Template decomposition and inversion over hexagonally sampled images

Dean Lucas; Laurie Gibson

doi:10.1117/12.46112

1 July 1991 Template decomposition and inversion over hexagonally sampled images

Dean Lucas, Laurie Gibson

Proceedings Volume 1568, Image Algebra and Morphological Image Processing II; (1991) https://doi.org/10.1117/12.46112
Event: San Diego, '91, 1991, San Diego, CA, United States

Abstract

The family of real-valued circulant templates on nXm rectangular images is isomorphic to a quotient ring of the ring of real polynomials in two variables. Template decomposition is equivalent to factoring the corresponding polynomial. Template invertibility corresponds to polynomial invertibility in the quotient ring. Factoring and inverting are more difficult for polynomials in two variables than for those in one. Hexagonally sampled images have properties which simplify these operations. Hexagons organize themselves naturally into a hierarchy of snowflake-shaped regions. These tile the plane and consequently yield a simple definition of circulancy. Unlike the circulancy of rectangles in the plane, which yields a toroidal topology, the hexagonal analogue yields the topology of a circle. As a result, circulant templates are mapped isomorphically into a quotient of the ring of polynomials in one variable. These polynomials are products of linear factors over the complex numbers. A polynomial will be invertible in the quotient ring whenever each of its linear factors is invertible. This results in a simple criterion for template invertibility.

Citation Download Citation

Dean Lucas and Laurie Gibson "Template decomposition and inversion over hexagonally sampled images", Proc. SPIE 1568, Image Algebra and Morphological Image Processing II, (1 July 1991); https://doi.org/10.1117/12.46112

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