Wavelet Transform 國立交通大學電子工程學系 陳奕安 2007.8.15. Outline Comparison of...
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Transcript of Wavelet Transform 國立交通大學電子工程學系 陳奕安 2007.8.15. Outline Comparison of...
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Wavelet Transform
國立交通大學電子工程學系陳奕安
2007.8.15
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Outline Comparison of Transformations Multiresolution Analysis Discrete Wavelet Transform Fast Wavelet Transform Lifting Scheme Beyond Wavelet Reference
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Outline Comparison of Transformations Multiresolution Analysis Discrete Wavelet Transform Fast Wavelet Transform Lifting Scheme Beyond Wavelet Reference
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Fourier Transform Frequency domain: Fourier Transform
(Joseph Fourier 1807 )
dtetxfX ftj 2)()(
Cannot provide simultaneously time and frequency information.
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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
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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
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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
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Comparison of Transformations
From http://www.cerm.unifi.it/EUcourse2001/Gunther_lecturenotes.pdf, p.10
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Wavelet Series Expansion Linear decomposition of a function:
Basis orthogonal:
Then the coefficients can be calculated by
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Outline Comparison of Transformations Multiresolution Analysis Discrete Wavelet Transform Fast Wavelet Transform Lifting Scheme Beyond Wavelet Reference
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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:
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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.
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Multiresolution Analysis
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Multiresolution Analysis Multiresolution Formulation.
( Scaling coefficients)
( Wavelet coefficients)
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Outline Comparison of Transformations Multiresolution Analysis Discrete Wavelet Transform Fast Wavelet Transform Lifting Scheme Beyond Wavelet Reference
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Discrete Wavelet Transform (DWT)
Discrete Wavelet Transform Calculation: Using Multiresolution Analysis:
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Outline Comparison of Transformations Multiresolution Analysis Discrete Wavelet Transform Fast Wavelet Transform Lifting Scheme Beyond Wavelet Reference
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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.
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Fast Wavelet Transform
(a) A two-stage or two-scale FWT analysis bank and
(b) its frequency splitting characteristics.
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Fast Wavelet Transform Fast Wavelet Transform
Inverse Fast Wavelet Transform
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Fast Wavelet Transform
A two-stage or two-scale FWT-1 synthesis bank.
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Outline Comparison of Transformations Multiresolution Analysis Discrete Wavelet Transform Fast Wavelet Transform Lifting Scheme Beyond Wavelet Reference
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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.
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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)
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Lifting Scheme A spatial domain construction of bi-orthogonal
wavelets, consists of the 4 operations:
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Lifting Scheme A spatial domain construction of bi-orthogonal
wavelets, consists of the 4 operations:
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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
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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
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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
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Outline Comparison of Transformations Multiresolution Analysis Discrete Wavelet Transform Fast Wavelet Transform Lifting Scheme Beyond Wavelet Reference
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Beyond Wavelet
Ridgelet TransformCurvelet Transform
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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
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Continuous Ridgelet Transform
The ridgelet coefficients of an object f are given by analysis of the Radon transform via:
dta
bttRAbaR ff )(),(),,(
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The Curvelet Transform
Decomposition of the original image into subbands .
Spatial partitioning of each subband.
Appling the ridgelet transform.
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Beyond Wavelet A standard multiscale decomposition into octav
e bands, where the lowpass channel is subsampled while the highpass is not.
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Outline Comparison of Transformations Multiresolution Analysis Discrete Wavelet Transform Fast Wavelet Transform Lifting Scheme Beyond Wavelet Reference
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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.
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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.