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Optimization based on mean-field theory, including mean-field annealing (MFA), is widely used for discrete optimization (label assignment) problems defined on the pixel sites of an image. One formulation of MFA is via maximum entropy, where one seeks the joint distribution over the (random) assignments subject to an average level of cost. MFA is obtained by assuming the assignments at each pixel are statistically independent, given the observed image. Alternatively, we make the less restrictive assumption of independent row labelings. The independence assumption means that at each step, MFA optimizes over only one pixel, whereas our method jointly optimizes over an entire row, i.e., our method is less greedy. In principle, an MFA extension could be developed that explicitly re-estimates the row labeling distributions, but such an approach is, in practice, infeasible. Even so, we can indirectly implement this re-estimation, by re-estimating quantities that determine the row labeling distributions. These quantities are the a posteriori site probabilities, re-estimable via the well-known forward/backward (F/B) algorithm. Thus, our algorithm, which descends in the ME Lagrangian/free energy, consists of iterative application of F/B to the image rows (columns). At convergence, maximum a posteriori site labeling is performed. Our method was applied to segmentation of both real and synthetic noise-corrupted images. It achieved lower Markov random field model potentials and better segmentations compared with other methods, and, in high noise, standard MFA.