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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.