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# A Review of the Split Bregman Method for L1 Regularized Problems

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### A Review of the Split Bregman Method for L1 Regularized Problems

1. 1. Introduction TV Denoising L1 Regularization Split Bregman Method Results A Review of the Split Bregman Method for L1 Regularized Problems Pardis Noorzad1 1 Department of Computer Engineering and IT Amirkabir University of Technology Khordad 1389 A Review of the Split Bregman Method for L1 Regularized Problems Pardis Noorzad
2. 2. Introduction TV Denoising L1 Regularization Split Bregman Method Results 1 Introduction 2 TV Denoising The ROF Model Iterated Total Variation 3 L1 Regularization Easy vs. Hard Problems 4 Split Bregman Method Split Bregman Formulation Bregman Iteration Applying SB to TV Denoising 5 Results Fast Convergence Acceptable Intermediate Results A Review of the Split Bregman Method for L1 Regularized Problems Pardis Noorzad
3. 3. Introduction TV Denoising L1 Regularization Split Bregman Method Results Image Restoration and Variational Models • Fundamental problem in image restoration: denoising • Denoising is an important step in machine vision tasks • Concern is to preserve important image features • edges, texture while removing noise • Variational models have been very successful A Review of the Split Bregman Method for L1 Regularized Problems Pardis Noorzad
4. 4. Introduction TV Denoising L1 Regularization Split Bregman Method Results Image Restoration and Variational Models • Fundamental problem in image restoration: denoising • Denoising is an important step in machine vision tasks • Concern is to preserve important image features • edges, texture while removing noise • Variational models have been very successful A Review of the Split Bregman Method for L1 Regularized Problems Pardis Noorzad
5. 5. Introduction TV Denoising L1 Regularization Split Bregman Method Results Image Restoration and Variational Models • Fundamental problem in image restoration: denoising • Denoising is an important step in machine vision tasks • Concern is to preserve important image features • edges, texture while removing noise • Variational models have been very successful A Review of the Split Bregman Method for L1 Regularized Problems Pardis Noorzad
6. 6. Introduction TV Denoising L1 Regularization Split Bregman Method Results Image Restoration and Variational Models • Fundamental problem in image restoration: denoising • Denoising is an important step in machine vision tasks • Concern is to preserve important image features • edges, texture while removing noise • Variational models have been very successful A Review of the Split Bregman Method for L1 Regularized Problems Pardis Noorzad
7. 7. Introduction TV Denoising L1 Regularization Split Bregman Method Results Image Restoration and Variational Models • Fundamental problem in image restoration: denoising • Denoising is an important step in machine vision tasks • Concern is to preserve important image features • edges, texture while removing noise • Variational models have been very successful A Review of the Split Bregman Method for L1 Regularized Problems Pardis Noorzad
8. 8. Introduction TV Denoising L1 Regularization Split Bregman Method Results TV-based Image Restoration • Total variation based image restoration models ﬁrst introduced by Rudin, Osher, and Fatemi [ROF92] • An early example of PDE based edge preserving denoising • Has been extended and solved in a variety of ways • Here, the Split Bregman method is introduced A Review of the Split Bregman Method for L1 Regularized Problems Pardis Noorzad
9. 9. Introduction TV Denoising L1 Regularization Split Bregman Method Results TV-based Image Restoration • Total variation based image restoration models ﬁrst introduced by Rudin, Osher, and Fatemi [ROF92] • An early example of PDE based edge preserving denoising • Has been extended and solved in a variety of ways • Here, the Split Bregman method is introduced A Review of the Split Bregman Method for L1 Regularized Problems Pardis Noorzad
10. 10. Introduction TV Denoising L1 Regularization Split Bregman Method Results TV-based Image Restoration • Total variation based image restoration models ﬁrst introduced by Rudin, Osher, and Fatemi [ROF92] • An early example of PDE based edge preserving denoising • Has been extended and solved in a variety of ways • Here, the Split Bregman method is introduced A Review of the Split Bregman Method for L1 Regularized Problems Pardis Noorzad
11. 11. Introduction TV Denoising L1 Regularization Split Bregman Method Results TV-based Image Restoration • Total variation based image restoration models ﬁrst introduced by Rudin, Osher, and Fatemi [ROF92] • An early example of PDE based edge preserving denoising • Has been extended and solved in a variety of ways • Here, the Split Bregman method is introduced A Review of the Split Bregman Method for L1 Regularized Problems Pardis Noorzad
