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Digital Geometry by Azriel Rosenfeld, Reinhard Klette

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CHAPTER 3

Metrics

This chapter discusses metric spaces [S, d], in which S is a subset of Rn or a subset of a grid; it also addresses metric spaces, in which S is a family of such subsets. It also discusses metrics on pictures in which distances depend on the pixel values.

3.1 Basics About Metrics

Measurement requires a metric space. In this section, we summarize facts about metric spaces that are relevant to digital geometry. The definition of a metric (distance function) based on properties M1 through M3 was given in Section 1.2.1.

3.1.1 The Euclidean metric

We first consider the metric that is used in Euclidean geometry: the distance between two points is equal to the length of the straight line segment defined by the two points. This metric ...

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