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Coherent Multiscale Image Processing using Quaternion Wavelets

Coherent Multiscale Image Processing using Quaternion Wavelets. Wai Lam Chan M.S. defense. Committee: Hyeokho Choi, Richard Baraniuk, Michael Orchard. Image Location Information. Edges:. “location” “orientation”. Goal : Encode/Estimate location information

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Coherent Multiscale Image Processing using Quaternion Wavelets

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  1. Coherent Multiscale Image Processing using Quaternion Wavelets Wai Lam Chan M.S. defense Committee: Hyeokho Choi, Richard Baraniuk, Michael Orchard

  2. Image Location Information Edges: “location” “orientation” Goal: Encode/Estimate location information from phase (coherent processing)

  3. Location and Phase d • Fourier phase to encode/analyze location • Linear phase change as signal shifts (Fourier Shift Theorem)

  4. Image Geometry and Phase Rich’s picture Rich’s phase-only picture Rich’s phase + Cameraman’s amplitude • Global phase (no localization)

  5. Local Fourier Analysis • Local Fourier analysis for “location” • Short time Fourier transform (Gabor analysis) • Local Fourier phase (relates to local geometry)

  6. But conventional discrete wavelets are “Real” Lack of phase to encode geometry! Wavelet Analysis • “Multiscale” analysis • Sparse representation of piecewise smooth signals • Orthonormal basis / tight frame • Fast computation by filter banks

  7. Short Time Fourier vs. Wavelet Short time Fourier Wavelet “Real”

  8. Phase in Wavelets • Development of dual-tree complex wavelet transform (DT-CWT) DWT [e.g., Daubechies] 1-D HT / analytic signal 1-D DT-CWT [Lina, Kingsbury, Selesnick,…]

  9. Phase in Wavelets • Development of DT-CWT and quaternion wavelet transform (DT-QWT) DWT [e.g., Daubechies] 1-D HT and analytic signal 1-D DT-CWT [Lina, Kingsbury, Selesnick,…] 2-D HT and analytic signal (complex / quaternion) DT-QWT 2-D DT-CWT [Chan, Choi, Baraniuk] [Kingsbury, Selesnick,…]

  10. Major Thesis Contributions • QWT Construction • QWT Properties • Magnitude-phase representation • Shift Theorem • QWT Applications • Edge Estimation • Image Flow Estimation

  11. Phase for Wavelets ? • Need to have quadrature component • phase shift of

  12. Complex Wavelet • Complex wavelet transform (CWT) • [Kingsbury,Selesnick,Lina]

  13. 1-D Complex Wavelet Transform (CWT) Complex (analytic) wavelet Hilbert Transform wavelet = + j* +j -j +2

  14. 2-D Complex Fourier Transform (CFT) • Phase ambiguity • cannot obtain from phase shift

  15. Quaternion Fourier Transform (QFT) [Bülow et al.] • Separate4 quadrature components • Organize as quaternion • Quaternions: • Multiplication rules: and

  16. QFT Phase • Quaternion phase angles: Shift theorem • QFT shift theorem: • invariant to signal shift • linear to signal shift • encodes mixing of signal orientations

  17. “Real” 2-D Wavelet Transform v u v u

  18. “Real” 2-D Wavelet Transform

  19. “Real” 2-D Wavelet Transform HH HL LH LL

  20. v v v v u u u u 2-D Hilbert Transform HT in u HT in v HT in both

  21. v v v v u u u u 2-D Hilbert Transform

  22. Quaternion Wavelets v u

  23. +j -j +j -j +1 -1 -j -j -1 +1 +j +j Quaternion Wavelets Hx Hy Hy Hx

  24. Quaternion Wavelet Transform (QWT) • Quaternion basis function (HH) • 3 subbands (HH, HL, LH) v v v HH subband HL subband LH subband u u u

  25. v QWT bases x u QWT Shift Theorem • Shift theorem approximately holds for QWT • where denotes the spectral center • Estimate (dx, dy) from • Edge estimation • Image flow estimation

  26. QWT Phase for Edges • non-unique (dx, dy) for edges • Phase shift • non-unique : • (no change) d dy  dx

  27.  QWT Magnitude for Edges v QWT basis  u  Edge model spectrum of edge HL subband magnitudes HH subband magnitudes

  28. QWT Edge Estimation • Edge parameter (offset/orientation) estimation •  edge offset • QWT magnitude  edge orientation

  29. Multiscale Image Flow Estimation • Disparity estimation in QWT domain • QWT Shift Theorem • Multiscale phase-wrap correction • Efficient computation (O(N))

  30. Image Flow Example dy dx Image Shifts Image Flow

  31. Multiscale Estimation Algorithm • Step 1: Estimate from change in QWT phase for each image block • Step 2: Estimate (dx, dy) for each scale • Bilinear Interpolation • Multiscale phase unwrapping algorithm • Average over previous scale and subband estimates to improve estimation

  32. Multiscale Estimation Advantages • Multiscale phase unwrapping algorithm • Combine scale and subband estimates • to improve estimation dy coarse scale dx fine scale

  33. Image Flow Estimation Result

  34. Summary • Development of DT-CWT and quaternion wavelet transform (DT-QWT) DWT [e.g., Daubechies] 1-D HT and analytic signal 1-D DT-CWT [Lina, Kingsbury, Selesnick,…] 2-D HT and analytic signal (complex / quaternion) DT-QWT 2-D DT-CWT [Chan, Choi, Baraniuk] [Kingsbury, Selesnick,…]

  35. Conclusions • Developed QWT for image analysis • Fast, “multiscale” • QWT phase and Shift Theorem • Multiscale flow estimation through QWT phase • Local QFT analysis (details in thesis) • Future Directions • Hypercomplex wavelets (3-D or higher) • Image compression [Ates,Orchard,…]

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