binaryerosion

Binary erosion is a fundamental operation in mathematical morphology used to shrink foreground regions in binary images. Given a binary image A, typically with foreground value 1 and background value 0, and a structuring element B, the erosion of A by B, denoted A ⊖ B, consists of all locations x where B translated to x fits entirely inside A.

In discrete form, (A ⊖ B)(x) = min{ A(x + b) : b ∈ B }. When A and B are binary, this means that every pixel of B, centered at x, must lie within the foreground region of A for the result at x to be 1; otherwise it is 0. The structuring element defines the neighborhood over which the image is examined, and shapes such as squares, disks, or lines are common. The operation tends to shrink objects, disconnect thin connections, and remove small objects that cannot accommodate the structuring element.

Properties and relationships: erosion is anti-extensive, meaning A ⊖ B ⊆ A. It is monotone with respect to A: if A ⊆ C, then A ⊖ B ⊆ C ⊖ B. Repeated erosion with the same B progressively erodes features. Erosion is related to dilation through De Morgan-type identities; for symmetric B, A ⊖ B can be expressed in terms of the complement of dilation of the complement.

Openings and closings are derived operations: opening is (A ⊖ B) ⊕ B, and closing is (A ⊕ B) ⊖ B, where ⊕ denotes dilation. Applications include noise removal, separation of connected components, and shape simplification; choices of B strongly influence the outcome and feature preservation.