Matrix Rounding under the Lp-Discrepancy Measure and Its Application to Digital Halftoning

Abstract

We study the problem of rounding a real-valued matrix into an integer-valued matrix to minimize an Lp-discrepancy measure between them. To define the Lp-discrepancy measure, we introduce a ${\cal F}$ of regions (rigid submatrices) of the matrix and consider a hypergraph defined by the family. The difficulty of the problem depends on the choice of the region ${\cal F}$. We first investigate the rounding problem by using integer programming problems with convex piecewise-linear objective functions and give some nontrivial upper bounds for the Lp discrepancy. We propose family for constructing a practical and well-solvable class of ${\cal F}$. Indeed, we show that the problem is solvable in polynomial time if ${\cal F}$ is the union of two laminar families. Finally, we show that the matrix rounding using L1 discrepancy for the union of two laminar families is suitable for developing a high-quality digital-halftoning software.