Outline

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