Morphological gradient

Let ${\displaystyle f:E\mapsto R}$ be a grayscale image, mapping points from a Euclidean space or discrete grid E (such as R² or Z²) into the real line. Let ${\displaystyle b(x)}$ be a grayscale structuring element. Usually, b is symmetric and has short-support, e.g.,

${\displaystyle b(x)=\left\{{\begin{array}{ll}0,&|x|\leq 1,\\-\infty ,&{\mbox{otherwise}}\end{array}}\right.}$.

Then, the morphological gradient of f is given by:

${\displaystyle G(f)=f\oplus b-f\ominus b}$,

where ${\displaystyle \oplus }$ and ${\displaystyle \ominus }$ denote the dilation and the erosion, respectively.

An internal gradient is given by:

${\displaystyle G_{i}(f)=f-f\ominus b}$,

and an external gradient is given by:

${\displaystyle G_{e}(f)=f\oplus b-f}$.

The internal and external gradients are "thinner" than the gradient, but the gradient peaks are located on the edges, whereas the internal and external ones are located at each side of the edges. Notice that ${\displaystyle G_{i}+G_{e}=G}$.

If ${\displaystyle b(0)\geq 0}$, then all the three gradients have non-negative values at all pixels.

• Image Analysis and Mathematical Morphology by Jean Serra, ISBN 0-12-637240-3 (1982)
• Image Analysis and Mathematical Morphology, Volume 2: Theoretical Advances by Jean Serra, ISBN 0-12-637241-1 (1988)
• An Introduction to Morphological Image Processing by Edward R. Dougherty, ISBN 0-8194-0845-X (1992)

• Morphological gradients, Centre de Morphologie Mathématique, École_des_Mines_de_Paris