### Abstract

Digital watermarking applications have a voracious demand for large sets of distinct 2D arrays of variable size that possess both strong auto-correlation and weak cross-correlation. We use the discrete Finite Radon Transform to construct “perfect” p × p arrays, for p any prime. Here the array elements are comprised of the integers {0,\pm 1,+2\}. Each array exhibits perfect periodic auto-correlation, having peak correlation value p^{2}, with all off-peak values being exactly zero. Each array, by design, contains just 3(p-1)/2 zero elements, the minimum number possible when using this “grey” alphabet. The grey alphabet and the low number of zero elements maximises the efficiency with which these perfect arrays can be embedded into discrete data. The most useful aspect of this work is that large families of such arrays can be constructed. Here the family size, M, is given by M = p^{2}-1. Each of the M(M-1)/2 intra-family periodic cross-correlations is guaranteed to have one of the three lowest possible merit factors for arrays with this alphabet. The merit factors here are given by v^{2}/(p^{2}-v^{2}), for v = 2, 3 and 4. Whilst the strength of the auto-correlation rises with array size p as p^{2}, the strength of the many (order p^{4}) cross-correlations between all M family members falls as 1/p^{2}.

Original language | English |
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Title of host publication | Discrete Geometry for Computer Imagery |

Subtitle of host publication | 20th IAPR International Conference, DGCI 2017, Proceedings |

Editors | Walter G Kropatsch, Nicole M Artner, Ines Janusch |

Place of Publication | Cham Switzerland |

Publisher | Springer |

Pages | 46-56 |

Number of pages | 11 |

Volume | 10502 LNCS |

ISBN (Electronic) | 9783319662725 |

ISBN (Print) | 9783319662718 |

DOIs | |

Publication status | Published - 2017 |

Event | International Conference on Discrete Geometry for Computer Imagery 2017 - Campus Gußhaus of TU Wien, Vienna, Austria Duration: 19 Sep 2017 → 21 Sep 2017 Conference number: 20th http://dgci2017.prip.tuwien.ac.at |

### Publication series

Name | Lecture Notes in Computer Science |
---|---|

Publisher | Springer International Publishing AG |

Volume | 10502 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | International Conference on Discrete Geometry for Computer Imagery 2017 |
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Abbreviated title | DGCI 2017 |

Country | Austria |

City | Vienna |

Period | 19/09/17 → 21/09/17 |

Other | The aim of the DGCI conference is to gather researchers in discrete geometry and topology, and discrete models, with applications in image analysis and image synthesis. Discrete geometry plays an expanding role in the fields of shape modelling, image synthesis, and image analysis. It deals with topological and geometrical definitions of digitized objects or digitized images and provides both a theoretical and computational framework for computer imaging. |

Internet address |

### Keywords

- Discrete projection
- Finite Radon Transform
- Low cross-correlation arrays
- Perfect arrays
- Watermarking

### Cite this

*Discrete Geometry for Computer Imagery: 20th IAPR International Conference, DGCI 2017, Proceedings*(Vol. 10502 LNCS, pp. 46-56). (Lecture Notes in Computer Science; Vol. 10502 LNCS). Cham Switzerland: Springer. https://doi.org/10.1007/978-3-319-66272-5_5

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*Discrete Geometry for Computer Imagery: 20th IAPR International Conference, DGCI 2017, Proceedings.*vol. 10502 LNCS, Lecture Notes in Computer Science, vol. 10502 LNCS, Springer, Cham Switzerland, pp. 46-56, International Conference on Discrete Geometry for Computer Imagery 2017, Vienna, Austria, 19/09/17. https://doi.org/10.1007/978-3-319-66272-5_5

**Large families of “grey” arrays with perfect auto-correlation and optimal cross-correlation.** / Svalbe, Imants; Ceko, Matthew; Tirkel, Andrew.

Research output: Chapter in Book/Report/Conference proceeding › Conference Paper › Research › peer-review

TY - GEN

T1 - Large families of “grey” arrays with perfect auto-correlation and optimal cross-correlation

AU - Svalbe, Imants

AU - Ceko, Matthew

AU - Tirkel, Andrew

PY - 2017

Y1 - 2017

N2 - Digital watermarking applications have a voracious demand for large sets of distinct 2D arrays of variable size that possess both strong auto-correlation and weak cross-correlation. We use the discrete Finite Radon Transform to construct “perfect” p × p arrays, for p any prime. Here the array elements are comprised of the integers {0,\pm 1,+2\}. Each array exhibits perfect periodic auto-correlation, having peak correlation value p2, with all off-peak values being exactly zero. Each array, by design, contains just 3(p-1)/2 zero elements, the minimum number possible when using this “grey” alphabet. The grey alphabet and the low number of zero elements maximises the efficiency with which these perfect arrays can be embedded into discrete data. The most useful aspect of this work is that large families of such arrays can be constructed. Here the family size, M, is given by M = p2-1. Each of the M(M-1)/2 intra-family periodic cross-correlations is guaranteed to have one of the three lowest possible merit factors for arrays with this alphabet. The merit factors here are given by v2/(p2-v2), for v = 2, 3 and 4. Whilst the strength of the auto-correlation rises with array size p as p2, the strength of the many (order p4) cross-correlations between all M family members falls as 1/p2.

AB - Digital watermarking applications have a voracious demand for large sets of distinct 2D arrays of variable size that possess both strong auto-correlation and weak cross-correlation. We use the discrete Finite Radon Transform to construct “perfect” p × p arrays, for p any prime. Here the array elements are comprised of the integers {0,\pm 1,+2\}. Each array exhibits perfect periodic auto-correlation, having peak correlation value p2, with all off-peak values being exactly zero. Each array, by design, contains just 3(p-1)/2 zero elements, the minimum number possible when using this “grey” alphabet. The grey alphabet and the low number of zero elements maximises the efficiency with which these perfect arrays can be embedded into discrete data. The most useful aspect of this work is that large families of such arrays can be constructed. Here the family size, M, is given by M = p2-1. Each of the M(M-1)/2 intra-family periodic cross-correlations is guaranteed to have one of the three lowest possible merit factors for arrays with this alphabet. The merit factors here are given by v2/(p2-v2), for v = 2, 3 and 4. Whilst the strength of the auto-correlation rises with array size p as p2, the strength of the many (order p4) cross-correlations between all M family members falls as 1/p2.

KW - Discrete projection

KW - Finite Radon Transform

KW - Low cross-correlation arrays

KW - Perfect arrays

KW - Watermarking

UR - http://www.scopus.com/inward/record.url?scp=85029493450&partnerID=8YFLogxK

U2 - 10.1007/978-3-319-66272-5_5

DO - 10.1007/978-3-319-66272-5_5

M3 - Conference Paper

SN - 9783319662718

VL - 10502 LNCS

T3 - Lecture Notes in Computer Science

SP - 46

EP - 56

BT - Discrete Geometry for Computer Imagery

A2 - Kropatsch, Walter G

A2 - Artner, Nicole M

A2 - Janusch, Ines

PB - Springer

CY - Cham Switzerland

ER -