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Complex Wavelets
Arnab Ghoshal
600.658 - Seminar on Shape Analysis and Retrieval
CS658: Seminar on Shape Analysis and Retrieval Complex Wavelets 1 of 37
Outline
Outline
1 Discrete Wavelet TransformBasics of DWTAdvantages and Limitations
2 Dual-Tree Complex Wavelet TransformThe Hilbert Transform ConnectionHilbert Transform Pairs of Wavelet Bases
3 Results1-D Signals2-D Signals
CS658: Seminar on Shape Analysis and Retrieval Complex Wavelets 2 of 37
DWTDual-Tree Wavelets
ResultsReference
Basics of DWTAdvantages and Limitations
Outline
1 Discrete Wavelet TransformBasics of DWTAdvantages and Limitations
2 Dual-Tree Complex Wavelet TransformThe Hilbert Transform ConnectionHilbert Transform Pairs of Wavelet Bases
3 Results1-D Signals2-D Signals
CS658: Seminar on Shape Analysis and Retrieval Complex Wavelets 3 of 37
DWTDual-Tree Wavelets
ResultsReference
Basics of DWTAdvantages and Limitations
How do Electrical Engineers Dream of Wavelets?
Filter banks
Use a set of filters to analyze thefrequency content of a signal
Lots of redundant information!
Can we do better?
Decimated filter banks
Subsample the frequency bands
Same length as that of the originalsignal (|γ|+ |λ| = |X | = |Y |)Theoretically possible to designfilters (h̃, g̃ , h, g) which will giveperfect reconstruction
CS658: Seminar on Shape Analysis and Retrieval Complex Wavelets 4 of 37
DWTDual-Tree Wavelets
ResultsReference
Basics of DWTAdvantages and Limitations
How do Electrical Engineers Dream of Wavelets?
Filter banks
Use a set of filters to analyze thefrequency content of a signal
Lots of redundant information!
Can we do better?
Decimated filter banks
Subsample the frequency bands
Same length as that of the originalsignal (|γ|+ |λ| = |X | = |Y |)Theoretically possible to designfilters (h̃, g̃ , h, g) which will giveperfect reconstruction
CS658: Seminar on Shape Analysis and Retrieval Complex Wavelets 5 of 37
DWTDual-Tree Wavelets
ResultsReference
Basics of DWTAdvantages and Limitations
How do Electrical Engineers Dream of Wavelets?
http://perso.wanadoo.fr/polyvalens/clemens/lifting
Filter banks
Use a set of filters to analyze thefrequency content of a signal
Lots of redundant information!
Can we do better?
Decimated filter banks
Subsample the frequency bands
Same length as that of the originalsignal (|γ|+ |λ| = |X | = |Y |)Theoretically possible to designfilters (h̃, g̃ , h, g) which will giveperfect reconstruction
CS658: Seminar on Shape Analysis and Retrieval Complex Wavelets 6 of 37
DWTDual-Tree Wavelets
ResultsReference
Basics of DWTAdvantages and Limitations
How do Electrical Engineers Dream of Wavelets?
http://perso.wanadoo.fr/polyvalens/clemens/lifting
Filter banks
Use a set of filters to analyze thefrequency content of a signal
Lots of redundant information!
Can we do better?
Decimated filter banks
Subsample the frequency bands
Same length as that of the originalsignal (|γ|+ |λ| = |X | = |Y |)Theoretically possible to designfilters (h̃, g̃ , h, g) which will giveperfect reconstruction
CS658: Seminar on Shape Analysis and Retrieval Complex Wavelets 7 of 37
DWTDual-Tree Wavelets
ResultsReference
Basics of DWTAdvantages and Limitations
Fully Decimated Wavelet Tree
[1]
CS658: Seminar on Shape Analysis and Retrieval Complex Wavelets 8 of 37
DWTDual-Tree Wavelets
ResultsReference
Basics of DWTAdvantages and Limitations
Fully Decimated Wavelet Tree
[1]
CS658: Seminar on Shape Analysis and Retrieval Complex Wavelets 9 of 37
DWTDual-Tree Wavelets
ResultsReference
Basics of DWTAdvantages and Limitations
Wavelet Bases
Orthogonal filter-bank
Even length filters (length = M + 1)
Shift orthogonal:∑
n ha0(n)ha
0(n + 2k) = δ(k)
Alternating flip: ha1(n) = (−1)nha
0(M − n)
Scaling and wavelet functions
Scaling function: φa(t) =√
2∑
n ha0(n)φa(2t − n)
Wavelet function: ψa(t) =√
2∑
n ha1(n)φa(2t − n)
Caveat
Wavelets need not be orthogonal!
