Application 1: Grain filter

Grains: small shapes. [Vincent 93], [Masnou 98]

For an image defined on a continuous domain:
$T_t u(\mathbf{x}) = \sup_{B\in\mathbf{x}+\mathcal{B}}\inf_{\mathbf{y}\in B}u(\mathbf{y})$
with $\mathcal B = \{B\subseteq X, \vert$sat$(B)\vert\geq t, \mathbf{O}\in B\cup$   holes of $B$ of area $\leq t\}$
$\leftrightarrow$ Pruning of tree

Summer School on Mathematical Problems in Image Processing, Trieste, Italy, September 4th-18th 2000