Morphological Operations on Delaunay Triangulations Nicolas Loménie (1)
Laurent Gallo (2)
Nicole Cambou (2)
Georges Stamon (1)
[email protected]
(1) Laboratoire SIP – CRIP5 – Université René Descartes –Paris 5 - 45 rue des Saints-Pères 75006 Paris – France. (1) Service Vision – Aérospatiale Matra – Branche Missiles – Site de Châtillon-Montrouge, France.
Abstract Algorithms are presented which allow to perform morphological transformations on unorganised sets of points represented by their Delaunay triangulation. The results show that these algorithms could behaves as morphological operators such as erosion, dilatation, opening do. As a matter of fact, they actually act as “shape filter”. These algorithms are applied to the problem of scene reconstruction by stereoscopy in which objets are represented by unstructured clouds of 3D points.
I.
After explaining what is an alpha-shape and how it is obtained in section 2, Section 3 defines our notions of alpha-erosion, alpha-dilatation, alpha-opening and section 4 illustrates how they can actually perform as classical morphological operators, by filtering operated shapes. Last, section 5 concludes with the expectation of an unified theory.
II.
α-Objects
We begin by explaining what an α-shape and its derived are and how they are obtained [6].
Introduction
Scientific computing often deals with data which is, in its abstract form, a finite point set in two or three dimensional space, and it is sometimes useful to compute what one might call the “shape of such a set”. It is the case in stereoscopic problems of reconstruction where scenes are available as a set of 3D points. The problem of segmenting such unorganised 3D point sets in terms of obstacles and navigable areas has already been addressed in [1] by using Fuzzy C-Means[2][3] improved in [4]. This resulted in a set of point sets sensed to represent objects of the scene which shape design is the last process to be performed in order to describe the scene. The use of 2D or 3D alpha-shapes first introduced by Edelsbrunner [5] gives a formal definition of the shape of point sets. They design shapes by sculpting Delaunay triangulation. But, it appears that the lack of accuracy due to real time preoccupations in stereoscopic vision does not allow to reconstruct the 3D shape of point sets. Thus, it is sufficient to reconstruct an approximation of the 2D shape of objets by projection of the clouds of points on approximating planes. It is the reason why we limit ourselves to 2D shape operations on unorganised point sets.
a)
b)
c)
d)
Figure 1. a) 2D point set. b) Delaunay triangulation Del or ∞complex c) 0.8-complex. d) 0.8-shape (region inside) and 0.8hull (edge).
First introduced by Edelsbrunner, the notion of α-shape gives a formal definition of what is the shape of a cloud of points. More precisely, it defines a discrete family of shapes whose detail level is regulated by the parameter α, controlling the maximum curvature allowed in shape description. We focus on α-complex of a point set S which can be viewed as a triangulation of the interior of the corresponding α-shape and which can be defined as a subgraph of Delaunay triangulation Del of S. Intuitively, once Delaunay triangulation is obtained [7], α-complex
acts as a spherical eraser deleting triangles of Del able to receive an open ball Bα of radius α not containing any point of S. Closely related to α-shape and α-complex is the notion of α-hull which is a generalisation of the convex hull of a point set. Figure 1 summarise all these structures for a synthesised 2D point set.
Then the following table 1 explains how to construct αobjets :
Let’s define an empty α-ball as a ball of radius α not containing any point of S. Then, α-shape is defined as the complement of the union of all empty α-balls. But, mathematical morphology is well known for its set relations : “B⊂S”, “B∩S≠∅” where S is the set to analyse and B is the structuring element whose shape depends on analysis need. As a matter of fact, these relations are the basis of elementary morphological operators such as erosion and dilatation. But definitions are quite different even though α-complex seems to erode ∞-complex corresponding to convex hull (fig. 1).
(µT ,∞[
Let’s define k-simplices σT = conv(T) that is convex hull of T, T⊆ S and |T|=k+1 for 0≤k≤2. Let’s define ρT as the radius of the circumscribe sphere of σT.For each simplex σT ∈ Del there is a single interval so that σT is a face of the α-shape Sα if and only if α is contained in this interval. Let up(σT) be the set of all faces incident to σT whose dimension is one higher than that of σT, that is up(σT) = {σT’ ∈ Del | T ⊂ T’ and |T’|=|T|+1 }. Then , for each σT, two values λT and µT are derived : if |T|=3, λT = µT = ρT; else λT = min ( {λT’ | σT’∈ up(σT)}) and µT = max ( {µT’ | σT’∈ up(σT)})
Last a simplex is said to be
interior if σ T ∉∂S α regular if σ T ∈ ∂S α and it bounds some higher − dimensional simplex in C α sin gular if σ T ∉∂Sα and it does not bound any higher − dimensional simplex in C α
σT is … Triangle Edges,∉∂conv(S) ∈∂conv(S) Vertex,∉∂conv(S) ∈∂conv(S)
Singular
(ρT ,λT) (ρT ,λT) [0,λT) [0,λT)
Regular (λT ,µT) (λT ,∞[ (λT ,µT) (λT ,∞[
Interior (ρT,∞[ (µT ,∞[
Table 1. Obtaining α-objects
Note that each edge belonging to ∂conv(S) is the edge of a triangle of infinite radius with a point at inifinite. Theorically, the α-complex Cα consists of all interior, regular, singular simplices for a given α value but we assimilate it to the interior of the α-shape Sα which is triangulated only by the interior triangles. The boundary of the α-shape is formed by the set of regular edges and their vertices.
III.
Erosion, Dilatation, Opening.
Once an optimal α-complex of S is obtained, for instance the one of minimum volume and containing all points of S, our purpose is to define morphological operators in order to filter its shape. From now, α-complex is assimilated to a triangulation of the interior of the αshape, that is to a sub-triangulation Tα obtained from Del. αk-erosion is defined as a sub-graph of Del obtained by propagating µT values to neighbour triangles. Thus, to each triangle conv(T) of Cα is associated values eTk defined by : eTk = max { eT’k-1 | T’∈ neighbour(T) } and eT0 = µT = ρT where neighbour(T) is the set of all triangles T’ of Del sharing at least one vertex with the triangle T that is : neighbour(T) = { T’∈ Del | T’∩T≠∅ and |T’|=|T|=3 }. The α-eroded of order k is defined as the reunion of all the triangles of Cα whose eTk is inferior to α that is
αk-eroded = { T’∈ Del | eT’ k
