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.

### Minkowski Functionals

Area

Perimeter

Euler characteristic

with

= curvature

## 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 .

### Minkowski Vectors

### Minkowski Tensors

## Irreducible representation (Circular Minkowski Tensors)

under construction

Consider a polygon with edges L:

Density of normals:

Fourier analysis:

(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)

: …

**Morphometric distance**

A morphometric distance of a polygon to a reference structure can be quantified by considering the pseudo distance function

.