Graphics Reference
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Figure 7.4: From left to right a cell decomposition [ 78 ], a triangle mesh, and a voxel grid.
in which all the cells are hypercubes. Triangle meshes are the most popular example of simplicial
models. Examples of regular models are 2D and 3D images, where the cells are known as pixels in
2D and voxels in 3D, respectively [ 116 ]. Simplicial meshes are usually based on a piecewise linear
interpolation of the shape geometry. Regular grids define a step-wise or analytical approximation
of the shape geometry, according to the type of interpolation associated with the hypercubes.
Embedding these decompositions in a Euclidean space of lowest dimension [ 203 ] is ex-
tremely important for practical data representation and computation. For instance, the Delaunay
complex D X is a geometric simplicial complex that is homotopy equivalent to a given subspace of
X2R n [ 65 ]. As a particular case, every orientable triangulated surface can be piecewise-linearly
embedded in R 3 [ 130 ].
Figure 7.4 shows three examples of data representations, from left to right a cell decompo-
sition, a triangle mesh and a voxel grid.
7.2
HOMOLOGY
Homology theory offers the theoretical background for translating the study of topological prop-
erties, such as the number of holes, cavities, etc., of a shape, into algebraic structures. Despite the
apparent complexity of the algebraic machinery behind homology, the advantage of using it is
mainly applicative: indeed, homology groups can be effectively computed and, differently from
homotopy, efficient algorithms for their computation exist [ 149 ].
e homology of a space is an algebraic object which reflects the topology of the space,
in some sense counting the number of holes. e homology of a space X is denoted by H .X/ ,
and is defined as a sequence of groups fH q .X/WqD0;1;2;:::g , where H q .X/ is called the q - th
homology group of X . In the literature there are various types of homologies [ 188 ]; the one we are
addressing here is (integer) simplicial homology , which is strictly related to the concept of simplicial
complex.
Let K be a simplicial complex in R n . For each q0 , a q - chain of K is a formal linear
combination
P
i a i i , of oriented q -simplices i , with integer coefficients a i . Two q -chains
P
P
P
aD
i a i i and bD
i b i i are added componentwise, that is to say aCbD
i .a i Cb i / i .
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