1. Vision Lab, Dept. of EE, NCTU Jui-Nan Chang 2009.4.6 1

2. Outline  Introduction  Basic Concepts  Properties of Fuzzy Transformation  Filter Generalization Using the FZT and Applications  Conclusion  References 2

3. Introduction (1/2)  Nonlinear signal processing methods - heavy tailed distribution or non-stationary statistics  Spatial & Rank (SR) orderings - center weighted median (CWM) - weighted median (WM) - permutation  Spatial correlation and rank order information crisp (binary) SR relations 3

4. Introduction (2/2)  Fuzzy SR relations - crisp SR relations sample spread (diversity) - fuzzy spatial samples - fuzzy order statistics - fuzzy spatial indexes - fuzzy rank crisp SR space fuzzy SR space fuzzy transformation 4

5. Basic Concepts (1/4) spatial sample crisp SR relations we get order statistic rank index spatial index 5

6. Basic Concepts (2/4)  Combined with spread information - membership functions Gaussian membership function Uniform membership function Triangular membership function Note: they are all monotonically non-decreasing function and 6

7. Basic Concepts (3/4)  Combined with spread information - fuzzy SR relations we get They are represented the weighted averages of the crisp order statistics, spatial samples,spatial indexes and rank indexes. row normalizedcolumn normalized 7

8. Basic Concepts (4/4) Example (Gaussian membership function) fuzzy SR space crisp SR space 8

9. Properties of Fuzzy Transformation Element Invariant Property - the crisp SR relations are fully preserved by the FZT Order Invariant Property - the fuzzy SR space contains SR information consistent with that in the crisp SR space Mean preserving an unbiased operator 9

10. Filter Generalization Using the FZT and Applications  Fuzzy identity filer - remove the blocking artifact with preserving edge - use Gaussian membership function - use MSE criteria to estimate the parameter 10

11. Filter Generalization Using the FZT and Applications  Fuzzy identity filer 11

12. Filter Generalization Using the FZT and Applications  Fuzzy identity filer blocking artifact QF=10result of fuzzy IF 12

13. Filter Generalization Using the FZT and Applications  LUM filter – impulse noise removal (lower-upper-middle) The LUM smoother may cause over smoothing when there are no outliers, or under smoothing when corrupted samples have ranks within the range [k,N-k+1 ] 13

14. Filter Generalization Using the FZT and Applications  FLUM filter – impulse noise removal (fuzzy lower-upper-middle)  The FLUM filter incorporates sample spread information, and thus more effectively identifies true outliers and improve filer performance 14

15. Filter Generalization Using the FZT and Applications  FLUM filter – impulse noise removal 15

16. Filter Generalization Using the FZT and Applications  FLUM filter – impulse noise removal 16

17. Filter Generalization Using the FZT and Applications  FLUM filter – impulse noise removal 5% impulse noise crisp LUM filter fuzzy LUM filter 17

18. Conclusion  FZT retains the consistent SR information of the samples  FZT effectively reflects sample spread information  The FZT is utilized to generalize conventional filters to exploit the joint spatial-rank-spread information  It has potential to be exploited in novel techniques for other signal processing applications 18

19. References  Yao Nie and K. E. Barner, "The fuzzy transformation and its applications in image processing," Image Processing, IEEE Transactions on, vol. 15, pp. 910-927, 2006. 19