2.2 Design of Aperiodic, Dispersed-Dot Screens

Stochastic Screening Topics General principles for design of stochastic screens Allebach and Stradling screen Rolleston and Cohen screen Mitsa and Parker screen Ulichney screen (Void and cluster) Stochastic screen design using DBS

Stochastic Screen Design Stochastic screens need to be large enough to hide visibility of fundamental period – typically 128x128 to 512x512 pixels For a 128x128 screen, each threshold value between 0 and 255 occurs approximately 64 times in the matrix. Stochastic screens are designed using search-based optimization strategies. The screen design is computationally intensive; but once the screen is obtained, images are still halftoned with one comparison/pixel. Macroscreens can be designed to have blue noise or green noise characteristics. These terms refer to the characteristics of the halftone textures generated by these screens, not the screens themselves.

Impact of Stochastic Screen Size As size of screen increases, visibility of fundamental period diminishes. Generally, a size between 128x128 and 256x256 is adequate. 16x16 pattern repeated 16x16 times 32x32 pattern repeated 8x8 times

Impact of Stochastic Screen Size (cont.) 64x64 pattern repeated 4x4 times 128x128 pattern repeated 2x2 times

Importance of incorporating wrap-around in screen design Level: 15/255 Level: 15/255 No wrap-around DBS Wrap-around DBS Level: 90/255 Level: 90/255

Spectral Properties of Blue Noise Mask - Highlight Texture Halftone texture Fourier spectrum

Spectral Properties of Blue Noise Masks - Midtone Texture Halftone texture Fourier spectrum

Spectral Properties of Green Noise Masks - Highlight Texture Halftone texture Fourier spectrum

Spectral Properties of Green Noise Masks - Midtone Texture Halftone texture Fourier spectrum

Iterative/Search-Based Methods for Screen Design Iterative/search-based methods for screen design may be categorized according to the domain and objective of the search: search domain dither(threshold) matrix dot profile function search objective minimize a cost function satisfy constraints The screen design methods that we will describe may be divided into four groups according to these descriptors. The intent of all these methods is to either directly or indirectly push the energy in the spectra of the binary textures away from the origin in the frequency domain, thus yielding a blue noise-like characteristic. [Spaulding, Miller, and Schildkraut, 1997] contains a good overview of some, but not all, of the screen designs discussed here.

Categorization of Methods for Screen Design

Search Domains Dither matrix – we directly look for the best spatial arrangement of the thresholds. In this way, we simultaneously optimize the entire dot profile function. This suggests a greater liklihood of finding a global optimum. In practice, however, these algorithms do not perform as well as those based on level-by-level design of the dot profile function. Dot profile function – We design the levels in the dot profile function one at a time. Each new level must satisfy the stacking constraint with respect to the patterns that already have been designed. The design is greedy, since each successive level is more highly constrained. Several different sequences of levels have been tried.

Stacking Constraint

Impact of order in which levels are designed for dot profile-based methods When any given level is to be designed, the next lower and next higher levels that have already been designed serve as constraint levels for application of the stacking constraint.

Pairwise Exchange Screen [Allebach and Stradling, 1979] Search for ordering of thresholds in dither matrix that minimizes a cost function Pairwise exchange search algorithm 1. generate random initial ordering of the thresholds 2. for k = 0, 1, ..., M2-1 for l = k+1, ..., M2-1 swap tk and tl if change decreases cost, keep it, otherwise restore tk and tl end Cost function Algorithm is very compute-intensive, and converges slowly. Dalton proposed a more efficient strategy based on level-by-level design of dot profile function [Dalton, 1989]

Rolleston and Cohen Screen Iterate back and forth between space domain and Fourier domain satisfying constraints on dither matrix in space domain and spectrum of dither matrix in frequency domain. This is an example of the classic Gerchberg-Saxton algorithm. We start with a randomly ordered dither matrix, and pull a solution when the iteration has converged satisfactorily. There is no known direct relation between the spectrum of the dither matrix, and the spectra of the binary textures at each level in the dot profile function.

