signal reconstruction from multiscale edges
DESCRIPTION
Signal reconstruction from multiscale edges. A wavelet based algorithm. Author Yen-Ming Mark Lai ( [email protected] ) Advisor Dr. Radu Balan [email protected] CSCAMM, MATH. Reference. “Characterization of Signals from Multiscale Edges” Stephane Mallat and Sifen Zhong - PowerPoint PPT PresentationTRANSCRIPT
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Signal reconstruction from multiscale edges
A wavelet based algorithm
![Page 2: Signal reconstruction from multiscale edges](https://reader035.vdocuments.us/reader035/viewer/2022081515/56813ad9550346895da31a7d/html5/thumbnails/2.jpg)
Author
Yen-Ming Mark Lai
Advisor
Dr. Radu Balan
CSCAMM, MATH
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Reference
• “Characterization of Signals from Multiscale Edges”
• Stephane Mallat and Sifen Zhong
• IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 14, pp 710-732, July 1992
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Motivation
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Input Signal (256 points)
Which points to save?
![Page 6: Signal reconstruction from multiscale edges](https://reader035.vdocuments.us/reader035/viewer/2022081515/56813ad9550346895da31a7d/html5/thumbnails/6.jpg)
Compressed Signal (37 points)
What else for reconstruction?
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Compressed Signal (37 points)
sharp one-sided edge
![Page 8: Signal reconstruction from multiscale edges](https://reader035.vdocuments.us/reader035/viewer/2022081515/56813ad9550346895da31a7d/html5/thumbnails/8.jpg)
Compressed Signal (37 points)
sharp two-sided edge
![Page 9: Signal reconstruction from multiscale edges](https://reader035.vdocuments.us/reader035/viewer/2022081515/56813ad9550346895da31a7d/html5/thumbnails/9.jpg)
Compressed Signal (37 points)
“noisy” edges
![Page 10: Signal reconstruction from multiscale edges](https://reader035.vdocuments.us/reader035/viewer/2022081515/56813ad9550346895da31a7d/html5/thumbnails/10.jpg)
Calculation
Reconstruction:
• edges
• edge type information
Original: (256 points)
(37 points)
(x points)
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37
Compression
edges edge type
+ x < 256
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Summary
Save edges
![Page 13: Signal reconstruction from multiscale edges](https://reader035.vdocuments.us/reader035/viewer/2022081515/56813ad9550346895da31a7d/html5/thumbnails/13.jpg)
Summary
Save edge type
sharp one-sided edge
sharp two-sided edge
“noisy” edges
![Page 14: Signal reconstruction from multiscale edges](https://reader035.vdocuments.us/reader035/viewer/2022081515/56813ad9550346895da31a7d/html5/thumbnails/14.jpg)
Summary
edges edge type reconstruct+ =
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Algorithm
Decomposition + Reconstruction
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Decomposition
Discrete Wavelet
Transform
Save edges e.g. local extrema
Input
“edges+edge type”
![Page 17: Signal reconstruction from multiscale edges](https://reader035.vdocuments.us/reader035/viewer/2022081515/56813ad9550346895da31a7d/html5/thumbnails/17.jpg)
Reconstruction
Find approximation
Inverse Wavelet
Transform
Output
local extrema “edges+edge type”
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What is Discrete Wavelet Transform?
Discrete Wavelet
Transform
Input
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What is DWT?
1) Choose mother wavelet
2) Dilate mother wavelet
3) Convolve family with input
DWT
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1) Choose mother wavelet
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2) Dilate mother waveletmother wavelet
dilate
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2) Dilate mother wavelet
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Convolve family with input
input
input
input
wavelet scale 1
wavelet scale 2
wavelet scale 4
=
=
=
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Convolve “family”
input
input
input
wavelet scale 1
wavelet scale 2
wavelet scale 4
=
=
=
DWT
multiscale
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What is DWT?
(mathematically)
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How to dilate?
)2(
2
1)(
2 jj
xxj
mother wavelet
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How to dilate?
)2(
2
1)(
2 jj
xxj
dyadic (powers of two)
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How to dilate?
)2(
2
1)(
2 jj
xxj
scale
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How to dilate?
)2(
2
1)(
2 jj
xxj z
halve amplitude
double support
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Mother Wavelet (Haar)scale 1, j=0
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Mother Wavelet (Haar)
scale 2, j=1
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Mother Wavelet (Haar)
scale 4, j=2
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What is DWT?
Convolution of dilates of mother wavelets against original signal.
)()2(
2
1)()(
2xf
xxfx
jjj
![Page 34: Signal reconstruction from multiscale edges](https://reader035.vdocuments.us/reader035/viewer/2022081515/56813ad9550346895da31a7d/html5/thumbnails/34.jpg)
What is DWT?
