University of Groningen Institute of Mathematics and Computing Science

1. Well Posed non-Iterative Edge and Corner Preserving Smoothing For Artistic Imaging Giuseppe Pápari, Nicolai Petkov, Patrizio Campisi University of Groningen Institute of Mathematics and Computing Science Universitá degli Studi di Roma Tre Dipartimento di Elettronica Applicata

2. Photographical image

3. Output of the proposed operator

4. Input image Gaussian smoothing Proposed operator Contents • Kuwahara Filter and Generalizations • Limitations • Proposed Operator • Results and Comparison • Discussion Smoothing out texture while preserving edges

5. Generic pixel of the input image • Four local averages: • Four local standard deviations: Kuwahara Filter and Generalizations Kuwahara output For each pixel, value of mi that corresponds to the minimum standard deviation

6. (x,y) = 1 Central pixel on the white side of the edge (x,y) = 0 Central pixel on the black side of the edge Kuwahara Filter and Generalizations Edge • Only the most homogeneous region is taken into account. No smoothing across the edge

7. Kuwahara Filter and Generalizations • Local averaging Smoothing • Flipping due toMinimum Variance Criterion  Edge Preserving

8. Artifacts on texture An example Kuwahara Filter and Generalizations Input image Kuwahara output

9. Number and shape of the sub-regions • Pentagons, hexagons, circles • Overlapping • Weighted local averages (reducing the Gibbs phenomenon) • Gaussian-Kuwahara • N local averages and local standard deviations (computed as convolutions) Generalizations Kuwahara Filter and Generalizations • New class of filters (Value and criterion filter structure) • Criterion: minimum standard deviation • Connections with the PDEs theory and morphological analysis

10. Kuwahara Filter and Generalizations • Limitations • Proposed Operator • Results and Comparison • Discussion

11. Not mathematically well defined ? Equal standard deviations si • Artifacts(partially eliminable with weighted averages) Limitations • Devastating instability in presence of noise

12. I(t) w1 w2 • 1D Kuwahara filtering  Two sub-windows w1 and w2 t* t Local averages • I(t) = kt I Input signal I(t)  tT t+T t Negative offset Simple one-dimensional example Limitations

13. I(t) w1 w2 • 1D Kuwahara filtering  Two sub-windows w1 and w2 t* t Local averages • I(t) = kt I Input signal I(t)  tT t+T t Simple one-dimensional example Limitations Negative offset Positive offset

14. I(t) w1 w2 • 1D Kuwahara filtering  Two sub-windows w1 and w2 t* t Local standard deviations • I(t) = kt I Input signal I(t) Local std. dev. s1(t), s2(t)  tT t+T t • Equal standard deviations Local averages m1(t), m2(t) Simple one-dimensional example Limitations

15. Proposed approach Synthetic two-dimensional example Limitations Input image Kuwahara filtering

16. Kuwahara Our approach Shadowed area Depleted edge Natural image example Limitations Gauss-Kuwahara Input image

17. We propose • Different weighting windowswi • A different selection criterion instead of the minimum standard deviation Ill-posedness of the minimum variance criterion. Devastating effects in presence of noisyshadowed areas. Limitations

18. Kuwahara Filter and Generalizations • Limitations • Proposed Operator • Results and Comparison • Discussion

19. N local averages and local standard deviations computed as convolutions Weighting windows Proposed Operator • Gaussian mask divided in N sectors  Nweighting windows

20. Normalization Selection criterion Proposed Operator • Output: • Weighted average of mi • Weights equal to proportional to (si)q(q is a parameter) • High variance  small coefficient (si)q • q Only the minimumsi survives  Criterion and value • No undetermination in case of equal standard deviations!

21. Equal standard deviations: s1 = s2 = … = sN Gaussian smoothing • One standard deviation is equal to zero: sk = 0 • Several values of si are equal to zero  = Arithmetic mean of the corresponding values of mi. Particular cases Proposed Operator

22. Edgeless areas: All std. dev. similar Gaussian smoothing (no Gibbs phenomenon) • EdgeHalf of the sectors have si = 0. The other ones are not considered • Corner preservation An example Proposed Operator • Automatic selection of the prominent sectors

23. Same combination rule with Not equivalent to apply the operator to each color component separately Color images Proposed Operator • 3 sets of local averages and local standard deviations, one for each color component

24. Input image RGB YCrCb L*a*b* Independence on the color space Proposed Operator

25. Local averages Linear transform.  independent Nonlinear transf. almost independent for homogeneous regions Why independence? Proposed Operator

26. Why independence? Proposed Operator • Local averages Linear transform.  independent Nonlinear transf. almost independent for homogeneous regions • Local standard deviations Low for homogeneous regions. The degree of homogeneity of a region does not depend on the color space.

27. Kuwahara Filter and Generalizations • Limitations • Proposed Operator • Results and Comparison • Discussion

28. Existing algorithm for comparison Results and comparison • Kuwahara filter and generalizations • Bilateral filtering • Morphological filters • Median filters

29. Input image

30. Proposed approach

31. Gauss-Kuwahara filter

32. Input image (blurred)

33. Proposed approach (deblurred)

34. Bilateral filtering (not deblurred)

35. Input image

36. Proposed approach

37. Morphological closing (Struct. elem.: Disk of radius 5px)

38. Morphological area open-closing

39. Input image

40. Morphological area open-closing

41. Proposed approach

42. Input image

43. Proposed approach

44. Kuwahara Filter

45. Morphological area open-closing

46. Input image

47. Proposed approach

48. Bilateral Filtering

49. Input image

50. proposed aproach