12. 12. Introduction TV Denoising L1 Regularization Split Bregman Method Results Denoising Decomposition f=u+v • f : Ω → R is the noisy image • Ω is the bounded open subset of R2 • u is the true signal • v ∼ N(0, σ2 ) is the white Gaussian noise A Review of the Split Bregman Method for L1 Regularized Problems Pardis Noorzad
13. 13. Introduction TV Denoising L1 Regularization Split Bregman Method Results Denoising Decomposition f=u+v • f : Ω → R is the noisy image • Ω is the bounded open subset of R2 • u is the true signal • v ∼ N(0, σ2 ) is the white Gaussian noise A Review of the Split Bregman Method for L1 Regularized Problems Pardis Noorzad
14. 14. Introduction TV Denoising L1 Regularization Split Bregman Method Results Denoising Decomposition f=u+v • f : Ω → R is the noisy image • Ω is the bounded open subset of R2 • u is the true signal • v ∼ N(0, σ2 ) is the white Gaussian noise A Review of the Split Bregman Method for L1 Regularized Problems Pardis Noorzad
15. 15. Introduction TV Denoising L1 Regularization Split Bregman Method Results Denoising Decomposition f=u+v • f : Ω → R is the noisy image • Ω is the bounded open subset of R2 • u is the true signal • v ∼ N(0, σ2 ) is the white Gaussian noise A Review of the Split Bregman Method for L1 Regularized Problems Pardis Noorzad
16. 16. Introduction TV Denoising L1 Regularization Split Bregman Method Results Denoising Decomposition f=u+v • f : Ω → R is the noisy image • Ω is the bounded open subset of R2 • u is the true signal • v ∼ N(0, σ2 ) is the white Gaussian noise A Review of the Split Bregman Method for L1 Regularized Problems Pardis Noorzad
17. 17. Introduction TV Denoising L1 Regularization Split Bregman Method Results Conventional Variational Model Easy to solve — results are dissapointing min (uxx + uyy )2 dxdy Ω such that udxdy = fdxdy Ω Ω (white noise is of zero mean) 1 (u − f)2 dxdy = σ2 , 0 2 (a priori information about v) A Review of the Split Bregman Method for L1 Regularized Problems Pardis Noorzad
18. 18. Introduction TV Denoising L1 Regularization Split Bregman Method Results Conventional Variational Model Easy to solve — results are dissapointing min (uxx + uyy )2 dxdy Ω such that udxdy = fdxdy Ω Ω (white noise is of zero mean) 1 (u − f)2 dxdy = σ2 , 0 2 (a priori information about v) A Review of the Split Bregman Method for L1 Regularized Problems Pardis Noorzad
19. 19. Introduction TV Denoising L1 Regularization Split Bregman Method Results Conventional Variational Model Easy to solve — results are dissapointing min (uxx + uyy )2 dxdy Ω such that udxdy = fdxdy Ω Ω (white noise is of zero mean) 1 (u − f)2 dxdy = σ2 , 0 2 (a priori information about v) A Review of the Split Bregman Method for L1 Regularized Problems Pardis Noorzad
20. 20. Introduction TV Denoising L1 Regularization Split Bregman Method Results Conventional Variational Model Easy to solve — results are dissapointing min (uxx + uyy )2 dxdy Ω such that udxdy = fdxdy Ω Ω (white noise is of zero mean) 1 (u − f)2 dxdy = σ2 , 0 2 (a priori information about v) A Review of the Split Bregman Method for L1 Regularized Problems Pardis Noorzad
21. 21. Introduction TV Denoising L1 Regularization Split Bregman Method Results The ROF Model Outline 1 Introduction 2 TV Denoising The ROF Model Iterated Total Variation 3 L1 Regularization Easy vs. Hard Problems 4 Split Bregman Method Split Bregman Formulation Bregman Iteration Applying SB to TV Denoising 5 Results Fast Convergence Acceptable Intermediate Results A Review of the Split Bregman Method for L1 Regularized Problems Pardis Noorzad
22. 22. Introduction TV Denoising L1 Regularization Split Bregman Method Results The ROF Model The ROF Model Diﬃcult to solve — successful for denoising min { u BV + λ f − u 2} 2 u∈BV(Ω) • λ > 0: scale parameter • BV(Ω): space of functions with bounded variation on Ω • . : BV seminorm or total variation given by, u BV = | u| Ω A Review of the Split Bregman Method for L1 Regularized Problems Pardis Noorzad
23. 23. Introduction TV Denoising L1 Regularization Split Bregman Method Results The ROF Model The ROF Model Diﬃcult to solve — successful for denoising min { u BV + λ f − u 2} 2 u∈BV(Ω) • λ > 0: scale parameter • BV(Ω): space of functions with bounded variation on Ω • . : BV seminorm or total variation given by, u BV = | u| Ω A Review of the Split Bregman Method for L1 Regularized Problems Pardis Noorzad
24. 24. Introduction TV Denoising L1 Regularization Split Bregman Method Results The ROF Model The ROF Model Diﬃcult to solve — successful for denoising min { u BV + λ f − u 2} 2 u∈BV(Ω) • λ > 0: scale parameter • BV(Ω): space of functions with bounded variation on Ω • . : BV seminorm or total variation given by, u BV = | u| Ω A Review of the Split Bregman Method for L1 Regularized Problems Pardis Noorzad
25. 25. Introduction TV Denoising L1 Regularization Split Bregman Method Results The ROF Model The ROF Model Diﬃcult to solve — successful for denoising min { u BV + λ f − u 2} 2 u∈BV(Ω) • λ > 0: scale parameter • BV(Ω): space of functions with bounded variation on Ω • . : BV seminorm or total variation given by, u BV = | u| Ω A Review of the Split Bregman Method for L1 Regularized Problems Pardis Noorzad
