sparse fourier transform
DESCRIPTION
Sparse Fourier Transform. By: Yanglet Date: 2013/3/6. Outline. Frequency-Sparsity Application Example: MobiCom 2012 Understanding of the DFT The FFT Algorithm Sparse Fourier Transform Possible Issues. Frequency-Sparsity. “frequency-sparsity” almost everywhere! newwu.jpg - PowerPoint PPT PresentationTRANSCRIPT
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Sparse Fourier TransformSparse Fourier Transform
By: Yanglet
Date: 2013/3/6
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Outline
Frequency-Sparsity
Application Example: MobiCom 2012
Understanding of the DFT
The FFT Algorithm
Sparse Fourier Transform
Possible Issues
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Frequency-Sparsity
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“frequency-sparsity” almost everywhere!― newwu.jpg― size: 1440 1152 3
1440 * 1152 pixels; RGB.
Pictures are sparse in the frequency domain.
(Just an example!)
• original picture frequencies (reshaped to 1D-plot)
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Frequency-Sparsity
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MobiCom 2012: Faster GPS via the Sparse Fourier Transform
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Frequency-Sparsity
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MobiCom 2012: Faster GPS via the Sparse Fourier Transform
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Framework : QuickSync
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Understanding of the DFT
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Discrete Fourier transform (DFT)
― DFT by matrix multiplication
How to understand DFT?
0 0 1 0 2 0 ( 1) 0
0 1 1 1 2 1 ( 1) 1
0 ( 1) 1 ( 1) 2 ( 1) ( 1) ( 1)
(0) (0)
(1)(1)
( 1)( 1)
N
N
N N N N N
W W W WF f
fW W W WF
f NF N W W W W
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The FFT Algorithm
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Why it is ever possible?
由于W具有周期性和对称性
9630
6420
3210
0000
WWWW
WWWW
WWWW
WWWWu=0
u=1
u=2
u=3
1010
0000
1010
0000
WWWW
WWWW
WWWW
WWWW
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The FFT Algorithm
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The calculation flow:
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The FFT Algorithm
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Optimal?
1.
2. For the “exact” case.
How to improve it?
Why we want to improve it? 1. To reduce runtime for big data (signal), e.g. real-time app., (the GPS sys.)
2. Sparse is everywhere, there is no need calculate all n-frequency in engineering tasks?
3. To save energy by using less calculations.
4. ubiquitous applications
“you don’t really study the Fourier transform for what it is,” says Laurent Demanet, an assistant professor of applied mathematics at MIT. “You take a class in signal processing, and there it is. You don’t have any choice.”
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A simple trial
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My way of improvement.
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sFFT & sIFFT
Piotr Indyk, Dina Katabi, Eric Price, Haitham Hassanieh
Haitham Hassanieh, Piotr Indyk, Dina Katabi, and Eric Price, "Nearly Optimal Sparse
Fourier Transform," STOC, 2012.
Haitham Hassanieh, Piotr Indyk, Dina Katabi, and Eric Price, "Simple and Practical
Algorithm for Sparse Fourier Transform," SODA, 2012.
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The Sparse Fourier Transform
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The Sparse Fourier Transform
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The Sparse Fourier Transform
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The Sparse Fourier Transform
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The Sparse Fourier Transform
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The Median Operator
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Median
Original one Noised one Denoised/Recovered one
Parameter d=0.2. Using median filter
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Near-Optimal?
Compressive Sensing Approach― Y is the random linear encoding results of K-sparse vector X
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~
1
1
X argmin X
. . XM M N Ns t Y A
Results
We need only to recovery Xlog( / )M CK N K N
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News
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•News : 100 Top Stories of 2013: 34. Better Math Makes Faster Data NetworksGillian Conahan January, 2013 •10 Emerging Technologies: A Faster Fourier Transform Mark Anderson, Technology Review May, 2012 •A Faster Fast Fourier Transform.David Schneider, IEEE Spectrum March, 2012 •News Hour BroadcastBBC World Service February, 2012 •Faster-Than-Fast Fourier TransformSlashdot January, 2012 •Better Mathematics Boosts Image-Processing AlgorithmJacob Aron, New Scientist January, 2012 •A New Faster Fourier Transform Can Speed One of IT‘s Fundamental AlgorithmsClay Dillow, Popular Science January, 2012 •The faster-than-fast Fourier transform.Larry Hardesty, MIT News Office January, 2012
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Joint Sparsity Models
JSM-1: Common Sparse Supports
Same support set, but with different coefficients.
JSM-2: Sparse Common component + sparse innovations
JSM-3: Nonsparse common component + sparse innovations
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1 2{ , ,..., }
, 1,2,...,
{1,2,...., }, .
J
j j
S X X X
X j J
Sparse Support N k
C
C 0
j 0
Z +Z , 1,2,..., ,
Z ,
Z , .
j j
C C C
j j j
X j J
k
k
C
C
j 0
Z +Z , 1,2,..., ,
Z ,
Z , .
j j
C
j j j
X j J
k
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Information Theoretic Framework
Sparsity: ― Sparsity:
― Joint sparsity: calculate collaboratively
― Conditional sparsity:
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Thank you!