motion estimation and compensation for very low bitrate video coding

190 Chapter D. Complements to Chapter 5
D.1.2 Delaunay Triangulation
A Delaunay triangulation D of V is the geometric dual of the Dirichlet
(or Vorono or Thiessen) tessellation P constructed on V . Such a
tessellation divides the plane into polygonal regions, called tiles. Each
tile Pi contains all the points of the plan closer to the tile generating
point vi than to other generating points vk (in the sense of the Euclidian
distance):
Pi = fN 2< 2 ; 8k 6= i; d(N; vi) d(N; vk)g: (D.1)
Graph
Delaunay triangulation
Figure D.1: Graph and associated (constrained) Delaunay triangulation
Delaunay triangulation is optimal from the interpolation point of view,
for its triangles are as much equiangular as possible (\locally equiangular").
It avoids having \ at" triangles which are not good for spline
interpolation where the approximation error depends on the triangle
\thickness." The Delaunay triangulation is de ned as:
De nition 3 The generalized Delaunay triangulation of G =(V;S)
is a triangulation T (G) =(V;S 0 ) for which the circumscribed circle of
every triangle 4vivjvk does not contain any other vertex visible from vi,
vj or vk. The edges of the set S 0 , S are called the Delaunay edges. A
vertex u is visible from a vertex v if the segment [u; v] does not cut any
edge of the set S on an interior point.
Based on this property of the circumscribed circle and on the fact that
a locally optimal triangulation in the sense of Delaunay is globally a