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Shift Theorem (2-D CWT vs QWT)

Shift Theorem (2-D CWT vs QWT). +1. +1. +j. -j. +1. +1. +j. -j. +1. -1. -j. -j. -1. +1. +j. +j. 2-D Hilbert Transform (wavelet). H x. H y. H y. +j. +1. -j. +1. +j. +1. -j. +1. H x. +1. -j. +1. +j. +1. +j. +1. -j. +1. -j. +1. +j. -j. +1. +1. +j.

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Shift Theorem (2-D CWT vs QWT)

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  1. Shift Theorem (2-D CWT vs QWT)

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

  3. +1 -j +1 +j +1 +j +1 -j +1 -j +1 +j -j +1 +1 +j 2-D complex wavelet • 2-D CWT basis functions 45 degree -45 degree

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

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

  6. QWT of real signals • QFT Plancharel Theorem: real window where • QFT inner product • Proof uses QFT convolution Theorem

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

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

  9. 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

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

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