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Sparse Fourier Transform

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

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Sparse Fourier Transform

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  1. Sparse Fourier Transform By: Yanglet Date: 2013/3/6

  2. Outline Frequency-Sparsity Application Example: MobiCom 2012 Understanding of the DFT The FFT Algorithm Sparse Fourier Transform Possible Issues

  3. Frequency-Sparsity • “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)

  4. Frequency-Sparsity • MobiCom 2012: Faster GPS via the Sparse Fourier Transform

  5. Frequency-Sparsity • MobiCom 2012: Faster GPS via the Sparse Fourier Transform

  6. Framework：QuickSync

  7. Understanding of the DFT • Discrete Fourier transform (DFT) • DFT by matrix multiplication • How to understand DFT?

  8. The FFT Algorithm • Why it is ever possible? • 由于W具有周期性和对称性 u=0 u=1 u=2 u=3

  9. The FFT Algorithm • The calculation flow:

  10. The FFT Algorithm • 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 byusing 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.”

  11. A simple trial • 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. 

  13. The Sparse Fourier Transform

  14. The Sparse Fourier Transform

  15. The Sparse Fourier Transform

  16. The Sparse Fourier Transform

  17. The Sparse Fourier Transform

  18. TheMedianOperator • Median Original one Noised one Denoised/Recovered one Parameter d=0.2. Using median filter

  19. Near-Optimal? • Compressive Sensing Approach • Y is the random linear encoding results of K-sparse vector X • Results • We need only to recovery X

  20. News

  21. 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

  22. Information Theoretic Framework • Sparsity: • Sparsity: • Joint sparsity: calculate collaboratively • Conditional sparsity:

  23. Thank you!

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