26. 26. Introduction TV Denoising L1 Regularization Split Bregman Method Results The ROF Model BV seminorm • It’s use is essential — allows image recovery with edges • What if ﬁrst term were replaced by Ω | u|p ? • Which is both diﬀerentiable and strictly convex • No good! For p > 1, its derivative has smoothing eﬀect in the optimality condition • For TV however, the operator is degenerate, and aﬀects only level lines of the image A Review of the Split Bregman Method for L1 Regularized Problems Pardis Noorzad
27. 27. Introduction TV Denoising L1 Regularization Split Bregman Method Results The ROF Model BV seminorm • It’s use is essential — allows image recovery with edges • What if ﬁrst term were replaced by Ω | u|p ? • Which is both diﬀerentiable and strictly convex • No good! For p > 1, its derivative has smoothing eﬀect in the optimality condition • For TV however, the operator is degenerate, and aﬀects only level lines of the image A Review of the Split Bregman Method for L1 Regularized Problems Pardis Noorzad
28. 28. Introduction TV Denoising L1 Regularization Split Bregman Method Results The ROF Model BV seminorm • It’s use is essential — allows image recovery with edges • What if ﬁrst term were replaced by Ω | u|p ? • Which is both diﬀerentiable and strictly convex • No good! For p > 1, its derivative has smoothing eﬀect in the optimality condition • For TV however, the operator is degenerate, and aﬀects only level lines of the image A Review of the Split Bregman Method for L1 Regularized Problems Pardis Noorzad
29. 29. Introduction TV Denoising L1 Regularization Split Bregman Method Results The ROF Model BV seminorm • It’s use is essential — allows image recovery with edges • What if ﬁrst term were replaced by Ω | u|p ? • Which is both diﬀerentiable and strictly convex • No good! For p > 1, its derivative has smoothing eﬀect in the optimality condition • For TV however, the operator is degenerate, and aﬀects only level lines of the image A Review of the Split Bregman Method for L1 Regularized Problems Pardis Noorzad
30. 30. Introduction TV Denoising L1 Regularization Split Bregman Method Results The ROF Model BV seminorm • It’s use is essential — allows image recovery with edges • What if ﬁrst term were replaced by Ω | u|p ? • Which is both diﬀerentiable and strictly convex • No good! For p > 1, its derivative has smoothing eﬀect in the optimality condition • For TV however, the operator is degenerate, and aﬀects only level lines of the image A Review of the Split Bregman Method for L1 Regularized Problems Pardis Noorzad
31. 31. Introduction TV Denoising L1 Regularization Split Bregman Method Results Iterated Total Variation Outline 1 Introduction 2 TV Denoising The ROF Model Iterated Total Variation 3 L1 Regularization Easy vs. Hard Problems 4 Split Bregman Method Split Bregman Formulation Bregman Iteration Applying SB to TV Denoising 5 Results Fast Convergence Acceptable Intermediate Results A Review of the Split Bregman Method for L1 Regularized Problems Pardis Noorzad
32. 32. Introduction TV Denoising L1 Regularization Split Bregman Method Results Iterated Total Variation Iterative Regularization Adding back the noise • In the ROF model, u − f is treated as error and discarded • In the decomposition of f into signal u and additive noise v • There exists some signal in v • And some smoothing of textures in u • Osher et al. [OBG+ 05] propose an iterated procedure to add the noise back A Review of the Split Bregman Method for L1 Regularized Problems Pardis Noorzad
33. 33. Introduction TV Denoising L1 Regularization Split Bregman Method Results Iterated Total Variation Iterative Regularization Adding back the noise • In the ROF model, u − f is treated as error and discarded • In the decomposition of f into signal u and additive noise v • There exists some signal in v • And some smoothing of textures in u • Osher et al. [OBG+ 05] propose an iterated procedure to add the noise back A Review of the Split Bregman Method for L1 Regularized Problems Pardis Noorzad
34. 34. Introduction TV Denoising L1 Regularization Split Bregman Method Results Iterated Total Variation Iterative Regularization Adding back the noise • In the ROF model, u − f is treated as error and discarded • In the decomposition of f into signal u and additive noise v • There exists some signal in v • And some smoothing of textures in u • Osher et al. [OBG+ 05] propose an iterated procedure to add the noise back A Review of the Split Bregman Method for L1 Regularized Problems Pardis Noorzad
35. 35. Introduction TV Denoising L1 Regularization Split Bregman Method Results Iterated Total Variation Iterative Regularization Adding back the noise • In the ROF model, u − f is treated as error and discarded • In the decomposition of f into signal u and additive noise v • There exists some signal in v • And some smoothing of textures in u • Osher et al. [OBG+ 05] propose an iterated procedure to add the noise back A Review of the Split Bregman Method for L1 Regularized Problems Pardis Noorzad
