For sufficiently smooth bodies , the Minkowski Functionals can be intuitively defined via (weighted) integrals over the volume or boundary of the body .
The scalar functionals can be interpreted as area, perimeter, or the Euler characteristic, which is a topological constant. The vectors are closely related to the centers of mass in either solid or hollow bodies. Accordingly, the second-rank tensors correspond the tensors of inertia, or they can be interpreted as the moment tensors of the distribution of the normals on the boundary.
Cartesian representation (Minkowski Tensors)
Using the position vector and the normal vector on the boundary, the Minkowski Vectors can be defined in the Cartesian representation.
The second-rank Minkowski tensors are defined using the symmetric tensor product .
Irreducible representation (Circular Minkowski Tensors)
Consider a polygon with edges L:
Density of normals:
(the circumference of the polygon)
(for a closed polygon)
The shape indexes are defined as
is the circumference of the object.
for closed polygons.
is the polar component of the normal density, it is related to
can detect anisotropy in a three-fold symmetric system
is the quadrupole component (suitable for detecting rectangles)
A morphometric distance of a polygon to a reference structure can be quantified by considering the pseudo distance function