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## Proximal Splitting Methods in Signal Processing

Citations: | 257 - 30 self |

### Citations

1025 | A fast iterative shrinkage-thresholding algorithm for linear inverse problems
- Beck, Teboulle
- 2009
(Show Context)
Citation Context ...], [35], [58], [89], [96], [97], [108], [112], [136]. Another type of convergence rate is that pertaining to the objective values (f1(xn)+f2(xn))n≥0. This rate has been investigated in several places =-=[13]-=-, [21], [80] and variants of (14) have been developed to improve it [13], [14], [81], [100], [123], [128] (note that the convergence of the sequence of iterates (xn)n≥0, which is often crucial in prac... |

941 |
Tsitsiklis, Parallel and distributed computation: numerical methods. Upper Saddle River
- Bertsekas, N
- 1989
(Show Context)
Citation Context ...in [23, 40, 48, 85, 130]. Let us note that, on the one hand, when f1 = 0, (17) reduces to the gradient method xn+1 = xn − γn∇f2(xn) (18) for minimizing a function with a Lipschitz continuous gradient =-=[19, 61]-=-. On the other hand, when f2 = 0, (17) reduces to the proximal point algorithm xn+1 = proxγnf1xn (19) for minimizing a nondifferentiable function [26, 48, 91, 98, 115]. The forward-backward algorithm ... |

617 | An algorithm for total variation minimization and applications
- Chambolle
(Show Context)
Citation Context ...ng Suppose that in Problem 6.1 f = h + ‖ · −r‖2/2, where h ∈ Γ0(RN ) and r ∈ RN . Then (41) becomes minimize x∈RN h(x) + g(Lx) + 1 2 ‖x− r‖2, (46) which models various signal recovery problems, e.g., =-=[33, 34, 51, 59, 112, 138]-=-. If (44) holds, proxg◦L is decomposable, and (46) can be solved with the Dykstra-like method of Section 5, where f1 = h + ‖ · −r‖2/2 (see Table 1.iv) and f2 = g ◦ L (see Table 1.x). Otherwise, we can... |

368 |
Zenios , Parallel Optimization: Theory, Algorithms, and Applications
- Censor, A
- 1997
(Show Context)
Citation Context ...C such that dC(x) = ‖x− PCx‖. 2 From projection to proximity operators One of the first widely used convex optimization splitting algorithms in signal processing is POCS (Projection Onto Convex Sets) =-=[31, 42, 141]-=-. This algorithm is employed to re2 cover/synthesize a signal satisfying simultaneously several convex constraints. Such a problem can be formalized within the framework of (1) by letting each functio... |

314 | On projection algorithms for solving convex feasibility problems
- Bauschke, Borwein
- 1996
(Show Context)
Citation Context ...xn. (3) When ⋂ m i=1 Ci = ∅ the sequence (xn)n≥0 thus produced converges to a solution to (2) [22]. Projection algorithms have been enriched with many extensions of this basic iteration to solve (2) =-=[8]-=-, [39], [41], [87]. Variants have also been proposed to solve more general problems, e.g., that of finding the projection of a signal onto an intersection of convex sets [19], [44], [129]. Beyond such... |

297 | A generalized Gaussian image model for edge-preserving MAP estimation
- Bouman, Sauer
- 1993
(Show Context)
Citation Context ... formulated as (9), where ‖·−y‖2/2 plays the role of a data fidelity term and where f models a priori knowledge about x. Such a formulation derives in particular from a Bayesian approach to denoising =-=[21, 124, 126]-=- in the presence of Gaussian noise and of a prior with a log-concave density exp(−f). 3 Forward-backward splitting In this section, we consider the case of m = 2 functions in (1), one of which is smoo... |

271 |
Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Space
- Bauschke, L
- 2011
(Show Context)
Citation Context ...ict ourselves to a finitedimensional setting to avoid technical digressions). We shall use the standard definitions and notation from convex analysis recalled below. DEFINITIONS FROM CONVEX ANALYSIS (=-=[11]-=-, [84], [107]) • Indicator function of a convex set C ⊂ R N : ιC : R N ( 0, if x ∈ C; → {0, +∞}: x ↦→ +∞, otherwise. • Support function of a nonempty set C ⊂ R N : σC : R N → ]−∞,+∞] : u ↦→ supx∈C x ⊤... |

