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. 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. In binary morphology, an image is viewed as a subset of a Euclidean space R d {displaystyle mathbb {R} ^{d}} or the integer grid Z d {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: where Bz is the translation of B by the vector z, i.e., B z = { b + z | b ∈ B } {displaystyle B_{z}={b+z|bin B}} , ∀ z ∈ E {displaystyle forall zin 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: A ⊖ B = ⋂ b ∈ B A − b {displaystyle Aominus B=igcap _{bin B}A_{-b}} , where A−b denotes the translation of A by -b.