Properties of grain filter

Monotonicity:$u\leq v \Rightarrow T_t u \leq T_t v$
Contrast invariance:$g$ u.s.c. $\nearrow$, $T_t\circ g = g\circT_t$
Affine invariance:$\forall A\in \operatorname{GA}(\mathbb {R}^n)$, $T_t(u\,\circ\,A)\:=\: (T_{t/\vert\det A\vert}u)\,\circ\,A$
$\rightarrow$Affine Morphological Scale-Space [Alvarez, Guichard, Lions, Morel 93], [Sapiro, Tannenbaum 93], [Moisan 98]
No regularity:$u\:C^2, \nabla u(\mathbf{x})\neq 0\Rightarrow$$\exists t>0$, $\forall h\leq t,\;T_h u(\mathbf{x}) = u(\mathbf{x})$ (``$\frac{\partialu}{\partial t} = 0$'')
Causality and idempotency:$\forall s,t,\,s\leq t$, $\existsT_{t,s}$ s.t. $T_t\:=\:T_{t,s}\,\circ\,T_s$, $T_{t,s} = T_t$
Selfduality:$\inf\sup = \sup\inf$, or equivalently $T_t(-u) =-T_t(u)$ if $u$ is continuous
No destruction of visual clues: preserves T-junctions, remaining curves do not move.
 

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