Geometric Description of Images as Topographic Maps

Abstract

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A Quasi-linear Algorithm to Compute the Tree of Shapes of nD Images

“…To construct the maxtree of a given image, we rely on a rooted tree defined by a parenthood function, named parent, and encoded as an nD image (so parent(p) is an nD point). When a node of the max-tree contains several points, we choose its first point 5 Due to limited place, this paper does not contain the following topics (they will be included into an extended version of this paper). A comparison of execution times of existing algorithms.…”

Discrete Set-Valued Continuity and Interpolation

“…The tree of shapes obtained with this interpolation is the one computed by all previous existing algorithms [7]. However, the max-or the min-interpolation do not lead to a self-dual filtering.…”

“…But all the shapes being open, the type of a shape can not be known from the shape itself (contrary to the case of the continuous framework [7]). Consider the family of sets:…”

“…21 is essentially the same as the one of Th. 2.6 in [7], adapted for the set-valued case of a plain map, and with a different ending that uses Lemma 20 and Lemma. 19.…”

“…This paper is the first in a series dedicated to the notion of the tree of shapes, which has been introduced [16,7] as a way to filter an image u : X ⊂ R n → R (n ≥ 2) in a self-dual way, meaning intuitively that processing either u or −u would give the same result. In its continuous definition, this tree is made by the connected components of upper level sets {x; u(x) ≥ λ} λ∈R and lower level sets {x; u(x) < λ} λ∈R .…”