Abstract
Given a set E, the partitions of E are usually ordered by merging of classes. In segmentation procedures, this ordering often generates small parasite classes. A new ordering, called ”grain building ordering”, or GBO, is proposed. It requires a connection over E and states that \(A\preccurlyeq B\), with A,B ⊆ E, when each connected component of B contains a connected component of A. TheGBO applies to sets, partitions, and numerical functions. Thickenings ψ with respect to the GBO are introduced as extensive idempotent operators that do not create connected components. The composition product ψγ of a connected opening by a thickening is still a thickening. Moreover, when {γ i ,i ∈ I} is a granulometric family, then the two sequences {ψγ i ,i ∈ I} and {γ i ψ,i ∈ I} generate hierarchies, from which semi-groups can be derived. In addition, the approach allows us to combine any set of partitions or of tessellations into a synthetic one.
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Serra, J. (2011). Grain Building Ordering. In: Soille, P., Pesaresi, M., Ouzounis, G.K. (eds) Mathematical Morphology and Its Applications to Image and Signal Processing. ISMM 2011. Lecture Notes in Computer Science, vol 6671. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-21569-8_4
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DOI: https://doi.org/10.1007/978-3-642-21569-8_4
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