Image transformation on hexagonal structure based on conversion between 1d and 2d coordinates

Spiral Architecture, a hexagonal image structure is a novel and powerful approach to machine vision system. The pixels on Spiral architecture are geometrically arranged using a 1D (Spiral) addressing scheme in an ascending order along a spiral-like curve. Spiral addition and Spiral multiplication are defined based on the Spiral addresses on Spiral Architecture. These two fundamental operations result in fast and easy translation, rotation and separation on images, and hence play very important roles for image processing on Spiral Architecture. Moreover, 2D coordinates according to rows and columns defined on Spiral Structure provide a good mapping to the ordinary 2D coordinates defined on the common square image structure. Therefore, how to convert the 1D Spiral addresses from and to the 2D coordinates on Spiral Architecture has become very important to apply the theory developed on a hexagonal image structure for image processing (e.g., rotation). In this paper, we perform a fast way to correctly locate any hexagonal pixel when its Spiral address is known, and compute the Spiral address of any hexagonal pixel when its location is known. As an illustration of the use of conversions, we demonstrate the accurate image translation and rotation using experimental results.