1. Oriented Wavelet 國立交通大學電子工程學系 陳奕安 2007.5.9

2. Outline Background Background Beyond Wavelet Beyond Wavelet Simulation Result Simulation Result Conclusion Conclusion

3. Outline Background Wavelet Review Wavelet Review The Failure of wavelet The Failure of wavelet Beyond Wavelet Beyond Wavelet Simulation Result Simulation Result Conclusion Conclusion

4. Wavelet Review Signal Decomposition: Signal Decomposition: Equal temporal and spatial resolutions Equal temporal and spatial resolutions “Natural” trade-off of temporal and spatial resolutions “Natural” trade-off of temporal and spatial resolutions(Wavelet)

5. Wavelet Review Wavelet Decomposition: Wavelet Decomposition: 1-D wavelet transform 1-D wavelet transform 2-D wavelet transform can be obtained from a 2-D wavelet transform can be obtained from a separable extension of 1-D transform separable extension of 1-D transform

6. The failure of wavelet 1-D: Wavelets are well adapted to singularities 1-D: Wavelets are well adapted to singularities 2-D: 2-D: Separable wavelets are only well adapted to point- singularity Separable wavelets are only well adapted to point- singularity However, in line- and curve-singularities … However, in line- and curve-singularities …

7. The inefficiency of wavelet Wavelet: fails to recognize that Wavelet: fails to recognize that boundary is smooth New: require challenging New: require challenging non-separable constructions

8. Outline Background Background Beyond Wavelet Curvelet Curvelet Contourlet Contourlet Bandelet Bandelet Oriented Wavelete Oriented Wavelete Simulation Result Simulation Result Conclusion Conclusion

9. Curvelets Curvelets can be interpreted as a grouping of nearby wavelet basis functions into linear structures so that they can capture the smooth discontinuity curve more efficiently Curvelets can be interpreted as a grouping of nearby wavelet basis functions into linear structures so that they can capture the smooth discontinuity curve more efficiently

10. Curvelets First, a standard multiscale decomposition is computed, where the low-pass channel is sub- sampled while the high-pass channel is not. Then, a directional decomposition with a DFB is applied to each high-pass channel.

11. Contourlet

12. 0 1 2 3

13. 7 6 5 4 3 2 1 0

14. 0 1 1 2 2 3 34 4 5 5 6 6 7 78 8 16 9 9 10 11 12 13 14 15 16 10 012 34 5 6 78 15161314 1112 9

15. Curvelets & Contourlet Pros: Pros: They do not require a geometric model of the image. They do not require a geometric model of the image. Cons: Cons: The discrete implementations of curvelet transforms are currently highly redundant. The discrete implementations of curvelet transforms are currently highly redundant.

16. Bandelets Using separable wavelet basis, if no geometric flow Using separable wavelet basis, if no geometric flow Using modified orthogonal wavelets in the flow direction, called bandelets Using modified orthogonal wavelets in the flow direction, called bandelets Quad-tree segmentation Quad-tree segmentation

17. Bandelets Bandelets use a geometric model to describe the discontinuities of the image; is theoretically more efficient than curvelets for compression purposes. Bandelets use a geometric model to describe the discontinuities of the image; is theoretically more efficient than curvelets for compression purposes. They are computationally intensive and have the problem of optimization of the bitrate allocation between the image geometry description and the wavelet coefficients. They are computationally intensive and have the problem of optimization of the bitrate allocation between the image geometry description and the wavelet coefficients.

18. Oriented Wavelete Applying the lifting steps of a 1D wavelet transform in the direction of the image contours Applying the lifting steps of a 1D wavelet transform in the direction of the image contours

19. Oriented Wavelete Using quincunx multi-resolution sampling, the image is filtered along horizontal and vertical or diagonal and anti-diagonal directions. Using quincunx multi-resolution sampling, the image is filtered along horizontal and vertical or diagonal and anti-diagonal directions.

20. Oriented Wavelete Horizontal (red) or vertical (green) filtering directions for the first decomposition level. Horizontal (red) or vertical (green) filtering directions for the first decomposition level. Diagonal '/' (blue) or anti-diagonal '\' (yellow) filtering directions for the second decomposition level. Diagonal '/' (blue) or anti-diagonal '\' (yellow) filtering directions for the second decomposition level.

21. Oriented Wavelete Use quad-tree structure to describe the geometry of the image leading to an efficient representation and a simpler rate-distortion optimization. Use quad-tree structure to describe the geometry of the image leading to an efficient representation and a simpler rate-distortion optimization.

22. Outline Background Background Beyond Wavelet Beyond Wavelet Simulation Result Image compression Image compression Denoising Denoising Conclusion Conclusion

23. Image compression Original lena and JPEG Compression (0.25 bpp) Original lena and JPEG Compression (0.25 bpp) JPEG PSNR 31.8 dB

24. Image compression Separable wavelets and oriented wavelets (0.25 bpp) Separable wavelets and oriented wavelets (0.25 bpp) Oriented wavelets PSNR 34.3 dBSeparable wavelets PSNR 34.3 dB

25. Denoising Noisy lena and separable wavelets Noisy lena and separable wavelets DWT PSNR 29.86 dBNoisy lena PSNR 20.24 dB

26. Denoising Noisy lena and separable wavelets Noisy lena and separable wavelets OWT PSNR 30.41 dBDWT PSNR 29.86 dB

27. Outline Background Background Beyond Wavelet Beyond Wavelet Simulation Result Simulation Result Conclusion

28. OWT has similar complexity as the separable wavelet transform while providing better energy compaction and staying critically sampled. OWT has similar complexity as the separable wavelet transform while providing better energy compaction and staying critically sampled. Filtering along the image contours allows to remove the noise more efficiently than anisotropic techniques like the ones based on separable wavelets. Filtering along the image contours allows to remove the noise more efficiently than anisotropic techniques like the ones based on separable wavelets.

29. Happy Birthday !!

30. Outline Background Background Wavelet Review Wavelet Review Failure of wavelet Failure of wavelet Beyond Wavelet Beyond Wavelet Curvelet Curvelet Contourlet Contourlet Bandelet Bandelet Oriented Wavelete Oriented Wavelete Simulation Result Simulation Result Image compression Image compression Denoising Denoising Conclusion Conclusion