Generalized Semimagic Squares for Digital Halftoning

TL;DR

This paper characterizes when certain integer arrangements with uniform regional sums are possible on a toroidal grid, extending previous work on digital halftoning, and proposes heuristic solutions for impossible cases.

Contribution

It fully determines the parameters allowing zero-discrepancy matrices for digital halftoning and introduces heuristic methods for cases where such matrices cannot exist.

Findings

Identified all parameter sets (m, n, k, l) permitting zero-discrepancy matrices.

Proved impossibility for specific parameter configurations.

Developed heuristic algorithms for near-zero discrepancy matrices in impossible cases.

Abstract

Completing Aronov et al.'s study on zero-discrepancy matrices for digital halftoning, we determine all (m, n, k, l) for which it is possible to put mn consecutive integers on an m-by-n board (with wrap-around) so that each k-by-l region holds the same sum. For one of the cases where this is impossible, we give a heuristic method to find a matrix with small discrepancy.