Digitizing the Plane with Cells of Nonuniform Size

A central problem in the field of geometric algorithms is to compare algorithms that process vector-type data specified in the continuous plane with algorithms that process raster-type data specified in the digitized plane. A set of criteria is proposed that a digitization of a collection of line segments should satisfy in order to be said to represent the same structure as their continuous plane counterpart. These criteria insure that the digitized collection has the same number of distinct features as the original data. In particular, attention is focussed on the correspondence of the precision with which the locations of endpoints on a continuous plane are specified to the number of grid cells in the appropriate digitized plane. It is shown that these criteria cannot be satisfied by grid cells of uniform size. The result leads to a reformulation of the digitization process that corresponds closely to a quadtree decomposition.