Erosion (morphology)

Binary erosion

In binary morphology, an image is viewed as a subset of a Euclidean space ${\displaystyle {\mathbb{R}}^d}$ or the integer grid ${\displaystyle {\mathbb{Z}}^d}$, for some dimension d.

The basic idea in binary morphology is to probe an image with a simple, pre-defined shape, drawing conclusions on how this shape fits or misses the shapes in the image. This simple "probe" is called structuring element, and is itself a binary image (i.e., a subset of the space or grid).

Let E be a Euclidean space or an integer grid, and A a binary image in E. The erosion of the binary image A by the structuring element B is defined by:

${\displaystyle A\ominus B=\{z\in E\mid B_z\subseteq A\}}$,

where B_z is the translation of B by the vector z, i.e., ${\displaystyle B_z=\{b+z\mid b\in B\}}$, ${\displaystyle \mathrm{\forall }z\in E}$.

When the structuring element B has a center (e.g., a disk or a square), and this center is located on the origin of E, then the erosion of A by B can be understood as the locus of points reached by the center of B when B moves inside A. For example, the erosion of a square of side 10, centered at the origin, by a disc of radius 2, also centered at the origin, is a square of side 6 centered at the origin.

The erosion of A by B is also given by the expression: ${\displaystyle A\ominus B=\bigcap _{b\in B}{A}_{-b}}$, where A_−b denotes the translation of A by -b.

This is more generally also known as a Minkowski difference.

Example

Suppose A is a 13 x 13 matrix and B is a 3 x 3 matrix:

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

Assuming that the origin B is at its center, for each pixel in A superimpose the origin of B, if B is completely contained by A the pixel is retained, else deleted.

Therefore, the Erosion of A by B is given by this 13 x 13 matrix.

0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 1 1 1 1 0 0 1 1 1 1 0 0 0 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0

This means that only when B is completely contained inside A that the pixels values are retained, otherwise it gets deleted or eroded.

Properties

• The erosion is translation invariant.
• It is increasing, that is, if ${\displaystyle A\subseteq C}$, then ${\displaystyle A\ominus B\subseteq C\ominus B}$.
• If the origin of E belongs to the structuring element B, then the erosion is anti-extensive, i.e., ${\displaystyle A\ominus B\subseteq A}$.
• The erosion satisfies ${\displaystyle (A\ominus B)\ominus C=A\ominus (B\oplus C)}$, where ${\displaystyle \oplus }$ denotes the morphological dilation.
• The erosion is distributive over set intersection