Matrix rounding under the Lp-Discrepancy measure and its application to digital halftoning

Tetsuo Asano, Naoki Katoh, Koji Obokata, Takeshi Tokuyama

Research output: Contribution to journalArticle

16 Citations (Scopus)

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 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 and give some nontrivial upper bounds for the Lp discrepancy. 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. 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.

Original languageEnglish
Pages (from-to)1423-1435
Number of pages13
JournalSIAM Journal on Computing
Volume32
Issue number6
DOIs
Publication statusPublished - 2003 Sep 1

Keywords

  • Approximation algorithm
  • Digital halftoning
  • Discrepancy
  • Linear programming
  • Matrix rounding
  • Network flow
  • Totally unimodular

ASJC Scopus subject areas

  • Computer Science(all)
  • Mathematics(all)

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