Proximal alternating linearized minimization for nonconvex and nonsmooth problems

Bolte, Jérôme; Sabach, Shoham; Teboulle, Marc

doi:10.1007/s10107-013-0701-9

Cited by 1,355 publications

(1,841 citation statements)

References 22 publications

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“…Inspired by the PALM (proximal alternating linearized minimization) algorithm [24], we propose to use a projected gradient method. For Equation (8) the following two steps are iterated for q = 1, 2, ... until convergence:…”

Section: Optimization Schemementioning

confidence: 99%

Hyperspectral Super-Resolution with Spectral Unmixing Constraints

2017

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Hyperspectral sensors capture a portion of the visible and near-infrared spectrum with many narrow spectral bands. This makes it possible to better discriminate objects based on their reflectance spectra and to derive more detailed object properties. For technical reasons, the high spectral resolution comes at the cost of lower spatial resolution. To mitigate that problem, one may combine such images with conventional multispectral images of higher spatial, but lower spectral resolution. The process of fusing the two types of imagery into a product with both high spatial and spectral resolution is called hyperspectral super-resolution. We propose a method that performs hyperspectral super-resolution by jointly unmixing the two input images into pure reflectance spectra of the observed materials, along with the associated mixing coefficients. Joint super-resolution and unmixing is solved by a coupled matrix factorization, taking into account several useful physical constraints. The formulation also includes adaptive spatial regularization to exploit local geometric information from the multispectral image. Moreover, we estimate the relative spatial and spectral responses of the two sensors from the data. That information is required for the super-resolution, but often at most approximately known for real-world images. In experiments with five public datasets, we show that the proposed approach delivers up to 15% improved hyperspectral super-resolution.

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Section: Optimization Schemementioning

confidence: 99%

Hyperspectral Super-Resolution with Spectral Unmixing Constraints

2017

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“…The key point of the proposed Algorithm 1 is that its convergence can be derived from Bolte et al (2014); Chouzenoux et al (2016). We present the convergence results in the following theorem: (u (k) 3 ) k ∈N be sequences generated by Algorithm 1.…”

Section: Convergence Resultsmentioning

confidence: 99%

A regularized tri-linear approach for optical interferometric imaging

Birdi¹,

Repetti²,

Wiaux³

2017

Monthly Notices of the Royal Astronomical Society

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In the context of optical interferometry, only undersampled power spectrum and bispectrum data are accessible. It poses an ill-posed inverse problem for image recovery. Recently, a trilinear model was proposed for monochromatic imaging, leading to an alternated minimization problem. In that work, only a positivity constraint was considered, and the problem was solved by an approximated Gauss-Seidel method. In this paper, we propose to improve the approach on three fundamental aspects. First, we define the estimated image as a solution of a regularized minimization problem, promoting sparsity in a fixed dictionary using either an ℓ 1 or a (re)weighted-ℓ 1 regularization term. Secondly, we solve the resultant non-convex minimization problem using a block-coordinate forward-backward algorithm. This algorithm is able to deal both with smooth and non-smooth functions, and benefits from convergence guarantees even in a non-convex context. Finally, we generalize our model and algorithm to the hyperspectral case, promoting a joint sparsity prior through an ℓ 2,1 regularization term. We present simulation results, both for monochromatic and hyperspectral cases, to validate the proposed approach.

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“…Therefore, alternative strategies based on proximal tools have been proposed which benefit from sounder convergence properties, particularly in the nonconvex setting. They consist of replacing, at each iteration, the minimization step by either a (single) proximal step [29,18] or a forward-backward step [30,31,32], giving rise, respectively, to the so-called proximal (resp. forward-backward) alternating algorithms.…”

Section: Minimization Strategymentioning

confidence: 99%

An alternating proximal approach for blind video deconvolution

Abboud¹,

Chouzenoux

Pesquet

et al. 2019

Signal Processing: Image Communication

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Blurring occurs frequently in video sequences captured by consumer devices, as a result of various factors such as lens aberrations, defocus, relative camerascene motion, and camera shake. When it comes to the contents of archive documents such as old films and television shows, the degradations are even more serious due to several physical phenomena happening during the sensing, transmission, recording, and storing processes. We propose in this paper a versatile formulation of blind video deconvolution problems that seeks to estimate both the sharp unknown video sequence and the underlying blur kernel from an observed video. This inverse problem is ill-posed, and an appropriate solution can be obtained by modeling it as a nonconvex minimization problem. We provide a novel iterative algorithm to solve it, grounded on the use of recent advances in convex and nonconvex optimization techniques, and having the ability of including numerous well-known regularization strategies.

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Proximal alternating linearized minimization for nonconvex and nonsmooth problems

Cited by 1,355 publications

References 22 publications

Hyperspectral Super-Resolution with Spectral Unmixing Constraints

Hyperspectral Super-Resolution with Spectral Unmixing Constraints

A regularized tri-linear approach for optical interferometric imaging

An alternating proximal approach for blind video deconvolution

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