36. 36. Introduction TV Denoising L1 Regularization Split Bregman Method Results Iterated Total Variation Iterative Regularization Adding back the noise • In the ROF model, u − f is treated as error and discarded • In the decomposition of f into signal u and additive noise v • There exists some signal in v • And some smoothing of textures in u • Osher et al. [OBG+ 05] propose an iterated procedure to add the noise back A Review of the Split Bregman Method for L1 Regularized Problems Pardis Noorzad
37. 37. Introduction TV Denoising L1 Regularization Split Bregman Method Results Iterated Total Variation Iterative Regularization The iteration Step 1: Solve the ROF model to obtain: u1 = arg min | u| + λ (f − u)2 u∈BV(Ω) Step 2: Perform a correction step: u2 = arg min | u| + λ (f + v1 − u)2 u∈BV(Ω) (v1 is the noise estimated by the ﬁrst step, f = u1 + v1 ) A Review of the Split Bregman Method for L1 Regularized Problems Pardis Noorzad
38. 38. Introduction TV Denoising L1 Regularization Split Bregman Method Results Iterated Total Variation Iterative Regularization The iteration Step 1: Solve the ROF model to obtain: u1 = arg min | u| + λ (f − u)2 u∈BV(Ω) Step 2: Perform a correction step: u2 = arg min | u| + λ (f + v1 − u)2 u∈BV(Ω) (v1 is the noise estimated by the ﬁrst step, f = u1 + v1 ) A Review of the Split Bregman Method for L1 Regularized Problems Pardis Noorzad
39. 39. Introduction TV Denoising L1 Regularization Split Bregman Method Results Deﬁnition L1 regularized optimization min Φ(u) 1 + H(u) u • Many important problems in imaging science (and other problems in engineering) can be posed as L1 regularized optimization problems • . 1 : the L1 norm • both Φ(u) 1 and H(u) are convex functions A Review of the Split Bregman Method for L1 Regularized Problems Pardis Noorzad
40. 40. Introduction TV Denoising L1 Regularization Split Bregman Method Results Deﬁnition L1 regularized optimization min Φ(u) 1 + H(u) u • Many important problems in imaging science (and other problems in engineering) can be posed as L1 regularized optimization problems • . 1 : the L1 norm • both Φ(u) 1 and H(u) are convex functions A Review of the Split Bregman Method for L1 Regularized Problems Pardis Noorzad
41. 41. Introduction TV Denoising L1 Regularization Split Bregman Method Results Deﬁnition L1 regularized optimization min Φ(u) 1 + H(u) u • Many important problems in imaging science (and other problems in engineering) can be posed as L1 regularized optimization problems • . 1 : the L1 norm • both Φ(u) 1 and H(u) are convex functions A Review of the Split Bregman Method for L1 Regularized Problems Pardis Noorzad
42. 42. Introduction TV Denoising L1 Regularization Split Bregman Method Results Deﬁnition L1 regularized optimization min Φ(u) 1 + H(u) u • Many important problems in imaging science (and other problems in engineering) can be posed as L1 regularized optimization problems • . 1 : the L1 norm • both Φ(u) 1 and H(u) are convex functions A Review of the Split Bregman Method for L1 Regularized Problems Pardis Noorzad
43. 43. Introduction TV Denoising L1 Regularization Split Bregman Method Results Easy vs. Hard Problems Outline 1 Introduction 2 TV Denoising The ROF Model Iterated Total Variation 3 L1 Regularization Easy vs. Hard Problems 4 Split Bregman Method Split Bregman Formulation Bregman Iteration Applying SB to TV Denoising 5 Results Fast Convergence Acceptable Intermediate Results A Review of the Split Bregman Method for L1 Regularized Problems Pardis Noorzad
44. 44. Introduction TV Denoising L1 Regularization Split Bregman Method Results Easy vs. Hard Problems Easy Instances 2 arg min Au − f 2 diﬀerentiable u 2 arg min u 1 + u−f 2 solvable by shrinkage u A Review of the Split Bregman Method for L1 Regularized Problems Pardis Noorzad
45. 45. Introduction TV Denoising L1 Regularization Split Bregman Method Results Easy vs. Hard Problems Easy Instances 2 arg min Au − f 2 diﬀerentiable u 2 arg min u 1 + u−f 2 solvable by shrinkage u A Review of the Split Bregman Method for L1 Regularized Problems Pardis Noorzad
46. 46. Introduction TV Denoising L1 Regularization Split Bregman Method Results Easy vs. Hard Problems Shrinkage or Soft Thresholding Solves the L1 problem of the form (H(.) is convex and diﬀerentiable): arg min µ u 1 + H(u) u Based on this iterative scheme 1 2 uk+1 → arg min µ u 1 + u − (uk − δk H(uk )) u 2δk A Review of the Split Bregman Method for L1 Regularized Problems Pardis Noorzad
47. 47. Introduction TV Denoising L1 Regularization Split Bregman Method Results Easy vs. Hard Problems Shrinkage or Soft Thresholding Solves the L1 problem of the form (H(.) is convex and diﬀerentiable): arg min µ u 1 + H(u) u Based on this iterative scheme 1 2 uk+1 → arg min µ u 1 + u − (uk − δk H(uk )) u 2δk A Review of the Split Bregman Method for L1 Regularized Problems Pardis Noorzad