179 | A new TwIST: Two-step iterative shrinkage/thresholding algorithms for image restoration
- Bioucas-Dias, Figueiredo
- 2007
(Show Context)
Citation Context ...satisfies (∀n ∈ Nr {0}) f1(xn) + f2(xn) ≤ f1(x) + f2(x) + 2β‖x0 − x‖ 2 (n+ 1)2 . (23) Other variations of the forward-backward algorithm have also been reported to yield improved convergence profiles =-=[20, 70, 97, 134, 135]-=-. Problem 3.1 and Proposition 3.3 cover a wide variety of signal processing problems and solution methods [55]. For the sake of illustration, let us provide a few examples. For notational convenience,... |

163 | Fast gradient-based algorithms for constrained total variation image denoising and deblurring problems
- Beck, Teboulle
- 2009
(Show Context)
Citation Context ...nce rate is that pertaining to the objective values (f1(xn)+f2(xn))n≥0. This rate has been investigated in several places [13], [21], [80] and variants of (14) have been developed to improve it [13], =-=[14]-=-, [81], [100], [123], [128] (note that the convergence of the sequence of iterates (xn)n≥0, which is often crucial in practice, is no longer guaranteed in general in such variants). Other variations h... |

123 | Fast image recovery using variable splitting and constrained optimization
- Afonso, Bioucas-Dias, et al.
- 2010
(Show Context)
Citation Context ...proposed [5, 39]. In image processing, ADMM was applied in [81] to an `1 regularization problem under the name “alternating split Bregman algorithm.” Further applications and connections are found in =-=[2, 69, 117, 143]-=-. 13 7 Problems with m ≥ 2 functions We return to the general minimization problem (1). Problem 7.1 Let f1,. . . ,fm be functions in Γ0(R N ) such that (ri dom f1) ∩ · · · ∩ (ri dom fm) 6= ∅ (52) and ... |

118 |
Image decomposition into a bounded variation component and an oscillating component
- Aujol, Aubert, et al.
(Show Context)
Citation Context ...n multicore architectures. A parallel proximal algorithm is also available to solve multicomponent signal processing problems [24]. This framework captures in particular problem formulations found in =-=[5]-=-, [6], [77], [85], [125]. Let us add that an alternative splitting framework applicable to (42) was recently proposed in [64]. VIII. CONCLUSION We have presented a panel of efficient convex optimizati... |

118 | Total variation minimization and a class of binary MRF models
- Chambolle
- 2005
(Show Context)
Citation Context ... to the projected gradient method xn+1 = PC ( xn − γn∇f2(xn) ) , ε ≤ γn ≤ 2/β − ε. (25) This algorithm has been used in numerous signal processing problems, in particular in total variation denoising =-=[34]-=-, in image deblurring [18], in pulse shape design [50], and in compressed sensing [73]. Example 3.9 (projected Landweber) In Example 3.8, setting f2 : x 7→ ‖Lx−y‖2/2, where L ∈ RM×N r {0} and y ∈ RM ,... |

117 | Regularized wavelet approximations
- Antoniadis, Fan
- 2001
(Show Context)
Citation Context ... in Table 2 closed-form expressions of the proximity operators of 3 various functions in Γ0(R) (in the case of functions such as | · |p, proximity operators implicitly appear in several places, e.g., =-=[3, 4, 35]-=-). From a signal processing perspective, proximity operators have a very natural interpretation in terms of denoising. Let us consider the standard denoising problem of recovering a signal x ∈ RN from... |

93 |
Dual norms and image decomposition models
- Aujol, Chambolle
- 2005
(Show Context)
Citation Context ...ticore architectures. A parallel proximal algorithm is also available to solve multicomponent signal processing problems [24]. This framework captures in particular problem formulations found in [5], =-=[6]-=-, [77], [85], [125]. Let us add that an alternative splitting framework applicable to (42) was recently proposed in [64]. VIII. CONCLUSION We have presented a panel of efficient convex optimization al... |

93 |
A method of finding projections onto the intersection of convex sets in Hilbert spaces
- Boyle, Dykstra
- 1986
(Show Context)
Citation Context ...is basic iteration to solve (6) [10, 43, 45, 90]. Variants have also been proposed to solve more general problems, e.g., that of finding the projection of a signal onto an intersection of convex sets =-=[22, 47, 137]-=-. Beyond such problems, however, projection methods are not appropriate and more general operators are required to tackle (1). Among the various generalizations of the notion of a convex projection op... |

