1. 2.1 Direct Binary Search (DBS)

2. Outline Overview of search-based halftoning methods DBS framework
DBS behavior
Efficient implementation for DBS
Dual interpretation for DBS
Optimal parameter choices

3. Overview of Search-Based Methods
Search-based methods use numerical optimization strategies to find the best halftone image.
These methods are usually iterative.
They can be used to design:
Constant gray-value texture patches
LUT texture stacks
Macroscreens
Optimal halftone images
Search methods include:
Linear programming
Simulated annealing
Gerchberg-Saxton iteration
Direct binary search (DBS)

4. DBS Framework [Analoui and Allebach, 1992]*
*Similar algorithms reported at same time by Pappas and Neuhoff and Mulligan

5. The Search Heuristic

6. DBS Convergence: 0, 1, 2, 4, 6, and 8 Iterations

7. Swaps vs. Toggles
Toggle only
Swap and toggle

8. Device model (linear, shift-invariant)
Image rendered on print or display
Digital image used to drive printer or display
Display/printer spot profile
Printer addressable resolution
Assumes spot overlap is additive if there is any overlap
Assume identical model for continuous-tone image
Example: ideal printer with no spot overlap

9. HVS model and error image

10. Impact of viewing geometry on spatial frequencies
Both arrows A and B generate same retinal image
For small ratio , the angle subtended at the retina in radians is

11. Spatial frequency conversion
To convert between (cycles/inch) viewed at distance (inches) and (cycles/degree) subtended at the retina, we thus have
For a viewing distance of 12 inches, this becomes

12. Spatial frequency filtering stage
Based on pyschophysical measurements of contrast sensitivity function
Use sinusoidal stimuli with modulation along achromatic, red-green, or blue-yellow axes
For any fixed spatial frequency, threshold of visibility is depends only on This is Weber’s Law.

13. Campbell’s contrast sensivity function on log-log axes

14. Dependence of sine wave visibility on contrast and spatial frequency

15. Models for achromatic spatial contrast sensitivty*
Author
Contrast sensitivity function
Constants
Campbell
1969
Mannos
1974
Nasanen
1984
Daly
1987
*Kim and Allebach, IEEE T-IP, March 2002

16. Achromatic spatial contrast sensitivity curves

17. Error metric

18. Impact of scale parameter
S = RD ; R = resolution in dpi, D=viewing distance in inches.
Role in DBS
S1=0.5S2
S2=300x9.5
S3=2.0S2

19. Efficient computation
Direct computation of effect of a trial change requires operations for filter containing pixels.

20. Efficient evaluation of trial changes
Change in digital halftone
Coefficients
Change in mean-squared error
correlation functions

21. Update required for accepted change
Computation is , but updates occur much less often than trial changes.

22. Dual interpretation of DBS [Lieberman and Allebach, 1999]
Consider toggle from to
Recall:
In this case, , and
Change in correlation

23. Summary of results Toggle from to Condition for acceptance
Change in correlation
Toggle from to

24. Statistics for

25. What is ?
Can show that
For Nasanen’s HVS model

26. Summary of dual interpretation
Minimize mean-squared error at distance
Minimize maximum error at distance

27. Illustration of Dual Interpretation
f[m] f[m]*p[m] f[m]*cpp[m] ~ ~~
g[m] g[m]*p[m] g[m]*cpp[m] ~ ~~

28. Tone reproduction with DBS

29. The impact of filter size on halftone comparisons
set1

30. Set 1 Swap only Swap neighborhood: 161x161, Block size: 5x4
Pattern size: 256x256
Scale factor: 2000
Radius: 4 pixels
Filter size: (4x4+1)=17
Scale factor: 2000
Radius: 6 pixels
Filter size: (6x4+1)=25
Scale factor: 2000
Radius: 13 pixels
Filter size: (13x4+1)=53
Based on this result, for scale factor 2000, radius of 13 pixels is the best, it covers 99% of the area.
Each pixel is represented by 3x3 pixels.

31. Set 1 log-magnitude of power spectra✓
K=7
K=7
K=7
Scale factor: 2000
Radius: 4 pixels
Filter size: (4x4+1)=17
Scale factor: 2000
Radius: 6 pixels
Filter size: (6x4+1)=25
Scale factor: 2000
Radius: 13 pixels
Filter size: (13x4+1)=53

32. Set 1 RAPS ✓ Scale factor: 2000 Radius: 4 pixels
Filter size: (4x4+1)=17
Scale factor: 2000
Radius: 6 pixels
Filter size: (6x4+1)=25
Scale factor: 2000
Radius: 13 pixels
Filter size: (13x4+1)=53 ✓

33. The impact of scale factor on halftone comparisons
set2

34. Set 2 ✓ Scale factor: 2000 Radius: 13 pixels Filter size: (13x4+1)=53

35. Set 2 log-magnitude of power spectra
Scale factor: 3000
K=7
Scale factor: 2000
K=7
Scale factor: 2500
K=7
Scale factor: 3500
K=7
Scale factor: 4000
K=7
Scale factor: 4500
K=7

36. Set 2 RAPS Scale factor: 2000 Radius: 13 pixels
Filter size: (13x4+1)=53
Scale factor: 2500
Radius: 17 pixels
Filter size: (17x4+1)=69
Scale factor: 3000
Radius: 20 pixels
Filter size: (20x4+1)=81

37. Set 2 RAPS ✓ Scale factor: 3500 Radius: 23 pixels
Filter size: (23x4+1)=93 ✓
Scale factor: 4000
Radius: 27 pixels
Filter size: (27x4+1)=109
Scale factor: 4500
Radius: 30 pixels
Filter size: (30x4+1)=121

38. Comparisons of swap neighborhood
Pattern size: 512x512
best
- 106 to 106
- 116 to 116
- 255 to 255
Swap only
We increase the pattern size so that we can swap in larger neighborhoods.

39. Comparisons of swap neighborhood log-magnitude of power spectra
-106 to 106
K=15
-116 to 116
K=15
-255 to 255
K=15

40. Comparisons of swap neighborhood RAPS
- 106 to 106
- 116 to 116
- 255 to 255 ✓