Convolution of dilates of mother wavelets against original signal.
)()2(
2
1)()(
2xf
xxfx
jjj
convolution
![Page 35: Signal reconstruction from multiscale edges](https://reader035.vdocuments.us/reader035/viewer/2022081515/56813ad9550346895da31a7d/html5/thumbnails/35.jpg)
What is DWT?
Convolution of dilates of mother wavelets against original signal.
)()2(
2
1)()(
2xf
xxfx
jjj
dilates
![Page 36: Signal reconstruction from multiscale edges](https://reader035.vdocuments.us/reader035/viewer/2022081515/56813ad9550346895da31a7d/html5/thumbnails/36.jpg)
What is DWT?
Convolution of dilates of mother wavelets against original signal.
)()2(
2
1)()(
2xf
xxfx
jjj
original signal
![Page 37: Signal reconstruction from multiscale edges](https://reader035.vdocuments.us/reader035/viewer/2022081515/56813ad9550346895da31a7d/html5/thumbnails/37.jpg)
What is convolution?(best match operation)
Discrete Wavelet
Transform
Input1)mother wavelet
2)dilation
3)convolution
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Convolution (best match operator)
dtgftgf )()()(
dummy variable
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Convolution (best match operator)
flip g around y axis
dtgftgf )()()(
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dtgftgf )()()(
Convolution (best match operator)
shifts g by t
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dtgftgf )()()(
do nothing to f
Convolution (best match operator)
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dtgftgf )()()(
Convolution (best match operator)
pointwise multiplication
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dtgftgf )()()(
Convolution (best match operator)
integrate over R
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)7.7(gf
flip g and shift by 7.7
dgf )7.7()(
Convolution (one point)
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)7.7(gf
do nothing to f
dgf )7.7()(
Convolution (one point)
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)7.7(gf
multiply f and g pointwise
dgf )7.7()(
Convolution (one point)
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)7.7(gf
integrate over R
dgf )7.7()(
Convolution (one point)
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)7.7(gf
Convolution (one point)
scalar
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Convolution of two boxes
f
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Convolution of two boxes
45.1g
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Convolution of two boxes
45.1t
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Convolution of two boxes
dgfgf )45.1()()45.1( 0
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Convolution of two boxes
f
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Convolution of two boxes
5.0g
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Convolution of two boxes
5.0t
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Convolution of two boxes
dgfgf )5.0()()5.0( 5.0
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Convolution of two boxes
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Convolution of two boxes
gf
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Why convolution?
Location of maximum best fit
![Page 60: Signal reconstruction from multiscale edges](https://reader035.vdocuments.us/reader035/viewer/2022081515/56813ad9550346895da31a7d/html5/thumbnails/60.jpg)
Where does red box most look like blue box?
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Why convolution?
Location of maximum best fit
maximum
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Why convolution?
Location of maximum best fit
maxima best fit location
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Where does exponential most look like box?
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Where does exponential most look like box?
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Where does exponential most look like box?
maximum
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Where does exponential most look like box?
maximum best fit location
![Page 67: Signal reconstruction from multiscale edges](https://reader035.vdocuments.us/reader035/viewer/2022081515/56813ad9550346895da31a7d/html5/thumbnails/67.jpg)
So what?
If wavelet is an edge, convolution detects location of edges
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Mother Wavelet (Haar)
![Page 69: Signal reconstruction from multiscale edges](https://reader035.vdocuments.us/reader035/viewer/2022081515/56813ad9550346895da31a7d/html5/thumbnails/69.jpg)
Mother Wavelet (Haar)
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Mother Wavelet (Haar)
![Page 71: Signal reconstruction from multiscale edges](https://reader035.vdocuments.us/reader035/viewer/2022081515/56813ad9550346895da31a7d/html5/thumbnails/71.jpg)
What is edge?
Local extrema of wavelet transform
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Summary of Decomposition
Discrete Wavelet
Transform
Save “edges” e.g. local extrema
Input
“edges+edge type”
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Summary of Decomposition
input
input
input
edge detection (scale 1)
edge detection (scale 2)
edge detection (scale 4)
=
=
=
![Page 74: Signal reconstruction from multiscale edges](https://reader035.vdocuments.us/reader035/viewer/2022081515/56813ad9550346895da31a7d/html5/thumbnails/74.jpg)
How to find approximation?
