On the number of digital convex polygons inscribed into an (m,m)-grid

Binary images of objects are digitized by coloring a pixel cell black if more than half of its area is within the interior of the object. For simplicity, the digitization is often modified by looking only at the center point of a cell to determine its pixel value. The digitized boundary curve consists of a sequence of 4-directional links, sometimes called a "crack" code since it follows the cracks or edges of the pixel cells. Of interest here is the entropy of digitized binary objects or planar curves on an m×m integer grid. Let D(m) denote the number of digital convex polygons which can be inscribed into an integer grid of size m×m. The asymptotic estimation of log D(m) is of interest in determining the entropy of digitized convex shapes. It is shown that log D(m) is of the order m^2/3