<p>Digital geometry is the geometry of digital images. Compared to Euclid’s geometry, which has been studied for more than two thousand years, this field is very young.</p><p>Efim Khalimsky’s topology on the integers, invented in the 1970s, is a digital counterpart of the Euclidean topology on the real line. The Khalimsky topology became widely known to researchers in digital geometry and computer imagery during the early 1990s.</p><p>Suppose that a continuous function is defined on a subspace of an <i>n-</i>dimensional Khalimsky space. One question to ask is whether this function can be extended to a continuous function defined on the whole space. We solve this problem. A related problem is to characterize the subspaces on which every continuous function can be extended. Also this problem is solved.</p><p>We generalize and solve the extension problem for integer-valued, Khalimsky-continuous functions defined on arbitrary smallest-neighborhood spaces, also called Alexandrov spaces.</p><p>The notion of a digital straight line was clarified in 1974 by Azriel Rosenfeld. We introduce another type of digital straight line, a line that respects the Khalimsky topology in the sense that a line is a topological embedding of the Khalimsky line into the Khalimsky plane.</p><p>In higher dimensions, we generalize this construction to digital Khalimsky hyperplanes, surfaces and curves by digitization of real objects. In particular we study approximation properties and topological separation properties. </p><p>The last paper is about Khalimsky manifolds, spaces that are locally homeomorphic to <i>n-</i>dimensional Khalimsky space. We study different definitions and address basic questions such as uniqueness of dimension and existence of certain manifolds.</p>

Digital geometry is the geometry of digital images. Compared to Euclid’s geometry, which has been studied for more than two thousand years, this field is very young. Efim Khalimsky’s topology on the integers, invented in the 1970s, is a digital counterpart of the Euclidean topology on the real line. The Khalimsky topology became widely known to researchers in digital geometry and computer imagery during the early 1990s. Suppose that a continuous function is defined on a subspace of an n-dimensional Khalimsky space. One question to ask is whether this function can be extended to a continuous function defined on the whole space. We solve this problem. A related problem is to characterize the subspaces on which every continuous function can be extended. Also this problem is solved. We generalize and solve the extension problem for integer-valued, Khalimsky-continuous functions defined on arbitrary smallest-neighborhood spaces, also called Alexandrov spaces. The notion of a digital straight line was clarified in 1974 by Azriel Rosenfeld. We introduce another type of digital straight line, a line that respects the Khalimsky topology in the sense that a line is a topological embedding of the Khalimsky line into the Khalimsky plane. In higher dimensions, we generalize this construction to digital Khalimsky hyperplanes, surfaces and curves by digitization of real objects. In particular we study approximation properties and topological separation properties. The last paper is about Khalimsky manifolds, spaces that are locally homeomorphic to n-dimensional Khalimsky space. We study different definitions and address basic questions such as uniqueness of dimension and existence of certain manifolds.

This thesis consists of two parts: digital geometry and discrete optimization. In the first part we study the structure of digital straight line segments. We also study digital curves from a combinatorial point of view. In Paper I we study the straightness in the 8-connected plane and in the Khalimsky plane by considering vertical distances and unions of two segments. We show that we can investigate the straightness of Khalimsky arcs by using our knowledge from the 8-connected plane. In Paper II we determine the number of Khalimsky-continuous functions with 2, 3 and 4 points in their codomain. These enumerations yield examples of known sequences as well as new ones. We also study the asymptotic behavior of each of them. In Paper III we study the number of Khalimsky-continuous functions with codomain Z and N. This gives us examples of Schröder and Delannoy numbers. As a byproduct we get some relations between these numbers. In Paper IV we study the number of Khalimsky-continuous functions between two points in a rectangle. Using a generating function we get a recurrence formula yielding this numbers.   In the second part we study an analogue of discrete convexity, namely lateral convexity. In Paper V we define by means of difference operators the class of lateral convexity. The functions have plus infinity in their codomain. For the real-valued functions we need to check the difference operators for a smaller number of points. We study the relation between this class and integral convexity. In Paper VI we study the marginal function of real-valued functions in this class and its generalization. We show that for two points with a certain distance we have a Lipschitz property for the points where the infimum is attained. We show that if a function is in this class, the marginal function is also in the same class. / At the time of the doctoral defense, the following papers were unpublished and had a status as follows: Paper 4: Submitted. Paper 5: Manuscript. Paper 6: Manuscript.