In fact, JPEG2000 uses 5/3 and 9/7 bi-orthogonal filters
CS658: Seminar on Shape Analysis and Retrieval Complex Wavelets 10 of 37
DWTDual-Tree Wavelets
ResultsReference
Basics of DWTAdvantages and Limitations
Wavelet Bases
Orthogonal filter-bank
Even length filters (length = M + 1)
Shift orthogonal:∑
n ha0(n)ha
0(n + 2k) = δ(k)
Alternating flip: ha1(n) = (−1)nha
0(M − n)
Scaling and wavelet functions
Scaling function: φa(t) =√
2∑
n ha0(n)φa(2t − n)
Wavelet function: ψa(t) =√
2∑
n ha1(n)φa(2t − n)
Caveat
Wavelets need not be orthogonal!
In fact, JPEG2000 uses 5/3 and 9/7 bi-orthogonal filters
CS658: Seminar on Shape Analysis and Retrieval Complex Wavelets 11 of 37
DWTDual-Tree Wavelets
ResultsReference
Basics of DWTAdvantages and Limitations
Wavelet Bases
Orthogonal filter-bank
Even length filters (length = M + 1)
Shift orthogonal:∑
n ha0(n)ha
0(n + 2k) = δ(k)
Alternating flip: ha1(n) = (−1)nha
0(M − n)
Scaling and wavelet functions
Scaling function: φa(t) =√
2∑
n ha0(n)φa(2t − n)
Wavelet function: ψa(t) =√
2∑
n ha1(n)φa(2t − n)
Caveat
Wavelets need not be orthogonal!
In fact, JPEG2000 uses 5/3 and 9/7 bi-orthogonal filters
CS658: Seminar on Shape Analysis and Retrieval Complex Wavelets 12 of 37
DWTDual-Tree Wavelets
ResultsReference
Basics of DWTAdvantages and Limitations
Fundamental Properties of Wavelets
Advantages
Non-redundant orthonormal bases, perfect reconstruction
Multiresolution decomposition, attractive for object matching
Fast O(n) algorithms with short filters, compact support
Limitations
Lack of adaptivity
Poor frequency resolution
Shift variance
Wavelet coefficients behave unpredictably when the signal isshiftedDWT lacks phase information
CS658: Seminar on Shape Analysis and Retrieval Complex Wavelets 13 of 37
DWTDual-Tree Wavelets
ResultsReference
Basics of DWTAdvantages and Limitations
Fundamental Properties of Wavelets
Advantages
Non-redundant orthonormal bases, perfect reconstruction
Multiresolution decomposition, attractive for object matching
Fast O(n) algorithms with short filters, compact support
Limitations
Lack of adaptivity
Poor frequency resolution
Shift variance
Wavelet coefficients behave unpredictably when the signal isshiftedDWT lacks phase information
CS658: Seminar on Shape Analysis and Retrieval Complex Wavelets 14 of 37
DWTDual-Tree Wavelets
ResultsReference
Basics of DWTAdvantages and Limitations
Shift Variance of DWT
CS658: Seminar on Shape Analysis and Retrieval Complex Wavelets 15 of 37
DWTDual-Tree Wavelets
ResultsReference
Basics of DWTAdvantages and Limitations
Shift Variance of DWT
CS658: Seminar on Shape Analysis and Retrieval Complex Wavelets 16 of 37
DWTDual-Tree Wavelets
ResultsReference
