Sparsity and Compressed Sensing

Talk at Cachan

1. Sparsity and Compressed Sensing Gabriel Peyré www.numerical-tours.com
2. Signals, Images and More
3. Signals, Images and More
4. Signals, Images and More
5. Signals, Images and More
6. Signals, Images and More
7. Overview • Approximation in an Ortho-Basis • Compression and Denoising • Compressed Sensing
8. Orthogonal basis { m }m of L2([0, 1]d) Continuous signal/image f L2([0, 1]d). Orthogonal Decompositions
9. Orthogonal basis { m }m of L2([0, 1]d) f = m f, m m ||f|| = |f(x)|2dx = m | f, m ⇥|2 Continuous signal/image f L2([0, 1]d). Orthogonal Decompositions
10. Orthogonal basis { m }m of L2([0, 1]d) f = m f, m m ||f|| = |f(x)|2dx = m | f, m ⇥|2 Continuous signal/image f L2([0, 1]d). Orthogonal Decompositions m
11. 1-D Wavelet Basis Wavelets: j,n (x) = 1 2j/2 x 2jn 2j Position n, scale 2j, m = (n, j).
12. 1-D Wavelet Basis Wavelets: j,n (x) = 1 2j/2 x 2jn 2j Position n, scale 2j, m = (n, j).
13. m1,m2 Basis { m1,m2 (x1, x2 )}m1,m2 of L2([0, 1]2) m1,m2 (x1, x2 ) = m1 (x1 ) m2 (x2 ) tensor product 2-D Fourier Basis Basis { m (x)}m of L2([0, 1]) m1 m2
14. m1,m2 Basis { m1,m2 (x1, x2 )}m1,m2 of L2([0, 1]2) m1,m2 (x1, x2 ) = m1 (x1 ) m2 (x2 ) tensor product f(x) f, m1,m2 Fourier transform 2-D Fourier Basis Basis { m (x)}m of L2([0, 1]) m1 m2 x m
15. 3 elementary wavelets { H, V , D}. Orthogonal basis of L2([0, 1]2): k j,n (x) = 2 j (2 jx n) k=H,V,D j<0,2j n [0,1]2 2-D Wavelet Basis V (x) H(x) D(x)
16. 3 elementary wavelets { H, V , D}. Orthogonal basis of L2([0, 1]2): k j,n (x) = 2 j (2 jx n) k=H,V,D j<0,2j n [0,1]2 2-D Wavelet Basis V (x) H(x) D(x)
17. wavelet f, k j,n Example of Wavelet Decomposition f(x) transform x (j, n, k)
18. Discrete Computations Discrete orthogonal basis { m } of CN . f = m f, m m
19. Fast Fourier Transform (FFT), O(N log(N)) operations. Discrete Computations Discrete orthogonal basis { m } of CN . m [n] = 1 N e2i N nm f = m f, m m
20. Fast Fourier Transform (FFT), O(N log(N)) operations. Fast Wavelet Transform, O(N) operations. Discrete Wavelet basis: no closed-form expression. Discrete Computations Discrete orthogonal basis { m } of CN . m [n] = 1 N e2i N nm f = m f, m m
21. Sparse Approximation in a Basis
22. Sparse Approximation in a Basis
23. Sparse Approximation in a Basis
24. Best basis Fastest error decay ||f fM ||2 log(||f fM ||) log(M) Efficiency of Transforms Fourier DCT Local DCT Wavelets
25. Overview • Approximation in an Ortho-Basis • Compression and Denoising • Compressed Sensing
26. JPEG-2000 vs. JPEG, 0.2bit/pixel
27. Compression by Transform-coding Image f Zoom on f f forward a[m] = ⇥f, m ⇤ R transform
28. Compression by Transform-coding Image f Zoom on f f forward a[m] = ⇥f, m ⇤ R transform Quantization: q[m] = sign(a[m]) |a[m]| T ⇥ Z ˜ a[m] T T 2T 2T a[m] Quantized q[m] bin T q[m] Z
29. Compression by Transform-coding Image f Zoom on f f forward a[m] = ⇥f, m ⇤ R coding transform Entropic coding: use statistical redundancy (many 0’s). Quantization: q[m] = sign(a[m]) |a[m]| T ⇥ Z ˜ a[m] T T 2T 2T a[m] Quantized q[m] bin T q[m] Z
30. Compression by Transform-coding Image f Zoom on f f forward a[m] = ⇥f, m ⇤ R coding decoding q[m] Z transform Entropic coding: use statistical redundancy (many 0’s). Quantization: q[m] = sign(a[m]) |a[m]| T ⇥ Z ˜ a[m] T T 2T 2T a[m] Quantized q[m] bin T q[m] Z
31. Compression by Transform-coding Image f Zoom on f f forward Dequantization: ˜ a[m] = sign(q[m]) |q[m] + 1 2 ⇥ T a[m] = ⇥f, m ⇤ R coding decoding q[m] Z ˜ a[m] dequantization transform Entropic coding: use statistical redundancy (many 0’s). Quantization: q[m] = sign(a[m]) |a[m]| T ⇥ Z ˜ a[m] T T 2T 2T a[m] Quantized q[m] bin T q[m] Z
32. Compression by Transform-coding Image f Zoom on f f , R =0.2 bit/pixel f forward Dequantization: ˜ a[m] = sign(q[m]) |q[m] + 1 2 ⇥ T a[m] = ⇥f, m ⇤ R coding decoding q[m] Z ˜ a[m] dequantization transform backward fR = m IT ˜ a[m] m transform Entropic coding: use statistical redundancy (many 0’s). Quantization: q[m] = sign(a[m]) |a[m]| T ⇥ Z ˜ a[m] T T 2T 2T a[m] Quantized q[m] bin T q[m] Z
