Shift theorem 2 d cwt vs qwt
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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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Shift theorem 2 d cwt vs qwt

Shift Theorem (2-D CWT vs QWT)


2 d hilbert transform wavelet

+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


2 d complex wavelet

+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


2 d cwt

Complex Wavelets

2-D CWT

[Kingsbury,Selesnick,...]

  • Other subbands for LH and HL (equation)

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


Challenge in coherent processing phase wrap around

Challenge in Coherent Processing – phase wrap-around

y

x

QFT phase

where


Qwt of real signals

QWT of real signals

  • QFT Plancharel Theorem:

real window

where

  • QFT inner product

  • Proof uses QFT convolution Theorem


Qwt as local qft analysis

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


Qwt edge response

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

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

QWT Third Phase

  • Behavior of third phase angle

  • Mixing of signal orientations

  • Texture analysis


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