Mitsa and Parker Screen This algorithm employs a level-by-level iteration similar to that of Rolleston and Cohen algorithm to design a mask that satisfies constraints. The spatial domain constraint is that the texture at each level must be binary. The frequency domain constraint is that the radially averaged power spectrum have the prescribed blue noise shape as postulated by Ulichney, including the correct principal frequency. They determined empirically that scaling the principal frequency from that specified by Ulichney for a rectangular dot arrangement by a factor of yielded improved textures.

Void and Cluster Screen [Ulichney, 1993] The algorithm employs a level-by-level search to minimize a cost function. It is based entirely in the spatial domain. Cost function is approximately The filter h[m,n] can be interpreted as the point spread function of the human visual system; Ulichney used a Gaussian filter with s = 1.5 pixels. Convolution is circular to account for the periodicity of the dither matrix

Swapping pixels c. swap hole closest to center of largest void with dot closest to center of tightest cluster.

DBS screen design: Example 1 Dual-metric DBS applied directly to image Dual-metric DBS screen (128x128)

DBS screen design: Example 2 DBS applied directly to image Dual-metric DBS screen (128x128)

Impact of screen size 128x128 DBS screen 64x64 DBS screen

Impact of screen size 64x64 DBS screen 32x32 DBS screen

Impact of screen size 128x128 DBS screen 64x64 DBS screen

Impact of screen size 64x64 DBS screen 32x32 DBS screen

Level by level screen design 1. Design dot profile for all levels p[m,n;b] in chosen sequence, where b = 0,1,…, max level (255) 2. Combine the dot profiles and form the screen g[m,n]: 3. Verify screen correctness. 4. Use the screen to halftone an image.

Patch generation sequence Investigated sequences (using DBS with toggle&swap): 1. Midtone to extreme tones (next slide) 2. Binary tree (not as good as 1st choice) Generate dot profiles of middle graylevels first. If generate from graylevel 1 to 254, the sequence is: For lower half(1~127) of dot profiles (upper half is similar): Start from level 127: 3. alternate sequence. Utpal’s sequence (it is better than 1st) 64 32 96 80 112 16 48 8 24 40 56 72 88 104 120 127 32,96 16,48,80,112 8,24,40,56,72,88,104,120

Midtone to extreme tones sequence Generate patches for neugebauer primaries W,K in sequence 1. Ordering: Lower half: sequentially decreasing; Upper half: sequentially increasing. W = , K = Constraints: W+K = 100% 256 total patches were generated. White dots are assigned 0. Black dots are assigned 1. W = 100% K = 0% W = 99.61% K = 1 - W W = 0.39% K = 1 - W W = 0% K = 100% White 1 127 128 254 255 Black White dot Black dot Total dot Total dot

Utpal’s alternate sequence 1  39, 255  216, 40  79, 215  128, 80  126, 129  175 (1) (3) (6) (5) (7) (4) (2) (1 39) (40 79) (80 126) (127 128) (129 175) (176 215) (216 255)

Stacking constraint Upper level Current level Lower level Can be white or black White dot Black dot

Screen generation DBS mono halftoning Screen size: 256x256 Sequence: Utpal’s alternate sequence Using swap only in a neighborhood of -127 to 127 Scale factor: 3500, psf radius: 99% coverage, 23 pixels DBS mono halftoning Both toggle and swap are enabled, swap neighborhood is 3x3. HVS filter: scale factor 3000, psf radius: 90% coverage, 6 pixels. (This configuration has better result)

Halftoned ramp image -- DBS monochrome Screened ramp image -- Alternate sequence Comparison Halftoned ramp image -- DBS monochrome

Halftoned ramp image -- DBS monochrome Screened ramp image -- Alternate sequence Halftoned ramp image -- DBS monochrome

Halftoned Ruiyi image -- DBS monochrome Screened Ruiyi image -- Alternate sequence Comparison Halftoned Ruiyi image -- DBS monochrome