33. Compression by Transform-coding Image f Zoom on f f , R =0.2 bit/pixel f forward Dequantization: ˜ a[m] = sign(q[m]) |q[m] + 1 2 ⇥ T a[m] = ⇥f, m ⇤ R coding decoding q[m] Z ˜ a[m] dequantization transform backward fR = m IT ˜ a[m] m transform Entropic coding: use statistical redundancy (many 0’s). Quantization: q[m] = sign(a[m]) |a[m]| T ⇥ Z ˜ a[m] T T 2T 2T a[m] Quantized q[m] bin T q[m] Z “Theorem:” ||f fM ||2 = O(M ) =⇥ ||f fR ||2 = O(log (R)R )
34. Noise in Images
35. Denoising
36. Denoising thresh. f = N 1 m=0 f, m ⇥ m ˜ f = | f, m ⇥|>T f, m ⇥ m
37. Denoising thresh. f = N 1 m=0 f, m ⇥ m ˜ f = | f, m ⇥|>T f, m ⇥ m In practice: T 3 for T = 2 log(N) Theorem: if ||f0 f0,M ||2 = O(M ), E(|| ˜ f f0 ||2) = O( 2 +1 )
38. Overview • Approximation in an Ortho-Basis • Compression and Denoising • Compressed Sensing
39. f[n] f0 (n/N) Sampling: ˜ f L2([0, 1]d) f RN Idealization: acquisition device Discretization
40. Data aquisition: Sensors Pointwise Sampling and Smoothness ˜ f L2 f RN f[i] = ˜ f(i/N)
41. Data aquisition: Sensors ˜ f(t) = i f[i]h(Nt i) Shannon interpolation: if Supp( ˆ ˜ f) [ N , N ] h(t) = sin( t) t Pointwise Sampling and Smoothness ˜ f L2 f RN f[i] = ˜ f(i/N)
42. Data aquisition: Sensors ˜ f(t) = i f[i]h(Nt i) Natural images are not smooth. Shannon interpolation: if Supp( ˆ ˜ f) [ N , N ] h(t) = sin( t) t Pointwise Sampling and Smoothness ˜ f L2 f RN f[i] = ˜ f(i/N)
43. Data aquisition: Sensors ˜ f(t) = i f[i]h(Nt i) Natural images are not smooth. But can be compressed e ciently. Shannon interpolation: if Supp( ˆ ˜ f) [ N , N ] 0,1,0,. . . h(t) = sin( t) t Sample and compress simultaneously? Pointwise Sampling and Smoothness ˜ f L2 f RN f[i] = ˜ f(i/N) JPEG-2k
44. Single Pixel Camera (Rice) y[i] = f0, i ⇥
45. Single Pixel Camera (Rice) y[i] = f0, i ⇥ f0, N = 2562 f , P/N = 0.16 f , P/N = 0.02
46. Physical hardware resolution limit: target resolution f RN . ˜ f L2 f RN y RP micro mirrors array resolution CS hardware K CS Hardware Model CS is about designing hardware: input signals ˜ f L2(R2).
47. Physical hardware resolution limit: target resolution f RN . ˜ f L2 f RN y RP micro mirrors array resolution CS hardware , ... K CS Hardware Model CS is about designing hardware: input signals ˜ f L2(R2). , ,
48. Physical hardware resolution limit: target resolution f RN . ˜ f L2 f RN y RP micro mirrors array resolution CS hardware , ... f Operator K K CS Hardware Model CS is about designing hardware: input signals ˜ f L2(R2). , ,
49. Need to solve y = Kf. More unknown than equations. dim(ker(K)) = N P is huge. Inversion and Sparsity f Operator K
50. Need to solve y = Kf. More unknown than equations. dim(ker(K)) = N P is huge. Prior information: f is sparse in a basis { m }m . J (f) = Card {m \ | f, m | > } is small. Inversion and Sparsity f Operator K f, m f
51. 0 reconstruction: Minimize subject to Kf = y y = f f, 1 f, 2 CS Reconstruction J0 (f) = Card {m \ f, m = 0}
52. 0 reconstruction: Minimize subject to Kf = y NP-hard to solve. y = f f, 1 f, 2 CS Reconstruction J0 (f) = Card {m \ f, m = 0}
53. 0 reconstruction: Minimize subject to Kf = y 1 reconstruction: m | f, m | Polynomial-time algorithms. NP-hard to solve. y = f f, 1 f, 2 CS Reconstruction J0 (f) = Card {m \ f, m = 0} Minimize subject to Kf = y
54. Theorem: [Candes, Romberg, Tao, Donoho, 2004] If f is k-sparse, i.e. J0 (f) k If P C log(N/k)k then 1-CS reconstruction is exact. Theoretical Performance Guaranties
55. Theorem: [Candes, Romberg, Tao, Donoho, 2004] If f is k-sparse, i.e. J0 (f) k If P C log(N/k)k then 1-CS reconstruction is exact. Extensions to: noisy observation y = Kf + approximate sparsity f = fk sparse + Theoretical Performance Guaranties
56. Theorem: [Candes, Romberg, Tao, Donoho, 2004] If f is k-sparse, i.e. J0 (f) k If P C log(N/k)k then 1-CS reconstruction is exact. Extensions to: noisy observation y = Kf + approximate sparsity f = fk sparse + Research problem: optimal value of C ? for N/k = 4, C log(N/k) 5. “CS is 5 less e cient than JPEG-2k” Theoretical Performance Guaranties
57. Conclusion
58. Conclusion
59. random acquisition. optimization for reconstruction. #measures sparsity Conclusion • Compressed sensing.