A Friendly Guide to the Frame Theory. and Its Application to Signal Processing

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1 A Friendly uide to the Frame Theory and Its Application to Signal Processing inh N. Do Department of Electrical and Computer Engineering University of Illinois at Urbana-Champaign minhdo

2 A Basic Problem Consider the following linear inverse problem: Ax = b placements where A R m n is fixed, b R m is given, and x R n is unknown. x b ˆb ˆx A S b Examples: Deconvolution, computerized tomography, transform coding,... Possible noise due to: model mismatch, measurement and/or transmission error, quantization, thresholding,... 1

3 Two Basic Questions placements Ax = b x b ˆb ˆx A S b 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

4 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) = σ 1(A) σ n (A). The smaller κ(a) (κ(a) 1), the more stable (or well-conditioned) the problem is. Intuition: σ 1 and σ n are the largest and smallest singular values of A. Thus, for all x: σ n x 2 Ax 2 σ 1 x 2 That means A should behavior modestly with respect to 2-norm! 3

5 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 = (A T A) 1 A T ˆx = 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

6 Frames: eneralize to Hilbert Spaces Consider A as a linear operator, A : R n R m, then Ax = b b i = x, a i, i = 1, 2,..., m where a T i are the rows of A. Def: A sequence {φ k } k Γ in a Hilbert space H is a frame if there exist two constants (frame bounds) α > 0 and β < such that for any x H α x 2 k Γ x, φ k 2 β x 2. Best case: α = β = tight frame Significance: {φ k } k Γ is a frame one can recover x H from { x, φ k } k Γ. 5

7 Dual Frame Frame operator: A : H l 2 (Γ) (Ax) k = x, φ k, for k Γ. Pseudo inverse: A = (A A) 1 A exists and bounded because {φ k } k Γ is a frame. Result: Reconstruction using pseudo inverse is related to a dual frame x = A Ax = k Γ x, φ k φ k where the dual frame is defined as φ k = (A A) 1 φ k. Easiest case: Tight frame (α = β), φ k = α 1 φ k. 6

8 Iterative Frame Reconstruction Algorithm Both pseudo-inverse and dual frame computations need the inversion of A A, where A Ax = k Γ x, φ k φ k Consider R = I 2 αβ A A, then because α I A A β I Thus, Iterative reconstruction: R β α β α 1. (A A) 1 = 2 α β (I R) 1 = 2 α β x n = x n 1 2 α β i=0 R i ( x, φ k x n 1, φ k )φ k k Γ 7

9 Application to eneralized Sampling Sampled data: s[k] = x, φ k, where φ k is the point spreading function (PSF) of the sensoring device at location t k. Sampling theorem : Function x(t) H can be recovered in a numerically stable way from samples s[k] if and only if {φ k } k Γ is a frame of H. Classical sampling: H = BL([ π, π]) and {φ k } = {sinc(t k)} k Z 8

10 Laplacian Pyramid: Burt Adelson,

Laplacian pyramid (LP) only generates one isometric detailed signal at each level.

11 Why Laplacian Pyramid Instead of Orthogonal Filter Banks? (2,2) Wavelet FB Laplacian Pyramid (2,2) (2,2) (2,2) (2,2) Even in higher dimensions, the Laplacian pyramid (LP) only generates one isometric detailed signal at each level. highpass (HP) LP has no frequency scrambling due to downsamplingpsfrag of thereplacements highpass channel: π downsampled HP π π π 10

12 eplacements Decomposition in the Laplacian Pyramid c x H p _ d H Coarse: Residual: c = Hx d = x Hx = (I H)x. Combining gives ( ) c d }{{} y = ( ) H x. I H }{{} A 11

13 Usual Reconstruction in the Laplacian Pyramid PSfrag replacements c ˆx d ˆx = ( I ) ( ) c }{{} d S 1 }{{} y Note that S 1 A = I (perfect reconstruction) for any H and. But... what about noisy pyramids: ŷ = 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

14 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, ˆx = Sŷ = S(y e) = x Se. The optimal left inverse (minimizing S ) is the pseudo-inverse of A: A = (A T A) 1 A T. If the noise is white, then among all left inverses, the pseudo-inverse minimizes the reconstruction SE. But... reconstruction using the pseudo-inverse might be computationally expensive, unless we have a tight frame. 13

15 A Tight Frame Case Orthogonal filters: g[ ], g[ n] = δ[n], and h[n] = g[ n], or H = T. Theorem. The Laplacian pyramid with orthogonal filters is a tight frame. Proof: Under the orthogonality condition: p[n] = x[ ], g[ k] PSfrag g[n replacements k]. }{{} k Z d c[k] Using the Pythagorean theorem: x 2 = p 2 d 2 = c 2 d 2. x p d V 14

16 Inspiring New Reconstruction Filter Bank As a result, pseudo-inverse of A is simply its transpose A = A T = ( H I T ) T = ( I T ). So the optimal reconstruction is eplacements ˆx = A y = c (I T )d = (c Hd) d. c x d p H _ ˆx 15

17 eneral Cases eplacements Consider the following filter bank for reconstruction c x d p H _ ˆx Theorem. 1. It is an inverse of the LP if and only if H and are biorthogonal filters, or H is a projector. 2. It is the pseudo-inverse if and only if H 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

18 H H x Comparing Two Reconstruction ethods H xc pc ˆx pd H _ ˆx d Usual reconstruction: PSfrag New reconstruction: replacements x 1 = c d x 2 = c (I H) d }{{} P W W P W ˆd ˆd ˆx 2 ˆx 1 d x p V 17

19 replacements Laplacian Pyramid as an Oversampled Filter Bank H c x K 0 d 0 F 0 ˆx K 1 d 1 F 1 d For the usual reconstruction method, synthesis filters F [1] i (delay) filters: F [1] i (z) = z k i For the proposed reconstruction method, synthesis filters F [1] i filters: F [2] i (z) = z k i (z)h i (z ). are all-pass are high-pass 18

20 ultilevel Laplacian Pyramids frag replacements 2 2 ˆx 2 2 F 0 2 F 0 F 1 2 F 1 Comparing frequency responses of equivalent synthesis filters (REC-1 vs. REC-2) agnitude agnitude Normalized Frequency ( π rad/sample) Normalized Frequency ( π rad/sample) 19

21 PSfrag replacements Experimental Results x LP dec. y e ŷ usual rec. new rec. ˆx 1 ˆx 2 With additive uniform white noise in [0, 0.1] (non-zero mean)... usual rec. SNR = 6.28 db new rec. SNR = db 20

22 Summary Frames are a powerful tool... eneralizes matrix inversions for general (possible infinite dimensional) vector spaces. eneralizes bases for overcomplete (redundant) systems. Framing pyramids lead to... New reconstruction algorithm with significant improvement over the usual method. Complete characterization of left inverses and the pseudo-inverse. Frames are everywhere... ive 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! 21

Multiresolution image processing

Multiresolution image processing Laplacian pyramids Some applications of Laplacian pyramids Discrete Wavelet Transform (DWT) Wavelet theory Wavelet image compression Bernd Girod: EE368 Digital Image Processing