Grain Buiding Ordering

International audienceGiven 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 4 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 with respect to the GBO are introduced as extensive idempotent operators that do not create connected components. The composition product of a connected opening by a thickening is still a thickening. Moreover, when {i, i ∈ I} is a granulometric family, then the two sequences { i, i ∈ I} and { i , 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

Flat zones filtering, connected operators, and filters by reconstruction

This correspondence deals with the notion of connected operators. Starting from the definition for operator acting on sets, it is shown how to extend it to operators acting on function. Typically, a connected operator acting on a function is a transformation that enlarges the partition of the space created by the flat zones of the functions. It is shown that from any connected operator acting on sets, one can construct a connected operator for functions (however, it is not the unique way of generating connected operators for functions). Moreover, the concept of pyramid is introduced in a formal way. It is shown that, if a pyramid is based on connected operators, the flat zones of the functions increase with the level of the pyramid. In other words, the flat zones are nested. Filters by reconstruction are defined and their main properties are presented. Finally, some examples of application of connected operators and use of flat zones are described.Peer ReviewedPostprint (published version