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# Signal Processing Course : Inverse Problems Regularization

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### Signal Processing Course : Inverse Problems Regularization

1. 1. Inverse Problems Regularization www.numerical-tours.com Gabriel Peyré
2. 2. Overview • Variational Priors • Gradient Descent and PDE’s • Inverse Problems Regularization
3. 3. J(f) = ||f||2 W 1,2 = Z R2 ||rf(x)||dxSobolev semi-norm: Smooth and Cartoon Priors | f|2
4. 4. J(f) = ||f||2 W 1,2 = Z R2 ||rf(x)||dxSobolev semi-norm: Total variation semi-norm: J(f) = ||f||TV = Z R2 ||rf(x)||dx Smooth and Cartoon Priors | f|| f|2
5. 5. J(f) = ||f||2 W 1,2 = Z R2 ||rf(x)||dxSobolev semi-norm: Total variation semi-norm: J(f) = ||f||TV = Z R2 ||rf(x)||dx Smooth and Cartoon Priors | f|| f|2
6. 6. Natural Image Priors
7. 7. Discrete Priors
8. 8. Discrete Priors
9. 9. Discrete Differential Operators
10. 10. Discrete Differential Operators
11. 11. Laplacian Operator
12. 12. Laplacian Operator
13. 13. Function: ˜f : x 2 R2 7! f(x) 2 R ˜f(x + ") = ˜f(x) + hrf(x), "iR2 + O(||"||2 R2 ) r ˜f(x) = (@1 ˜f(x), @2 ˜f(x)) 2 R2 Gradient: Images vs. Functionals
14. 14. Function: ˜f : x 2 R2 7! f(x) 2 R Discrete image: f 2 RN , N = n2 f[i1, i2] = ˜f(i1/n, i2/n) rf[i] ⇡ r ˜f(i/n) ˜f(x + ") = ˜f(x) + hrf(x), "iR2 + O(||"||2 R2 ) r ˜f(x) = (@1 ˜f(x), @2 ˜f(x)) 2 R2 Gradient: Images vs. Functionals
15. 15. Function: ˜f : x 2 R2 7! f(x) 2 R Discrete image: f 2 RN , N = n2 f[i1, i2] = ˜f(i1/n, i2/n) Functional: J : f 2 RN 7! J(f) 2 R J(f + ⌘) = J(f) + hrJ(f), ⌘iRN + O(||⌘||2 RN ) rf[i] ⇡ r ˜f(i/n) ˜f(x + ") = ˜f(x) + hrf(x), "iR2 + O(||"||2 R2 ) r ˜f(x) = (@1 ˜f(x), @2 ˜f(x)) 2 R2 rJ : RN 7! RN Gradient: Images vs. Functionals
16. 16. Function: ˜f : x 2 R2 7! f(x) 2 R Discrete image: f 2 RN , N = n2 f[i1, i2] = ˜f(i1/n, i2/n) Functional: J : f 2 RN 7! J(f) 2 R Sobolev: rJ(f) = (r⇤ r)f = f J(f) = 1 2 ||rf||2 J(f + ⌘) = J(f) + hrJ(f), ⌘iRN + O(||⌘||2 RN ) rf[i] ⇡ r ˜f(i/n) ˜f(x + ") = ˜f(x) + hrf(x), "iR2 + O(||"||2 R2 ) r ˜f(x) = (@1 ˜f(x), @2 ˜f(x)) 2 R2 rJ : RN 7! RN Gradient: Images vs. Functionals
17. 17. rJ(f) = div ✓ rf ||rf|| ◆ If 8 n, rf[n] 6= 0, If 9n, rf[n] = 0, J not di↵erentiable at f. Total Variation Gradient ||rf|| rJ(f)
18. 18. rJ(f) = div ✓ rf ||rf|| ◆ If 8 n, rf[n] 6= 0, Sub-di↵erential: If 9n, rf[n] = 0, J not di↵erentiable at f. Cu = ↵ 2 R2⇥N (u[n] = 0) ) (↵[n] = u[n]/||u[n]||) @J(f) = { div(↵) ; ||↵[n]|| 6 1 and ↵ 2 Crf } Total Variation Gradient ||rf|| rJ(f)
19. 19. −10 −8 −6 −4 −2 0 2 4 6 8 10 −2 0 2 4 6 8 10 12 −10 −8 −6 −4 −2 0 2 4 6 8 10 −2 0 2 4 6 8 10 12 p x2 + "2 |x| Regularized Total Variation ||u||" = p ||u||2 + "2 J"(f) = P n ||rf[n]||"
20. 20. −10 −8 −6 −4 −2 0 2 4 6 8 10 −2 0 2 4 6 8 10 12 −10 −8 −6 −4 −2 0 2 4 6 8 10 −2 0 2 4 6 8 10 12 rJ"(f) = div ✓ rf ||rf||" ◆ p x2 + "2 |x| rJ" ⇠ /" when " ! +1 Regularized Total Variation ||u||" = p ||u||2 + "2 J"(f) = P n ||rf[n]||" rJ"(f)
