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L’échantillonnage compressé

Gabriel Peyré
November 02, 2016

L’échantillonnage compressé

Conférence "Hommage à Claude Shannon, le père de la théorie de l'information" au CIRM

Gabriel Peyré

November 02, 2016
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  1. Gabriel Peyré
    www.numerical-tours.com
    L’échantillonnage
    compressé
    É C O L E N O R M A L E
    S U P É R I E U R E

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  2. Echantillonner puis compresser
    0,1,0,. . .
    Lena Lena.bmp Lena.jpg

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  3. Echantillonner puis compresser
    0,1,0,. . .
    Lena Lena.bmp Lena.jpg

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  4. Echantillonner puis compresser
    0,1,0,. . .
    Lena Lena.bmp Lena.jpg

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  5. Appareil photo à pixel unique
    •  Digital Micromirror Device (DMD) by Texas Instruments
    http://dsp.rice.edu/cscamera

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  6. Appareil photo à pixel unique
    •  Digital Micromirror Device (DMD) by Texas Instruments
    http://dsp.rice.edu/cscamera

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  7. Modèle mathématique de l’acquisition

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  8. Modèle mathématique de l’acquisition

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  9. Modèle mathématique de l’acquisition

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  10. Modèle mathématique de l’acquisition

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  11. Système linéaire

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  12. Système linéaire

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  13. Système linéaire

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  14. Compressibilité et reconstruction
    Implication of sparsity: image “compression”
    1 Compute 1,000,000 wavelet coe cients of mega-pixel image
    2 Set to zero all but the 25,000 largest coe cients
    3 Invert the wavelet transform
    original image after zeroing out smallest coe cients
    This principle underlies modern lossy coders (sound, still-picture, video)
    Implication of sparsity: image “compression”
    1 Compute 1,000,000 wavelet coe cients of mega-pixel image
    2 Set to zero all but the 25,000 largest coe cients
    3 Invert the wavelet transform
    original image after zeroing out smallest coe cients
    This principle underlies modern lossy coders (sound, still-picture, video)

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  15. Compressibilité et reconstruction
    Implication of sparsity: image “compression”
    1 Compute 1,000,000 wavelet coe cients of mega-pixel image
    2 Set to zero all but the 25,000 largest coe cients
    3 Invert the wavelet transform
    original image after zeroing out smallest coe cients
    This principle underlies modern lossy coders (sound, still-picture, video)
    Implication of sparsity: image “compression”
    1 Compute 1,000,000 wavelet coe cients of mega-pixel image
    2 Set to zero all but the 25,000 largest coe cients
    3 Invert the wavelet transform
    original image after zeroing out smallest coe cients
    This principle underlies modern lossy coders (sound, still-picture, video)

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  16. La théorie de Candès et al.
    mplication of sparsity: image “compression”
    1 Compute 1,000,000 wavelet coe cients of mega-pixel image
    2 Set to zero all but the 25,000 largest coe cients
    3 Invert the wavelet transform
    original image after zeroing out smallest coe cients
    his principle underlies modern lossy coders (sound, still-picture, video)
    Implication of sparsity: image “compression”
    1 Compute 1,000,000 wavelet coe cients of mega-pixel image
    2 Set to zero all but the 25,000 largest coe cients
    3 Invert the wavelet transform
    original image after zeroing out smallest coe cients
    This principle underlies modern lossy coders (sound, still-picture, video)
    Implication of sparsity: image “compression”
    1 Compute 1,000,000 wavelet coe cients of mega-pixel image
    2 Set to zero all but the 25,000 largest coe cients
    3 Invert the wavelet transform
    original image after zeroing out smallest coe cients
    This principle underlies modern lossy coders (sound, still-picture, video)
    Implication of sparsity: image “compression”
    1
    Compute 1,000,000 wavelet coe
    cients of mega-pixel image
    2
    Set to zero all but the 25,000 largest coe
    cients
    3
    Invert the wavelet transform
    original image
    after zeroing out smallest coe
    cients
    This principle underlies modern lossy coders (sound, still-picture, video)

    View Slide

  17. La théorie de Candès et al.
    mplication of sparsity: image “compression”
    1 Compute 1,000,000 wavelet coe cients of mega-pixel image
    2 Set to zero all but the 25,000 largest coe cients
    3 Invert the wavelet transform
    original image after zeroing out smallest coe cients
    his principle underlies modern lossy coders (sound, still-picture, video)
    Implication of sparsity: image “compression”
    1 Compute 1,000,000 wavelet coe cients of mega-pixel image
    2 Set to zero all but the 25,000 largest coe cients
    3 Invert the wavelet transform
    original image after zeroing out smallest coe cients
    This principle underlies modern lossy coders (sound, still-picture, video)
    Implication of sparsity: image “compression”
    1 Compute 1,000,000 wavelet coe cients of mega-pixel image
    2 Set to zero all but the 25,000 largest coe cients
    3 Invert the wavelet transform
    original image after zeroing out smallest coe cients
    This principle underlies modern lossy coders (sound, still-picture, video)
    Implication of sparsity: image “compression”
    1
    Compute 1,000,000 wavelet coe
    cients of mega-pixel image
    2
    Set to zero all but the 25,000 largest coe
    cients
    3
    Invert the wavelet transform
    original image
    after zeroing out smallest coe
    cients
    This principle underlies modern lossy coders (sound, still-picture, video)

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  18. Problèmes Inverses

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  19. Problèmes Inverses

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  20. Conclusion
    Claude Shannon, p`
    ere fondateur de la th´
    eorie de l’information.

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  21. Conclusion
    Claude Shannon, p`
    ere fondateur de la th´
    eorie de l’information.

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  22. Conclusion
    Claude Shannon, p`
    ere fondateur de la th´
    eorie de l’information.

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