Optimal projections onto grids and finite resolution images

The problem of determining nice (regular, simple, minimum crossing, and monotonic) and non-degenerate (with distinct x-coordinate, non-collinear, non-cocircular, and non-parallel) orthogonal and perspective images of a set of points or a set of disjoint line segments has been studied extensively in the literature for the theoretical case of infinite resolution images [J. Vis. Commun. Image Represent. 10 (2) (1999) 155; Int. J. Math. Algorithms 2 (2001) 227; J. Vis. Commun. Image Represent. 12 (4) (2001) 387; J. Vis. Commun. Image Represent.]. In this paper we propose to extend the study of this type of problems to the case where the images have finite resolution. Applications dealing with such images are common in practice, in fields such as computer graphics and computer vision. We derive algorithms that solve three related problems, both exactly and approximately. Given a set P in the plane or in the space, find a graduated line, a line partitioned into unit length intervals, so that the maximum number of points of P that are projected into a single interval is minimized. In space we also study the variant where we want to project P onto a graduated plane, a plane partitioned into unit squares.