This is a book on three distinct but related branches of science: digital geometry, mathematical morphology, and discrete optimization. They are united by a certain way of thinking as well as by the many applications where they can serve together. In addition to being useful, they are also intellectually challenging.The book contains a systematic study of inverses of mappings between ordered sets, and so offers a common foundation for various phenomena related to duality.To prepare the ground, there are chapters on convexity in real vector spaces, in anticipation of the many challenging problems of digital convexity. There is also a chapter on classical topology, preparing for the newer topologies introduced to serve in discrete spaces.Key Features: oThe concepts of upper and lower inverse of a mapping between preordered sets are completely elementary ones. They unify and generalize many procedures used in several fields of knowledge, starting with the Galois connections in algebra two centuries ago and with significant later instances, maybe the most prominent being the Fenchel transformation in convexity theoryoIn later chapters the inverses are used to define dualities for convolution equations. This is essential since there is no general associative convolution multiplicationoDifference operators are the simplest convolution operators and yet serve as a most useful tool in the study of digital straightness and digital convexity, also using the duality just mentioned

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