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