Wavelet Transform

國立交通大學電子工程學系陳奕安

2007.8.15

Outline Comparison of Transformations Multiresolution Analysis Discrete Wavelet Transform Fast Wavelet Transform Lifting Scheme Beyond Wavelet Reference

Outline Comparison of Transformations Multiresolution Analysis Discrete Wavelet Transform Fast Wavelet Transform Lifting Scheme Beyond Wavelet Reference

Fourier Transform Frequency domain: Fourier Transform

(Joseph Fourier 1807 )

dtetxfX ftj 2)()(

Cannot provide simultaneously time and frequency information.

Short Time Fourier Transform (STFT)

Time-Frequency analysis: STFT(Dennis Gabor 1946) Windowed Fourier transform

dtetttxft ftj

t

2*X ,STFT

function window the:tA function of time

and frequency

Short Time Fourier Transform (STFT)

Frequency and time resolutions are fixed: Narrow (Wide) window for poor freq. (time) resolution

Via Narrow Window Via Wide Window

The two figures were from Robi Poliker, 1994

Continuous Wavelet Transform

Width of the window is changed as the transform is computed for every spectral components.

Altered resolutions are placed.

dts

ttx

sss xx

*1

, ,CWT

Translation

(The location of the window) Scale Mother Wavelet

Comparison of Transformations

From http://www.cerm.unifi.it/EUcourse2001/Gunther_lecturenotes.pdf, p.10

Wavelet Series Expansion Linear decomposition of a function:

Basis orthogonal:

Then the coefficients can be calculated by

Outline Comparison of Transformations Multiresolution Analysis Discrete Wavelet Transform Fast Wavelet Transform Lifting Scheme Beyond Wavelet Reference

Multiresolution Analysis Idea: If a set of signals can be represented by a

weighted sum of φ(t-k), a larger set (including the original), can be represented by a weighted sum of φ (2t-k).

Increase the size of the subspace changing the time scale of the scaling functions:

Multiresolution Analysis The spanned spaces are nested:

Wavelets span the differences between spaces wi.

Wavelets and scaling functions should be orthogonal: simple calculation of coefficients.

Multiresolution Analysis

Multiresolution Analysis Multiresolution Formulation.

( Scaling coefficients)

( Wavelet coefficients)

Outline Comparison of Transformations Multiresolution Analysis Discrete Wavelet Transform Fast Wavelet Transform Lifting Scheme Beyond Wavelet Reference

Discrete Wavelet Transform (DWT)

Discrete Wavelet Transform Calculation: Using Multiresolution Analysis:

Outline Comparison of Transformations Multiresolution Analysis Discrete Wavelet Transform Fast Wavelet Transform Lifting Scheme Beyond Wavelet Reference

Fast Wavelet Transform Basic idea of Fast Wavelet Transform

(Mallat’s herringbone algorithm): Pyramid algorithm provides an efficient calculation.

DWT (direct and inverse) can be thought of as a filtering process.

After filtering, half of the samples can be eliminated: subsample the signal by two.

Subsampling: Scale is doubled. Filtering: Resolution is halved.

Fast Wavelet Transform

(a) A two-stage or two-scale FWT analysis bank and

(b) its frequency splitting characteristics.

Fast Wavelet Transform Fast Wavelet Transform

Inverse Fast Wavelet Transform

Fast Wavelet Transform

A two-stage or two-scale FWT-1 synthesis bank.

Outline Comparison of Transformations Multiresolution Analysis Discrete Wavelet Transform Fast Wavelet Transform Lifting Scheme Beyond Wavelet Reference

Lifting Scheme The lifting scheme is an alternative method of

computing the wavelet coefficients.

Advantages of the lifting scheme:Requires less computation and less memory. Linear, nonlinear, and adaptive wavelet

transform is feasible, and the resulting transform is invertible and reversible.

Lifting Scheme A spatial domain construction of bi-orthogonal

wavelets, consists of the 4 operations:

Split : sk(0)=x2i

(0), dk(0)=x2i+1

(0)

Predict : dk(r)= dk

(r-1) – pj(r) sk+j

(r-1)

Update : sk(r)= sk(r-1) + uj

(r) dk+j(r)

Scaling : sk(R)=K0sk

(R), dk(R)=K1dk

(R)

Lifting Scheme A spatial domain construction of bi-orthogonal

wavelets, consists of the 4 operations:

Lifting Scheme A spatial domain construction of bi-orthogonal

wavelets, consists of the 4 operations:

Lifting Scheme Example: Conventional 5/3 filter

C0 = (4*x[0]+2*x[0]+2*(x[-1]+x[1])-(x[2]+x[-2]) )/8C1 = x[0]- (x[1]+x[-1])/2 Number of operations per pixel = 9+3 = 12

Lifting Scheme Example: (2,2) lifting scheme Prediction rule : interpolation : [1,1]/2 Update rule: preservation of average (moments)

of the signal : [1,1]/4

Lifting Scheme

Conventional 5/3 filterC0=(4*x[0]+2*x[0]+2*(x[-1]+x[1])-(x[2]+x[-2]))/8

C1= x[0]- (x[1]+x[-1])/2Number of operations per pixel = 9+3 = 12

The (2,2) liftingD[0] = x[0]- (x[1]+x[-1])/2S[0] = x[0] + (D[0]+D[1])/4Number of operations per pixel = 6

Outline Comparison of Transformations Multiresolution Analysis Discrete Wavelet Transform Fast Wavelet Transform Lifting Scheme Beyond Wavelet Reference

Beyond Wavelet

Ridgelet TransformCurvelet Transform

Continuous Ridgelet Transform

Ridgelet Transform (Candes, 1998):

Ridgelet function:

The function is constant along lines.Transverse to these ridges, it is a wavelet.

R f a,b, a,b, x f x dx

a,b, x a1

2 x1 cos() x2 sin() b

a

Continuous Ridgelet Transform

The ridgelet coefficients of an object f are given by analysis of the Radon transform via:

dta

bttRAbaR ff )(),(),,(

The Curvelet Transform

Decomposition of the original image into subbands .

Spatial partitioning of each subband.

Appling the ridgelet transform.

Beyond Wavelet A standard multiscale decomposition into octav

e bands, where the lowpass channel is subsampled while the highpass is not.

Outline Comparison of Transformations Multiresolution Analysis Discrete Wavelet Transform Fast Wavelet Transform Lifting Scheme Beyond Wavelet Reference

Reference [1] P. P. Vaidyanathan, "Multirate systems and filter

banks,“pp.457-538 1992. [2] Howard L. Resnikoff, Raymond O. Wells, "Wavelet A

nalysis: The Scalable Structure of Information", Springer, 1998

[3] Martin Vetterli, "Wavelets, approximation and compression," IEEE Sig. Proc. Mag., Sept. 2001.

[4] Sweldens W. "The lifting scheme: A custom-design construction of biorthogonal wavelets." Applied and Computational Harmonic Analysis, 1996,3(2):186~200.

[5] E. L. Pennec, S. Mallat, "Sparse geometric image representations with bandelets," July 2003.

Reference [6] Candes, E. Ridgelets: theory and applications, Ph. D.

thesis, Department of Statistics, Stanford University, 1998.

[7] J.L. Starck, E.J. Candès and D.L. Donoho, The curvelet transform for image denoising, IEEE Transactions on Image Processing 11 (2002) (6), pp. 670–684.