48. 48. Introduction TV Denoising L1 Regularization Split Bregman Method Results Easy vs. Hard Problems Shrinkage Continued Since unknown u is componentwise separable, each component can be independently obtained: uk+1 = shrink((uk − δk H(uk ))i , µδk ), i = 1, . . . , n, i  y − α, y ∈ (α, ∞),  shrink(y, α) := sgn(y) max{|y| − α, 0} = 0, y ∈ [−α, α],   y + α, y ∈ (−∞, −α). A Review of the Split Bregman Method for L1 Regularized Problems Pardis Noorzad
49. 49. Introduction TV Denoising L1 Regularization Split Bregman Method Results Easy vs. Hard Problems Shrinkage Continued Since unknown u is componentwise separable, each component can be independently obtained: uk+1 = shrink((uk − δk H(uk ))i , µδk ), i = 1, . . . , n, i  y − α, y ∈ (α, ∞),  shrink(y, α) := sgn(y) max{|y| − α, 0} = 0, y ∈ [−α, α],   y + α, y ∈ (−∞, −α). A Review of the Split Bregman Method for L1 Regularized Problems Pardis Noorzad
50. 50. Introduction TV Denoising L1 Regularization Split Bregman Method Results Easy vs. Hard Problems Shrinkage Continued Since unknown u is componentwise separable, each component can be independently obtained: uk+1 = shrink((uk − δk H(uk ))i , µδk ), i = 1, . . . , n, i  y − α, y ∈ (α, ∞),  shrink(y, α) := sgn(y) max{|y| − α, 0} = 0, y ∈ [−α, α],   y + α, y ∈ (−∞, −α). A Review of the Split Bregman Method for L1 Regularized Problems Pardis Noorzad
51. 51. Introduction TV Denoising L1 Regularization Split Bregman Method Results Easy vs. Hard Problems Hard Instances 2 arg min Φ(u) 1 + u−f 2 u 2 arg min u 1 + Au − f 2 u What makes these problems hard? The coupling between the L1 and L2 terms. A Review of the Split Bregman Method for L1 Regularized Problems Pardis Noorzad
52. 52. Introduction TV Denoising L1 Regularization Split Bregman Method Results Easy vs. Hard Problems Hard Instances 2 arg min Φ(u) 1 + u−f 2 u 2 arg min u 1 + Au − f 2 u What makes these problems hard? The coupling between the L1 and L2 terms. A Review of the Split Bregman Method for L1 Regularized Problems Pardis Noorzad
53. 53. Introduction TV Denoising L1 Regularization Split Bregman Method Results Easy vs. Hard Problems Hard Instances 2 arg min Φ(u) 1 + u−f 2 u 2 arg min u 1 + Au − f 2 u What makes these problems hard? The coupling between the L1 and L2 terms. A Review of the Split Bregman Method for L1 Regularized Problems Pardis Noorzad
54. 54. Introduction TV Denoising L1 Regularization Split Bregman Method Results Easy vs. Hard Problems Hard Instances 2 arg min Φ(u) 1 + u−f 2 u 2 arg min u 1 + Au − f 2 u What makes these problems hard? The coupling between the L1 and L2 terms. A Review of the Split Bregman Method for L1 Regularized Problems Pardis Noorzad
55. 55. Introduction TV Denoising L1 Regularization Split Bregman Method Results Split Bregman Formulation Outline 1 Introduction 2 TV Denoising The ROF Model Iterated Total Variation 3 L1 Regularization Easy vs. Hard Problems 4 Split Bregman Method Split Bregman Formulation Bregman Iteration Applying SB to TV Denoising 5 Results Fast Convergence Acceptable Intermediate Results A Review of the Split Bregman Method for L1 Regularized Problems Pardis Noorzad
56. 56. Introduction TV Denoising L1 Regularization Split Bregman Method Results Split Bregman Formulation Split the L1 and L2 components To solve the general regularization problem: arg min Φ(u) 1 + H(u) u Introduce d = Φ(u) and solve the constrained problem arg min d 1 + H(u) such that d = Φ(u) u,d A Review of the Split Bregman Method for L1 Regularized Problems Pardis Noorzad
57. 57. Introduction TV Denoising L1 Regularization Split Bregman Method Results Split Bregman Formulation Split the L1 and L2 components To solve the general regularization problem: arg min Φ(u) 1 + H(u) u Introduce d = Φ(u) and solve the constrained problem arg min d 1 + H(u) such that d = Φ(u) u,d A Review of the Split Bregman Method for L1 Regularized Problems Pardis Noorzad
58. 58. Introduction TV Denoising L1 Regularization Split Bregman Method Results Split Bregman Formulation Split the L1 and L2 components To solve the general regularization problem: arg min Φ(u) 1 + H(u) u Introduce d = Φ(u) and solve the constrained problem arg min d 1 + H(u) such that d = Φ(u) u,d A Review of the Split Bregman Method for L1 Regularized Problems Pardis Noorzad
59. 59. Introduction TV Denoising L1 Regularization Split Bregman Method Results Split Bregman Formulation Split the L1 and L2 components To solve the general regularization problem: arg min Φ(u) 1 + H(u) u Introduce d = Φ(u) and solve the constrained problem arg min d 1 + H(u) such that d = Φ(u) u,d A Review of the Split Bregman Method for L1 Regularized Problems Pardis Noorzad
60. 60. Introduction TV Denoising L1 Regularization Split Bregman Method Results Split Bregman Formulation Split the L1 and L2 components Continued Add an L2 penalty term to get an unconstrained problem λ 2 arg min d 1 + H(u) + d − Φ(u) u,d 2 • Obvious way is to use the penalty method to solve this • However, as λk → ∞, the condition number of the Hessian approaches inﬁnity, making it impractical to use fast iterative methods like Conjugate Gradient to approximate the inverse of the Hessian. A Review of the Split Bregman Method for L1 Regularized Problems Pardis Noorzad
