Neighbor Finding Techniques for

Image representation plays an important role in image processing applications. Recently there has been a considerable interest in the use of quadtrees. This has led to the development of algorithms for performing image processing tasks as well as for performing converting between the quadtree and other representations. Common to these algorithms is a traversal of the tree and the performance of a given computation at each node. These computations typically require the ability to examine adjacencies between neighboring nodes. Algorithms are given for determining such adjacencies in the horizontal, vertical, and diagonal directions. The execution times of the algorithms are analyzed using a suitably defined model. Region representation is an important aspect of image processing with numerous representations finding use. Recently, there has emerged a considerable amount of interest in the quadtree [3-8, 111. This stems primarily from its hierarchical nature, which lends itself to a compact representation. It is also quite efficient for a number of traditional image processing operations such as computing perimeters [14], labeling connected components [ 131, finding the genus of an image [I], and computing centroids and set properties [ 181. Development of algorithms to convert between the quadtree representation and other representations such as chain codes [2, lo], rasters [12, 171, binary arrays [ll], and medial axis transforms [15, 16, 191 lend further support to this importance. In this paper we discuss methods for moving between adjacent blocks in the quadtree. We first show how transitions are made between blocks of equal size and then generalize our result to blocks of different size, where the destination block is either of larger or smaller size than the source block. Such blocks are termed neighbors. Note that the transitions that we discuss also include those along diagonal, as well as horizontal and vertical, directions. The importance of these methods lies in their being the cornerstone of many of the quadtree algorithms (e.g., [ 1, 2, 10, 12-19]), since they are basically tree traversals with a " visit " at each node. More often than not these visits involve probing a node's neighbors. The significance of our methods lies in the fact that they do not use coordinate information, knowledge of the size of the image, or storage in excess of that imposed by the nature of the quadtree data structure.