  This paper introduces a new approach for the automated reconstruction -
reassembly of fragmented objects having one surface near to plane, on the basis
of the 3D representation of their constituent fragments. The whole process
starts by 3D scanning of the available fragments. The obtained representations
are properly processed so that they can be tested for possible matches. Next,
four novel criteria are introduced, that lead to the determination of pairs of
matching fragments. These criteria have been chosen so as the whole process
imitates the instinctive reassembling method dedicated scholars apply. The
first criterion exploits the volume of the gap between two properly placed
fragments. The second one considers the fragments' overlapping in each possible
matching position. Criteria 3,4 employ principles from calculus of variations
to obtain bounds for the area and the mean curvature of the contact surfaces
and the length of contact curves, which must hold if the two fragments match.
The method has been applied, with great success, both in the reconstruction of
objects artificially broken by the authors and, most importantly, in the
virtual reassembling of parts of wall paintings belonging to the Mycenaic
civilization (c. 1300 B.C.), excavated in a highly fragmented condition in
Tyrins, Greece.
