Linear time distance transforms for quadtrees

The (region) quadtree, shown in Fig. 1, is a widely studied data structure for representing digitized images. An extensive survey of quadtrees and their use in image processing and graphics appears in Samet [ 1,2]. Unfortunately, the two-dimensional nature of the information they store makes quadtree usage more subtle than, say, binary search tree usage, and efficient algorithms often demand special quadtree techniques. This problem is intensified by the fact that there are at least three distinct representations for the quadtree structure reported in the literature. Each has unique advantages and disadvantages, with the result that each representation has applications for which it is most suited. The pointer-bused quadtree representation maintains the explicit tree structure as illustrated by the tree of Fig. lb. The linear quadtree [3] replaces the tree structure with a sorted linear list containing only the leaf nodes from the original tree. l The sort key is obtained by assigning to each leaf node an address derived by interleaving the bits of the x and y coordinates of the upper left pixel for the corresponding block in the image. The resulting records appear in the list in the same order as they would be visited by a depth-