21. 21. Overview • Variational Priors • Gradient Descent and PDE’s • Inverse Problems Regularization
22. 22. f(k+1) = f(k) ⌧krJ(f(k) ) f(0) is given. Gradient Descent
23. 23. and 0 < ⌧ < 2/L, then f(k) k!+1 ! f? a solution of min f J(f). If f is convex, C1 , rf is L-Lipschitz,Theorem: f(k+1) = f(k) ⌧krJ(f(k) ) f(0) is given. Gradient Descent
24. 24. and 0 < ⌧ < 2/L, then f(k) k!+1 ! f? a solution of min f J(f). If f is convex, C1 , rf is L-Lipschitz,Theorem: f(k+1) = f(k) ⌧krJ(f(k) ) f(0) is given. Optimal step size: ⌧k = argmin ⌧2R+ J(f(k) ⌧rJ(f(k) )) Proposition: One has hrJ(f(k+1) ), rJ(f(k) )i = 0 Gradient Descent
25. 25. Gradient Flows and PDE’s f(k+1) f(k) ⌧ = rJ(f(k) )Fixed step size ⌧k = ⌧:
26. 26. Gradient Flows and PDE’s f(k+1) f(k) ⌧ = rJ(f(k) )Fixed step size ⌧k = ⌧: Denote ft = f(k) for t = k⌧, one obtains formally as ⌧ ! 0: 8 t > 0, @ft @t = rJ(ft) and f0 = f(0)
27. 27. J(f) = R ||rf(x)||dxSobolev ﬂow: @ft @t = ftHeat equation: Explicit solution: Gradient Flows and PDE’s f(k+1) f(k) ⌧ = rJ(f(k) )Fixed step size ⌧k = ⌧: Denote ft = f(k) for t = k⌧, one obtains formally as ⌧ ! 0: 8 t > 0, @ft @t = rJ(ft) and f0 = f(0)
28. 28. Total Variation Flow
29. 29. @ft @t = rJ(ft) Noisy observations: y = f + w, w ⇠ N(0, IdN ). and ft=0 = y Application: Denoising
30. 30. Optimal choice of t: minimize ||ft f|| ! not accessible in practice. SNR(ft, f) = 20 log10 ✓ ||f ft|| ||f|| ◆ Optimal Parameter Selection t t
31. 31. Overview • Variational Priors • Gradient Descent and PDE’s • Inverse Problems Regularization
32. 32. Inverse Problems
33. 33. Inverse Problems
34. 34. Inverse Problems
35. 35. Inverse Problems
36. 36. Inverse Problems
37. 37. Inverse Problem Regularization
38. 38. Inverse Problem Regularization
39. 39. Inverse Problem Regularization
40. 40. Sobolev prior: J(f) = 1 2 ||rf||2 f? = argmin f2RN E(f) = ||y f||2 + ||rf||2 (assuming 1 /2 ker( )) Sobolev Regularization
41. 41. Sobolev prior: J(f) = 1 2 ||rf||2 f? = argmin f2RN E(f) = ||y f||2 + ||rf||2 rE(f? ) = 0 () ( ⇤ )f? = ⇤ yProposition: ! Large scale linear system. (assuming 1 /2 ker( )) Sobolev Regularization
42. 42. Sobolev prior: J(f) = 1 2 ||rf||2 f? = argmin f2RN E(f) = ||y f||2 + ||rf||2 rE(f? ) = 0 () ( ⇤ )f? = ⇤ yProposition: ! Large scale linear system. Gradient descent: (assuming 1 /2 ker( )) where ||A|| = max(A) ! Slow convergence. Sobolev Regularization Convergence: ⇥ < 2/||⇥ ⇥ ||
43. 43. Mask M, = diagi(1i2M ) Example: InpaintingFigure 3 shows iterations of the algorithm 1 to solve the inpainting problem on a smooth image using a manifold prior with 2D linear patches, as deﬁned in 16. This manifold together with the overlapping of the patches allow a smooth interpolation of the missing pixels. Measurements y Iter. #1 Iter. #3 Iter. #50 Fig. 3. Iterations of the inpainting algorithm on an uniformly regular image. 5 Manifold of Step Discontinuities In order to introduce some non-linearity in the manifold M, one needs to go log10(||f(k) f( ) ||/||f0||) k k E(f(k) ) M ( f)[i] = ⇢ 0 if i 2 M, f[i] otherwise.
