ℓ p − ℓ q minimization methods for image restoration CMIPI 2018 16th July 2018 A. Buccini 1 L. Reichel 1 1 Department of Mathematical Sciences, Kent State Univeristy, Kent OH, USA
Outline ℓ p - ℓ q minimization methods for image restoration Introduction Discrete ill-posed inverse problems Introduction ℓ p − ℓ q regularization Discrete ill-posed inverse problems ℓ p − ℓ q regularization Majorization-Minimization in Generalized Krylov Subspaces MM-GKS General idea General idea Algorithm Algorithm Theoretical Results Selection of the Theoretical Results regularization parameter Discrepancy Principle Selection of the regularization parameter Cross Validation Modified Cross Validation Discrepancy Principle Numerical Results Cross Validation Conclusions & Future Modified Cross Validation work Numerical Results Conclusions & Future work Dep. of Mathematical Sc. Kent State Univeristy Ohio, USA 31
Introduction Discrete ill-posed inverse problems ℓ p - ℓ q minimization Consider linear system of equations methods for image restoration A x = b , (1) Introduction 1 Discrete ill-posed inverse problems A may be rank deficient and is severely ill-conditioned, i.e., its ℓ p − ℓ q regularization singular values decay to 0 rapidly and without any gap. MM-GKS General idea Algorithm Theoretical Results 10 0 Selection of the regularization parameter Discrepancy Principle 10 -5 Cross Validation Modified Cross Validation 10 -10 Numerical Results Conclusions & Future work 10 -15 0 100 200 300 400 500 600 700 800 900 1000 Figure: Singular values of the shaw matrix. Dep. of Mathematical Sc. Kent State Univeristy Ohio, USA 31
Introduction Discrete ill-posed inverse problems (continued) ℓ p - ℓ q minimization methods for image We only have a noise contaminated right-hand side b δ . We will restoration assume that the error in b δ is made up of impulse noise and/or Introduction Gaussian noise. 2 Discrete ill-posed inverse problems ℓ p − ℓ q regularization MM-GKS General idea Algorithm Theoretical Results Selection of the regularization parameter Discrepancy Principle Cross Validation Modified Cross Validation Numerical Results Conclusions & Future work Dep. of Mathematical Sc. Kent State Univeristy Ohio, USA 31
Introduction Discrete ill-posed inverse problems (continued) ℓ p - ℓ q minimization methods for image We only have a noise contaminated right-hand side b δ . We will restoration assume that the error in b δ is made up of impulse noise and/or Introduction Gaussian noise. 2 Discrete ill-posed inverse problems ℓ p − ℓ q regularization Impulse noise affects only a certain percentage of the entries MM-GKS General idea of b and leaves the other entries unchanged. Algorithm Theoretical Results � d i with probability σ, Selection of the b δ i = regularization with probability 1 − σ, b i parameter Discrepancy Principle Cross Validation where the d i are identically and uniformly distributed random Modified Cross Validation Numerical Results numbers in the dynamic range of b [ d min , d max ] . If Conclusions & Future d i ∈ { d min , d max } impulse noise is commonly referred to as work salt-and-pepper noise. Dep. of Mathematical Sc. Kent State Univeristy Ohio, USA 31
Introduction Discrete ill-posed inverse problems (continued) ℓ p - ℓ q minimization methods for image We only have a noise contaminated right-hand side b δ . We will restoration assume that the error in b δ is made up of impulse noise and/or Introduction Gaussian noise. 2 Discrete ill-posed inverse problems ℓ p − ℓ q regularization Impulse noise affects only a certain percentage of the entries MM-GKS General idea of b and leaves the other entries unchanged. Algorithm Theoretical Results � d i with probability σ, Selection of the b δ i = regularization with probability 1 − σ, b i parameter Discrepancy Principle Cross Validation where the d i are identically and uniformly distributed random Modified Cross Validation Numerical Results numbers in the dynamic range of b [ d min , d max ] . If Conclusions & Future d i ∈ { d min , d max } impulse noise is commonly referred to as work salt-and-pepper noise. Noise and ill-conditioning make impossible to directly solve system (1), we need regularization methods. Dep. of Mathematical Sc. Kent State Univeristy Ohio, USA 31
