This paper is about a new concept for the description of 3D smooth sur
faces: the extremal mesh. In previous works, we have shown how to extr
act the extremal lines from 3D images, which are the lines where one o
f the two principal surface curvatures is locally extremal. We have al
so shown how to extract the extremal points, which are specific points
where the two principal curvatures are both extremal. The extremal me
sh is the graph of the surface whose vertices are the extremal points
and whose edges are the extremal lines: it is invariant with respect t
o rigid transforms. The good topological properties of this graph are
ensured with a new local geometric invariant of 3D surfaces, that we c
all the Gaussian extremality, and which allows to overcome orientation
problems encountered with previous definitions of the extremal lines
and points. This paper presents also an algorithm to extract the extre
mal mesh from 3D images, and experiments with synthetic and real 3D me
dical images show that this graph can be extremely precise and stable.