44. 44. Symmetric linear system: Conjugate Gradient Ax = b () min x2Rn E(x) = 1 2 hAx, xi hx, bi
45. 45. Symmetric linear system: x(k+1) = argmin E(x) s.t. x x(k) 2 span(rE(x(0) ), . . . , rE(x(k) )) Intuition: Conjugate Gradient Ax = b () min x2Rn E(x) = 1 2 hAx, xi hx, bi Proposition: 8 ` < k, hrE(xk ), rE(x` )i = 0
46. 46. Symmetric linear system: Initialization: x(0) 2 RN , r(0) = b Ax(0) , p(0) = r(0) r(k) = hrE(x(k) ), d(k) i hAd(k), d(k)i d(k) = rE(x(k) ) + ||v(k) || ||v(k 1)|| d(k 1) v(k) = rE(x(k) ) = Ax(k) b x(k+1) = x(k) r(k) d(k) Iterations: x(k+1) = argmin E(x) s.t. x x(k) 2 span(rE(x(0) ), . . . , rE(x(k) )) Intuition: Conjugate Gradient Ax = b () min x2Rn E(x) = 1 2 hAx, xi hx, bi Proposition: 8 ` < k, hrE(xk ), rE(x` )i = 0
47. 47. TV" regularization: (assuming 1 /2 ker( )) f? = argmin f2RN E(f) = 1 2 || f y|| + J" (f) Total Variation Regularization ||u||" = p ||u||2 + "2 J"(f) = P n ||rf[n]||"
48. 48. TV" regularization: (assuming 1 /2 ker( )) f(k+1) = f(k) ⌧krE(f(k) ) rE(f) = ⇤ ( f y) + rJ"(f) rJ"(f) = div ✓ rf ||rf||" ◆ Convergence: requires ⌧ ⇠ ". Gradient descent: f? = argmin f2RN E(f) = 1 2 || f y|| + J" (f) Total Variation Regularization ||u||" = p ||u||2 + "2 J"(f) = P n ||rf[n]||"
49. 49. TV" regularization: (assuming 1 /2 ker( )) f(k+1) = f(k) ⌧krE(f(k) ) rE(f) = ⇤ ( f y) + rJ"(f) rJ"(f) = div ✓ rf ||rf||" ◆ Convergence: requires ⌧ ⇠ ". Gradient descent: f? = argmin f2RN E(f) = 1 2 || f y|| + J" (f) Newton descent: f(k+1) = f(k) H 1 k rE(f(k) ) where Hk = @2 E"(f(k) ) Total Variation Regularization ||u||" = p ||u||2 + "2 J"(f) = P n ||rf[n]||"
50. 50. k Large" TV vs. Sobolev ConvergeSmall"
51. 51. Observations y Sobolev Total variation Inpainting: Sobolev vs. TV
52. 52. Noiseless problem: f? 2 argmin f J" (f) s.t. f 2 H Contraint: H = {f ; f = y}. f(k+1) = ProjH ⇣ f(k) ⌧krJ"(f(k) ) ⌘ ProjH(f) = argmin g=y ||g f||2 = f + ⇤ ( ⇤ ) 1 (y f) Inpainting: ProjH(f)[i] = ⇢ f[i] if i 2 M, y[i] otherwise. Projected gradient descent: f(k) k!+1 ! f? a solution of (?). (?) Projected Gradient Descent Proposition: If rJ" is L-Lipschitz and 0 < ⌧k < 2/L,
53. 53. TV Priors: Non-quadratic better edge recovery. =) Conclusion Sobolev
54. 54. TV Priors: Non-quadratic better edge recovery. =) Optimization Variational regularization: () – Gradient descent. – Newton. – Projected gradient. – Conjugate gradient. Non-smooth optimization ? Conclusion Sobolev