87 | A framelet-based image inpainting algorithm
- Cai, Chan, et al.
(Show Context)
Citation Context ...k,γnωk] ( xn + γnL >(y − Lxn) )> bk ε ≤ γn ≤ 2/‖L‖2 − ε, (34) converges to a solution to (32). Additional applications of the forward-backward algorithm in signal and image processing can be found in =-=[30, 28, 29, 32, 36, 37, 53, 55, 57, 74]-=-. 4 Douglas-Rachford splitting The forward-backward algorithm of Section 3 requires that one of the functions be differentiable, with a Lipschitz continuous gradient. In this section, we relax this as... |

84 |
The method of successive projection for finding a common point of convex sets
- Bregman
- 1965
(Show Context)
Citation Context ...a nonempty closed convex set Ci modeling a constraint. This reduces (1) to the classical convex feasibility problem [31, 42, 44, 86, 93, 121, 122, 128, 141] find x ∈ m⋂ i=1 Ci. (6) The POCS algorithm =-=[25, 141]-=- activates each set Ci individually by means of its projection operator PCi . It is governed by the updating rule xn+1 = PC1 · · ·PCmxn. (7) When ⋂m i=1 Ci 6= ∅ the sequence (xn)n∈N thus produced conv... |

73 | Combettes, A weak-to-strong convergence principle for Fejer-monotone methods
- Bauschke, L
(Show Context)
Citation Context ... however, projection methods are not appropriate and more general operators are required to tackle (1). Among the various generalizations of the notion of a convex projection operator that exist [8], =-=[9]-=-, [40], [87], proximity operators are best suited for our purposes. The projection PCix of x ∈ R N onto the nonempty closed convex set Ci ⊂ R N is the solution to the problem minimize y∈R N ιCi(y) + 1... |

57 |
Optima and Equilibria. An Introduction to Nonlinear Analysis
- Aubin
- 1991
(Show Context)
Citation Context ...essing methods were essentially linear, as they were based on classical functional analysis and linear algebra. With the development of nonlinear analysis in mathematics in the late 1950s/early 1960s =-=[4]-=-, [134] and the availability of faster computers, nonlinear techniques have slowly become prevalent. In particular, convex optimization has been shown to provide efficient algorithms for computing rel... |

52 | Linear convergence of iterative soft-thresholding
- Bredies, Lorenz
(Show Context)
Citation Context ...ing of Example 3.11 below, linear convergence of the iterates (xn)n∈N generated by Algorithm 3.2 fails [9, 139]. Nonetheless it can be achieved at the expense of additional assumptions on the problem =-=[10, 24, 40, 61, 92, 99, 100, 115, 119, 144]-=-. Another type of convergence rate is that pertaining to the objective values (f1(xn) + f2(xn))n∈N. This rate has been investigated in several places [16, 24, 83] and variants of Algorithm 3.2 have be... |

51 |
Dykstra’s alternating projection algorithm for two sets
- Bauschke, Borwein
- 1994
(Show Context)
Citation Context ...are available for the forward-backward algorithm. In general, the answer is negative: even in the simple setting of Example 6 below, linear convergence of the iterates (xn)n≥0 generated by (14) fails =-=[7]-=-, [131]. Nonetheless it can be achieved at the expense of additional assumptions on the problem [8], [35], [58], [89], [96], [97], [108], [112], [136]. Another type of convergence rate is that pertain... |

50 |
A ℓ1-unified variational framework for image restoration
- Bect, Féraud, et al.
(Show Context)
Citation Context ...• • • • • This type of formulation arises in signal recovery problems in which y is the observed signal and the original signal is known to have a sparse representation in the basis (bk)1≤k≤N , e.g., =-=[15]-=-, [17], [53], [55], [69], [70], [119]. We observe that (23) is a special case of (9) with { f1: x ↦→ ∑ 1≤k≤N ωk ∣ ⊤ x bk∣ (24) f2: x ↦→ ‖Lx − y‖ 2 /2. Since prox γf1 : x ↦→ ∑ 1≤k≤N soft [−γωk,γωk] (x ... |

40 |
Lions: Produits infinis de résolvantes
- Brézis, P-L
- 1978
(Show Context)
Citation Context ...function with a Lipschitz continuous gradient [19, 61]. On the other hand, when f2 = 0, (17) reduces to the proximal point algorithm xn+1 = proxγnf1xn (19) for minimizing a nondifferentiable function =-=[26, 48, 91, 98, 115]-=-. The forward-backward algorithm can therefore be considered as a combination of these two basic schemes. The following version incorporates relaxation parameters (λn)n∈N. 4 Algorithm 3.2 (Forward-bac... |