Hilbert TransformComplex Wavelet Bases
Outline
1 Discrete Wavelet TransformBasics of DWTAdvantages and Limitations
2 Dual-Tree Complex Wavelet TransformThe Hilbert Transform ConnectionHilbert Transform Pairs of Wavelet Bases
3 Results1-D Signals2-D Signals
CS658: Seminar on Shape Analysis and Retrieval Complex Wavelets 17 of 37
DWTDual-Tree Wavelets
ResultsReference
Hilbert TransformComplex Wavelet Bases
Frequency Ananlysis of Real Signals
Every real signal has equal amounts of positive and negativefrequency components
cos(ωt) = e jωt+e−jωt
2
sin(ωt) = e jωt−e−jωt
2
Hilbert transform: y(t) = H{x(t)}, iff
Y (ω) = −jX (ω), ω > 0 and Y (ω) = jX (ω), ω < 0y(θ) = x(θ − π
2 ), θ > 0 and y(θ) = x(θ + π2 ), θ < 0
The signal z(t) = x(t) + jH{x(t)} has no negativefrequencies (analytic signal)
Fourier Transform: Real sin and cos form a Hilberttransform pair!
CS658: Seminar on Shape Analysis and Retrieval Complex Wavelets 18 of 37
DWTDual-Tree Wavelets
ResultsReference
Hilbert TransformComplex Wavelet Bases
Frequency Ananlysis of Real Signals
Every real signal has equal amounts of positive and negativefrequency components
cos(ωt) = e jωt+e−jωt
2
sin(ωt) = e jωt−e−jωt
2
Hilbert transform: y(t) = H{x(t)}, iff
Y (ω) = −jX (ω), ω > 0 and Y (ω) = jX (ω), ω < 0y(θ) = x(θ − π
2 ), θ > 0 and y(θ) = x(θ + π2 ), θ < 0
The signal z(t) = x(t) + jH{x(t)} has no negativefrequencies (analytic signal)
Fourier Transform: Real sin and cos form a Hilberttransform pair!
CS658: Seminar on Shape Analysis and Retrieval Complex Wavelets 19 of 37
DWTDual-Tree Wavelets
ResultsReference
Hilbert TransformComplex Wavelet Bases
Frequency Ananlysis of Real Signals
Every real signal has equal amounts of positive and negativefrequency components
cos(ωt) = e jωt+e−jωt
2
sin(ωt) = e jωt−e−jωt
2
Hilbert transform: y(t) = H{x(t)}, iff
Y (ω) = −jX (ω), ω > 0 and Y (ω) = jX (ω), ω < 0y(θ) = x(θ − π
2 ), θ > 0 and y(θ) = x(θ + π2 ), θ < 0
The signal z(t) = x(t) + jH{x(t)} has no negativefrequencies (analytic signal)
Fourier Transform: Real sin and cos form a Hilberttransform pair!
CS658: Seminar on Shape Analysis and Retrieval Complex Wavelets 20 of 37
DWTDual-Tree Wavelets
ResultsReference
Hilbert TransformComplex Wavelet Bases
Frequency Ananlysis of Real Signals
Every real signal has equal amounts of positive and negativefrequency components
cos(ωt) = e jωt+e−jωt
2
sin(ωt) = e jωt−e−jωt
2
Hilbert transform: y(t) = H{x(t)}, iff
Y (ω) = −jX (ω), ω > 0 and Y (ω) = jX (ω), ω < 0y(θ) = x(θ − π
2 ), θ > 0 and y(θ) = x(θ + π2 ), θ < 0
The signal z(t) = x(t) + jH{x(t)} has no negativefrequencies (analytic signal)
Fourier Transform: Real sin and cos form a Hilberttransform pair!