Find approximation
local extrema“edges+edge type”
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Find approximation (iterative)
Alternate projections between two spaces
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Find approximation (iterative)
2'2
22:LnLnn ffnf
![Page 77: Signal reconstruction from multiscale edges](https://reader035.vdocuments.us/reader035/viewer/2022081515/56813ad9550346895da31a7d/html5/thumbnails/77.jpg)
Find approximation (iterative)
2'2
22:LnLnn ffnf
H_1 Sobolev Norm
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Find approximation (iterative)
functions that interpolate given local maxima points
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Find approximation (iterative)
dyadic wavelet transforms of L^2 functions
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Find approximation (iterative)
intersection = space of solutions
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Find approximation (iterative)
Start at zero element to minimize solution’s norm
![Page 82: Signal reconstruction from multiscale edges](https://reader035.vdocuments.us/reader035/viewer/2022081515/56813ad9550346895da31a7d/html5/thumbnails/82.jpg)
Q: Why minimize over K?
A: Interpolation points act like local extrema
2'2
22:LnLnn ffnf
![Page 83: Signal reconstruction from multiscale edges](https://reader035.vdocuments.us/reader035/viewer/2022081515/56813ad9550346895da31a7d/html5/thumbnails/83.jpg)
Reconstruction
Find approximation(minimization
problem)
Inverse Wavelet
Transform
Output
![Page 84: Signal reconstruction from multiscale edges](https://reader035.vdocuments.us/reader035/viewer/2022081515/56813ad9550346895da31a7d/html5/thumbnails/84.jpg)
Example
Input of 256 points
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Input Signal (256 points)
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Input Signal (256 points)
major edges
![Page 87: Signal reconstruction from multiscale edges](https://reader035.vdocuments.us/reader035/viewer/2022081515/56813ad9550346895da31a7d/html5/thumbnails/87.jpg)
Input Signal (256 points)
minor edges (many)
![Page 88: Signal reconstruction from multiscale edges](https://reader035.vdocuments.us/reader035/viewer/2022081515/56813ad9550346895da31a7d/html5/thumbnails/88.jpg)
Discrete Wavelet Transform
fW dj2
Dyadic (powers of 2)
= DWT of “f” at scale 2^j
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DWT (9 scales, 256 points each)
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DWT (9 scales, 256 points each)
major edges
![Page 91: Signal reconstruction from multiscale edges](https://reader035.vdocuments.us/reader035/viewer/2022081515/56813ad9550346895da31a7d/html5/thumbnails/91.jpg)
Input Signal (256 points)
major edges
![Page 92: Signal reconstruction from multiscale edges](https://reader035.vdocuments.us/reader035/viewer/2022081515/56813ad9550346895da31a7d/html5/thumbnails/92.jpg)
DWT (9 scales, 256 points each)
minor edges (many)
![Page 93: Signal reconstruction from multiscale edges](https://reader035.vdocuments.us/reader035/viewer/2022081515/56813ad9550346895da31a7d/html5/thumbnails/93.jpg)
Input Signal (256 points)
minor edges (many)
![Page 94: Signal reconstruction from multiscale edges](https://reader035.vdocuments.us/reader035/viewer/2022081515/56813ad9550346895da31a7d/html5/thumbnails/94.jpg)
Decomposition
Discrete Wavelet
Transform
Save “edges” e.g. local extrema
Input
![Page 95: Signal reconstruction from multiscale edges](https://reader035.vdocuments.us/reader035/viewer/2022081515/56813ad9550346895da31a7d/html5/thumbnails/95.jpg)
DWT (9 scales, 256 points each)
![Page 96: Signal reconstruction from multiscale edges](https://reader035.vdocuments.us/reader035/viewer/2022081515/56813ad9550346895da31a7d/html5/thumbnails/96.jpg)
Save Local Maxima
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Local Maxima of Transform
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Local Maxima of Transform
low scale most sensitive
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Mother Wavelet (Haar)
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Local Maxima of Transform
high scale least sensitive
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Mother Wavelet (Haar)
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Decomposition
Discrete Wavelet
Transform
Save “edges” e.g. local extrema
Input
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Local Maxima of Transform
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Find approximation (iterative)
Alternate projections between two spaces
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Reconstruction
Find approximation(minimization
problem)
Inverse Wavelet
Transform
Output
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Mallat’s Reconstruction (20 iterations)
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original
reconstruction (20 iterations)
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Implementation
Language: MATLAB – Matlab wavelet toolbox
Complexity: convergence criteria
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Databases
• Baseline signals
– sinusoids, Gaussians, step edges, Diracs
• Audio signals
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Validation
• Unit testing of components
– DWT/IDWT
– Local extrema search
– Projection onto interpolation space (\Gamma)
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Testing
• L2 norm of the error (sum of squares)
versus iterations
• Saturation point in iteration (knee)
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Schedule (Coding)
• October/November – code Alternate
Projections (8 weeks)
• December – write up mid-year report (2
weeks)
• January – code local extrema search (1
week)
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Schedule (Testing)
• February/March – test and debug entire
system (8 weeks)
• April – run code against database (4
weeks)
• May – write up final report (2 weeks)
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Milestones
• December 1, 2010 – Alternate Projections
code passes unit test
• February 1, 2011 – local extrema search
code passes unit test
• April 1, 2011 - codes passes system test
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Deliverables
• Documented MATLAB code
• Testing results (reproducible)
• Mid-year report/Final report
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Summary
Discrete Wavelet
Transform
Save edges e.g. local extrema
Input
Find approximation
Inverse Wavelet
TransformOutput
Questions?