35 |
Eds., “A -unified variational framework for image restoration
- Bect, Blanc-Féraud, et al.
(Show Context)
Citation Context ...orithm. 8 This type of formulation arises in signal recovery problems in which y is the observed signal and the original signal is known to have a sparse representation in the basis (bk)1≤k≤N , e.g., =-=[17, 20, 56, 58, 72, 73, 125, 127]-=-. We observe that (32) is a special case of (15) with{ f1 : x 7→ ∑ 1≤k≤N ωk|x>bk| f2 : x 7→ ‖Lx− y‖2/2. (33) Since proxγf1 : x 7→ ∑ 1≤k≤N soft[−γωk,γωk](x >bk) bk (see Table 1.viii and Table 2.ii), it... |

34 |
Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces
- Bauschke
- 2011
(Show Context)
Citation Context ...ven in Section 8. Notation. We denote by RN the usualN -dimensional Euclidean space, by ‖·‖ its norm, and by I the identity matrix. Standard definitions and notation from convex analysis will be used =-=[13, 87, 114]-=-. The domain of a function f : RN → ]−∞,+∞] is dom f = {x ∈ RN |f(x) < +∞}. Γ0(R N ) is the class of lower semicontinuous convex functions from RN to ]−∞,+∞] such that dom f 6= ∅. Let f ∈ Γ0(RN ). The... |

31 | The asymptotic behavior of the composition of two resolvents
- Bauschke, Combettes, et al.
(Show Context)
Citation Context ... 1 asserts that the sequence (xn)n≥0 generated by the backward-backward algorithm xn+1 = prox f (prox g xn) (20) converges to a solution to (19). Detailed analyses of this scheme can be found in [1], =-=[12]-=-, [45], [102]. Example 6 (alternating projections) In Example 5, let f and g be respectively the indicator functions of nonempty closed convex sets C and D, one of which is bounded. Then (19) amounts ... |

29 |
Wavelet thresholding for some classes of non-Gaussian noise
- Antoniadis, Leporini, et al.
- 2002
(Show Context)
Citation Context ... in Table 2 closed-form expressions of the proximity operators of 3 various functions in Γ0(R) (in the case of functions such as | · |p, proximity operators implicitly appear in several places, e.g., =-=[3, 4, 35]-=-). From a signal processing perspective, proximity operators have a very natural interpretation in terms of denoising. Let us consider the standard denoising problem of recovering a signal x ∈ RN from... |

27 | Convergence analysis of tight framelet approach for missing data recovery
- Cai, Chan, et al.
(Show Context)
Citation Context ...k,γnωk] ( xn + γnL >(y − Lxn) )> bk ε ≤ γn ≤ 2/‖L‖2 − ε, (34) converges to a solution to (32). Additional applications of the forward-backward algorithm in signal and image processing can be found in =-=[30, 28, 29, 32, 36, 37, 53, 55, 57, 74]-=-. 4 Douglas-Rachford splitting The forward-backward algorithm of Section 3 requires that one of the functions be differentiable, with a Lipschitz continuous gradient. In this section, we relax this as... |

20 |
Convergence d’un schéma de minimisation alternée
- Acker, Prestel
- 1980
(Show Context)
Citation Context ... γn ≡ 1 asserts that the sequence (xn)n≥0 generated by the backward-backward algorithm xn+1 = prox f (prox g xn) (20) converges to a solution to (19). Detailed analyses of this scheme can be found in =-=[1]-=-, [12], [45], [102]. Example 6 (alternating projections) In Example 5, let f and g be respectively the indicator functions of nonempty closed convex sets C and D, one of which is bounded. Then (19) am... |

17 | Combettes, A Dykstra-like algorithm for two monotone operators
- Bauschke, L
- 2008
(Show Context)
Citation Context ...= ∅. DYKSTRA-LIKE PROXIMAL ALGORITHM Set x0 = r, p0 = 0, q0 = 0 For n = 0, 1, . . . 666664 yn = prox g(xn + pn) pn+1 = xn + pn − yn xn+1 = prox f (yn + qn) qn+1 = yn + qn − xn+1. (31) Proposition 11 =-=[10]-=- Every sequence (xn)n≥0 generated by the Dykstra-like proximal algorithm (31) converges to the solution to Problem 10. Example 12 (best approximation) Let f and g be the indicator functions of closed ... |