61. 61. Introduction TV Denoising L1 Regularization Split Bregman Method Results Split Bregman Formulation Split the L1 and L2 components Continued Add an L2 penalty term to get an unconstrained problem λ 2 arg min d 1 + H(u) + d − Φ(u) u,d 2 • Obvious way is to use the penalty method to solve this • However, as λk → ∞, the condition number of the Hessian approaches inﬁnity, making it impractical to use fast iterative methods like Conjugate Gradient to approximate the inverse of the Hessian. A Review of the Split Bregman Method for L1 Regularized Problems Pardis Noorzad
62. 62. Introduction TV Denoising L1 Regularization Split Bregman Method Results Split Bregman Formulation Split the L1 and L2 components Continued Add an L2 penalty term to get an unconstrained problem λ 2 arg min d 1 + H(u) + d − Φ(u) u,d 2 • Obvious way is to use the penalty method to solve this • However, as λk → ∞, the condition number of the Hessian approaches inﬁnity, making it impractical to use fast iterative methods like Conjugate Gradient to approximate the inverse of the Hessian. A Review of the Split Bregman Method for L1 Regularized Problems Pardis Noorzad
63. 63. Introduction TV Denoising L1 Regularization Split Bregman Method Results Split Bregman Formulation Split the L1 and L2 components Continued Add an L2 penalty term to get an unconstrained problem λ 2 arg min d 1 + H(u) + d − Φ(u) u,d 2 • Obvious way is to use the penalty method to solve this • However, as λk → ∞, the condition number of the Hessian approaches inﬁnity, making it impractical to use fast iterative methods like Conjugate Gradient to approximate the inverse of the Hessian. A Review of the Split Bregman Method for L1 Regularized Problems Pardis Noorzad
64. 64. Introduction TV Denoising L1 Regularization Split Bregman Method Results Bregman Iteration Outline 1 Introduction 2 TV Denoising The ROF Model Iterated Total Variation 3 L1 Regularization Easy vs. Hard Problems 4 Split Bregman Method Split Bregman Formulation Bregman Iteration Applying SB to TV Denoising 5 Results Fast Convergence Acceptable Intermediate Results A Review of the Split Bregman Method for L1 Regularized Problems Pardis Noorzad
65. 65. Introduction TV Denoising L1 Regularization Split Bregman Method Results Bregman Iteration Analog of adding the noise back The optimization problem is solved by iterating λ 2 (uk+1 , dk+1 ) = arg min d 1 + H(u) + d − Φ(u) − bk u,d 2 bk+1 = bk + (Φ(u) − dk ) The iteration in the ﬁrst line can be done separately for u and d. A Review of the Split Bregman Method for L1 Regularized Problems Pardis Noorzad
66. 66. Introduction TV Denoising L1 Regularization Split Bregman Method Results Bregman Iteration Analog of adding the noise back The optimization problem is solved by iterating λ 2 (uk+1 , dk+1 ) = arg min d 1 + H(u) + d − Φ(u) − bk u,d 2 bk+1 = bk + (Φ(u) − dk ) The iteration in the ﬁrst line can be done separately for u and d. A Review of the Split Bregman Method for L1 Regularized Problems Pardis Noorzad
67. 67. Introduction TV Denoising L1 Regularization Split Bregman Method Results Bregman Iteration Analog of adding the noise back The optimization problem is solved by iterating λ 2 (uk+1 , dk+1 ) = arg min d 1 + H(u) + d − Φ(u) − bk u,d 2 bk+1 = bk + (Φ(u) − dk ) The iteration in the ﬁrst line can be done separately for u and d. A Review of the Split Bregman Method for L1 Regularized Problems Pardis Noorzad
68. 68. Introduction TV Denoising L1 Regularization Split Bregman Method Results Bregman Iteration 3-step Algorithm λ 2 Step 1: uk+1 = arg minu H(u) + 2 dk − Φ(u) − bk 2 λ 2 Step 2: dk+1 = arg mind d 1 + 2 dk − Φ(u) − bk 2 Step 3: bk+1 = bk + Φ(uk+1 ) − dk+1 • Step 1 is now a diﬀerentiable optimization problem, we’ll solve with Gauss Seidel • Step 2 can be solved eﬃciently with shrinkage • Step 3 is an explicit evaluation A Review of the Split Bregman Method for L1 Regularized Problems Pardis Noorzad
69. 69. Introduction TV Denoising L1 Regularization Split Bregman Method Results Bregman Iteration 3-step Algorithm λ 2 Step 1: uk+1 = arg minu H(u) + 2 dk − Φ(u) − bk 2 λ 2 Step 2: dk+1 = arg mind d 1 + 2 dk − Φ(u) − bk 2 Step 3: bk+1 = bk + Φ(uk+1 ) − dk+1 • Step 1 is now a diﬀerentiable optimization problem, we’ll solve with Gauss Seidel • Step 2 can be solved eﬃciently with shrinkage • Step 3 is an explicit evaluation A Review of the Split Bregman Method for L1 Regularized Problems Pardis Noorzad
70. 70. Introduction TV Denoising L1 Regularization Split Bregman Method Results Bregman Iteration 3-step Algorithm λ 2 Step 1: uk+1 = arg minu H(u) + 2 dk − Φ(u) − bk 2 λ 2 Step 2: dk+1 = arg mind d 1 + 2 dk − Φ(u) − bk 2 Step 3: bk+1 = bk + Φ(uk+1 ) − dk+1 • Step 1 is now a diﬀerentiable optimization problem, we’ll solve with Gauss Seidel • Step 2 can be solved eﬃciently with shrinkage • Step 3 is an explicit evaluation A Review of the Split Bregman Method for L1 Regularized Problems Pardis Noorzad
