Theory of projection onto the narrow quantization constraint set and its application

Since the postprocessing of coded images using a priori information depends on the constraints imposed on the coded images, it is important to utilize constraints that are best suited to postprocessing techniques. Among the constraint sets, the quantization constraint set (QCS) is commonly used in the iterative algorithms that are especially based on the theory of projections onto convex sets (POCS). The converged image in the iteration is usually a boundary point of the QCS. But, we can easily conjecture that the possible location of the original image is inside the QCS. In order to obtain an image inside the QCS, we proposed a new convex constraint set, a subset of the QCS called narrow QCS (NQCS) as a substitute for the QCS. In order to demonstrate that the NQCS works better than the QCS on natural images, we present mathematical analysis with examples and simulations by reformulating the iterative algorithm of the constrained minimization problem or of the POCS using the probability theory. Since the initial image of the iteration is the centroid of the QCS, we reach a conclusion that the first iteration is enough to recover the coded image, which implies no need of any theories that guarantee the convergences.