1. Error Diffusion (ED) Li Yang Campus Norrköping (ITN), University of Linköping

2. Fundamental concepts Threshold error feedback Input -> threshold -> error -> input ->... It is adaptive algorithm; It takes neiborghood information into account to determine the output value. Different from dither matrix.

3. A flow chart of ED

4. A historical review Sigma-delta modulation :Analog-to- digital conversion of 1-D audio signal (Inose and Yasuda, 1963); Error diffusion: 2-D for halftoning (Floyd and Steinberg, 1975); Massive of following studies: theoretical studies and practical applications about ED.

5. Two ways for error diffusion (descriptions) Standard ED: error is diffused from p(i,j) to its neighbours directly after its halftoning -> modified input …; Systematic error compensation: Halftone for the original input, collect the error from its neighbours and modify the output of the pixel according to ED filter. They are mathematically equivalent.

6. Two ways for error diffusion (error manipulation)

7. Two ways for error diffusion (process diagram)

8. Mathematical description of error diffusion (spatial domain)

9. Mathematical description of error diffusion (frequency domain)

10. Characteristics of the error filter, is a high pass filter: it lets only high spatial frequency components of the texture noise in the error spectrum pass into the output spectrum,

11. Some examples of error filters Floyd-Steinberg Filter X 7/16 3/16 5/16 1/16 Stucki error filter X 8/42 4/42 2/42 4/42 8/42 4/42 2/42 1/42 2/42 4/42 2/42 1/42

12. Applications and problems Worm artifacts

14. Topics of research Optimum error filter design; Stochastic error filter perturbation; Modification of raster direction and space filling-path; Threshold modulation; Image adaptive error diffusion; Model based error diffusion;

15. Optimum error filter design Goal: to minimize the difference between the input- and output-images in a human vision perspective; Mathematics:

16. Stochastic error filter perturbation Add random noise to the weights of the error filter(Schreiber 1981, Woo 1984); Some examples

19. Modification of raster direction

20. Various space filling-path

21. Threshold modulation Adopt to non-constant threshold values; Add a set of random values to the threshold: t=0.5  0.5+t(m,n); Varying the threshold spatially;

23. Image adaptive error diffusion Based on the observation: the error spectrum distribution depends on the local tone values of the input image (Zeggel and Bryngdahl, 1994) See examples

24. Image is scaled between 0 and 1

25. Image is scaled between 0 and 0.1

26. Image is scaled between 0.2 and 0.3

27. Image adaptive error diffusion (cont.)