Implementation of the Hilbert scanning algorithm and its application to data compression

The Hilbert curve is one of the simplest curves which pass through all points in a space. Many researchers have worked on this curve from the engineering point of view, such as for an expression of two-dimensional patterns, for data compression in an image or in color space, for pseudo color image displays, etc. A computation algorithm of this curve is usually based on a look-up table instead of a recursive algorithm. In such algorithm, a large memory is required for the path look-up table, and the memory size becomes proportional to the image size. In this paper, we present an implementation of a fast sequential algorithm that requires little memory for two and three dimensional Hilbert curves. Our method is based on some rules of quad-tree traversal in two dimensional space, and oct-tree traversal in three dimensional space. The two dimensional Hilbert curve is similar to the scanning of a DF (Depth First) expression, which is a quad-tree expression of an image. The important feature is that it scans continuously from one quadrant, which is obtained by quad tree splitting, to the next adjacent one in two dimensional space. From this point, if we consider run-lengths of black and white pixels during the scan, the run-lengths of the Hilbert scan tend to be longer than those of the raster scan and the DF expression scanning. We discuss the application to date compression using binary images and three dimensional data.