Mathematical morphology

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

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

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

Based on these operations, further operations such as opening , closing , thinning, outline extraction or, for example, 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 compared to classical (linear) signal processing, in which images are understood 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 principle of amplification and the principle of superposition. ${\displaystyle \varphi }$

${\displaystyle \varphi \left[\sum _{i}a_{i}S_{i}\right]=\sum _{i}a_{i}\varphi \left[S_{i}\right]}$

It can be shown that all shift invariant operators that satisfy this equation can be represented as linear filters. If the eigenfunctions of the vector space are chosen for the functions , then it is the Fourier spectrum of the operator. ${\displaystyle S_{i}}$ ${\displaystyle \varphi \left[S_{i}\right]}$

The basic links of an association are the formation of infimum ( ) and supremum ( ). Apart from the trivial identity mapping, however, there is no operator that is invariant with regard to both links. Accordingly, there are two basic operators, namely dilation and erosion , for which the following properties are required : ${\displaystyle \wedge _{i}S_{i}}$ ${\displaystyle \vee _{i}S_{i}}$ ${\displaystyle \delta }$ ${\displaystyle \varepsilon }$

• ${\displaystyle \delta \left[\bigvee _{i}S_{i}\right]=\bigvee _{i}\delta \left[S_{i}\right]}$
• ${\displaystyle \varepsilon \left[\bigwedge _{i}S_{i}\right]=\bigwedge _{i}\varepsilon \left[S_{i}\right]}$.

A dilation (or erosion) is an operator that is invariant with regard to the formation of the supremum (or formation of the infimum). This clearly means that (in the case of dilation) the image can be broken down into individual structures, each one dilated and the respective resultant images superimposed again using the supremum formation. The dual statement applies to erosion.

Topological approach

For the topological approach, the neighborhood (the environment filter ) is defined by a structuring element. In this case, open and close are the two basic dual operators. Opening an image with a structuring element is the largest subset of that is open with respect to the topology defined by . The same applies to closing. In the topological interpretation, the erosion of with represents the maximum amount of image points whose defined environment is completely contained in. The dilation of with is in turn the minimum amount of image points that contains for all points of the environment defined by . ${\displaystyle A}$ ${\displaystyle X}$ ${\displaystyle A}$ ${\displaystyle X}$ ${\displaystyle A}$ ${\displaystyle X}$ ${\displaystyle X}$ ${\displaystyle A}$ ${\displaystyle A}$ ${\displaystyle X}$ ${\displaystyle A}$ ${\displaystyle X}$

Morphological image processing

Morphological image processing is a branch of computer-aided image processing and can be understood as a technique for analyzing structures in images.

Morphology is the study of shape or form. This non-linear image processing method is used to analyze and influence the structure of images. It is a concept that on the set theory , the topology and the lattice theory is based. Both binary and gray-scale images are permitted, since binary images can already reproduce the shape and shape of an object. On the one hand, one goal of morphological image processing can be a new image that highlights what is relevant. Another goal can be a list that is filled with measured variables determined from the image.

It is important not to confuse morphological image processing with morphing . In the literature it can also be found under the term mathematical morphology.

In morphology, an image is understood as a subset of Euclidean space or a discrete lattice of dimension . ${\displaystyle A}$ ${\displaystyle \mathbb {R} ^{d}}$ ${\displaystyle \mathbb {Z} ^{d}}$ ${\displaystyle d}$

Structural element

A structure element is a structure set of the two-dimensional, discrete basic set. It consists of the original pixel and other randomly arranged pixels. The original pixel is usually also the reference point to which the filtering refers. The reference point is identified by the symbol . ${\displaystyle B\subset \mathbb {Z} ^{2}}$ ${\displaystyle \otimes }$

Examples of frequently used structural elements for images from : ${\displaystyle B}$ ${\displaystyle A}$ ${\displaystyle \mathbb {Z} ^{2}}$

• Four-neighborhood : ; ${\displaystyle B={\begin{bmatrix}0&1&0\\1&\otimes &1\\0&1&0\end{bmatrix}}}$
• Eight neighborhood : ; ${\displaystyle B={\begin{bmatrix}1&1&1\\1&\otimes &1\\1&1&1\end{bmatrix}}}$
• An approximation of the circle with radius 2 . ${\displaystyle B={\begin{bmatrix}0&0&1&0&0\\0&1&1&1&0\\1&1&\otimes &1&1\\0&1&1&1&0\\0&0&1&0&0\end{bmatrix}}}$

