Z-trees and Peano Rasters for Scan Adaptive Image Processing

Given a specific image processing task and an image, its effectiveness is influenced by proximity driven properties captured in the image scan. One could exploit this effect and maximize the performance by adapting the scan and/or suitably adjusting the original operator. This two part sequel explores the balance between the effort required in scan adaptation and operator reformulation. The definition, basic properties and the application of Peano rasters and Z-trees are presented. Peano raster is a specific space filling curve generated using bit splicing of its grid coordinates. The Z-tree is a binary tree whose terminal nodes follow a Peano raster. A rotation operator is designed to swap specific grand children of a node in the Z-tree, which would effectively adapt the scan locally. Analytical expressions are presented to fully characterize the mapping between the exact coordinates of a pixel and its rank in the image scan before and after such rotations. The effectiveness of scan adaptation as a preprocessing step is illustrated in a number of applications, as we develop a framework of data-driven performance maximization of the underlying image processing task. This approach can be extended to 3-D images and higher dimensional grids as are the Peano rasters and Z-trees. Part 2 of the sequel is focused on generalizing the scans made of 12-distinct motifs and thus the Z-tree into *-trees. Received: May 26, 2007 c © 2007, Academic Publications Ltd. §Correspondence author 124 B. Zavidovique, G. Seetharaman AMS Subject Classification: 68U10