34 Geometric Reconstruction Problems

Many problems from mathematics and engineering can be described in terms of reconstruction from geometric information. Reconstruction is the algorithmic problem of combining the results of measurements of some aspect of a physical or mathematical object to obtain desired information about the object. In this chapter, we consider three different classes of geometric reconstruction problems. In Section 34.1, we examine static reconstruction problems, where we are given a geometric structure derived from an original structure, and seek to invert this transformation. In Section 34.2, we consider interactive reconstruction problems, where we are permitted to repeatedly “probe” an unknown object at arbitrary places and seek to reconstruct the object using the fewest such probes. Finally, in Section 34.3, we turn to exploration and mapping of unknown environments, where we take the perspective of a mobile agent that locally observes its surroundings and aims to infer information about the global layout of the environment. Our focus in this chapter is on exact, theoretical reconstruction problems from a perspective of computational geometry. In contrast, a significant body of theoretical work in computational geometry is concerned with the approximate reconstruction of shapes and surfaces (see Chapter 35). Practical reconstruction problems beyond the scope of this chapter arise in many fields, with examples including computer vision, computer-aided tomography, and the reconstruction of 3D objects from 2D images. In these settings, quality criteria for good solutions are not always mathematically well-defined and may rely on aesthetics, practical applicability, or consistency with reference data.