Morphological Closing. Formula understanding

Question

I am trying to do use the morphological closing operation on a greyscaled image. As I found out closing is a combination of dilation and erosion. Now I am trying to understand the formulas of dilation and erosion. The formula of dilation is $$ \left( A \bigoplus X \right)\left( x,y \right) = max \left\{ A \left( x+s,y+t \right)+ X\left(s,t \right) \right\} $$ I dont unterstand what s and t should be? Is it a max over s and t? And s and t are coordinates of image pixels? But if so, is it a max over the whole image or just over the neighborhood of (x,y)?

Answer 1

The notation is incomplete. Let $D$ the domain of $X$, then

$$\left(A\bigoplus X\right)(x,y)=\max_{(s,t)\in D}\{ A(x+s,y+t)+X(s,t)\}.$$

This function is defined over all pixels of $A$, but some special convention must be specified for coordinates outside the image.