A Friendly Guide to the Frame Theory

and Its Application to Signal Processing

Minh N. Do

Department of Electrical and Computer EngineeringUniversity of Illinois at Urbana-Champaign

www.ifp.uiuc.edu/∼minhdominhdo@uiuc.edu

A Basic Problem

Consider the following linear inverse problem:

Ax = b

where A ∈ Rm×n is fixed, b ∈ R

m is given, and x ∈ Rn is unknown.

+

PSfrag replacements

x b xb

∆b

A S

Examples: Deconvolution, computerized tomography, transform coding,...

Possible noise due to: model mismatch, measurement and/or transmissionerror, quantization, thresholding,...

1

Two Basic Questions

Ax = b

+

PSfrag replacements

x b xb

∆b

A S

We have two questions:

1. Can we reconstruct x in a numerically stable way from b?

2. Which is the “optimal” reconstruction algorithm in the presence of noise?

2

Linear Inverse Problem Ax = b: First Question

Question: Can we reconstruct x in a numerically stable way from b?

Answer: It depends on the condition number of A:

κ(A) =σ1(A)

σn(A).

The smaller κ(A) (κ(A) ≥ 1), the more stable (or well-conditioned) theproblem is.

Intuition: σ1 and σn are the largest and smallest singular values of A.Thus, for all x:

σn‖x‖2 ≤ ‖Ax‖2 ≤ σ1‖x‖2

That means A should behavior modestly with respect to 2-norm!

3

Linear Inverse Problem Ax = b: Second Question

Question: When the system A is overcomplete, there are many (infinite)ways to reconstruct x from b. Which one is “optimal”?

Answer: Use the pseudo-inverse A† = (ATA)−1AT

x = A†b

Special properties of the pseudo-inverse:

• A† provides the least-squares solution⇒ Eliminates the influence of errors orthogonal to the range of A.

• A† has a minimum spectral norm among all left inverse of A

⇒ Recovers x and but doesn’t blow up the noise.

4

Frames: Generalize to Hilbert Spaces

Consider A as a linear operator, A : Rn → R

m, then

Ax = b ⇐⇒ bi = 〈x, ai〉, i = 1, 2, . . . , m

where aTi are the rows of A.

Def: A sequence {φk}k∈Γ in a Hilbert space H is a frame if there exist twoconstants (frame bounds) α > 0 and β < ∞ such that for any x ∈ H

α‖x‖2 ≤∑

k∈Γ

|〈x, φk〉|2 ≤ β‖x‖2.

Best case: α = β =⇒ tight frame

Significance: {φk}k∈Γ is a frame ⇐⇒ one can recover x ∈ H from{〈x, φk〉}k∈Γ.

5

Dual Frame

Frame operator: A : H → l2(Γ)

(Ax)k = 〈x, φk〉, for k ∈ Γ.

Pseudo inverse: A† = (A∗A)−1A∗ exists and bounded because {φk}k∈Γ

is a frame.

Result: Reconstruction using pseudo inverse is related to a dual frame

x = A†Ax =∑

k∈Γ

〈x, φk〉φk

where the dual frame is defined as φk = (A∗A)−1φk.

Easiest case: Tight frame (α = β), φk = α−1φk.

6

Iterative Frame Reconstruction Algorithm

Both pseudo-inverse and dual frame computations need the inversion ofA∗A, where

A∗Ax =∑

k∈Γ

〈x, φk〉φk

Consider R = I − 2α+β

A∗A, then because α · I ≤ A∗A ≤ β · I

‖R‖ ≤β − α

β + α≤ 1.

Thus,

(A∗A)−1 =2

α + β(I − R)−1 =

2

α + β

∞∑

i=0

Ri

Iterative reconstruction:

xn = xn−1 +2

α + β

∑

k∈Γ

(〈x, φk〉 − 〈xn−1, φk〉)φk

7

Application to Generalized Sampling

• Sampled data: s[k] = 〈x, φk〉, where φk is the point spreading function(PSF) of the sensoring device at location tk.

• “Sampling theorem”: Function x(t) ∈ H can be recovered in anumerically stable way from samples s[k] if and only if {φk}k∈Γ is aframe of H.

• Classical sampling: H = BL([−π, π]) and {φk} = {sinc(t − k)}k∈Z

8

Laplacian Pyramid: Burt Adelson, 1983

9

Why Laplacian Pyramid Instead of Orthogonal Filter

Banks?

(2,2)

(2,2)

(2,2)

(2,2)

Wavelet FB Laplacian Pyramid(2,2)

Even in higher dimensions, the Laplacian pyramid (LP) only generates oneisometric detailed signal at each level.

LP has no “frequency scrambling”due to downsampling of the highpasschannel:

highpass (HP)

downsampled HPPSfrag replacements

π

π

−π

−π

10

Decomposition in the Laplacian Pyramid

+_

PSfrag replacements

M MH G

H G

x p

c

d

Coarse: c = Hx

Residual: d = x − GHx = (I − GH)x.

