1. 1.3 Error Diffusion – Basic Concepts

2. Synopsis Error diffusion architecture Error diffusion textures
Edge enhancement effect of error diffusion
Spectral analysis of error diffusion
Variations on a theme
Tone-dependent error diffusion

3. Error Diffusion Architecture

4. Description of the Algorithm

5. 2-D Error Diffusion Weighting Filters

6. 1-D Example

7. 2-D Error Diffusion – Floyd-Steinberg Weights Texture Artifacts in Midtones

8. 2-D Error Diffusion – Floyd-Steinberg Weights Texture Artifacts in Highlights

9. Error Diffusion Characteristics
At each step, error diffusion preserves local average over part of image that has been binarized and part that is yet to be binarized
No fixed number of quantization levels
Requires more computation than screening
Excellent detail rendition (sharpens image)
Generally good texture with some exceptions:
Texture contouring
Worm-like patterns in highlights and shadows
Texture cliques in midtones
Texture used to render a given gray level may be context-dependent

10. Two Views of Error Diffusion

11. Fourier Analysis Combining (1) and (3), In Fourier domain,
Rearranging, we get
where is a high-pass filter
This shows that the spectrum of the binary image consists of the spectrum of the continuous-tone image plus a high-pass filtered version of the spectrum of the quantization error.

12. Knox’s Empirical Model for Quantization Error*
We cannot find an analytical expression for
Knox proposed the following model
Correlation coefficient –
Residual – (generally some image-dependence)
ED Weights c
1-D
0.0
Floyd and Steinberg
0.55
Jarvis, Judice, and Ninke
0.80
*K. T. Knox, “Error Image in Error Diffusion,” SPIE Vol (1992)

13. Application to Fourier Analysis
Combining (4) and (5), we get
The first term shows the origins of the sharpening effect that is characteristic of error diffusion.
To the extent that the residual is uncorrelated with the original image, the second term will look like blue noise.
High-emphasis filter
High-pass noise

14. Knox’s Experimental Results
Halftone Image
Quantized Error Image
The parameter in the upper right-hand corner is the correlation coefficient c
1-D
Floyd-Steinberg
Jarvis, Judice, Ninke

15. Spectral Characteristics of Error Diffusion
clustered dot dither
continuous tone
error diffusion
Bayer dither

16. Synopsis Error diffusion architecture Error diffusion textures
Edge enhancement effect of error diffusion
Spectral analysis of error diffusion
Variations on a theme
Tone-dependent error diffusion

17. Variations on a Theme Non-lexicographic scan raster Serpentine raster
Peano scan
Block-based scans
Modifications to weights
Randomized weights
Adaptive weights
Tone-dependent weights
Threshold modulation
Control sharpening
Encourage clustering
Serpentine Raster

18. Early Work
Knox (1993) developed a spectral analysis for serpentine raster, and showed that it eliminates the asymmetry in the error weighting frequency response that is due to the causal nature of the filter.
Ulichney (1986, 1987) experimented with many different combinations of scan rasters, weight sets, and randomization techniques, and concluded that a serpentine raster, Floyd-Steinberg coefficients, and 50% randomization yielded the best results.

19. Threshold Modulation Basic equation for threshold modulation
Image-independent threshold modulation where is a periodic screen function – encourages periodic clustered-dot behavior (Billotet-Hoffman and Bryngdahl, 1983)
Image-dependent threshold modulation where is a constant that controls degree of edge enhancement (Eschbach, 1991)
Approximately cancels edge enhancement intrinsic to standard error diffusion
Yields standard error diffusion
Results in more edge enhancement than is provided by standard error diffusion

20. Tone-Dependent Error Diffusion Introduction
Previous work related to TDED:
R. Eschbach, JEI 1993
J. Shu, SID 1996
V. Ostromoukhov, US Patent 1998
V. Ostromoukhov, SIGGRAPH 2001
Our approach:
The weights and thresholds are optimized based on a HVS model.
Use variable weight locations to reduce worm like artifacts.

21. Conventional Error Diffusion
f[m,n]
u[m,n]
g[m,n] +
Q(•) - + -
wk,l
d[m,n]

22. Tone Dependent Error Diffusion [Li and Allebach, 2004]
Weights and thresholds vary depending on the input
f[m,n]
u[m,n]
g[m,n] +
Q(•) - + -
d[m,n]
wk,l(f[m,n]) X

23. Tone Dependent Error Diffusion
Quantization process:
p[m,n;0.5]
Midtone halftone pattern generated with direct binary search (DBS)
Problem:
How to optimize tU, tL and wk,l for each gray level.

24. Optimization of TDED parameters
DBS
|DFT|2 +
Constant
Patch
(absorptance a)
Normalized MSE -
TDED
|DFT|2
Update weights and thresholds
Cost function

25. Optimal weights and thresholdsX

26. Floyd-Steinberg vs TDED

27. Floyd-Steinberg vs TDED

28. TDED vs DBS
TDED
DBS

29. Efficient Implementation of Error Diffusion
Reduced Lookup-table error diffusion
Removal of DBS midtone pattern – direct binary flipping (DBF)
Interpolation of weights and thresholds
Memory-efficient/parallelizable approaches to error diffusion
Block-based error diffusion
Block-interlaced pinwheel error diffusion
Serial block-based error diffusion
Error-quantized error diffusion
Compressible error diffusion

30. Prototypical System Architecture
High Speed
Memory
High Speed
Memory
High Speed
Memory
. . .
Processor
Processor
Processor
Bus
There may be only one of these
I/O
Low Speed
Memory

31. Implementation Issues – Parallelizability and Memory Access
Conventional error diffusion is inherently a serial process.
Cannot process any given pixel until all pixels that feed error to it have already been processed.
This limits opportunities for parallelism.
Must have access to the entire previous/next line of continuous-tone error/image data.
This impacts memory and/or bandwidth requirements.
It is a problem when we want to go very cheap or very fast.

32. Memory Requirements for Raster Implementationx
storage of errors for next line
storage of errors from previous line
storage of errors from previous line and current pixel n m
m+1
n-1
n+1
For low cost or large format printers, on-chip memory is not sufficient to buffer error information for one entire image row
All errors have to be written out to off-chip memory

33. Strategies for Artifact Reduction: Serpentine Raster
Conventional Raster
Serpentine Raster

34. Block Interlaced Pinwheel Error Diffusion (BIP-ED) with Serpentine Raster
Process outward spiral (black) blocks first. Then process inward spiral (green) blocks.

35. Application of TDED to BIP-ED
Adapt tone dependent error diffusion to pinwheel architecture.
Use different TDED filters for different block types and different locations within blocks.
Randomly initialize outward spiral ED to break up periodic patterns.
Initialize inward spiral ED using errors diffused from neighboring outward spiral blocks to eliminate boundary artifacts.

36. BIP-TDED
Block size 8x8
Block size 32x32

37. Pinwheel Error Diffusion Conventional Tone Dependent
BIP-TDED vs TDED
Pinwheel Error Diffusion
Block size 128x128
Conventional Tone Dependent
Error Diffusion

38. Stages for Strip-based Processing4 2 1 6 3 7 5 8
Outward spiral block
Inward spiral block