An $$\ell _{1}$$ minimization algorithm for non-smooth regularization in image processing

Ramírez, Carlos; Argaez, Miguel

doi:10.1007/s11760-013-0454-1

Cited by 9 publications

(10 citation statements)

References 24 publications

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“…The main difference of this formulation and the one presented in (Ramirez & Argaez, 2013) is that the latter assumes normality in the distribution of residuals r i = a T i x − b i whereas in this work the residual vector r is modeled as sparse vector due to the presence of spikes in the measurements b. Now, notice that for any μ k > 0, Problem (2) is strictly convex since the Hessian matrix (2) is nonempty and has at most one element.…”

Section: Methodsmentioning

confidence: 96%

“…In order to solve (1), we extend our ideas reported in (Ramirez & Argaez, 2013) where the non differentiability of the 1 norm is overcome by approximating the absolute value with a continuously differentiable and strictly convex function that depends on a regularization parameter μ k . More specifically, |x| ≈ x 2 + μ k where μ k → 0 as k → ∞.…”

Section: Methodsmentioning

confidence: 99%

“…We propose a modification of the Fixed Point Least Squares-Preconditioned Conjugate Gradient (FPLS PCG) algorithm presented in (Ramirez & Argaez, 2013) for solving Problem (1). Step 5.…”

Section: The Algorithmmentioning

confidence: 99%

“…Firstly, it is important to realize that the main difference between the FPLS PCG algorithm presented here and the one in (Ramirez & Argaez, 2013) is in the Steps 6 and 7. In these Steps there is an extra re-weighting matrix D r associated with the residual vector r = Ax − b which accounts for the non-Gaussian noise present in the measurement vector b. Secondly, although any monotonically decreasing sequence {μ k } k is theoretically a guarantee for convergence to the optimal set of (1) as proved in next section, numerical experiments have demonstrated that the choice μ k = 1 2 k+15 performs well in practice for a broad set of problems.…”

Section: End-if Step 12 End-formentioning

confidence: 99%

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Interpolating Sparsely Corrupted Signals in Micrometeorology

Ramírez¹,

Argaez²,

Jaimes³

et al. 2014

JMR

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In real applications where data acquisition is carried out under extreme conditions, post-processing techniques for systematic corrections are of critical importance. In micrometeorological studies, it is often the case that acquired data contains both missing information and impulse noise due to instrumentation failure, data transmission and data rejection for quality assurance. In this work, we propose a simple algorithm based on an 1 − 1 variational formulation that simultaneously suppresses impulse noise and interpolates missing information. Our approach consists of relaxing the objective function in the variational formulation with a strictly convex and continuously differentiable function that depends on a regularization parameter. We solve a sequence of strictly convex optimization subproblems as the regularization parameter goes to zero, converging to the solution of the original problem. Numerical experiments on real micrometeorological datasets are conducted showing the effectiveness of the proposed approach. Furthermore, a convergence analysis is presented providing theoretical guaranties of our method.

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Section: Methodsmentioning

confidence: 96%

Section: Methodsmentioning

confidence: 99%

“…We propose a modification of the Fixed Point Least Squares-Preconditioned Conjugate Gradient (FPLS PCG) algorithm presented in (Ramirez & Argaez, 2013) for solving Problem (1). Step 5.…”

Section: The Algorithmmentioning

confidence: 99%

Section: End-if Step 12 End-formentioning

confidence: 99%

See 2 more Smart Citations

Interpolating Sparsely Corrupted Signals in Micrometeorology

Ramírez¹,

Argaez²,

Jaimes³

et al. 2014

JMR

Self Cite

View full text Add to dashboard Cite

show abstract

“…In this venue, numerous applications took place, including signal deconvolution, signal and image inpainting, multiresolution, image separation, and the new sampling theory of compressive sensing (see, e.g., Argáez, Ramirez, and Sanchez ; Figueiredo, Nowak, and Wright ; Shen et al . ; Hale, Yin, and Zhang ; Ramirez and Argáez ). In the present, it has been demonstrated that the sparse representation model (at least in signal and image processing) has achieved state‐of‐the‐art results compared with other signal models such as partial differential equation or variational methods.…”

Section: Introductionmentioning

confidence: 99%

Salt body detection from seismic data via sparse representation

Ramírez¹,

Larrazábal²,

González³

2015

Geophysical Prospecting

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A B S T R A C TIn seismic interpretation and seismic data analysis, it is of critical importance to effectively identify certain geologic formations from very large seismic data sets. In particular, the problem of salt characterization from seismic data can lead to important savings in time during the interpretation process if solved efficiently and in an automatic manner. In this work, we present a novel numerical approach that is able to automatically segmenting or identifying salt structures from a post-stack seismic data set with a minimum intervention from the interpreter. The proposed methodology is based on the recent theory of sparse representation and consists in three major steps: first, a supervised learning assisted by the user which is performed only once, second a segmentation process via unconstrained 1 optimization, and finally a post-processing step based on signal separation. Furthermore, since the second step only depends upon local information at each time, the whole process greatly benefits from parallel computing platforms. We conduct numerical experiments in a synthetic 3D seismic data set demonstrating the viability of our method. More specifically, we found that the proposed approach matches up to 98.53% with respect to the corresponding 3D velocity model available in advance. Finally, in appendixes A and B, we present a convergence analysis providing theoretical guarantees for the proposed method.

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Parametric PSF estimation based on predicted-SURE with $$\ell _1$$ ℓ 1 -penalized sparse deconvolution

Xue

Liu

2018

SIViP

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An $$\ell _{1}$$ minimization algorithm for non-smooth regularization in image processing

Cited by 9 publications

References 24 publications

Interpolating Sparsely Corrupted Signals in Micrometeorology

Interpolating Sparsely Corrupted Signals in Micrometeorology

Salt body detection from seismic data via sparse representation

Parametric PSF estimation based on predicted-SURE with $$\ell _1$$ ℓ 1 -penalized sparse deconvolution

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