Matrix Rounding and Related Problems with Application to Digital Halftoning

Abstract

AbstractIn this paper, we first study the problem of rounding a real-valued matrix into an integer-valued matrix to minimize an L p-discrepancy measure between them. To define the L p-discrepancy measure, we introduce a family 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 family F. We first investigate the rounding problem by using integer programming problems with convex piecewise-linear objective functions. Then, we propose “laminar family” for constructing a practical and well-solvable class of F. Indeed, we show that the problem is solvable in polynomial time if F is the union of two laminar families. We shall present experimental results. We also give some nontrivial upper bounds for the L p-discrepancy. We then foucs on the number of global roundings defined on a hypergraph H G = (V, P G ) which corresponds to a set of shortest paths for a weighted graph G = (V,E). For a given real assignment a on V satisfying 0 ≤ a(v) ≤ 1, a global rounding a with respect to H G is a binary assignment satisfying that |Σν ∈ F a(ν) − α(ν)| < 1 for every F ∈ P G . We conjecture that there are at most |V| + 1 global roundings for H G