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Oriented Wavelet

Oriented Wavelet. 國立交通大學電子工程學系 陳奕安 2007.5.9. Outline. Background Beyond Wavelet Simulation Result Conclusion. Outline. Background Wavelet Review The Failure of wavelet Beyond Wavelet Simulation Result Conclusion. Wavelet Review. Signal Decomposition:

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Oriented Wavelet

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

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

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

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

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

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

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

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

  9. Curvelets • Curvelets can be interpreted as a grouping of nearby wavelet basis functions into linearstructures 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. Contourlet 3 2 1 0

  13. Contourlet 0 1 2 3 4 5 6 7

  14. Contourlet 11 12 7 8 9 10 11 12 16 13 5 6 15 1 2 14 10 5 8 7 4 3 0 3 4 7 8 1 9 6 3 4 14 2 15 6 5 13 16 0 2 1 12 11 10 9 14 13 15 16

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

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

  19. Oriented Wavelete • 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 . • 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.

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

  23. Image compression • 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 waveletsPSNR 34.3 dB Oriented waveletsPSNR 34.3 dB

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

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

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

  28. Conclusion • 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.

  29. Happy Birthday !!

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

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