Sparse Fourier TransformSparse 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

3

“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)

Frequency-Sparsity

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MobiCom 2012: Faster GPS via the Sparse Fourier Transform

Frequency-Sparsity

5

MobiCom 2012: Faster GPS via the Sparse Fourier Transform

6

Framework ： QuickSync

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

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

The FFT Algorithm

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The calculation flow:

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.”

A simple trial

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My way of improvement.

12

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. 

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

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

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 

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

Information Theoretic Framework

Sparsity: ― Sparsity:

― Joint sparsity: calculate collaboratively

― Conditional sparsity:

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Thank　you!