Introduction                  TV Denoising                   L1 Regularization     Split Bregman Method          Results

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Introduction                  TV Denoising                   L1 Regularization   Split Bregman Method          Results



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Introduction                  TV Denoising                   L1 Regularization   Split Bregman Method          Results



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Introduction                  TV Denoising                   L1 Regularization   Split Bregman Method          Results



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Introduction                  TV Denoising                   L1 Regularization   Split Bregman Method          Results



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Introduction                  TV Denoising                   L1 Regularization   Split Bregman Method          Results



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Introduction                  TV Denoising                   L1 Regularization   Split Bregman Method          Results



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Introduction                  TV Denoising                   L1 Regularization   Split Bregman Method          Results



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Introduction                  TV Denoising                   L1 Regularization   Split Bregman Method          Results



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Introduction                  TV Denoising                   L1 Regularization   Split Bregman Method          Results



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Introduction                  TV Denoising                   L1 Regularization   Split Bregman Method          Results



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Introduction                  TV Denoising                   L1 Regularization   Split Bregman Method          Results



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Introduction                  TV Denoising                   L1 Regularization   Split Bregman Method          Results



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Introduction                  TV Denoising                   L1 Regularization   Split Bregman Method          Results



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Introduction                  TV Denoising                   L1 Regularization   Split Bregman Method          Results



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Introduction                  TV Denoising                   L1 Regularization   Split Bregman Method          Results



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Introduction                  TV Denoising                    L1 Regularization           Split Bregman Method          Re...
Introduction                  TV Denoising                    L1 Regularization           Split Bregman Method          Re...
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Introduction                  TV Denoising                    L1 Regularization           Split Bregman Method          Re...
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Introduction                  TV Denoising                   L1 Regularization              Split Bregman Method          ...
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Introduction                  TV Denoising                   L1 Regularization   Split Bregman Method          Results



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Introduction                  TV Denoising                   L1 Regularization   Split Bregman Method          Results



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Introduction                  TV Denoising                   L1 Regularization   Split Bregman Method          Results



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Introduction                  TV Denoising                   L1 Regularization   Split Bregman Method          Results



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Introduction                  TV Denoising                   L1 Regularization   Split Bregman Method          Results



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Introduction                  TV Denoising                   L1 Regularization   Split Bregman Method          Results



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Introduction                  TV Denoising                   L1 Regularization   Split Bregman Method          Results



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Introduction                  TV Denoising                   L1 Regularization   Split Bregman Method          Results



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Introduction                  TV Denoising                   L1 Regularization   Split Bregman Method          Results



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Introduction                  TV Denoising                   L1 Regularization   Split Bregman Method          Results



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Introduction                  TV Denoising                   L1 Regularization     Split Bregman Method          Results

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Introduction                  TV Denoising                   L1 Regularization     Split Bregman Method          Results

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Introduction                  TV Denoising                   L1 Regularization        Split Bregman Method          Result...
Introduction                  TV Denoising                   L1 Regularization        Split Bregman Method          Result...
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Introduction                  TV Denoising                   L1 Regularization   Split Bregman Method          Results



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Introduction                  TV Denoising                   L1 Regularization   Split Bregman Method          Results



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Introduction                  TV Denoising                   L1 Regularization            Split Bregman Method          Re...
Introduction                  TV Denoising                   L1 Regularization            Split Bregman Method          Re...
Introduction                  TV Denoising                   L1 Regularization            Split Bregman Method          Re...
Introduction                  TV Denoising                   L1 Regularization            Split Bregman Method          Re...
Introduction                  TV Denoising                    L1 Regularization           Split Bregman Method          Re...
Introduction                  TV Denoising                    L1 Regularization           Split Bregman Method          Re...
Introduction                  TV Denoising                    L1 Regularization           Split Bregman Method          Re...
Introduction                  TV Denoising                    L1 Regularization           Split Bregman Method          Re...
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Introduction                  TV Denoising                   L1 Regularization       Split Bregman Method              Res...
Introduction                  TV Denoising                   L1 Regularization       Split Bregman Method              Res...
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Introduction                  TV Denoising                   L1 Regularization           Split Bregman Method          Res...
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Introduction                  TV Denoising                   L1 Regularization   Split Bregman Method          Results



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Introduction                  TV Denoising                   L1 Regularization   Split Bregman Method          Results



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Introduction                      TV Denoising               L1 Regularization   Split Bregman Method          Results



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Introduction                      TV Denoising               L1 Regularization   Split Bregman Method          Results



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References


                Tom Goldstein and Stanley Osher, The split bregman method for
                l1-regularized ...
References


                Tom Goldstein and Stanley Osher, The split bregman method for
                l1-regularized ...
References


                Tom Goldstein and Stanley Osher, The split bregman method for
                l1-regularized ...
Thank you for your attention. Any questions?




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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

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