### Abstract

Random fields in two dimensions may be specified on 2 times 2 elements such that the probabilities of finite configurations and the entropy may be calculated explicitly. The Pickard random field is one example where probability of a new (non-boundary) element is conditioned on three previous elements. To extend the concept we consider extending such a field such that a vector or block of elements is conditioned on a larger set of previous elements. Given a stationary model defined on 2 times 2 elements, iterative scaling is used to define the extended model. The extended model may be used for models of two-dimensional constraints and as examples we apply it to the hard-square constraint and the no isolated bits (n.i.b) constraint. The iterative scaling can ensure that the entropy of the extension is optimized and that the entropy is increased compared to the initial model defined on 2 times 2 elements. Application to a simple stationary model with hidden states is also outlined. For the n.i.b constraint, the initial model is based on elements defined by blocks of (1 times 2) binary symbols.

Original language | English |
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Title of host publication | Proceedings, ISIT |

Publisher | IEEE |

Publication date | 2009 |

ISBN (Print) | 978-1-4244-4312-3 |

DOIs | |

Publication status | Published - 2009 |

Event | 2009 IEEE International Symposium on Information Theory - Seoul, Korea, Republic of Duration: 28 Jun 2009 → 3 Jul 2009 http://www.isit2009.info/ |

### Conference

Conference | 2009 IEEE International Symposium on Information Theory |
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Country | Korea, Republic of |

City | Seoul |

Period | 28/06/2009 → 03/07/2009 |

Internet address |

### Bibliographical note

Copyright: 2009 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE## Cite this

Forchhammer, S. (2009). Extending models for two-dimensional constraints. In

*Proceedings, ISIT*IEEE. https://doi.org/10.1109/ISIT.2009.5206052