1. Tone Dependent Color Error Diffusion Halftoning
Multi-Dimensional DSP Project
Vishal Monga,
April 30, 2003

2. Grayscale Error Diffusion
2- D sigma delta modulation [Anastassiou, 1989]
Shape quantization noise into high frequencies
Linear Gain Model [Kite, Evans, Bovik, 1997]
Replace quantizer by scalar gain Ks and additive noise image + _
e(m)
b(m)
x(m)
difference
threshold
compute error
shape error
u(m)
current pixel
weights
Transfer functions
3/16
7/16
5/16
1/16

3. Direct Binary Search Used in screen design
[Analoui, Allebach 1992]
Computationally too expensive for real-time applns. viz. printing
Used in screen design
Serves as a practical upper bound for achievable halftone quality

4. Tone Dependent Error Diffusion
b(m) + _
e(m)
x(m)
Tone dependent error filter
Tone dependent threshold modulation
Train error diffusion weights and threshold modulation [Li & Allebach, 2002]
DBS pattern
for graylevel x
Halftone pattern
FFT
Midtone regions
Highlights and shadows
FFT
Graylevel patch x
Halftone pattern
for graylevel x
FFT

5. Tone Dependent Color Error Diffusion
Color TDED, Goal
Obtain optimal (in visual quality) error filters with filter weights dependent on input RGB triplet (or 3-tuple)
Extension to color is non-trivial
Applying grayscale TDED independently to the 3 color channels ignores the correlation amongst them
Choice of error filter
Separable error filters for each color channel
Matrix valued filters [Damera-Venkata, Evans 2001]
Design of error filter key to quality
Take human visual system (HVS) response into account

6. Tone Dependent Color Error Diffusion
Problem(s):
Criterion for error filter design ?
(256)3 possible input RGB tuples
Solution
Train error filters to minimize the visually weighted squared error between the magnitude spectra of a “constant” RGB image and its halftone pattern
Design error filters along the diagonal line of the color cube i.e. (R,G,B) = {(0,0,0) ; (1,1,1) …(255,255,255)}
Color screens are designed in this manner
256 error filters for each of the 3 color planes

7. Perceptual Error Metric
Input RGB Patch
FFT
Color Transformation
sRGB  Yy Cx Cz
(Linearized CIELab) 
FFT
Halftone Pattern

8. Perceptual Error Metric
HVS Chrominance
Frequency Response
HVS Luminance
Total Squared Error (TSE) 
Yy
Cx
Cz
Find optimal error filters that minimize TSE subject to diffusion and non-negativity constraints, m = r,g,b; a  (0,255)
(Floyd-Steinberg)

9. Linear CIELab Color Space Transformation
Linearize CIELab space about D65 white point [Flohr, Kolpatzik, R.Balasubramanian, Carrara, Bouman, Allebach, 1993]
Yy = 116 Y/Yn – L = 116 f (Y/Yn) – 116
Cx = 200[X/Xn – Y/Yn] a = 200[ f(X/Xn ) – f(Y/Yn ) ]
Cz = 500 [Y/Yn – Z/Zn] b = 500 [ f(Y/Yn ) – f(Z/Zn ) ]
where
f(x) = 7.787x + 16/ ≤ x <
f(x) = x1/ ≤ x ≤ 1
Decouples incremental changes in Yy, Cx, Cz at white point on (L,a,b) values
Transformation is sRGB  CIEXYZ  YyCx Cz

10. HVS Filtering Filter chrominance channels more aggressively
Luminance frequency response [Näsänen and Sullivan, 1984]
L average luminance of display
weighted radial spatial frequency
Chrominance frequency response [Kolpatzik and Bouman, 1992]
Chrominance response allows more low frequency chromatic error not to be perceived vs. luminance response

11. Search Algorithm [Li, Allebach 2002]
Let p be the vector of “filter weights”
a  (0,255), k = (k1, k2),
define the neighborhood of
Set p(0) to be the optimal value from the last designed “3 tuple”
(First choice: p(0) Floyd-Steinberg)
hw = 1/16 , i = 0
while (p(i)  p(i-1)) {
find p(i+1)  Nhw (p(i)) that minimizes the total squared error (TSE)
i  i + 1 }
while (hw  1/256) {
hw  hw/2
find p(i+1)  Nhw (p(i)) that minimizes TSE

12. a) Original b) FS Halftone c) TDED Serpentine
Results a) b) c)
a) Original b) FS Halftone c) TDED Serpentine

13. d) TDED Raster e) TDED 2-row serp f) Detail of FS (left) and TDED

14. Original
House Image

15. Floyd Steinberg Halftone

16. TDED Halftone

17. Conclusion Color TDED Scan path choice Future Work
Worms and other directional artifacts removed
False textures eliminated
Visibility of “halftone-pattern” minimized (HVS model)
More accurate color rendering at extreme levels
Scan path choice
Serpentine scan gives best results (not parallelizable)
2-row serpentine gives comparable quality
Future Work
Design “optimum” matrix valued filters ?
Look for better HVS models/transformations

18. Back Up Slides
HVS model details, Monochrome images  Yy, Cx planes of color halftones

19. Floyd Steinberg Yy component

20. Floyd Steinberg Cx component

21. TDED Yy component

22. TDED Cx component

23. HVS Filtering contd…. frequency [Sullivan, Ray, Miller 1991]
Role of frequency weighting
weighting by a function of angular spatial
frequency [Sullivan, Ray, Miller 1991]
where p = (u2+v2)1/2 and
w – symmetry parameter
reduces contrast sensitivity at odd multiples of 45 degrees
equivalent to dumping the luminance error across the diagonals
where the eye is least sensitive.