Mathematical morphology

The mathematical morphology ( MM) is a theoretical model for digital images and is based on lattice theory and topology.

The morphology is a branch of image processing that deals with the processing of binary images (raster graphics). Binary raster images are images whose picture elements ( pixels) can have only one of two different color values.

Basic operations in morphology are dilation, erosion, union, intersection, and difference formation formation.

Based on these operations, other operations such as opening, closing, thinning, contour extraction or for example the skeletonization can be constructed.

Basic Concepts

Interpretation as an association

In mathematical morphology, image signals are interpreted as elements of a (complete ) association. This is a paradigm shift in comparison to the classical ( linear ) signal processing, to be taken in the images as elements of a vector space. In both cases, one is interested in operators that preserve the underlying structure. In the case of the vector space, these are the gain and the superposition principle.

It can be shown that all shift- operators satisfy this equation can be represented as a linear filter. If we choose for the functions the eigenfunctions of the vector space, then it is in order, the Fourier spectrum of the operator.

The basic links of an association are the formation of infimum () and supremum (). Apart from the trivial identity mapping, however, there is no operator, which is invariant with respect to both links. Accordingly, there are two basic operators, namely dilation and erosion, is demanded for the following properties:

• .

A dilatation (or erosion) thus refers to an operator that is invariant with respect to the Supremumsbildung (or Infimumsbildung ). Clearly, this means that ( in the case of dilatation) can decompose the image into separate structures, each dilated for themselves and superimposes the respective result images using the Supremumsbildung again. For the erosion of the dual statement holds.

Topological approach

For the topological approach, the neighborhood is ( the neighborhood filter ) is defined by a structuring element. In this case, open and close are the two basic dual operators. Opening an image with a be patterned element is the largest subset of which is open with respect to the topology defined by. The same applies for the dual Close. The erosion of with is in the topological interpretation represents the maximum amount of pixels whose by -defined environment is fully contained. The dilation of turn with the minimum amount of pixels that contains all points of the by -defined environment.