The tree of shapes of an image

¹ Univ. Pompeu-Fabra, Passeig de Circumvalació 8, 08003 Barcelona, Spain; coloma.ballester@tecn.upf.es. vicent.caselles@tecn.upf.es.
² CMLA, ENS Cachan, 61 avenue du Président Wilson, 94235 Cachan Cedex, France; monasse@cmla.ens-cachan.fr.

Received: 1 February 2002

Abstract

In [CITE], Kronrod proves that the connected components of isolevel sets of a continuous function can be endowed with a tree structure. Obviously, the connected components of upper level sets are an inclusion tree, and the same is true for connected components of lower level sets. We prove that in the case of semicontinuous functions, those trees can be merged into a single one, which, following its use in image processing, we call “tree of shapes”. This permits us to solve a classical representation problem in mathematical morphology: to represent an image in such a way that maxima and minima can be computationally dealt with simultaneously. We prove the finiteness of the tree when the image is the result of applying any extrema killer (a classical denoising filter in image processing). The shape tree also yields an easy mathematical definition of adaptive image quantization.