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

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

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

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

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

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Wavelet Analysis 1.“Multiscale” analysis 2.Sparse representation of piecewise smooth signals 3.Orthonormal basis / tight frame 4.Fast computation by filter banks But conventional discrete wavelets are “Real” Lack of phase to encode geometry!

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

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

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

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

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

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

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

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

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Quaternion Fourier Transform (QFT) Separate 4 quadrature components Organize as quaternion Quaternions: Multiplication rules: and [Bülow et al.]

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QFT Phase Quaternion phase angles: Shift theorem QFT shift theorem: 1. invariant to signal shift 2. linear to signal shift encodes mixing of signal orientations

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“Real” 2-D Wavelet Transform v u v u

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HH LL LH HL

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

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2-D Hilbert Transform u v u v u v u v

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Quaternion Wavelets v u

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HxHx HyHy +1 +j -j +j -j HyHy +j -j HxHx

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

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Estimate (d x, d y ) from Edge estimation Image flow estimation QWT Shift Theorem Shift theorem approximately holds for QWT where denotes the spectral center v u QWT bases x

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QWT Phase for Edges non-unique (d x, d y ) for edges Phase shift non-unique : (no change) d dxdx dydy

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v u QWT basis QWT Magnitude for Edges Edge model HL subband magnitudesHH subband magnitudes spectrum of edge

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QWT Edge Estimation Edge parameter (offset/orientation) estimation – edge offset –QWT magnitude edge orientation

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Multiscale Image Flow Estimation Disparity estimation in QWT domain 1.QWT Shift Theorem 2.Multiscale phase-wrap correction 3.Efficient computation (O(N))

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Image Flow Example Image Shifts dxdx dydy Image Flow

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

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Multiscale Estimation Advantages Multiscale phase unwrapping algorithm Combine scale and subband estimates to improve estimation dxdx dydy coarse scale fine scale

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Image Flow Estimation Result

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

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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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