Positivity 12 (2008), 55–73
2007 Birkh¨auser Verlag Basel/Switzerland
1385-1292/010055-19, published online October 29, 2007
Constructing the Tree of Shapes of an Image
by Fusion of the Trees of Connected
Components of Upper and Lower Level Sets
Vicent Caselles, Enric Meinhardt and Pascal Monasse
Devoted to the memory of Professor H.H. Schaefer
Abstract. The tree of shapes of an image is an ordered structure which
permits an eﬃcient manipulation of the level sets of an image, modeled as
a real continuous function deﬁned on a rectangle of IR
, N ≥ 2. In this paper
we construct the tree of shapes of an image by fusing both trees of connected
components of upper and lower level sets. We analyze the branch structure
of both trees and we construct the tree of shapes by joining their branches in
a suitable way. This was the algorithmic approach for 2D images introduced
by F. Guichard and P. Monasse in their initial paper, though other eﬃcient
approaches were later developed in this case. In this paper, we prove the well-
foundedness of this approach for the general case of multidimensional images.
This approach can be eﬀectively implemented in the case of 3D images and
can be applied for segmentation, but this is not the object of this paper.
Mathematics Subject Classiﬁcation (2000). 54F05; 54C30; 06B23; 68U10.
Keywords. Tree of shapes, Order complete, Level sets.
Ordered tree structures play an important role in several contexts in image
processing. They permit a hierarchical organization of information and the devel-
opment of fast algorithms. Let us review some of them.
In most image processing based applications, an image is usually viewed as a
set of pixels placed on a rectangular grid. The pixel provides a very local informa-
tion: taking it as elementary unit places the scale of representation far from the
interpretation or decision scale. In recent years, an increasing number of applica-
tions rely on more structured image representations. For instance, region-based or
level-lines image representations oﬀer two advantages with respect to pixel based
ones: the number of regions or level-lines is usually much lower than the number