A variational formulation for frame-based inverse problems

Chaux, Caroline; Combettes, Patrick L.; Pesquet, Jean-Christophe; Wajs, Valérie R.

doi:10.1088/0266-5611/23/4/008

Cited by 188 publications

(235 citation statements)

References 36 publications

(65 reference statements)

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“…If C is a nonempty closed convex set of H, then prox ιC reduces to the projection Π C onto C. Note that this operator possesses numerous mathematical properties [14,15].…”

Section: Proximity Operatorsmentioning

confidence: 99%

“…Closed form expressions of the considered power functions are indeed available for typical values of the exponent [14].…”

Section: A Deeper Look At the Criterion And Proximity Operatorsmentioning

confidence: 99%

“…, K}, p k ≡ 2 and subband-adaptive values of κ k,1 and κ k,2 are set. The constraints C 1 and C 2 are chosen according to (14) and (15), where ε 1,p = 0.1 and ε 2,p = 0.07 for every p.…”

Section: Simulationsmentioning

confidence: 99%

See 2 more Smart Citations

Seismic multiple removal with a primal-dual proximal algorithm

Pham

Chaux

Duval

et al. 2013

2013 IEEE International Conference on Acoustics, Speech and Signal Processing

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Both random and structured perturbations affect seismic data. Their removal, to unveil meaningful geophysical information, requires additional priors. Seismic multiples are one form of structured perturbations related to wave-field bouncing. In this paper, we model these undesired signals through a timevarying filtering process accounting for inaccuracies in amplitude, time-shift and average frequency of available templates. We recast the problem of jointly estimating the filters and the signal of interest (primary) in a new convex variational formulation, allowing the incorporation of knowledge about the noise statistics. By making some physically plausible assumptions about the slow time variations of the filters, and by adopting a potential promoting the sparsity of the primary in a wavelet frame, we design a primal-dual algorithm which yields good performance in the provided simulation examples.

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“…If C is a nonempty closed convex set of H, then prox ιC reduces to the projection Π C onto C. Note that this operator possesses numerous mathematical properties [14,15].…”

Section: Proximity Operatorsmentioning

confidence: 99%

“…Closed form expressions of the considered power functions are indeed available for typical values of the exponent [14].…”

Section: A Deeper Look At the Criterion And Proximity Operatorsmentioning

confidence: 99%

See 1 more Smart Citation

Seismic multiple removal with a primal-dual proximal algorithm

Pham

Chaux

Duval

et al. 2013

2013 IEEE International Conference on Acoustics, Speech and Signal Processing

Self Cite

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“…One can show (see [12] for more details, or [32] for a complete proof) that for any h > 0 the following problem always has a unique solution :…”

Section: Proximal Operatormentioning

confidence: 99%

Some proximal methods for Poisson intensity CBCT and PET

Anthoine¹,

Aujol²,

Boursier³

et al. 2012

Inverse Problems &Amp; Imaging

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Cone-Beam Computerized Tomography (CBCT) and Positron Emission Tomography (PET) are two complementary medical imaging modalities providing respectively anatomic and metabolic information of a patient. In the context of public health, one must address the problem of dose reduction of the potentially harmful quantities related to each exam protocol : X-rays for CBCT and radiotracer for PET. Two demonstrators based on a technological breakthrough (acquisition devices work in photon-counting mode) have been developed and we investigate in this paper the two related tomographic reconstruction problems. We formulate separately the CBCT and the PET problems in two general frameworks that encompass the physics of the acquisition devices and the specific discretization of the object to reconstruct. These objects may be observed from a limited number of angles of views and we take into account the specificity of the Poisson noise. We propose various fast numerical schemes based on proximal methods to compute the solution of each problem. In particular, we show that primal-dual approaches are well suited in the PET case when considering non differentiable regularizations such as Total Variation. Experiments on numerical simulations and real data are in favor of the proposed algorithms when compared with the well-established methods.

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“…Let us notice that unified approaches to both these models for sparse solution of inverse problems appear, e.g., in [15,18,21,22]. Moreover, the near-equivalence of constraints promoting sparse wavelet coefficients and BV norm was established in [16,17].…”

Section: The General Settingmentioning

confidence: 99%

The application of joint sparsity and total variation minimization algorithms to a real-life art restoration problem

2008

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On March 11, 1944, the famous Eremitani Church in Padua (Italy) was destroyed in an Allied bombing along with the inestimable frescoes by Andrea Mantegna et al. contained in the Ovetari Chapel. In the last 60 years, several attempts have been made to restore the fresco fragments by traditional methods, but without much success. One of the authors contributed to the development of an efficient pattern recognition algorithm to map the original position and orientation of the fragments, based on comparisons with an old gray level image of the fresco prior to the damage. This innovative technique allowed for the partial reconstruction of the frescoes. Unfortunately, the surface covered by the colored fragments is only 77 m 2 , while the original area was of several hundreds. This means that we can reconstruct only a fraction (less than 8%) of this inestimable artwork. In particular the original color of the blanks is not known. This begs the question of whether it is possible to estimate mathematically the original colors of the frescoes by making use of the potential information given by the available fragments and the gray level of the pictures taken before the damage. Moreover, is it possible to estimate how faithful such a restoration is? In this paper we retrace the development of the recovery of the frescoes as an inspiring and challenging real-life problem for the development of new mathematical methods. Then we shortly review two models recently studied independently by the authors for the recovery of vector valued functions from incomplete data, with applications to the recolorization problem. The models are based on the minimization of a functional which is formed by the discrepancy with respect to the data and additional regularization constraints. The latter refer to joint sparsity measures with respect to frame expansions, in particular wavelet or curvelet expansions, for the first functional and functional total variation for the second. We establish relations between these two models. As a major contribution of this work we perform specific numerical test on the real-life problem of the A. Mantegna's frescoes and we compare the results due to the two methods.MSC: 15A29, 47A52, 68U10, 94A08, 94A40

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A variational formulation for frame-based inverse problems

Cited by 188 publications

References 36 publications

Seismic multiple removal with a primal-dual proximal algorithm

Seismic multiple removal with a primal-dual proximal algorithm

Some proximal methods for Poisson intensity CBCT and PET

The application of joint sparsity and total variation minimization algorithms to a real-life art restoration problem

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