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Orthogonal Wavelet Frames and Vector-valued Discrete Wavelet Transforms Brody Dylan Johnson Saint Louis University Joint work with: Eric Weber (IA-State) & Ghanshyam Bhatt (Rose-Hulman) 1
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Orthogonal Wavelet Frames and Vector-valued Discrete Wavelet

Feb 12, 2022

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Page 1: Orthogonal Wavelet Frames and Vector-valued Discrete Wavelet

Orthogonal Wavelet Frames

and Vector-valued

Discrete Wavelet Transforms

Brody Dylan Johnson

Saint Louis University

Joint work with:Eric Weber (IA-State) & Ghanshyam Bhatt (Rose-Hulman)

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Page 2: Orthogonal Wavelet Frames and Vector-valued Discrete Wavelet

Outline:• Motivation – 1 slide

• Preliminaries – 2 slides

• Orthogonal Frames – 2 slides

• Orthogonal Wavelet Frames (OWFs) – 6 slides

• Vector-valued Discrete Wavelet Transform (VDWT) – 4 slides

• Compression with the VDWT – 1 slide

? Color Images – 6 slides

? Stereo Audio (Voice) – 2 slides

• Conclusion – 1 slide

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Page 3: Orthogonal Wavelet Frames and Vector-valued Discrete Wavelet

Motivation:• There are many situations in which correlated multichannel data

occurs naturally, e.g., color images, stereo audio, etc.

• One can always apply a standard wavelet transform to each chan-nel, but this fails to take advantage of any correlation betweenthe channels.

• The primary goal of this work is to develop a vector-valued dis-crete wavelet transform (VDWT) allowing for simultaneous pro-cessing of multichannel data.

• By using orthogonal wavelet frames for each channel, one can ac-tually sum the “high-pass” components of the associated DWTsin hopes of achieving a more efficient representation of the signal.

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Page 4: Orthogonal Wavelet Frames and Vector-valued Discrete Wavelet

Preliminaries: (1 of 2)

• Fourier transform: f ∈ L1 ∩ L2(R)

f(ξ) =∫

Rf(x)e−2πixξ dx.

• Translation operator: T : L2(R) → L2(R)

Tf(x) = f(x− 1).

• Dilation operator: D : L2(R) → L2(R)

Df(x) =√

2f(2x).

• Affine systems: Given Ψ = {ψ1, . . . ψr} ⊂ L2(R)

X(Ψ) ={DjT kψ` : j, k ∈ Z, 1 ≤ ` ≤ r

}.

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Page 5: Orthogonal Wavelet Frames and Vector-valued Discrete Wavelet

Preliminaries: (2 of 2)

• Frame: X := {xj}j∈J ⊂ H is a frame for H if there exist con-stants 0 < C1 ≤ C2 < ∞ such that for all x ∈ H,

C1‖x‖2 ≤∑

j∈J

|〈x, xj〉|2 ≤ C2‖x‖2.

Parseval frames occur when one may choose C1 = C2 = 1.

• Analysis operator: ΘX : H→ `2(J) given by

ΘXx = {〈x, xj〉}j∈J .

• Synthesis operator: Θ∗X : `2(J) → H given by

Θ∗X{cj}j∈J =∑

j∈J

cjxj .

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Page 6: Orthogonal Wavelet Frames and Vector-valued Discrete Wavelet

Orthogonal Frames: (1 of 2)

• Orthogonality: Let X = {xj}j∈J and Y = {yj}j∈J be Besselsequences, then X and Y are orthogonal if

Θ∗Y ΘX =∑

j∈J

〈·, xj〉yj = 0.

• Reconstruction: If X and Y are orthogonal Parseval frames thenfor all f1, f2 ∈ H,

Θ∗Y (ΘXf1 + ΘY f2) = Θ∗Y ΘY f2 = f2.

• In order that X and Y are pairwise orthogonal, each Parsevalframe must provide a redundant representation of H, e.g., noticethat

x =∑

j∈J

〈x, xj + yj〉xj =∑

j∈J

〈x, xj〉xj .

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Page 7: Orthogonal Wavelet Frames and Vector-valued Discrete Wavelet

Orthogonal Frames: (2 of 2)

• Application to multiple channels: Signal f = f1 ⊕ · · · ⊕ fN .

? Start with pairwise orthogonal Parseval frames: X1, . . . , XN .

? Apply ΘXkto fk and sum the result:

f 7→ ΘXf :=N∑

k=1

ΘXkfk.

