Erosion (morphology)

Erosion (usually represented by ⊖) is one of two fundamental operations (the other being dilation) in morphological image processing from which all other morphological operations are based.

It was originally defined for binary images, later being extended to grayscale images, and subsequently to complete lattices.

The erosion operation usually uses a structuring element for probing and reducing the shapes contained in the input image.

In binary morphology, an image is viewed as a subset of a Euclidean space

, 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: where Bz is the translation of B by the vector z, i.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.

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

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

In grayscale morphology, images are functions mapping a Euclidean space or grid E into

is an element larger than any real number, and

Denoting an image by f(x) and the grayscale structuring element by b(x), where B is the space that b(x) is defined, the grayscale erosion of f by b is given by where "inf" denotes the infimum.

In other words the erosion of a point is the minimum of the points in its neighborhood, with that neighborhood defined by the structuring element.

Complete lattices are partially ordered sets, where every subset has an infimum and a supremum.

be a complete lattice, with infimum and supremum symbolized by

The erosion of the dark-blue square by a disk, resulting in the light-blue square.
Example of erosion on a grayscale image using a 5x5 flat structuring element. The top figure demonstrates the application of the structuring element window to the individual pixels of the original image. The bottom figure shows the resulting eroded image.