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Supplemental Slides
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Similar Idea to JPEG
• Discrete Fourier Transform
• Save largest coefficients
• Inverse Discrete Fourier Transform
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Comparison of algorithms
• DFT
• Save largest coefficients
• IDFT
• DWT (Redundant)
• Save local maxima on each scale
• Find approximation
• IDWT
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Problem Definition
• Given:– positions– values
of local maxima of |W_2^j f(x)| at each scale
• Find:– Approximation h(x) of f(x)– or equivalently W_{2^j} h(x)
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Maxima Constraint I
At each scale 2^j, for each local maximum located at x_n^j,
e.g. W h(x) = W f(x) at given set of local maxima points (interpolation problem) at each scale
)()(22
jn
jn xfWxhW jj
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Maxima Constraint II
At each scale 2^j, the local maxima of |W_2^j h(x)| are located at the abscissa (x_n^j)_n \in Z
e.g Local maxima of |W h(x)| are local maxima of |W f(x)| at each scale
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Maxima Constraint II
• Constraint not convex (difficult to analyze)
• Use convex constraint instead:
local maxima of |W h(x)| at certain points
|W h(x)|^2 and |d W h(x)|^2 small as possible on average
Original Approximation
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Maxima Constraint “II”
• Minimize |W h(x)|^2 creates local maxima at specified positions
• Minimize |d W h|^2 minimize modulus maxima outside specified positions
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Maxima Constraint “II”
• Solve minimization problem
|||h|||^2= \sum_j ( ||W_{2^j} h ||^2 + 2^{2j} || dW_{2^j}h/ dx ||^2)
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Solve minimization problem
Let K be the space of all sequences of fucntions (g_j^(x) )_j \in Z such that
|(g_j(x))_j \in Z|^2 = \sum_j ( \| g_j\| ^2 + 2^{2j} \| \frac{dg_j}{dx} \|^2 < \infty
(inspired by condition “II”)
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V space
• Let V be the space of all dyadic wavelet tranforms of L^2(R).
• V \subset K
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\Gamma space
Let \Gamma be the affine space of sequences of functions (g_j(x))_j \in Z \in K such that for any index j and all maxima positions x_n^j
g_j(x_n^j)= W_{2^j} f(x_n^j)
(inspired by Condition I)
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\Gamma space
• One can prove that \Gamma is closed in K
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Recap
• K: sequence of functions whose sum of – norm of each element– norm of each element’s derivative
is finite
• V: dyadic wavelet transform of L^2
• \Gamma: sequences of functions whose value match those of |W f| at local maxima
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Condition I in K
Dyadic wavelet transforms that satisfy Condition I
\Lambda = V \cap \Gamma
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Condition I + “II” in K
• Find element of \Lambda = V \cap \Gamma whose norm is minimum
• Use alternate projections on V and \Gamma
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Projection on V
P_V = W \circ W^{-1}
In other words,
First:W^-1 dyadic inverse wavelet transform
Then:W dyadic wavelet transform
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Projection on \Gamma
At each scale 2^j, add discrete signal e_j^d = ( e_j(n))_{1 \leq n \leq N} that is computed from
e_j(x) = \alpha e^{2^{-jx}} + \beta e^{-2^{-jx}}
e.g. Add piecewise exponential curves to each function of sequence
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How to compute coefficients?
• Solve system of equations given by equation (110)
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Where to start?
Answer:
zero element of K e.g.
g_j(x) = 0 for all j \in Z
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Converge to what?
Answer:
Alternate projections converge to – element of \Lambda (Condition I) – whose norm is minimum (Condition “II”)
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How fast is convergence?
Answer:
If (\sqrt{2^j} \psi_{2^j}(x_n^j-x)) is frame and there exists constant 0< D <=1 such that at all scales 2^j the distances between any two consecutive maxima satisfy
|x_n^j – x_{n-1}^j | \geq D2^j
then, convergence is exponential
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Convex constraint?