2005 Fiscal Year Final Research Report Summary
Research on data rounding based on discrete systems.
| Project/Area Number |
16500001
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Fundamental theory of informatics
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| Research Institution | Tohoku University |
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Principal Investigator |
TOKUYAMA Takeshi Tohoku University, Graduate School of Information Sciences, Professor, 大学院・情報科学研究科, 教授 (40312631)
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| Co-Investigator(Kenkyū-buntansha) |
KAWARABAYASHI Kenichi Tohoku University, Graduate School of Information Sciences, Research Associate, 大学院・情報科学研究科, 助手 (40361159)
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| Project Period (FY) |
2004 – 2005
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| Keywords | Algorithms / combinatorics / graph theory / Discrepancy / digitization / computational geometry |
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| Research Abstract |
The research pursue the generation of high-quality digitization (rounding) of analogue data with respect to quality measures based on discrete mathematical concept of graphs and hypergraphs.
Such research has important applications to digital halftoning that is a major topic in image processing, as well as other many digitization applications.
In particular, we consider discrepancy distances associated with hypergraphs and investigate on the space of global roundings that have the discrepancy distance at most one from the input.
The most important outputs are given in the category of mathematical investigation of the combinatorial structure of the space of global roundings. We have given a conjecture that for the hypergraph generated from the shortest path metric of a connected graph, the space of global roundings is indeed a simplex. We have affirmatively proven this conjecture for several classes of graphs such as paths, cycles, mesh, series-parallel graphs, uniform k-trees, and so on.
As byproducts, we have designed efficient polynomial-time algorithm for enumerating global roundings for such hypergraphs. Also we have proven a similar property for several range spaces considered in computational geometry.
Moreover, we have investigated several properties/algorithms on graphs defined by minors, since it seems that our conjecture will be proven on such graphs by using graph-minor theory. We have published 10 international journal papers and several cnference papers during the period.
We have also investigated lower/upper bounds of discrepancies of hypergraphs corresponding to submatrices of a matrix that are important in application to digital halftoning. We have done some experiments on digital halftoning instances by using our methodology.
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