Reconstruction

$\displaystyle u(\mathbf{x})$	$\displaystyle = \max\{\lambda, \mathbf{x}\in [u\geq\lambda]\}$
	$\displaystyle = \inf\{\mu, \mathbf{x}\in [u < \mu]\}$

From the cc's: $u(\mathbf{x})=$ level of the smallest cc containing $\mathbf{x}$
Monotonicity characterizes level sets: [Guichard, Morel 2000]

If $(X_\lambda)_{\lambda\in \mathbb {R}}\:\searrow$ , and if we define

$u(\mathbf{x}) = \sup\{\lambda, \mathbf{x}\in X_\lambda\}$

then

$\fbox {$[u \geq\lambda] = X_\lambda$\ a.e.}$

If moreover $\bigcap_{\mu <\lambda} X_\mu = X_\lambda$ ,

then $\fbox {$[u \geq\lambda] = X_\lambda$}$ everywhere

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