Optimal Quadtrees for Image Segments

Quadtrees are compact hierarchical representations of images. In this paper, we define the efficiency of quadtrees in representing image segments and derive the relationship between the size of the enclosing rectangle of an image segment and its optimal quadtree. We show that if an image segment has an enclosing rectangle having sides of lengths x and y, such that 2
^N-1
× max (x, y) ⩽ 2
^N
, then the optimal quadtree may be the one representing an image of size 2
^N
× 2
^N
or 2
^N+1
× 2
^N+1
. It is shown that in some situations the quadtree corresponding to the larger image has fewer nodes. Also, some necessary conditions are derived to identify segments for which the larger image size results in a quadtree which is no more expensive than the quadtree for the smaller image size.