Researchers in this area should consider this well-written paper. The paper provides two models, which are seen to be equivalent, for digital halftoning, one based on maximum-entropy Gibbs measures and the other based on reversible Markov chains. Two algorithms that are associated with these models are also presented. One of the algorithms is based on neural networks, while the other is based on simulated annealing.

The paper starts by describing the digital halftoning problem. The failure resulting from the use of the round algorithm, the most obvious choice for the digital halftoning problem, is discussed. The second section presents three of the most commonly used halftoning algorithms, the ordered dither algorithm, the Floyd-Steinberg error diffusion algorithm, and the dot diffusion algorithm. The maximum-entropy Gibbs measure model is developed in the third section. The following section describes a neural network–based algorithm for selecting a halftone representation with the minimum energy, which was the result induced from the maximum-entropy Gibbs measure model. Section 5 presents the reversible Markov chain model and an associated simulated annealing algorithm for it. The pseudocode for the neural network and the simulated annealing algorithms is included in appendices to the paper. Three different images were used to test the neural network and the simulated annealing algorithms and compare them with the commonly used halftoning algorithms. The quality of the halftone images produced using the different algorithms was measured both subjectively and objectively. The subjective measure was a combination of the sharpness of image detail and the smoothness of the grayscale simulation, while the objective measure was the developed Gibbs measure or, equivalently, the image energy.

The final section contains the conclusions and the authors’ current research directions. The issues discussed in this section can open new directions for interested researchers.