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

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

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

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

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

6. QWT of real signals QFT Plancharel Theorem: where QFT inner product
real window
where
People will ask!
QFT inner product
Proof uses QFT convolution Theorem

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

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