71. 71. Introduction TV Denoising L1 Regularization Split Bregman Method Results Bregman Iteration 3-step Algorithm λ 2 Step 1: uk+1 = arg minu H(u) + 2 dk − Φ(u) − bk 2 λ 2 Step 2: dk+1 = arg mind d 1 + 2 dk − Φ(u) − bk 2 Step 3: bk+1 = bk + Φ(uk+1 ) − dk+1 • Step 1 is now a diﬀerentiable optimization problem, we’ll solve with Gauss Seidel • Step 2 can be solved eﬃciently with shrinkage • Step 3 is an explicit evaluation A Review of the Split Bregman Method for L1 Regularized Problems Pardis Noorzad
72. 72. Introduction TV Denoising L1 Regularization Split Bregman Method Results Bregman Iteration 3-step Algorithm λ 2 Step 1: uk+1 = arg minu H(u) + 2 dk − Φ(u) − bk 2 λ 2 Step 2: dk+1 = arg mind d 1 + 2 dk − Φ(u) − bk 2 Step 3: bk+1 = bk + Φ(uk+1 ) − dk+1 • Step 1 is now a diﬀerentiable optimization problem, we’ll solve with Gauss Seidel • Step 2 can be solved eﬃciently with shrinkage • Step 3 is an explicit evaluation A Review of the Split Bregman Method for L1 Regularized Problems Pardis Noorzad
73. 73. Introduction TV Denoising L1 Regularization Split Bregman Method Results Bregman Iteration 3-step Algorithm λ 2 Step 1: uk+1 = arg minu H(u) + 2 dk − Φ(u) − bk 2 λ 2 Step 2: dk+1 = arg mind d 1 + 2 dk − Φ(u) − bk 2 Step 3: bk+1 = bk + Φ(uk+1 ) − dk+1 • Step 1 is now a diﬀerentiable optimization problem, we’ll solve with Gauss Seidel • Step 2 can be solved eﬃciently with shrinkage • Step 3 is an explicit evaluation A Review of the Split Bregman Method for L1 Regularized Problems Pardis Noorzad
74. 74. Introduction TV Denoising L1 Regularization Split Bregman Method Results Applying SB to TV Denoising Outline 1 Introduction 2 TV Denoising The ROF Model Iterated Total Variation 3 L1 Regularization Easy vs. Hard Problems 4 Split Bregman Method Split Bregman Formulation Bregman Iteration Applying SB to TV Denoising 5 Results Fast Convergence Acceptable Intermediate Results A Review of the Split Bregman Method for L1 Regularized Problems Pardis Noorzad
75. 75. Introduction TV Denoising L1 Regularization Split Bregman Method Results Applying SB to TV Denoising Anisotropic TV µ arg min | x u| +| y u| + u−f 2 2 u 2 A Review of the Split Bregman Method for L1 Regularized Problems Pardis Noorzad
76. 76. Introduction TV Denoising L1 Regularization Split Bregman Method Results Applying SB to TV Denoising Anisotropic TV The steps Step 1: uk+1 = G(uk ) 1 Step 2: dk+1 = shrink( x xu k+1 + bk , λ ) x 1 Step 3: dk+1 = shrink( y yu k+1 + bk , λ ) y Step 4: bk+1 = bk + ( x x xu − x) Step 5: bk+1 = bk + ( y y yu − y) • G(uk ): result of one Gauss-Seidel sweep for the corresponding L2 optimization • This algorithm is cheap — each step is a few operations per pixel A Review of the Split Bregman Method for L1 Regularized Problems Pardis Noorzad
77. 77. Introduction TV Denoising L1 Regularization Split Bregman Method Results Applying SB to TV Denoising Anisotropic TV The steps Step 1: uk+1 = G(uk ) 1 Step 2: dk+1 = shrink( x xu k+1 + bk , λ ) x 1 Step 3: dk+1 = shrink( y yu k+1 + bk , λ ) y Step 4: bk+1 = bk + ( x x xu − x) Step 5: bk+1 = bk + ( y y yu − y) • G(uk ): result of one Gauss-Seidel sweep for the corresponding L2 optimization • This algorithm is cheap — each step is a few operations per pixel A Review of the Split Bregman Method for L1 Regularized Problems Pardis Noorzad
78. 78. Introduction TV Denoising L1 Regularization Split Bregman Method Results Applying SB to TV Denoising Anisotropic TV The steps Step 1: uk+1 = G(uk ) 1 Step 2: dk+1 = shrink( x xu k+1 + bk , λ ) x 1 Step 3: dk+1 = shrink( y yu k+1 + bk , λ ) y Step 4: bk+1 = bk + ( x x xu − x) Step 5: bk+1 = bk + ( y y yu − y) • G(uk ): result of one Gauss-Seidel sweep for the corresponding L2 optimization • This algorithm is cheap — each step is a few operations per pixel A Review of the Split Bregman Method for L1 Regularized Problems Pardis Noorzad
79. 79. Introduction TV Denoising L1 Regularization Split Bregman Method Results Applying SB to TV Denoising Anisotropic TV The steps Step 1: uk+1 = G(uk ) 1 Step 2: dk+1 = shrink( x xu k+1 + bk , λ ) x 1 Step 3: dk+1 = shrink( y yu k+1 + bk , λ ) y Step 4: bk+1 = bk + ( x x xu − x) Step 5: bk+1 = bk + ( y y yu − y) • G(uk ): result of one Gauss-Seidel sweep for the corresponding L2 optimization • This algorithm is cheap — each step is a few operations per pixel A Review of the Split Bregman Method for L1 Regularized Problems Pardis Noorzad
80. 80. Introduction TV Denoising L1 Regularization Split Bregman Method Results Applying SB to TV Denoising Anisotropic TV The steps Step 1: uk+1 = G(uk ) 1 Step 2: dk+1 = shrink( x xu k+1 + bk , λ ) x 1 Step 3: dk+1 = shrink( y yu k+1 + bk , λ ) y Step 4: bk+1 = bk + ( x x xu − x) Step 5: bk+1 = bk + ( y y yu − y) • G(uk ): result of one Gauss-Seidel sweep for the corresponding L2 optimization • This algorithm is cheap — each step is a few operations per pixel A Review of the Split Bregman Method for L1 Regularized Problems Pardis Noorzad