Introduction ℓ p − ℓ q regularization ℓ p - ℓ q minimization A regularization technique consists in solving an ℓ p - ℓ q methods for image restoration minimization problem of the form � A x − b δ � � 1 p + µ Introduction x ∗ = arg min � p q � L x � q 0 < p , q ≤ 2 Discrete ill-posed inverse q p problems x ℓ p − ℓ q regularization 3 where MM-GKS General idea N ( A ) ∩ N ( L ) = { 0 } . Algorithm Theoretical Results Selection of the regularization parameter Discrepancy Principle Cross Validation Modified Cross Validation Numerical Results Conclusions & Future work Dep. of Mathematical Sc. Kent State Univeristy Ohio, USA 31
Introduction ℓ p − ℓ q regularization ℓ p - ℓ q minimization A regularization technique consists in solving an ℓ p - ℓ q methods for image restoration minimization problem of the form � � A x − b δ � 1 p + µ Introduction x ∗ = arg min � p q � L x � q 0 < p , q ≤ 2 Discrete ill-posed inverse q p problems x ℓ p − ℓ q regularization 3 where MM-GKS General idea N ( A ) ∩ N ( L ) = { 0 } . Algorithm Theoretical Results Selection of the ◮ p = 2 and q = 2 yield to the classical Tikhonov regularization parameter regularization (Bai, Chan, Donatelli, Fenu, Gazzola, Discrepancy Principle Cross Validation Hayami, Hanke, Hansen, Nagy, Ramlau, Reichel, Modified Cross Validation Rodriguez,. . . ); Numerical Results Conclusions & Future work Dep. of Mathematical Sc. Kent State Univeristy Ohio, USA 31
Introduction ℓ p − ℓ q regularization ℓ p - ℓ q minimization A regularization technique consists in solving an ℓ p - ℓ q methods for image restoration minimization problem of the form � � A x − b δ � 1 p + µ Introduction x ∗ = arg min � p q � L x � q 0 < p , q ≤ 2 Discrete ill-posed inverse q p problems x ℓ p − ℓ q regularization 3 where MM-GKS General idea N ( A ) ∩ N ( L ) = { 0 } . Algorithm Theoretical Results Selection of the ◮ p = 2 and q = 2 yield to the classical Tikhonov regularization parameter regularization (Bai, Chan, Donatelli, Fenu, Gazzola, Discrepancy Principle Cross Validation Hayami, Hanke, Hansen, Nagy, Ramlau, Reichel, Modified Cross Validation Rodriguez,. . . ); Numerical Results Conclusions & Future ◮ 1 ≤ p , q < 2 yield to a convex minimization problem work (Chan, Chung, Donatelli, Estatico, Gazzola, Hansen, Huang, Nagy, Reichel,. . . ); Dep. of Mathematical Sc. Kent State Univeristy Ohio, USA 31
Introduction ℓ p − ℓ q regularization ℓ p - ℓ q minimization A regularization technique consists in solving an ℓ p - ℓ q methods for image restoration minimization problem of the form � � A x − b δ � 1 p + µ Introduction x ∗ = arg min � p q � L x � q 0 < p , q ≤ 2 Discrete ill-posed inverse q p problems x ℓ p − ℓ q regularization 3 where MM-GKS General idea N ( A ) ∩ N ( L ) = { 0 } . Algorithm Theoretical Results Selection of the ◮ p = 2 and q = 2 yield to the classical Tikhonov regularization parameter regularization (Bai, Chan, Donatelli, Fenu, Gazzola, Discrepancy Principle Cross Validation Hayami, Hanke, Hansen, Nagy, Ramlau, Reichel, Modified Cross Validation Rodriguez,. . . ); Numerical Results Conclusions & Future ◮ 1 ≤ p , q < 2 yield to a convex minimization problem work (Chan, Chung, Donatelli, Estatico, Gazzola, Hansen, Huang, Nagy, Reichel,. . . ); ◮ 0 < q < 1 or 0 < p < 1 yield to a non-convex minimization problem (Chan, Huang, Lanza, Morigi, Reichel, Dep. of Mathematical Sc. Sgallari,. . . ). Kent State Univeristy Ohio, USA 31
Introduction ℓ p − ℓ q regularization (continued) ℓ p - ℓ q minimization In many situations it is useful to impose sparsity to improve the methods for image restoration quality of the computed reconstructions Introduction Discrete ill-posed inverse problems ℓ p − ℓ q regularization 4 MM-GKS General idea Algorithm Theoretical Results Selection of the regularization parameter Discrepancy Principle Cross Validation Modified Cross Validation Numerical Results Conclusions & Future work Dep. of Mathematical Sc. Kent State Univeristy Ohio, USA 31
Introduction ℓ p − ℓ q regularization (continued) ℓ p - ℓ q minimization In many situations it is useful to impose sparsity to improve the methods for image restoration quality of the computed reconstructions To enhance sparsity, we may consider using the so-called Introduction ℓ 0 -norm. Discrete ill-posed inverse problems ℓ p − ℓ q regularization 4 MM-GKS General idea Algorithm Theoretical Results Selection of the regularization parameter Discrepancy Principle Cross Validation Modified Cross Validation Numerical Results Conclusions & Future work Dep. of Mathematical Sc. Kent State Univeristy Ohio, USA 31
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