1. 2.2 Design of Aperiodic, Dispersed-Dot Screens

2. 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

3. 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.

4. 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

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

6. 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

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

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

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

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

11. 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.

12. Categorization of Methods for Screen Design

13. 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.

14. Stacking Constraint

15. 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.

16. 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]

17. 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.

18. 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.

19. 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

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

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

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

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

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

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

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

27. 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.

28. 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

29. 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

30. 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)

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

32. 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)

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

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

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