Signal Processing Course: Signal Processing in Ortho-bases

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1. Signal and Image Processing with Orthogonal Decompositions Gabriel Peyré www.numerical-tours.com
2. Signals, Images and More Continuous signal: f 2 L 2 ([0 , 1]).
3. Continuous image: f 2 L 2 ([0 , 1] 2 ). Signals, Images and More Continuous signal: f 2 L 2 ([0 , 1]).
4. Continuous image: f 2 L 2 ([0 , 1] 2 ). General setting: f : [0, 1]d ! Rs Videos: d = 3 , s = 1. Signals, Images and More Continuous signal: f 2 L 2 ([0 , 1]).
5. Continuous image: f 2 L 2 ([0 , 1] 2 ). General setting: f : [0, 1]d ! Rs Videos: d = 3 , s = 1. Color image: d = 2 , s = 3. Signals, Images and More Continuous signal: f 2 L 2 ([0 , 1]).
6. Continuous image: f 2 L 2 ([0 , 1] 2 ). General setting: f : [0, 1]d ! Rs Videos: d = 3 , s = 1. Color image: d = 2 , s = 3. Multi-spectral: d = 2, s 3. Signals, Images and More Continuous signal: f 2 L 2 ([0 , 1]).
7. Overview •Orthogonal Representations •Linear and Non-linear Approximations •Compression and Denoising
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. Overview •Orthogonal Representations •Linear and Non-linear Approximations •Compression and Denoising
22. Linear Approximation
23. Linear Approximation
24. Linear Approximation
25. Non-Linear Approximation
26. Coe cients Non-Linear Approximation
27. Coe cients Non-Linear Approximation
28. Best basis Fastest error decay ||f fM ||2 log(||f fM ||) log(M) Efficiency of Transforms Fourier DCT Local DCT Wavelets
29. Efficient Approximation
30. Efficient Approximation
31. Efficient Approximation
32. Efficient Approximation
33. Efficient Approximation
34. Efficient Approximation
35. Efficient Approximation
36. Overview •Orthogonal Representations •Linear and Non-linear Approximations •Compression and Denoising
37. JPEG-2000 vs. JPEG, 0.2bit/pixel
38. Compression by Transform-coding Image f Zoom on f f forward a[m] = ⇥f, m ⇤ R transform
39. 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
40. 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
41. 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
42. 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
43. 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
44. 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 ||f fM ||2 = O(M ) =⇥ ||f fR ||2 = O(log (R)R ) Theorem:
45. Noise in Images
46. Denoising
47. Denoising thresh. f = N 1 m=0 f, m ⇥ m ˜ f = | f, m ⇥|>T f, m ⇥ m
48. 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 )
49. Inverse Problems
50. Inverse Problems
51. Inverse Problems
52. Inverse Problems
53. Inverse Problems
54. Restoration with Sparsity
55. Restoration with Sparsity
56. Restoration with Sparsity
57. Conclusion
58. Conclusion
59. Conclusion