Shape approximation using quadtrees

-The quadtree representation encodes a 2" by 2" binary image as a set of maximal blocks of l's or O's whose sizes and positions are powers of 2. With the aid of the quadtree, a hierarchy of approximations to the image can be defined. Several ways of doing this are described. The accuracy of these approximations is empirically evaluated by studying how fast estimates of the first few moments of the image, computed from the approximations, converge to the true values, using a database of 112 airplane silhouettes. Approaches to the problem of fast shape matching using these approximations are also discussed. Image representations Quadtrees Shape approximation Matching Moments I. I N T R O D U C T I O N In recent years there has been rapidly growing interest in "quadtree" representations for binary images." ~5~ Given a 2" by 2" binary image I, we construct its quadtree as follows: The root node of the tree corresponds to all of I. If I consists of all O's or all l's, we label the root node 0 or 1, and it is all of the tree. Otherwise, the root node has four sons corresponding to the four quadrants of / , and we repeat the process for each of these quadrants. When this construction is complete, the leaf nodes of the tree correspond to blocks (-sub ... subquadrants of I) consisting entirely of 0's or l's. A node at level k (where the root is at level n) corresponds to a block of size 2 k by 2 k, in a position whose coordinates are multiples of 2 k. F rom now on we will call a leaf node"whi te" o r"b lack" if it is labelled 0 or 1, respectively, and we will refer to nonleaf nodes as "gray". An example of a binary image of an airplane and its quadtree is shown in Fig. 1. In this example we have n = 6 (i.e. the binary image is 64 by 64), so that the tree has seven levels (including the root); there are no black leaf nodes at levels 6, 5, 4 or 3. With the aid of the quadtree, we can define a hierarchy of approximations to the given image I. This can be done in various ways, as discussed in Section 2. To test the accuracy of these approximations, Section 3 presents an empirical investigation of how fast estimates of the first few moments of I, computed from the approximations, converge to their true values, for a set of 112 binary images of airplanes. Approximations should also be useful for matching purposes, since they should make it possible to reject mismatches rapidly. For shapes that are all similar to one another, however, e.g. for airplanes, the savings inherent in this approach may not be very great; the * All correspondence with Dr. Rosenfeld use of quadtrees for matching binary (or arbitrary) images is discussed in Section 4. 2. A P P R O X I M A T I O N S Given the quadtree representation of a binary image I, we can define several kinds of approximations to I :