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

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

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

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

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

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

Modèle mathématique de l’acquisition

Modèle mathématique de l’acquisition

Modèle mathématique de l’acquisition

Modèle mathématique de l’acquisition

Système linéaire

Système linéaire

Système linéaire

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)

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)

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)

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)

Problèmes Inverses

Problèmes Inverses

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

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

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