Combining gives (c

d

)

︸ ︷︷ ︸y

=

(H

I − GH

)

︸ ︷︷ ︸A

x.

11

Usual Reconstruction in the Laplacian Pyramid

+

PSfrag replacements

M Gxc

d

x =(

G I)

︸ ︷︷ ︸S1

(c

d

)

︸ ︷︷ ︸y

.

Note that S1A = I (perfect reconstruction) for any H and G.

But... what about noisy pyramids: y = y + e ?

The most serious disadvantage of the LP for coding applications[Simoncelli & Adelson, 1991]: “...the errors from highpass subbands of

a multilevel LP do not remain in these subbands but appear as broadband

noise in the reconstructed signal...”

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Frame Analysis

• LP is a frame operator (A) with redundancy.

• It admits an infinite number of left inverses.

• Let S be an arbitrary left inverse of A,

x = Sy = S(y + e) = x + Se.

• The optimal left inverse (minimizing ‖S‖) is the pseudo-inverse of A:

A† = (ATA)−1AT .

• If the noise is white, then among all left inverses, the pseudo-inverseminimizes the reconstruction MSE.

• But... reconstruction using the pseudo-inverse might be computationallyexpensive, unless we have a tight frame.

13

A Tight Frame Case

Orthogonal filters:

〈g[•], g[• − Mn]〉 = δ[n], and

h[n] = g[−n], or H = GT .

Theorem. The Laplacian pyramid with orthogonal filters is a tight frame.

Proof: Under the orthogonality condition:

p[n] =∑

k∈Zd

〈x[•], g[• − Mk]〉︸ ︷︷ ︸

c[k]

g[n − Mk].

Using the Pythagorean theorem:

‖x‖2 = ‖p‖2 + ‖d‖2 = ‖c‖2 + ‖d‖2.

PSfrag replacements x

p

d

V

14

Inspiring New Reconstruction Filter Bank

As a result, pseudo-inverse of A is simply its transpose

A† = AT =

(H

I − GGT

)T

=(

G I − GGT).

So the optimal reconstruction is

x = A†y = Gc + (I − GGT )d = G(c − Hd) + d.

++_

PSfrag replacements

M MH G

xx

p

c

d

15

General Cases

Consider the following filter bank for reconstruction

++_

PSfrag replacements

M MH G

xx

p

c

d

Theorem.

1. It is an inverse of the LP if and only if H and G are biorthogonal filters,or GH is a projector.

2. It is the pseudo-inverse if and only if GH is an orthogonal projector.

Recall: A linear operator P is a projector if P 2 = P .Furthermore, if P = P T then P is an orthogonal projector.

16

Comparing Two Reconstruction Methods

+

PSfrag replacements

M

H

G

H

G

x

xpc

d

++_

PSfrag replacements

M MH G

H

G

x

xp

c

d

Usual reconstruction: x1 = Gc + d

New reconstruction: x2 = Gc + (I − GH)︸ ︷︷ ︸

PW

dPSfrag replacements

V

W

xd

p

dPW d

x1x2

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Laplacian Pyramid as an Oversampled Filter Bank

M M

MM

+

MM

PSfrag replacements

H G

K0

K|M |−1

F0

F|M |−1

x x

c

d

d0

d|M |−1

For the usual reconstruction method, synthesis filters F[1]i are all-pass

(delay) filters: F[1]i (z) = z−ki

For the proposed reconstruction method, synthesis filters F[1]i are high-pass

filters: F[2]i (z) = z−ki − G(z)Hi(z

M).

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Multilevel Laplacian Pyramids

2 +

2

2

+

2

2

2PSfrag replacements GG

F0 F0

F1 F1

x

Comparing frequency responses of equivalent synthesis filters (REC-1 vs.REC-2)

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3

3.5

Normalized Frequency (×π rad/sample)

Mag

nitu

de

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3

3.5

Normalized Frequency (×π rad/sample)

Mag

nitu

de

19

Experimental Results

dec.LP

usualrec.

rec.new

+

PSfrag replacements

x y

e

y

x1

x2

With additive uniform white noise in [0, 0.1] (non-zero mean)...

usualrec.SNR =6.28 dB

new rec.SNR =

17.42 dB

20

Summary

• Frames are a powerful tool...

– Generalizes matrix inversions for general (possible infinite dimensional)vector spaces.

– Generalizes bases for overcomplete (redundant) systems.

• Framing pyramids lead to...

– New reconstruction algorithm with significant improvement over theusual method.

– Complete characterization of left inverses and the pseudo-inverse.

• Frames are everywhere...

– Give me a linear operator with a bounded inverse, I’ll frame it!– If you have to deal with an overcomplete system, consider the frame

theory!

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