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Shift Theorem (2-D CWT vs QWT)

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- 2-D CWT basis functions

45 degree

-45 degree

Complex Wavelets

[Kingsbury,Selesnick,...]

- Other subbands for LH and HL (equation)
- Six directional subbands (15,45,75 degrees)

y

x

QFT phase

where

- QFT Plancharel Theorem:

real window

where

- QFT inner product

- Proof uses QFT convolution Theorem

v

LH subband

HH subband

HL subband

u

- For quaternion basis function :

quaternion bases

where

- Single-quadrant QFT
- inner product

v

QWT basis

- Edge QFT:

u

QFT spectrum of edge

- QFT inner product with QWT bases

- Spectral center:

- 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

- Behavior of third phase angle
- Mixing of signal orientations
- Texture analysis