Morphological Erosion

Definition

Morphological erosion is one of the four basic operators from the field of Mathematical Morphology (MM). This is a dual operator to morphological dilation. Depending on the domain of application, the definition varies. In the standard case of discrete binary images, given a binary image X, and a structuring element B (cross-ref), the erosion of X is defined as

$$ {\upepsilon}_B(X)=\bigcap \limits_{b\in \hat{B}}{X}_b $$

(1)

Here $\hat{B}$ denotes the reflection of B – $x\in B\iff -x\in \hat{B}$. X_b denotes the image X translated by b for each b belonging to the $\hat{B}$. It is also a common practice to write ϵ_B(X) = X ⊖ B. Observe that the definition in (1) extends to continuous binary images as well.

In the case of gray-scale images, let f denote the gray-scale image and g denote the structuring function. Then, morphological erosion is defined by

$$ {\upepsilon}_g(f)(x)=\underset{y\in E}{\operatorname{inf}}\left\{\, ,f(y)-g\left(x-y\right)\right\} $$

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