CS658: Seminar on Shape Analysis and Retrieval Complex Wavelets 21 of 37
DWTDual-Tree Wavelets
ResultsReference
Hilbert TransformComplex Wavelet Bases
Frequency Ananlysis of Real Signals
Every real signal has equal amounts of positive and negativefrequency components
cos(ωt) = e jωt+e−jωt
2
sin(ωt) = e jωt−e−jωt
2
Hilbert transform: y(t) = H{x(t)}, iff
Y (ω) = −jX (ω), ω > 0 and Y (ω) = jX (ω), ω < 0y(θ) = x(θ − π
2 ), θ > 0 and y(θ) = x(θ + π2 ), θ < 0
The signal z(t) = x(t) + jH{x(t)} has no negativefrequencies (analytic signal)
Fourier Transform: Real sin and cos form a Hilberttransform pair!
CS658: Seminar on Shape Analysis and Retrieval Complex Wavelets 22 of 37
DWTDual-Tree Wavelets
ResultsReference
Hilbert TransformComplex Wavelet Bases
Hilbert Transform Pairs of Wavelet Bases
We want to find ψb(t) = H{ψa(t)}Very hard to find ψb(t) having finite support
Design both the wavelets simultaneously [2]
Assume Hb0 (ω) = Ha
0 (ω)e−jθ(ω)
Hilbert transform condition on wavelets and scaling functionsis satisfied by: Hb
0 (ω) = Ha0 (ω)e−j ω
2 ⇔ hb0(n) = ha
0(n − 12 )
The half-sample delay can only be approximated with finitelength filters [1]
CS658: Seminar on Shape Analysis and Retrieval Complex Wavelets 23 of 37
DWTDual-Tree Wavelets
ResultsReference
Hilbert TransformComplex Wavelet Bases
Hilbert Transform Pairs of Wavelet Bases
We want to find ψb(t) = H{ψa(t)}Very hard to find ψb(t) having finite support
Design both the wavelets simultaneously [2]
Assume Hb0 (ω) = Ha
0 (ω)e−jθ(ω)
Hilbert transform condition on wavelets and scaling functionsis satisfied by: Hb
0 (ω) = Ha0 (ω)e−j ω
2 ⇔ hb0(n) = ha
0(n − 12 )
The half-sample delay can only be approximated with finitelength filters [1]
CS658: Seminar on Shape Analysis and Retrieval Complex Wavelets 24 of 37
DWTDual-Tree Wavelets
ResultsReference
Hilbert TransformComplex Wavelet Bases
Hilbert Transform Pairs of Wavelet Bases
We want to find ψb(t) = H{ψa(t)}Very hard to find ψb(t) having finite support
Design both the wavelets simultaneously [2]
Assume Hb0 (ω) = Ha
0 (ω)e−jθ(ω)
Hilbert transform condition on wavelets and scaling functionsis satisfied by: Hb
0 (ω) = Ha0 (ω)e−j ω
2 ⇔ hb0(n) = ha
0(n − 12 )
The half-sample delay can only be approximated with finitelength filters [1]
CS658: Seminar on Shape Analysis and Retrieval Complex Wavelets 25 of 37
DWTDual-Tree Wavelets
ResultsReference
Hilbert TransformComplex Wavelet Bases
Hilbert Transform Pairs of Wavelet Bases
We want to find ψb(t) = H{ψa(t)}Very hard to find ψb(t) having finite support
Design both the wavelets simultaneously [2]
Assume Hb0 (ω) = Ha
0 (ω)e−jθ(ω)
Hilbert transform condition on wavelets and scaling functionsis satisfied by: Hb
0 (ω) = Ha0 (ω)e−j ω
2 ⇔ hb0(n) = ha
0(n − 12 )
The half-sample delay can only be approximated with finitelength filters [1]
CS658: Seminar on Shape Analysis and Retrieval Complex Wavelets 26 of 37
DWTDual-Tree Wavelets
ResultsReference
Hilbert TransformComplex Wavelet Bases
Hilbert Transform Pairs of Wavelet Bases
