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

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

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

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

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

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!

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

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]

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]

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

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

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

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

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

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

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

“Real” 2-D Wavelet Transform v u v u

HH LL LH HL

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

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

Quaternion Wavelets v u

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

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

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

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

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

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

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

Image Flow Example Image Shifts dxdx dydy Image Flow

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

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

Image Flow Estimation Result

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)

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,…]