Tone Dependent Color Error Diffusion Halftoning

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

2. current pixel difference threshold u(m) x(m) b(m) _ + Transfer functions 7/16 _ + 3/16 5/16 1/16 e(m) weights shape error compute error 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

3. Direct Binary Search [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 threshold modulation b(m) x(m) _ + _ + Tone dependent error filter Midtone regions e(m) FFT DBS pattern for graylevel x Halftone pattern for graylevel x FFT Tone Dependent Error Diffusion • Train error diffusionweights and thresholdmodulation[Li & Allebach, 2002] 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. Yy HVS Luminance Frequency Response Total Squared Error (TSE)  Cx HVS Chrominance Frequency Response HVS Chrominance Frequency Response Cz Perceptual Error Metric • 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 – 116 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 + 16/116 0 ≤ x < 0.008856 f(x) = x1/3 0.008856 ≤ 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 thevector of “filter weights” a  (0,255), k = (k1, k2), 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 i  i + 1 } define the neighborhood of

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

13. d) e) f) 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 • 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…. • 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.