Generalized total variation denoising via augmented Lagrangian cycle spinning with Haar wavelets

Kamilov, Ulugbek S.; Bostan, Emrah; Unser, Michaël

doi:10.1109/icassp.2012.6288032

Cited by 5 publications

(2 citation statements)

References 12 publications

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“…The first term is the negative log-likelihood for Poisson random variables [12,37,14,58] and the second term is the 1D total variation penalty, also called fused lasso in the statistics literature [52,48,53]. 1D total variation denoising has been studied in [2,35,34,56,23]. For simplicity, assume d is odd.…”

Section: Numerical Examplesmentioning

confidence: 99%

Uniqueness of DRS as the 2 operator resolvent-splitting and impossibility of 3 operator resolvent-splitting

Ryu¹

2019

Math. Program.

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Given the success of Douglas-Rachford splitting (DRS), it is natural to ask whether DRS can be generalized. Are there other 2 operator resolvent-splittings sharing the favorable properties of DRS? Can DRS be generalized to 3 operators? This work presents the answers: no and no. In a certain sense, DRS is the unique 2 operator resolvent-splitting, and generalizing DRS to 3 operators is impossible without lifting, where lifting roughly corresponds to enlarging the problem size. The impossibility result further raises a question. How much lifting is necessary to generalize DRS to 3 operators? This work presents the answer by providing a novel 3 operator resolvent-splitting with provably minimal lifting that directly generalizes DRS.

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Section: Numerical Examplesmentioning

confidence: 99%

Uniqueness of DRS as the 2 operator resolvent-splitting and impossibility of 3 operator resolvent-splitting

Ryu¹

2019

Math. Program.

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“…Non-Gaussian models arise in numerous applications in inverse problems [34][35][36][37]. In this context, the posterior distribution is non-Gaussian and does not generally follow a standard probability model.…”

Section: Designing Efficient Proposals In Mh Algorithmsmentioning

confidence: 99%

An Auxiliary Variable Method for Markov Chain Monte Carlo Algorithms in High Dimension

Marnissi

Chouzenoux

Benazza-Benyahia

et al. 2018

Entropy

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Abstract:In this paper, we are interested in Bayesian inverse problems where either the data fidelity term or the prior distribution is Gaussian or driven from a hierarchical Gaussian model. Generally, Markov chain Monte Carlo (MCMC) algorithms allow us to generate sets of samples that are employed to infer some relevant parameters of the underlying distributions. However, when the parameter space is high-dimensional, the performance of stochastic sampling algorithms is very sensitive to existing dependencies between parameters. In particular, this problem arises when one aims to sample from a high-dimensional Gaussian distribution whose covariance matrix does not present a simple structure. Another challenge is the design of Metropolis-Hastings proposals that make use of information about the local geometry of the target density in order to speed up the convergence and improve mixing properties in the parameter space, while not being too computationally expensive. These two contexts are mainly related to the presence of two heterogeneous sources of dependencies stemming either from the prior or the likelihood in the sense that the related covariance matrices cannot be diagonalized in the same basis. In this work, we address these two issues. Our contribution consists of adding auxiliary variables to the model in order to dissociate the two sources of dependencies. In the new augmented space, only one source of correlation remains directly related to the target parameters, the other sources of correlations being captured by the auxiliary variables. Experiments are conducted on two practical image restoration problems-namely the recovery of multichannel blurred images embedded in Gaussian noise and the recovery of signal corrupted by a mixed Gaussian noise. Experimental results indicate that adding the proposed auxiliary variables makes the sampling problem simpler since the new conditional distribution no longer contains highly heterogeneous correlations. Thus, the computational cost of each iteration of the Gibbs sampler is significantly reduced while ensuring good mixing properties.

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The STATiX pipeline for the detection of X-ray transients in three dimensions

Ruiz,

Georgakakis,

Georgantopoulos

et al. 2023

Monthly Notices of the Royal Astronomical Society

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The recent serendipitous discovery of a new population of short duration X-ray transients, thought to be associated with collisions of compact objects or stellar explosions in distant galaxies, has motivated efforts to build up statistical samples by mining X-ray telescope archives. Most searches to date, however, do not fully exploit recent developments in the signal and imaging processing research domains to optimize searches for short X-ray flashes. This paper addresses this issue by presenting a new source detection pipeline, STATiX (Space and Time Algorithm for Transients in X-rays), which directly operates on three-dimensional X-ray data cubes consisting of two spatial and one temporal dimension. The algorithm leverages wavelet transforms and the principles of sparsity to denoise X-ray observations and then detect source candidates on the denoised data cubes. The light curves of the detected sources are then characterized using the Bayesian blocks algorithm to identify flaring periods. We describe the implementation of STATiX in the case of XMM–Newton data, present extensive validation and performance verification tests based on simulations and also apply the pipeline to a small subset of seven XMM–Newton observations, which are known to contain transients sources. In addition to known flares in the selected fields, we report a previously unknown short duration transient found by our algorithm that is likely associated with a flaring Galactic star. This discovery demonstrates the potential of applying STATiX to the full XMM–Newton archive.

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Generalized total variation denoising via augmented Lagrangian cycle spinning with Haar wavelets

Cited by 5 publications

References 12 publications

Uniqueness of DRS as the 2 operator resolvent-splitting and impossibility of 3 operator resolvent-splitting

Uniqueness of DRS as the 2 operator resolvent-splitting and impossibility of 3 operator resolvent-splitting

An Auxiliary Variable Method for Markov Chain Monte Carlo Algorithms in High Dimension

The STATiX pipeline for the detection of X-ray transients in three dimensions

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