17 |
A forward-backward splitting algorithm for the minimization of non-smooth convex functionals
- Bredies
(Show Context)
Citation Context ...m finds its roots in the projected gradient method [94] and in decomposition methods for solving variational inequalities [99, 119]. More recent forms of the algorithm and refinements can be found in =-=[23, 40, 48, 85, 130]-=-. Let us note that, on the one hand, when f1 = 0, (17) reduces to the gradient method xn+1 = xn − γn∇f2(xn) (18) for minimizing a function with a Lipschitz continuous gradient [19, 61]. On the other h... |

15 | Simultaneously inpainting in image and transformed domains, Num
- Cai, Chan, et al.
(Show Context)
Citation Context ...k,γnωk] ( xn + γnL >(y − Lxn) )> bk ε ≤ γn ≤ 2/‖L‖2 − ε, (34) converges to a solution to (32). Additional applications of the forward-backward algorithm in signal and image processing can be found in =-=[30, 28, 29, 32, 36, 37, 53, 55, 57, 74]-=-. 4 Douglas-Rachford splitting The forward-backward algorithm of Section 3 requires that one of the functions be differentiable, with a Lipschitz continuous gradient. In this section, we relax this as... |

14 | Combettes. Convex variational formulation with smooth coupling for multicomponent signal decomposition and recovery
- Briceño-Arias, L
(Show Context)
Citation Context ...n. This parallel structure is useful when the algorithms are implemented on multicore architectures. A parallel proximal algorithm is also available to solve multicomponent signal processing problems =-=[27]-=-. This framework captures in particular problem formulations found in [7, 8, 80, 88, 133]. Let us add that an alternative splitting framework applicable to (53) was recently proposed in [67]. 8 Conclu... |

13 |
Nonnegative least-squares image deblurring: improved gradient projection approaches, Inverse Problems 26 (2) (2010) 25004–25021. Shiming Xiang received the B.S. degree in mathematics from Chongqing Normal Un the Ph.D. degree from the Institute of Computin
- Benvenuto, Zanella, et al.
- 1992
(Show Context)
Citation Context ... method xn+1 = PC ( xn − γn∇f2(xn) ) , ε ≤ γn ≤ 2/β − ε. (25) This algorithm has been used in numerous signal processing problems, in particular in total variation denoising [34], in image deblurring =-=[18]-=-, in pulse shape design [50], and in compressed sensing [73]. Example 3.9 (projected Landweber) In Example 3.8, setting f2 : x 7→ ‖Lx−y‖2/2, where L ∈ RM×N r {0} and y ∈ RM , yields the constrained le... |

9 |
Augmented Lagrangian and proximal alternating direction methods of multipliers in Hilbert spaces. Applications to games, PDE’s and control
- Attouch, Soueycatt
- 2009
(Show Context)
Citation Context ... the dual of (32). This analysis was pursued in [63], where the convergence of (xn)n≥0 to a solution to (32) is shown. Variants of the method relaxing the requirements on L in (39) have been proposed =-=[3]-=-, [36]. In image processing, ADMM was applied in [78] to an ℓ1 regularization problem under the name “alternating split Bregman algorithm.” Further applications and connections are found in [2], [66],... |

6 |
P.L.: A weak-to-strong convergence principle for Fejér-monotone methods in Hilbert space
- Bauschke, Combettes
- 2001
(Show Context)
Citation Context ...lems, however, projection methods are not appropriate and more general operators are required to tackle (1). Among the various generalizations of the notion of a convex projection operator that exist =-=[10, 11, 44, 90]-=-, proximity operators are best suited for our purposes. The projection PCx of x ∈ RN onto the nonempty closed convex set C ⊂ RN is the solution to the problem minimize y∈RN ιC(y) + 1 2 ‖x− y‖2. (8) Un... |

5 |
J.M.: Dykstra’s alternating projection algorithm for two sets
- Bauschke, Borwein
- 1994
(Show Context)
Citation Context ...e for the forward-backward algorithm. In general, the answer is negative: even in the simple setting of Example 3.11 below, linear convergence of the iterates (xn)n∈N generated by Algorithm 3.2 fails =-=[9, 139]-=-. Nonetheless it can be achieved at the expense of additional assumptions on the problem [10, 24, 40, 61, 92, 99, 100, 115, 119, 144]. Another type of convergence rate is that pertaining to the object... |

2 |
A.: An iterative method for parallel MRI SENSE-based reconstruction in the wavelet domain
- Chaâri, Pesquet, et al.
- 2009
(Show Context)
Citation Context |