Digital Squares

Image processing and pattern recognition are mainly concerned with classifying shapes or patterns which appear in pictures. To do such tasks by digital computers, the pictures must be digitized, i.e., converted into arrays of lattice points. Discrete geometry is proposed by such a motivation. Recently, this research field provides various interesting topics. The following problem is one of these topics: How does a real figure relate to the corresponding lattice points? For example, how do we characterize sets of lattice points which are the digital images of real straight line segments? There have been published many papers which discuss such problems (e.g., see [l-8]). In [5, 81, we have also given the digital characterizations of some geometric pictures such as circles, rectangles, equilateral triangles, and isosceles right triangles. But, in Section 5 of [8], we left open the problems of some other figures like squares, rhombi, and so on. Since these elementary figures are concerned with distances, the problem seems more difficult. The purpose of this paper is to give a solution for the case of squares. Namely, we give an algorithm to decide whether or not an arbitrary set of lattice points is the digitization of a real square. In the last section, we shall give some remarks on other problems.