We want to find ψb(t) = H{ψa(t)}Very hard to find ψb(t) having finite support
Design both the wavelets simultaneously [2]
Assume Hb0 (ω) = Ha
0 (ω)e−jθ(ω)
Hilbert transform condition on wavelets and scaling functionsis satisfied by: Hb
0 (ω) = Ha0 (ω)e−j ω
2 ⇔ hb0(n) = ha
0(n − 12 )
The half-sample delay can only be approximated with finitelength filters [1]
CS658: Seminar on Shape Analysis and Retrieval Complex Wavelets 27 of 37
DWTDual-Tree Wavelets
ResultsReference
Hilbert TransformComplex Wavelet Bases
Hilbert Transform Pairs of Wavelet Bases
We want to find ψb(t) = H{ψa(t)}Very hard to find ψb(t) having finite support
Design both the wavelets simultaneously [2]
Assume Hb0 (ω) = Ha
0 (ω)e−jθ(ω)
Hilbert transform condition on wavelets and scaling functionsis satisfied by: Hb
0 (ω) = Ha0 (ω)e−j ω
2 ⇔ hb0(n) = ha
0(n − 12 )
The half-sample delay can only be approximated with finitelength filters [1]
CS658: Seminar on Shape Analysis and Retrieval Complex Wavelets 28 of 37
DWTDual-Tree Wavelets
ResultsReference
Hilbert TransformComplex Wavelet Bases
1-D Dual-Tree Wavelets
Picture courtesy: Dr. Trac D. Tran (520.646 Lecture notes)
CS658: Seminar on Shape Analysis and Retrieval Complex Wavelets 29 of 37
DWTDual-Tree Wavelets
ResultsReference
1-D Signals2-D Signals
Outline
1 Discrete Wavelet TransformBasics of DWTAdvantages and Limitations
2 Dual-Tree Complex Wavelet TransformThe Hilbert Transform ConnectionHilbert Transform Pairs of Wavelet Bases
3 Results1-D Signals2-D Signals
CS658: Seminar on Shape Analysis and Retrieval Complex Wavelets 30 of 37
DWTDual-Tree Wavelets
ResultsReference
1-D Signals2-D Signals
Level 2 Wavelets for Shifted Step Functions
CS658: Seminar on Shape Analysis and Retrieval Complex Wavelets 31 of 37
DWTDual-Tree Wavelets
ResultsReference
1-D Signals2-D Signals
Original Signal
CS658: Seminar on Shape Analysis and Retrieval Complex Wavelets 32 of 37
DWTDual-Tree Wavelets
ResultsReference
1-D Signals2-D Signals
DWT: Level 1
CS658: Seminar on Shape Analysis and Retrieval Complex Wavelets 33 of 37
DWTDual-Tree Wavelets
ResultsReference
1-D Signals2-D Signals
Complex Wavelets: Level 1
CS658: Seminar on Shape Analysis and Retrieval Complex Wavelets 34 of 37
DWTDual-Tree Wavelets
ResultsReference
1-D Signals2-D Signals
DWT: Level 3
CS658: Seminar on Shape Analysis and Retrieval Complex Wavelets 35 of 37
DWTDual-Tree Wavelets
ResultsReference
1-D Signals2-D Signals
Complex Wavelets: Level 3
CS658: Seminar on Shape Analysis and Retrieval Complex Wavelets 36 of 37
DWTDual-Tree Wavelets
ResultsReference
References
N. G. KingsburyComplex wavelets for shift invariant analysis and filtering of signals
Journal of Applied and Computational Harmonic Analysis,
10(3):234-253,2001.
I. W. SelesnickHilbert Transform Pairs of Wavelet Bases
IEEE Signal Processing Letters, 8(6):170-173, June 2001.
I. DaubechiesTen Lectures on Wavelets
CBMS-NSF Regional Conference Series in Applied Mathematics,
SIAM, Vol. 61, Philadelphia, PA 1992.
CS658: Seminar on Shape Analysis and Retrieval Complex Wavelets 37 of 37