Optimizing prediction gain in axial symmetric scans

Though most lossless image coding techniques use a raster scan to order the pixels for context-based predictive coding, other scans, such as the Hilbert or Peano scan, have been proposed as alternatives with potentially better performance. However, a general understanding of the merits of different scans has been lacking. In previous work, the authors had presented a framework in which the effect of pixel scan order on lossless compression can be quantitatively analyzed, so that comparisons of different scans can be made. Assuming a quantized-Gaussian and isotropic image model with contexts consisting of previously scanned adjacent pixels in a distance constrained neighborhood, it was found that the raster scan is better than the Hilbert scan. In this paper we further develop our arguments and show that for a large class of scans, which we call axial symmetric scans, the raster scan is indeed optimal. We would like to note that many common scans including the Hilbert scan fall under the class of axial symmetric scans.