81. 81. Introduction TV Denoising L1 Regularization Split Bregman Method Results Applying SB to TV Denoising Anisotropic TV The steps Step 1: uk+1 = G(uk ) 1 Step 2: dk+1 = shrink( x xu k+1 + bk , λ ) x 1 Step 3: dk+1 = shrink( y yu k+1 + bk , λ ) y Step 4: bk+1 = bk + ( x x xu − x) Step 5: bk+1 = bk + ( y y yu − y) • G(uk ): result of one Gauss-Seidel sweep for the corresponding L2 optimization • This algorithm is cheap — each step is a few operations per pixel A Review of the Split Bregman Method for L1 Regularized Problems Pardis Noorzad
82. 82. Introduction TV Denoising L1 Regularization Split Bregman Method Results Applying SB to TV Denoising Anisotropic TV The steps Step 1: uk+1 = G(uk ) 1 Step 2: dk+1 = shrink( x xu k+1 + bk , λ ) x 1 Step 3: dk+1 = shrink( y yu k+1 + bk , λ ) y Step 4: bk+1 = bk + ( x x xu − x) Step 5: bk+1 = bk + ( y y yu − y) • G(uk ): result of one Gauss-Seidel sweep for the corresponding L2 optimization • This algorithm is cheap — each step is a few operations per pixel A Review of the Split Bregman Method for L1 Regularized Problems Pardis Noorzad
83. 83. Introduction TV Denoising L1 Regularization Split Bregman Method Results Applying SB to TV Denoising Isotropic TV With similar steps 2 2 µ 2 arg min ( x u)i +( y u)i + u−f 2 u 2 i A Review of the Split Bregman Method for L1 Regularized Problems Pardis Noorzad
84. 84. Introduction TV Denoising L1 Regularization Split Bregman Method Results Fast Convergence Outline 1 Introduction 2 TV Denoising The ROF Model Iterated Total Variation 3 L1 Regularization Easy vs. Hard Problems 4 Split Bregman Method Split Bregman Formulation Bregman Iteration Applying SB to TV Denoising 5 Results Fast Convergence Acceptable Intermediate Results A Review of the Split Bregman Method for L1 Regularized Problems Pardis Noorzad
85. 85. Introduction TV Denoising L1 Regularization Split Bregman Method Results Fast Convergence Split Bregman is fast Intel Core 2 Duo desktop (3 GHz), compiled with g++ A Review of the Split Bregman Method for L1 Regularized Problems Pardis Noorzad
86. 86. Introduction TV Denoising L1 Regularization Split Bregman Method Results Fast Convergence Split Bregman is fast Compared to Graph Cuts A Review of the Split Bregman Method for L1 Regularized Problems Pardis Noorzad
87. 87. Introduction TV Denoising L1 Regularization Split Bregman Method Results Acceptable Intermediate Results Outline 1 Introduction 2 TV Denoising The ROF Model Iterated Total Variation 3 L1 Regularization Easy vs. Hard Problems 4 Split Bregman Method Split Bregman Formulation Bregman Iteration Applying SB to TV Denoising 5 Results Fast Convergence Acceptable Intermediate Results A Review of the Split Bregman Method for L1 Regularized Problems Pardis Noorzad
88. 88. Introduction TV Denoising L1 Regularization Split Bregman Method Results Acceptable Intermediate Results Intermediate images are smooth A Review of the Split Bregman Method for L1 Regularized Problems Pardis Noorzad
89. 89. Introduction TV Denoising L1 Regularization Split Bregman Method Results Acceptable Intermediate Results Intermediate images are smooth A Review of the Split Bregman Method for L1 Regularized Problems Pardis Noorzad
90. 90. References Tom Goldstein and Stanley Osher, The split bregman method for l1-regularized problems, SIAM Journal on Imaging Sciences 2 (2009), no. 2, 323–343. Stanley Osher, Martin Burger, Donald Goldfarb, Jinjun Xu, and Wotao Yin, An iterative regularization method for total variation-based image restoration, Multiscale Modeling and Simulation 4 (2005), 460–489. Leonid I. Rudin, Stanley Osher, and Emad Fatemi, Nonlinear total variation based noise removal algorithms, Physica D 60 (1992), no. 1-4, 259–268. A Review of the Split Bregman Method for L1 Regularized Problems Pardis Noorzad
91. 91. References Tom Goldstein and Stanley Osher, The split bregman method for l1-regularized problems, SIAM Journal on Imaging Sciences 2 (2009), no. 2, 323–343. Stanley Osher, Martin Burger, Donald Goldfarb, Jinjun Xu, and Wotao Yin, An iterative regularization method for total variation-based image restoration, Multiscale Modeling and Simulation 4 (2005), 460–489. Leonid I. Rudin, Stanley Osher, and Emad Fatemi, Nonlinear total variation based noise removal algorithms, Physica D 60 (1992), no. 1-4, 259–268. A Review of the Split Bregman Method for L1 Regularized Problems Pardis Noorzad
92. 92. References Tom Goldstein and Stanley Osher, The split bregman method for l1-regularized problems, SIAM Journal on Imaging Sciences 2 (2009), no. 2, 323–343. Stanley Osher, Martin Burger, Donald Goldfarb, Jinjun Xu, and Wotao Yin, An iterative regularization method for total variation-based image restoration, Multiscale Modeling and Simulation 4 (2005), 460–489. Leonid I. Rudin, Stanley Osher, and Emad Fatemi, Nonlinear total variation based noise removal algorithms, Physica D 60 (1992), no. 1-4, 259–268. A Review of the Split Bregman Method for L1 Regularized Problems Pardis Noorzad
93. 93. Thank you for your attention. Any questions? A Review of the Split Bregman Method for L1 Regularized Problems Pardis Noorzad