1. 1 Inzell, Germany, September 17-21, 2007 Agnieszka Lisowska University of Silesia Institute of Informatics Sosnowiec, POLAND alisow@ux2.math.us.edu.pl Second Order Wedgelets – Efficient Tool in Image Processing

2. 2 Outline Introduction Geometrical wavelets – preliminaries Second order wedgelets...... and their applications in Image coding Denoising Edge detection Summary

3. 3 Geometrical wavelets Wavelets equation (classical wavelets) Wavelets equation (geometrical wavelets)

4. 4 Beamlet, wedgelet – geometrical wavelets

5. 5 Wedgelets’ dictionary (Donoho D., 1999) M W (S i,j ) – number of straight wedgelets on S i,j

6. 6 Beamlets (Donoho D., Huo X., 2000) Platelets (Willett R.M., Nowak R.D., 2001) Modifications of dictionary (1)

7. 7 Surflets (Chandrasekaran V. et al., 2004) Arclets (Führ H. et al., 2005) Modifications of dictionary (2)

8. 8 Conic curves parabola ellipse hyperbola Second order curves:

9. 9 New modification – generalization (2003) M W (S i,j ) – number of straight wedgelets on S i,j D – the number of bits used to code parameter d Second Order Wedgelets

10. 10 Comparison of different kinds of approximation a) wavelets b) wedgelets c) second order wed.  Original image and its decompositions:

11. 11 Optimal approximation is the solution of the problem: Optimal image approximation (1) Solving method: - For every quadtree element the optimal wedgelet function is found from among the given node - Using the bottom-up tree pruning algorithm the optimal subtree is found

12. 12 Optimal image approximation (2) Full quadtree Optimal quadtree Bottom-up tree prunning algorithm Processing of all nodes Wedgelet ensuring the smallest error

13. 13 Speed up of computations 1) Find the best wedgelet w1 within the smaller set of beamlets 1) 2) 2) Find the best wedgelet w2 in neighbourhood of w1 (for example +/- 5 pixels)

14. 14 Fast computation of second order wedgelet 1) Find the best wedgelet w1 2) Find the best second order wedgelet w2 in neighbourhood of w1 (for example +/- 5 pixels from the wedgelet w1) and changing the value of parameter d 1) 2)

15. 15 level 1 level 2 optimal approximation level 3 level 5 quadtree partition Optimal image approximation – example (second order wedgelets)

16. 16 Image coding

17. 17 Image coding with wedgelets no information – internal node – undecorated node – decorated by straight wedgelet

18. 18 Image coding with second order wedgelets no information – internal node – undecorated node – decorated by straight wedgelet – decorated by curved wedgelet

19. 19 Experimental results- coding Artificial image coding: Still image coding ->

20. 20 Experimaental results

21. 21 original image straight wedg. second order wedg. Experimental results - coding PSNR: 31.39 dB 31.45 dB Number of wedg.: 5821 5695 Number of bytes: 14211 14185

22. 22 Denoising

23. 23 Image denoising But, in the case of noisy images, instead of F we have Z:

24. 24

25. 25 Experimental results – denoising (1)

26. 26 Experimental results – denoising (2)

27. 27 Edge detection

28. 28 Edge detection - geometry

29. 29 Edge detection - multiresolution

30. 30 Edge detection – noise resistance

31. 31 The adventages of image coding and processing with the use of second order wedgelets: Improvement of coding effectiveness (0-25% in the case of artificial images and ~1.44% in the case of still images) Better denoising effectiveness in comparison to other known methods (up to 0.5dB) Geometrical multiresolution noise resistant tool in edge detection Summary

32. 32 Main publications [1] Lisowska A. Effective coding of images with the use of geometrical wavelets, Proceedings of Decision Support Systems Conference, Zakopane, Poland, (2003). [2] Lisowska A., Extended Wedgelets - Geometrical Wavelets in Efficient Image Coding, Machine Graphics & Vision, Vol. 13, No. 3, pp. 261-274, (2004). [3] Lisowska A., Bent Beamlets - Efficient Tool in Image Coding, Annales UMCS Informatica AI, Vol. 2, pp. 217-225, (2004). [4] Lisowska A., Intrinsic Dimensional Selective Operator Based on Geometrical Wavelets, Journal of Applied Computer Science, Vol. 12, No. 2, pp.99-112, (2005). [5] Lisowska A., Second Order Wedgelets in Image Coding, Proceedings of EUROCON '07 Conference, Warsaw, Poland, (2007). [6] Lisowska A. Image Denoising with Second Order Wedgelets, Special Issue on "Denoising" of International Journal of Signal and Imaging Systems Engineering, accepted (2007).

33. 33 Bibliography [1] Do M. N., Directional Multiresolution Image Representations, Ph.D. Thesis, Department of Communication Systems, Swiss Federal Institute of Technology Lausanne, November (2001). [2] Donoho D. L., Wedgelets: Nearly-minimax estimation of edges, Annals of Statistics, Vol. 27, pp. 859–897, (1999). [3] Donoho D. L., Huo X., Beamlet Pyramids: A New Form of Multiresolution Analysis, Suited for Extracting Lines, Curves and Objects from Very Noisy Image Data, Proceedings of SPIE, Vol. 4119, (2000). [4] Willet R. M., Nowak R. D., Platelets: A Multiscale Approach for Recovering Edges and Surfaces in Photon Limited Medical Imaging, Technical Report TREE0105, Rice University, (2001). [5] Zetzsche C., Barth E., Fundamental Limits of Linear Filters in the Visual Processing of Two-Dimensional Signals, Vision Research, Vol. 30, pp. 1111- 1117, (1990).

34. 34 And finally... Thank you for your attention http://www.math.us.edu.pl/al