Optimal Halftoning over Hexagonal Grids

Optimal Halftoning over Hexagonal Grids J. Bacca Rodr´ıgueza , A. J. Gonz´alez Lozanoa , G. R. Arcea and D. L .Laub a Department

of Electrical and Computer Engineering University of Delaware, Newark, DE 19716 b Department of Electrical and Computer Engineering University of Kentucky, Lexington, KY 40506 ABSTRACT

Halftoning approaches to image rendering on binary devices have traditionally relied on rectangular grids for dot placement. This practice has been followed mainly due to restrictions on printer hardware technology. However, recent advances on printing devices coupled with the availability of efficient interpolation and resampling algorithms are making the implementation of halftone prints over alternate dot placement tessellations feasible. This is of particular interest since blue noise dithering principles indicate that the visual artifacts at several tone densities, which appear in rectangular-grid halftones, can be overcome through the use of hexagonal tessellations. While the spectral analysis of blue noise dithering provides the desired spectral characteristics one must attain, it does not provide the dithering structures needed to achieve these. In this paper, these optimal dithering mechanisms are developed through modifications of the Direct Binary Search (DBS) algorithm extensively used for rectangular grids. Special attention is given to the effects of the new geometry on the Human Visual System (HVS) models and on the efficient implementation of the hexagonal-grid DBS. This algorithm provides the best possible output at the expense of high computational complexity, and while the DBS algorithm is not practical in most applications, it provides a performance benchmark for other more practical algorithms. Finally, a tonedependent, hexagonal-grid, error-diffusion algorithm is developed, where the DBS algorithm is used to optimize the underlying filter weights. The characteristics of the HVS are thus implicitly used in the optimization. Extensive simulations show that hexagonal grids do indeed reduce disturbing artifacts, providing smoother halftone textures over the entire gray-scale region. Results also show that tone-dependent error-diffusion can provide comparable results to that of the DBS algorithms but at a significantly lower computational complexity. Keywords: Digital Halftoning, Direct Binary Search, Error Diffusion, Hexagonal Grids

1. INTRODUCTION Digital Halftoning is the technique to produce the illusion of continuous tone images in displays able to reproduce only black and white dots. These dots are arranged in such a way that lighter colors are represented by largely spaced dots while darker tones require arrangements with smaller distances. Halftoning, as well as other image processing techniques, has been restricted to the use of rectangular grids for several reasons. The availability of signal processing tools in such grids, the lack of proper hardware to reproduce images in other tessellations, and some theoretical results showing arguments in favor of rectangular grids. Recently, the hardware limitations have been overcome thanks to advances in printing technology. Theoretical restrictions have also been reevaluated in the new blue-noise theory for hexagonal grids.1 These developments have generated renewed interest in the problem of halftoning on hexagonal grids. This paper starts by describing the new theory of blue-noise for hexagonal grids and other advantages of working with them in contrast to the classic rectangular grids. It also includes a summary of previous approaches to halftoning on hexagonal grids with their results. Section 3 describes the DBS algorithm for rectangular grids. The properties of this algorithm as the best generator of blue noise patterns are well-known. The extension of DBS to hexagonal grids is described in section 4 where it is shown that the attained halftones exhibit superior robustness and other interesting properties of the tessellation. Finally, the results obtained with DBS are used to design a tone-dependant error-diffusion algorithm for hexagonal grids. This is a computationally effective alternative to DBS that can be implemented in printing mechanisms with close to optimal results. Section 6 provides the conclusions.

2. HALFTONING ON HEXAGONAL GRIDS The first mention of the application of hexagonal grids in halftoning was presented by Stevenson and Arce. 2 They introduced a 12-tap error diffusion filter. This filter was analyzed by Ulichney 3 who also introduced a two-tap and a four-tap (a hexagonal version of the Floyd-Steinberg filter) filters that included weight randomization. The filters, and examples of the results obtained with them using serpentine scan are shown in Fig. 1. Other approaches to hexagonal grid halftones include an attempt to perform tone-dependant error diffusion by Jodoin and Ostromoukhov.4 They used different 3-tap filters for each gray level. These filters were calculated based on an optimization algorithm that compared the spectra of their halftone patterns with an ideal analytic function based on the old blue-noise theory. Another halftoning algorithm designed for hexagonal sampling is the generation of clustered-dot ordered dither patterns by Turbek,5 later refined and analyzed by Cholewo.6 An analysis of screen tessellations for hexagonal grids was developed by Rao et. al. 7 Ulichney3 showed the inconvenience of using hexagonal grids to generate blue-noise patterns. He characterized blue-noise patterns as the ones having a principal frequency defined by: ( √ ( 2√ g g 1 √ 0 < g ≤ 21 f or0 < g ≤ 3S S 2 √ √ fb = f = b 2 √1−g 1−g 1 (1) f or 21 < g < 1 S 2