? Recover fk0 by applying Θ∗Xk0to ΘXf :

Θ∗Xk0ΘXf =

N∑

k=1

Θ∗Xk0ΘXk

fk = Θ∗Xk0ΘXk0

fk0 = fk0 .

• The multichannel analysis operator, ΘX :N⊕

k=1

H→ `2(J), pro-

cesses the components of f simultaneously.

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Page 8: Orthogonal Wavelet Frames and Vector-valued Discrete Wavelet

Orthogonal Wavelet Frames: (1 of 6)

• Characterizing dual wavelet frames: (Ron and Shen ‘97)

Theorem 1. Suppose {ψ1, . . . , ψr} and {η1, . . . , ηr} generatewavelet frames in L2(R). The frames are dual if and only if

r∑

k=1

j∈Zψk(2jξ)ηk(2jξ) = 1, a.e. ξ ∈ R,

and for every q ∈ Z \ 2Z,

r∑

k=1

∞∑

j=0

ψk(2jξ)ηk(2j(ξ + q)) = 0, a.e. ξ ∈ R.

In particular, {ψ1, . . . , ψr} generates a Parseval wavelet frame ifthe two equations hold for ηk = ψk.

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Page 9: Orthogonal Wavelet Frames and Vector-valued Discrete Wavelet

Orthogonal Wavelet Frames: (2 of 6)

• Characterizing orthogonal wavelet frames: (Weber ‘04)

Theorem 2. Suppose {ψ1, . . . , ψr} and {η1, . . . , ηr} generateaffine Bessel sequences in L2(R), then they are orthogonal if andonly if

r∑

k=1

j∈Zψk(2jξ)ηk(2jξ) = 0, a.e. ξ ∈ R,

and for every q ∈ Z \ 2Z,

r∑

k=1

∞∑

j=0

ψk(2jξ)ηk(2j(ξ + q)) = 0, a.e. ξ ∈ R.

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Page 10: Orthogonal Wavelet Frames and Vector-valued Discrete Wavelet

Orthogonal Wavelet Frames: (3 of 6)

• Construction of wavelet frames from a scaling function and fil-ters:

? Let ϕ ∈ L2(R) be a refinable function, with low pass filterm(ξ), satisfying:1. limξ→0 ϕ(ξ) = 1;2.

∑l∈Z |ϕ(ξ + l)|2 ∈ L∞(R).

? Let m1(ξ), . . . , mr(ξ) ∈ L∞([0, 1)) and define

M(ξ) =

m(ξ) m(ξ + 1/2)

m1(ξ) m1(ξ + 1/2)

......

mr(ξ) mr(ξ + 1/2)

M(ξ) =

m1(ξ) m1(ξ + 1/2)

......

mr(ξ) mr(ξ + 1/2)

.

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Page 11: Orthogonal Wavelet Frames and Vector-valued Discrete Wavelet

Orthogonal Wavelet Frames: (4 of 6)

• Unitary Extension Principle: (Daubechies, B. Han, Ron, andShen ‘03)

Theorem 3. Suppose ϕ ∈ L2(R) is a refinable function as de-scribed above. Let m1(ξ), . . . , mr(ξ) ∈ L∞([0, 1)) such that thematrix M(ξ) satisfies

M∗(ξ)M(ξ) = I2, a.e. ξ ∈ R.

Then, the affine system generated by {ψ1, . . . , ψr}, where

ψk(2ξ) = mk(ξ)ϕ(ξ), k = 1, . . . , r,

is a Parseval wavelet frame.

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Page 12: Orthogonal Wavelet Frames and Vector-valued Discrete Wavelet

Orthogonal Wavelet Frames: (5 of 6)

Theorem 4 (Bhatt, J–, Weber ‘06). Let ϕ ∈ L2(R) be a refinablefunction as described above. Let M = {m0(ξ),m1(ξ), . . . , mr(ξ)}and N = {n0(ξ), n1(ξ), . . . , nr(ξ)} be filter banks with m0 = n0 = m.Suppose that the following matrix equations hold:

1. M∗(ξ)M(ξ) = I2 for almost every ξ;

2. N∗(ξ)N(ξ) = I2 for almost every ξ;

3. M∗(ξ)N(ξ) = 0 for almost every ξ.

Let ψk(2ξ) = mk(ξ)ϕ(ξ) and ηk(2ξ) = nk(ξ)ϕ(ξ), 1 ≤ k ≤ r. Then{ψ1, . . . , ψr} and {η1, . . . , ηr} generate orthogonal Parseval waveletframes.

