Bayesian Denoising by Entropy Reduction

The proposed strategy relies on several pieces of technology that interact to provide accurate, practical models of image statistics. For clarity, the discussion begins at a high level and successive sections discuss how each of these pieces is developed from the input data.

Given the noisy image ${\bf\tilde x}$, our goal is to find the maximum-a-posteriori (MAP) estimate ${\bf x}^*$ of the true image ${\bf x}$:

$${\bf x}^* = \mathop{\mbox{argmax }}_{{\bf x}} P ({\bf x} \vert {\bf\tilde x}).$$ (114)

Writing the posterior as

$$P ( {\bf x} \vert {\bf\tilde x} ) = P ( x_t \vert \{ x_u \}_{u \i... ...} ) P ( \{ x_u \}_{u \in \mathcal{T} \setminus \{ t \} } \vert {\bf\tilde x} ),$$ (115)

where $t$ is an arbitrary pixel, motivates us to employ an iterative restoration scheme where, starting from some initial image estimate, we update the estimate pixel-wise so that the posterior never decreases. Besag's ICM algorithm [16] gives one such strategy that updates $x_t$ to the mode of the PDF $P ( x_t \vert \{ x_u \}_{u \in \mathcal{T} \setminus \{ t \}}, {\bf\tilde x} )$. Finding modes of PDFs, however, is not always straightforward or computationally efficient. Therefore, we propose a new algorithm that updates $x_t$ by moving it closer to the local mode of $P ( x_t \vert \{ x_u \}_{u \in \mathcal{T} \setminus \{ t \}}, {\bf\tilde x} )$. The proposed algorithm is similar in spirit to the ICM algorithm, but relies on entropy reduction on the PDF that updates pixel intensities by performing a gradient ascent on the logarithm of the PDF--hence called iterated conditional entropy reduction (ICER). The relationship between reducing Shannon's entropy of Parzen-window PDFs and gradient ascent on the logarithm of the posterior PDF is described in detail in [9,6]. It follows that by updating intensities $x_t$ to reduce the entropy $h ( x_t \vert \{ x_u \}_{u \in \mathcal{T} \setminus \{ t \}}, {\bf\tilde x} )$ and bringing them closer to their local modes, we can guarantee nondecreasing values for $P ( x_t \vert \{ x_u \}_{u \in \mathcal{T} \setminus \{ t \}}, {\bf\tilde x} )$ and, thereby, convergence.

Let us assume for simplicity that, given the true image ${\bf x}$, the RVs in the MRF ${\bf\tilde X}$ are conditionally independent ( 2.86). Subsequently, Bayes rule gives [16]

$$\mathop{\mbox{argmax }}_{x_t} P ( x_t \vert \{ x_u \}_{u \in \mat... ...hop{\mbox{argmax }}_{x_t} P ( x_t \vert {\bf y}_t ) P ( \tilde x_t \vert x_t ),$$ (116)

where $P ( x_t \vert {\bf y}_t )$ is the unknown prior PDF and $P (\tilde x_t \vert x_t)$ is the likelihood as determined from the Rician noise model. We model the prior using nonparametric Parzen-window density estimates with Gaussian kernels. The next section describes a method for adaptively inferring the prior based on the input data and the knowledge of the noise model.