TY - GEN

T1 - Matrix rounding under the Lp-discrepancy measure and its application to digital halftoning

AU - Asano, Tetsuo

AU - Katoh, Naoki

AU - Obokata, Koji

AU - Tokuyama, Takeshi

PY - 2002/1/1

Y1 - 2002/1/1

N2 - In this paper we study the problem of rounding a real-valued matrix into an integer-valued matrix to rmmmize 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 J-. 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. Then, we propose "laminar family" for constructing a practical and well-solvable class of T. Indeed, we show that the problem is solvable in polynomial time if T is a union of two laminar families. Finally, we show that the matrix rounding using Li-discrepancy for a union of two laminar families is suitable for developing a high-quality digital-halftoning software.

AB - In this paper we study the problem of rounding a real-valued matrix into an integer-valued matrix to rmmmize 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 J-. 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. Then, we propose "laminar family" for constructing a practical and well-solvable class of T. Indeed, we show that the problem is solvable in polynomial time if T is a union of two laminar families. Finally, we show that the matrix rounding using Li-discrepancy for a union of two laminar families is suitable for developing a high-quality digital-halftoning software.

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M3 - Conference contribution

AN - SCOPUS:84902475989

T3 - Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms

SP - 896

EP - 904

BT - Proceedings of the 13th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2002

PB - Association for Computing Machinery

T2 - 13th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2002

Y2 - 6 January 2002 through 8 January 2002

ER -