The reflection of the structural element is with in: . The choice of structural element depends on the problem and is therefore normally facilitated by existing prior knowledge. ${\displaystyle B}$ ${\displaystyle {\check {B}}}$ ${\displaystyle {\check {B}}=\{-b|b\in B\};B\subseteq \mathbb {E} }$

Standard Morphological Operators

Left: binary image of a chestnut; Middle left: erosion; Middle right: dilation. Right: opening. The effects of the morphological operations on the binary image are marked in blue.

The standard morphological operators are erosion and dilation . The combination of these results in the opening and closing . The standard operators are closely related to the Minkowski sum and form the basis of morphological image processing.

The erosion of an image with the structural element removes the edge of the objects. A result of this can be that initially connected object structures are separated. ${\displaystyle A\ominus B}$ ${\displaystyle A}$ ${\displaystyle B}$

Similarly, the dilation expands the object structures in the image. This can also lead to previously separate objects merging. ${\displaystyle A\oplus B}$

The relationship between erosion and dilatation is called duality. For binary images, and (Central) symmetrical structure elements applies: . It is the complement to , so . ${\displaystyle A^{c}\ominus B=(A\oplus B)^{c}}$ ${\displaystyle A^{c}}$ ${\displaystyle A}$ ${\displaystyle A^{c}=\{x\in \mathbb {E} |x\notin A\}}$

The opening of the image with the structural element consists of two steps: erosion of with , then dilation of the result with . Interpreted geometrically, the opening can be used to smooth outer corners, to remove thin webs or "spikes" and to remove small external objects. For example, the spines of a chestnut can be removed while the shape of the fruit is largely retained. ${\displaystyle A\circ B}$ ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle B}$

Similar to opening, closing consists of the same steps in reverse order. First, the picture is with dilated to get the result again with erode. Because of the duality the closure can also be formulated alternatively: . The closure has a geometrical effect through the smoothing of inner corners, the bridging of small distances and especially the eponymous closure of inner holes. ${\displaystyle A\bullet B}$ ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle B}$ ${\displaystyle A\bullet B=(A^{c}\circ B^{s})^{c}}$

Binary image of gears before and after morphological closure. You can see that the holes are closed, but the shape is retained.

Properties of the standard operators

• Erosion is monotonously increasing: ${\displaystyle A_{1}\subseteq A_{2}\Rightarrow (A_{1}\ominus B)\subseteq (A_{2}\ominus B)}$
• Dilatation is monotonically increasing: ${\displaystyle A_{1}\subseteq A_{2}\Rightarrow (A_{1}\oplus B)\subseteq (A_{2}\oplus B)}$
• Dilation is extensive; H. if B contains the origin ${\displaystyle A\subseteq A_{1}\oplus B}$
• Erosion is anti-extensive, i. H. if B contains the origin ${\displaystyle A\ominus B\subseteq A}$
• If A is convex, so is ${\displaystyle A\ominus B}$
• Translation invariance: ${\displaystyle A\ominus B_{z}=(A\ominus B)_{z}}$

Other operators and applications

Filtering

• Morphological gradients (edge ​​filter)
• Contrast enhancement
• Granulometry
• Skeletonization

segmentation

• Watershed transformation
• Marker-controlled segmentation

classification

• Cluster analysis

application areas

The areas of application for morphological image processing are varied. Examples are industrial quality control, document processing, image coding and medical image processing. The technology is also used in the geosciences, materials science and security control.

literature

• P. Soille, Morphological Image Processing, Springer-Verlag, Berlin Heidelberg, 1998, doi : 10.1007 / 978-3-642-72190-8
• Bernd Jähne , digital image processing, Springer-Verlag, Berlin Heidelberg, 2012, doi : 10.1007 / 978-3-642-04952-1
• J. Beyerer, F. Puente León, C. Frese, Automatic visual inspection, Springer-Verlag, Berlin Heidelberg, 2012, ISBN 978-3-642-23966-3

Web links

Commons : Mathematical morphology  - collection of images, videos and audio files

• Mathematical Morphology (English)
• ImageJ plugin for morphological image processing (English)