On global and local convergence of half-quadratic algorithms

Allain, Marc; Idier, Jérôme; Goussard, Yves

doi:10.1109/icip.2002.1040080

Cited by 21 publications

(55 citation statements)

References 32 publications

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“…Obviously, such an acceleration requires that the preconditioner B is wisely chosen. For our joint Blind-SIM problem, the preconditioning matrix is derived from Geman and Yang semi-quadratic construction [50], [51,Eq. (6)]…”

Section: Resolution Of the Joint Blind-sim Sub-problemmentioning

confidence: 99%

Joint Reconstruction Strategy for Structured Illumination Microscopy With Unknown Illuminations

Labouesse

Negash

Idier

et al. 2017

IEEE Trans. on Image Process.

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The blind structured illumination microscopy strategy proposed by Mudry et al. is fully re-founded in this paper, unveiling the central role of the sparsity of the illumination patterns in the mechanism that drives super-resolution in the method. A numerical analysis shows that the resolving power of the method can be further enhanced with optimized one-photon or two-photon speckle illuminations. A much improved numerical implementation is provided for the reconstruction problem under the image positivity constraint. This algorithm rests on a new preconditioned proximal iteration faster than existing solutions, paving the way to 3D and real-time 2D reconstruction.

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Section: Resolution Of the Joint Blind-sim Sub-problemmentioning

confidence: 99%

Joint Reconstruction Strategy for Structured Illumination Microscopy With Unknown Illuminations

Labouesse

Negash

Idier

et al. 2017

IEEE Trans. on Image Process.

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“…Several choices have been proposed in the literature for matrices (D n ) n∈N * . On the one hand, if, for every n ∈ N * , rank(D n ) = N , Algorithm (5) becomes equivalent to a half-quadratic method with unit stepsize [13,23,24]. Half-quadratic algorithms are known to be effective optimization methods, but the resolution of the minimization subproblem involved in (5) requires the inversion of matrix A n (h n ) which may have a high computational cost.…”

Section: Subspace Algorithmmentioning

confidence: 99%

“…In particular, it seems that the best performance is obtained for the memory gradient subspace [12], spanned by the current gradient and the previous direction, leading to the so-called MM Memory Gradient (3MG) algorithm. However, only an analysis concerning the convergence rates of half-quadratic algorithms (corresponding to the case when the subspace spans the whole Euclidean space) is available [13,14]. Section 2 describes the general form of the MM subspace algorithm and its main known properties.…”

Section: Introductionmentioning

confidence: 99%

Convergence Rate Analysis of the Majorize–Minimize Subspace Algorithm

Chouzenoux

Pesquet

2016

IEEE Signal Process. Lett.

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State-of-the-art methods for solving smooth optimization problems are nonlinear conjugate gradient, low memory BFGS, and Majorize-Minimize (MM) subspace algorithms. The MM subspace algorithm which has been introduced more recently has shown good practical performance when compared with other methods on various optimization problems arising in signal and image processing. However, to the best of our knowledge, no general result exists concerning the theoretical convergence rate of the MM subspace algorithm. This paper aims at deriving such convergence rates both for batch and online versions of the algorithm and, in particular, discusses the influence of the choice of the subspace.

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“…In the sequel, we focus on a discretized formulation of the observation model (1). Solving the two-dimensional (2D) joint Blind-SIM reconstruction problem is equivalent to finding a joint solution ( ρ, { I m } M m=1 ) to the following constrained minimisation problem [14]:…”

Section: Blind-sim Problem Reformulationmentioning

confidence: 99%

“…with ζ (k) := ∇g(q (k) ) + ω (k) and where the preconditioning matrix B is chosen from the Geman and Yang semi-quadratic construction [7], [1,Eq. (6)]…”

Section: A New Optimization Strategymentioning

confidence: 99%