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Combinatorial Pyramids and discrete geometry for energy minimizing segmentation

M. Braure de Calignon &
Luc Brun &
Jacques Olivier Lachaud.
The scale set theory allows to define a hierarchy of
segmentations according to a scale parameter. This
theory closely related to the Bayesian and the Minimum
description Length(MDL) frameworks describes the
energy of a partition as the sum of two terms : a
goodness to fit and a regularisation term. This last
term May be interpreted as the encoding cost of the
model associated to the partition. It usually includes
the total length of the partition's boundaries and is
simply computed as the number of lignels between the
regions of the partition. We propose to use a better
estimation of the total length of the boundaries by
using discrete length estimators. We state the basic
properties which must be fulfilled by these estimators
and show their influence on the partitition's energy.