Elena 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).

On mean values of some arithmetic functions in number fields

We obtain, for quadratic and cyclotimic fields, asymptotic formulas for two arithmetic functions, which are similar to divisor function.

Cones of Weighted and Partial Metrics

A partial semimetric on V_n={1, ..., n} is a function f=((f_{ij})): V_n^2 -> R_>=0 satisfying f_ij=f_ji >= f_ii and f_ij+f_ik-f_jk-f_ii >= 0 for all i,j,k in V_n. The function f is a weak partial semimetric if f_ij >= f_ii is dropped, and it is a strong partial semimetric if f_ij >= f_ii is complemented by f_ij <= f_ii+f_jj. We describe the cones of weak and strong partial semimetrics via corresponding weighted semimetrics and list their 0,1-valued elements, identifying when they belong to extreme rays. We consider also related cones, including those of partial hypermetrics, weighted hypermetrics, l_1-quasi semimetrics and weighted/partial cuts.

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.

Length Measures and Scales

The term length has many meanings: distance, extent, linear measure, span, reach, end, limit, etc.; for example, the length of a train, a meeting, a book, a trip, a shirt, a vowel, a proof. The length of an object is its linear extent, while the height is the vertical extent, and width (or breadth) is the side-to-side distance at 90∘ to the length, wideness. The depth is the distance downward, distance inward, deepness, profundity, drop.

Riemannian and Hermitian Metrics

Riemannian Geometry is a multidimensional generalization of the intrinsic geometry of two-dimensional surfaces in the Euclidean space \(\mathbb{E}^{3}\). It studies real smooth manifolds equipped with Riemannian metrics, i.e., collections of positive-definite symmetric bilinear forms ((g ij )) on their tangent spaces which vary smoothly from point to point. The geometry of such (Riemannian) manifolds is based on the line element ds 2=∑ i,j g ij dx i dx j . This gives, in particular, local notions of angle, length of curve, and volume.

Distances in Physics and Chemistry

Physics studies the behavior and properties of matter in a wide variety of contexts, ranging from the submicroscopic particles from which all ordinary matter is made (Particle Physics) to the behavior of the material Universe as a whole (Cosmology).