect while for the other cases, the fit is close to perfect. We use a Levenberg-Marquardt curve-fitting technique [137] to fit Gaussians to the output corrupted PDFs.

Figure 5.1: These graphs depict the Rician corruption process in $1$D with $\sigma = 5$ and $\sigma _R$ = 5. The input Gaussian PDF is corrupted by Rician noise resulting in the output corrupted PDF. We fit a Gaussian to approximate this corrupted PDF. The graphs show this process for different means of the input Gaussian: (a) $x_u$ = 1, (b) $x_u$ = 5, (c) $x_u$ = 15, and (d) $x_u$ = 20. We have numerically found that the maximum relative error between the output and its Gaussian approximation is always less than $0.1$.

Given the uncorrupted PDF $P_\mathcal{U} (\cdot)$ and the Rician noise level $\sigma _R$, we can approximate the corrupted-signal Markov PDF as

$$\hat P_C ({\bf\tilde z}) \approx \frac {1} {\vert\mathcal{U}\vert} \sum_{u \in \mathcal{U}} G ({\bf\tilde z} - {\bf z'}_u, \Psi'_u),$$ (119)

where we define the $i$-th component of the neighborhood-intensity vector ${\bf z'}_u$ as

$${\bf z'}_u (i) = \mathcal{L}_{\mu} ({\bf z}_u (i), \sigma, \sigma_R)$$ (120)

and the entry on the $i$-th row of the diagonal covariance matrix $\Psi'_u$ as

$$\Psi'_u (i,i) = \mathcal{L}_{\sigma} ({\bf z}_u (i), \sigma, \sigma_R).$$ (121)