shift theorem 2 d cwt vs qwt n.
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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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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)
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