Shift Theorem (2-D CWT vs QWT)
1
2-D Hilbert Transform (wavelet)
+1
+1
+j
-j
+1
+1
+j
-j
Hx
Hy
Hy
-j
-j
+1
+1
+j
+1
-j
-1
+j
+j
+1
+j
-1
+1
+1
-j
Hx
2
2-D complex wavelet
• 2-D CWT basis functions
+1
-j
45 degree
+1
+j
+1
+1
+j
+1
-j
+1
+j
+1
+1
-j
-j
-45 degree
+j
3
2-D CWT
[Kingsbury,Selesnick,...]
• Other subbands for LH and HL (equation)
• Six directional subbands (15,45,75 degrees)
Complex Wavelets
4
Challenge in Coherent Processing –
phase wrap-around
y
x
QFT phase
where
5
QWT of real signals
• QFT Plancharel Theorem:
real window
where
• QFT inner product
• Proof uses QFT convolution Theorem
6
QWT as Local QFT Analysis
• For quaternion basis function
:
quaternion bases
where
v
LH subband
HH subband
• Single-quadrant QFT
inner product
HL subband
u
7
QWT Edge response
v


• Edge QFT:
QWT basis
u
QFT spectrum of edge
• QFT inner product with QWT bases
• Spectral center:
8
QWT Phase for Edges
• Behavior of third phase angle:
•
denotes energy ratio between positive and
leakage quadrant
v
• Frequency leakage / aliasing
positive
quadrant S1
• Shift theorem unaffected
u
leakage
leakage
quadrant
9
QWT Third Phase
• Behavior of third phase angle
• Mixing of signal orientations
• Texture analysis
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