Shift Theorem (2-D CWT vs QWT)

2-D Hilbert Transform (wavelet) +1 +j -j Hx Hy Hy +1 -1 +j +1 +1 -j +j -j +1 +j -j +1 Hx

2-D complex wavelet 2-D CWT basis functions +1 +j -j +1 +j -j +j -j 45 degree +1 +j -j +j -j -45 degree

2-D CWT Other subbands for LH and HL (equation) [Kingsbury,Selesnick,...] Other subbands for LH and HL (equation) Six directional subbands (15,45,75 degrees) Complex Wavelets

Challenge in Coherent Processing – phase wrap-around y x QFT phase where

QWT of real signals QFT Plancharel Theorem: where QFT inner product real window where People will ask! QFT inner product Proof uses QFT convolution Theorem

QWT as Local QFT Analysis For quaternion basis function : quaternion bases where v u HH subband HL subband LH subband Single-quadrant QFT inner product

QWT Edge response   v u Edge QFT: QFT inner product with QWT bases QWT basis  u  QFT spectrum of edge Edge QFT: QFT inner product with QWT bases Spectral center:

QWT Phase for Edges Behavior of third phase angle: denotes energy ratio between positive and leakage quadrant Frequency leakage / aliasing Shift theorem unaffected v positive quadrant S1 u leakage quadrant leakage

QWT Third Phase Behavior of third phase angle Mixing of signal orientations Texture analysis