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Page 13: Orthogonal Wavelet Frames and Vector-valued Discrete Wavelet

Orthogonal Wavelet Frames: (6 of 6)

Theorem 5 (Bhatt, J–, Weber ‘06). Suppose K(ξ) is a 1/2-periodic r × r matrix which is unitary for a.e. ξ; let Kj(ξ) denotethe j-th column. Suppose m0 and m1 are low and high pass filters,respectively, for an orthonormal wavelet basis with scaling functionϕ. For j = 1, . . . , r, define new filters via

nj1(ξ)...

njr(ξ)

= Kj(ξ)m1(ξ).

Then, for j = 1, . . . , r, the affine systems generated by {ψjl : l =

1, . . . , r} obtained via

ψjl (2ξ) = nj

l (ξ)ϕ(ξ) (1)

are Parseval frames and are pairwise orthogonal.

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Page 14: Orthogonal Wavelet Frames and Vector-valued Discrete Wavelet

Discrete Implementation of OWFs: (1 of 4)

• Begin with scaling function ϕ and wavelet ψ for an orthonormalwavelet (filters m(ξ) and n(ξ), respectively).

• Choose unitary matrix K(ξ) and construct orthogonal waveletframes as in Theorem 5.

• Analysis/Reconstruction of discrete data is accomplished usingthe associated filter banks.

• This leads to a notion of orthogonal filter banks that will beapplied to the high-pass filters.

• Filter banks: M = {m0,m1, . . . , mr}, N = {n0, n1, . . . , nr}.

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Page 15: Orthogonal Wavelet Frames and Vector-valued Discrete Wavelet

Discrete Implementation of OWFs: (2 of 4)

f - m0 ±°²¯↓2 - g0

Analysis︷ ︸︸ ︷ Synthesis︷ ︸︸ ︷

- m1 ±°²¯↓2 - g1

..

....

..

....

..

....

- mr ±°²¯↓2 - gr

- ↑2±°²¯

n0-

- ↑2±°²¯

n1-

6

- ↑2±°²¯

nr

6...

i+

i+- n×2 -f

Figure 1: Filter bank block diagram.

ˆf(ξ) =

r∑

`=0

[m`(ξ)n`(ξ)f(ξ) + m`(ξ + 1/2)n`(ξ)f(ξ + 1/2)

]

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Page 16: Orthogonal Wavelet Frames and Vector-valued Discrete Wavelet

Discrete Implementation of OWFs: (3 of 4)

• M andN are orthogonal if, for any input vector, the compositionof the analysis stage of M with the synthesis stage of N yields0, i.e. for any input f ∈ `2(Z), f = 0.

• When m0 = n0 = m is a low-pass filter it is impossible for Mand N to be orthogonal. (assuming the remaining filters arehigh-pass)

• The high-pass portions of the filter banks are orthogonal if andonly if M∗(ξ)N(ξ) = 0. (as for OWFs.)

• Each filter bank has the perfect reconstruction property if andonly if M∗(ξ)M(ξ) = N∗(ξ)N(ξ) = I2. (as for OWFs)

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Page 17: Orthogonal Wavelet Frames and Vector-valued Discrete Wavelet

Discrete Implementation of OWFs: (4 of 4)

f1j

- m0l↓2

- m1l↓2

- m2l↓2

- n1l↓2

- n2l↓2

n0l↓2f2

j-

f1j+1

JJJJ

JJ

­­­

­­­

g+g+

gj+1,1

gj+1,2

f2j+1

- l↑2

- l↑2- l↑2

- l↑2

m0

m1

m2

n0

n2

n1

g+

g+

g+g+

@@

@@@

@

¡¡

¡

¡¡

¡

k×2 f1j

k×2 f2j

Figure 2: Two-Channel VDWT as in Stereo Audio Context.

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Page 18: Orthogonal Wavelet Frames and Vector-valued Discrete Wavelet

Image Compression:• Hard thresholding was applied over four scales of the associated

discrete wavelet transform. Recall that with hard thresholdingonly the coefficients greater than a chosen threshold T > 0 arekept for reconstruction. (No quantizing/encoding is done here.)

• The benefit and cost of thresholding are quantified by:

Compression Factor :=Total # of pixels × 3

# of coefficients ≥ threshold,

SNR := 20 log10

( ‖Original‖2‖Original− Reconstruction‖2

).

