  The present paper introduces the $\eta$ and {\eta} connections in order to
add regional information on $\lambda$-flat zones, which only take into account
a local information. A top-down approach is considered. First $\lambda$-flat
zones are built in a way leading to a sub-segmentation. Then a finer
segmentation is obtained by computing $\eta$-bounded regions and $\mu$-geodesic
balls inside the $\lambda$-flat zones. The proposed algorithms for the
construction of new partitions are based on queues with an ordered selection of
seeds using the cumulative distance. $\eta$-bounded regions offers a control on
the variations of amplitude in the class from a point, called center, and
$\mu$-geodesic balls controls the "size" of the class. These results are
applied to hyperspectral images.
