Michel-Marie Deza

Image and Audio Distances

Image Processing treats signals such as photographs, video, or tomographic output. In particular, Computer Graphics consists of image synthesis from some abstract models, while Computer Vision extracts some abstract information: say, the 3D description of a scene from video footage of it. From about 2000, analog image processing (by optical devices) gave way to digital processing, and, in particular, digital image editing (for example, processing of images taken by popular digital cameras).

Distances and Similarities in Data Analysis

A data set is a finite set comprising m sequences \((x^{j}_{1},\ldots, x^{j}_{n})\), j∈{1,…,m}, of length n. The values \(x^{1}_{i},\ldots, x^{m}_{i}\) represent an attribute S i . It can be numerical, including continuous (real numbers) and binary (presence/absence expressed by 1/0), ordinal (numbers expressing rank only), or nominal (which are not ordered).

Distances in Applied Social Sciences

In this chapter we present selected distances used in real-world applications of Human Sciences. In this and next chapter, expression of distances ranges from numeric (say, in meters) to ordinal (as a degree assigned according to some rule). Depending on context, the distances are either practical ones, used in daily life and work outside of science, or those used as a metaphor for remoteness (the fact of being apart, being unknown, coldness of manner, etc.). • Approximative human-scale distances The arms length is a distance (about 0.7 m, i.e., within personal distance) discouraging familiarity or conflict (analogs: Italian braccio, Turkish pik, and Old Russian sazhen). The reach distance is the difference between maximum reach and arms length distance. Keep someone/something at arms length (or keep your distance from someone/something means to avoid becoming too friendly or emotionally involved with. The striking distance is a short distance (say, through which an object can be reached by striking). The spitting distance is a very close distance. The shouting distance is short, easily reachable distance. The stones throw is a distance about 25 fathoms (46 m). The hailing distance is the distance within which the human voice can be heard. The walking distance is the distance normally (depending on the context) reach-able by walking. For example, some UK high schools define 2 and 3 miles as statutory walking distance for children before and after 11 years. Similar distance idiom is go the (full) distance, i.e., to continue to do something until it is successfully completed. • Distances between people In [Hall69], four interpersonal bodily distances were introduced: the intimate distance for embracing or whispering (15 − 45 cm), the personal-casual distance for 389

Metrics on Normed Structures

In this chapter we consider a special class of metrics defined on some normed structures, as the norm of the difference between two given elements. This structure can be a group (with a group norm), a vector space (with a vector norm or, simply, a norm), a vector lattice (with a Riesz norm), a field (with a valuation), etc.

Distances in Graph Theory

A graph is a pair G = (V, E), where V is a set, called the set of vertices of the graph G, and E is a set of unordered pairs of vertices, called the edges of the graph G. A directed graph (or digraph) is a pair D = (V, E), where V is a set, called the set of vertices of the digraph D, and E is a set of ordered pairs of vertices, called arcs of the digraph D.

Zigzags and Railroads of Spheres 3_v and 4_v

We consider the zigzag and railroad structures of \(3\)-regular plane graphs and, especially, graphs \(a_v\), i.e., \(v-vertex\) \((\{a,6\},3)\)-spheres, where \(a=2\), \(3\), or \(4\). The case \(a=5\) has been treated in previous Chapter.

Zigzags of Fullerenes and c-Disk-Fullerenes

The fullerenes , i.e., the maps \((\{5,6\},3)\)-\(\mathbb {S}^2\), are of particular interest in Carbon Chemistry. Denote by \(F_v(G)\) any \(v\)-vertex fullerene of symmetry \(G\). Denote by \(C_v\) and call IP fullerene any \(F_v\) with isolated (i.e., no two of them are adjacent) \(5\)-gons. A number of \(C_v\)’s with \(60\le v<100\), including \(C_{60}(I_h)\), \(C_{70}(D_{5h})\), \(C_{76}(D_2)\) and some with \(v=78,82,84\) have been characterized as all-carbon molecular cages.

Goldberg–Coxeter Construction and Parametrization

In this chapter, we consider parametrization and, especially, one with \(1\) complex parameter, i.e., the Goldberg–Coxeter construction \(GC_{k,l}(G_0)\) (a generalization of a simplicial subdivision of Dodecahedron considered in [Gold37] and [Cox71]), producing a plane graph from any \(3\)- or \(4\)-regular plane graph \(G_0\) for integer parameters \(k,l\ge 0\). See the main features of \(GC\)-construction in Table 6.1.

Zigzags of Polytopes and Complexes

In this chapter, based mainly on [DeDu04], we focus on generalization of zigzags for higher dimension. Inspired by Coxeter’s notion of Petrie polygon for \(d\)-polytopes (see [Cox73]), we generalize the notion of zigzag circuits on complexes and compute the zigzag structure for several interesting families of \(d\)-polytopes, including semiregular, regular-faced, Wythoff Archimedean ones, Conway’s \(4\)-polytopes, half-cubes, and folded cubes.

ZC-Circuits of 5- and 6-Regular Spheres

In this chapter, based mainly on [DDS13a] and [DeDu12], we consider \((\{3,4\}, 5)\)-spheres (named icosahedrites) and \((\{1, 2,3\}, 6)\)-spheres. Both cases allow to consider zigzags. But in contrast to the \(3\)- and \(4\)-regular cases, the second structure of edges appear: weak zigzags for \(5\)- and central circuits for \(6\)-regular graphs.