A higher SNR corresponds to a smaller ‖ · ‖2 error.

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Page 19: Orthogonal Wavelet Frames and Vector-valued Discrete Wavelet

Preliminary Results:

Picture Method Threshhold Comp. Ratio SNR

Lena D4, none 15 9.36 30.64

Lena D4, scalar 15 10.96 30.93

Lena D4, poly. 15 9.88 30.64

Lena D4, none 50 28.77 26.14

Lena D4, scalar 50 34.62 26.75

Lena D4, poly. 50 30.58 26.37

Pepper D4, none 15 10.71 31.41

Pepper D4, scalar 15 12.14 32.06

Pepper D4, poly. 15 10.83 31.66

Table 1: Image compression using orthogonal wavelet frames.

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Page 20: Orthogonal Wavelet Frames and Vector-valued Discrete Wavelet

Example:

50 100 150 200 250 300 350 400 450 500

50

100

150

200

250

300

350

400

450

500

Original 512× 512 Lena image.

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Page 21: Orthogonal Wavelet Frames and Vector-valued Discrete Wavelet

Example:

50 100 150 200 250 300 350 400 450 500

50

100

150

200

250

300

350

400

450

500

Ordinary DWT: Reconstructed Image.

D4 filters, Threshold=15: C.R. ≈ 9.36 & SNR ≈ 30.64.

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Page 22: Orthogonal Wavelet Frames and Vector-valued Discrete Wavelet

Example:

50 100 150 200 250 300 350 400 450 500

50

100

150

200

250

300

350

400

450

500

VDWT: Reconstructed Image.

D4 filters, Threshold=15: C.R. ≈ 10.96 & SNR ≈ 30.93.

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Page 23: Orthogonal Wavelet Frames and Vector-valued Discrete Wavelet

Example:

120× 120 section of the Lena image.

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Page 24: Orthogonal Wavelet Frames and Vector-valued Discrete Wavelet

Example:

Ordinary DWT: Reconstructed Image.

D4 filters, Threshold=15: C.R. ≈ 9.36 & SNR ≈ 30.64.

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Page 25: Orthogonal Wavelet Frames and Vector-valued Discrete Wavelet

Example:

VDWT: Reconstructed Image.

D4 filters, Threshold=15: C.R. ≈ 10.96 & SNR ≈ 30.93.

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Page 26: Orthogonal Wavelet Frames and Vector-valued Discrete Wavelet

Example:

50 100 150 200 250 300 350 400 450 500

50

100

150

200

250

300

350

400

450

500

Ordinary DWT: After thresholding.

D4 filters, Threshold=15: C.R. ≈ 9.36 & SNR ≈ 30.64.

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Page 27: Orthogonal Wavelet Frames and Vector-valued Discrete Wavelet

Example:

50 100 150 200 250 300 350 400 450 500

50

100

150

200

250

300

350

400

450

500

Scalar Orthogonalization VDWT: After thresholding.

D4 filters, Threshold=15: C.R. ≈ 10.96 & SNR ≈ 30.93.

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Page 28: Orthogonal Wavelet Frames and Vector-valued Discrete Wavelet

Audio Example:

2.5 2.55 2.6 2.65

x 104

−0.4

−0.2

0

0.2

0.4

0.6

left

ch

an

ne

l

originalapproximation

2.5 2.55 2.6 2.65

x 104

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

rig

ht

ch

an

ne

l

originalapproximation

Ordinary DWT: Comparison of Left/Right Channels.

Shannon filters (2000 coeff.), Threshold=0.0175: C.R. ≈ 10.90 &SNR ≈ 21.05.

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Page 29: Orthogonal Wavelet Frames and Vector-valued Discrete Wavelet

Audio Example:

2.5 2.55 2.6 2.65

x 104

−0.4

−0.2

0

0.2

0.4

0.6

left

ch

an

ne

l

originalapproximation

2.5 2.55 2.6 2.65

x 104

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

rig

ht

ch

an

ne

l

originalapproximation

VDWT: Comparison of Left/Right Channels.

Shannon filters (2000 coeff.), Threshold=0.0175: C.R. ≈ 11.71 &SNR ≈ 21.06.

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Page 30: Orthogonal Wavelet Frames and Vector-valued Discrete Wavelet

Conclusion:• Orthogonal wavelet frames and the VDWT may provide a viable

means for dealing with multichannel data.

• Future work:

? consideration of quantization/encoding issues

? optimization of the choice of unitary in construction of OWFs

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