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Inzell, Germany, September 17-21, 2007

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

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Inzell, Germany, September 17-21, 2007

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

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

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

  4. Beamlet, wedgelet – geometrical wavelets

  5. Wedgelets’ dictionary (Donoho D., 1999) MW(Si,j) – number of straight wedgelets on Si,j

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

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

  8. Conic curves parabola Second order curves: ellipse hyperbola

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

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

  11. Optimal image approximation (1) Optimal approximation is the solution of the problem: 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. Wedgelet ensuring the smallest error Processing of all nodes Bottom-up tree prunning algorithm Full quadtree Optimal quadtree Optimal image approximation (2)

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

  16. Image coding

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

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

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

  20. Experimaental results

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

  22. Denoising

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

  24. Experimental results – denoising (1)

  25. Experimental results – denoising (2)

  26. Edge detection

  27. Edge detection - geometry

  28. Edge detection - multiresolution

  29. Edge detection – noise resistance

  30. Summary 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

  31. 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" ofInternational Journal of Signal and Imaging Systems Engineering, accepted (2007).

  32. 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).

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

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