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 octtree 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 data compression using binary images and three dimensional data.

In previous work we have proposed a steganographic technique for gray scale images called BPCS-Steganography. We also apply this technique to full color images by decomposing the image into its three color component images and treating each as a gray scale image. This paper proposes a method to apply BPCS-Steganography to palette-based images. In palette-based images, the image data can be decomposed into color component images similar to those of full color images. We can then embed into one or more of the color component images. However, even if only one of the color component images is used for embedding, the number of colors in the palette after embedding can be over the maximum number allowed. In order to represent the image data in palette-based format, color quantization is therefore needed. We cannot change the pixel values of the color component image that contains the embedded information, but can only change the pixel values of the other color component images. We assume that the degrading of the color component2 image with information embedded is smaller than that of the color component images that are used for color reduction. We therefore embed secret information into the G component image, because the human visual system is more sensitive to changes the luminance of a color, and G has the largest contribution to luminance of the three color components. In order to reduce the number of colors, the R and B component images are then changed in a way that minimizes the square error.