Shift Theorem (2-D CWT vs QWT) - PowerPoint PPT Presentation

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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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2-D complex wavelet

• 2-D CWT basis functions

45 degree

-45 degree

Complex Wavelets

2-D CWT

[Kingsbury,Selesnick,...]

• Other subbands for LH and HL (equation)

• Six directional subbands (15,45,75 degrees)

y

x

QFT phase

where

QWT of real signals

• QFT Plancharel Theorem:

real window

where

• QFT inner product

• Proof uses QFT convolution Theorem

v

LH subband

HH subband

HL subband

u

QWT as Local QFT Analysis

• For quaternion basis function :

quaternion bases

where

• inner product

QWT Edge response

v

QWT basis

• Edge QFT:

u

QFT spectrum of edge

• 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

u

leakage