Iterated Conditional Entropy Reduction (ICER)

At each pixel $t$, the prior PDF is

$$P ( x_t \vert {\bf y}_t ) = \frac { \sum_{u \in \mathcal{U}} G ({... ...- x_u, \sigma) } { \sum_{u \in \mathcal{U}} G ({\bf y}_t - {\bf y}_u, \sigma) }$$ (131)

and the likelihood PDF is

$$P ( \tilde x_t \vert x_t ) = \frac {1} {\eta (\tilde x_t, \sigma_... ... 2 \sigma_R^2} \Bigg) % I_0 \Bigg( \frac {\tilde x_t x_t} {\sigma_R^2} \Bigg),$$ (132)

where $\eta (\tilde x_t, \sigma_R)$ is the normalization factor that depends on the observed value $\tilde x_t$ and the noise level $\sigma _R$. We propose updating pixel intensities $x_t$, to increase the posterior probability $P ( x_t \vert \{ x_u \}_{u \in \mathcal{T} \setminus \{ t \}}, {\bf\tilde x} )$ in ( 5.2), by performing a gradient ascent on the logarithm of the posterior. In [9,6], we showed the equivalence between a gradient ascent on the logarithm of a PDF and entropy reduction using the Shannon's entropy measure. Entropy reduction on this posterior PDF results in the following update rule for all pixel intensities $x_t$

$$x_t \leftarrow x_t - \frac {\partial h (x_t \vert {\bf y}_t, \tilde x_t)} {\partial x_t}$$

$$= x_t + \Bigg[ \frac {\partial \log P ( x_t \vert {\bf y}_t )} {\pa... ...l x_t} + \frac {\partial \log P ( \tilde x_t \vert x_t )} {\partial x_t} \Bigg]$$

$$= x_t$$

$$+ \frac { \sum_{u \in \mathcal{U}} G ({\bf y}_t - {\bf y}_u, \sig... ...m_{u \in \mathcal{U}} G ({\bf y}_t - {\bf y}_u, \sigma) G (x_t - x_u, \sigma) }$$

$$- \frac {\hat x_t^m} {\sigma^2} + \frac {\tilde x_t} {\sigma^2} \... ...\tilde x_t \hat x_t^m / \sigma^2) } { I_0 (\tilde x_t \hat x_t^m / \sigma^2) },$$ (133)

where $I_1 (\cdot)$ is the first-order modified Bessel function of the first kind. ( The expression for the gradient of the logarithm of the Rician likelihood PDF appears in [11]. ) These sequence of updates leads to image estimates with nondecreasing posterior probabilities and, hence, guarantee convergence to a local maximum of the posterior PDF. We call this novel proposed algorithm for performing Bayesian estimation on MRFs as the iterated conditional entropy reduction (ICER).