François Orieux

Bayesian Estimation for Optimized Structured Illumination Microscopy

Structured illumination microscopy is a recent imaging technique that aims at going beyond the classical optical resolution by reconstructing high-resolution (HR) images from low-resolution (LR) images acquired through modulation of the transfer function of the microscope. The classical implementation has a number of drawbacks, such as requiring a large number of images to be acquired and parameters to be manually set in an ad-hoc manner that have, until now, hampered its wide dissemination. Here, we present a new framework based on a Bayesian inverse problem formulation approach that enables the computation of one HR image from a reduced number of LR images and has no specific constraints on the modulation. Moreover, it permits to automatically estimate the optimal reconstruction hyperparameters and to compute an uncertainty bound on the estimated values. We demonstrate through numerical evaluations on simulated data and examples on real microscopy data that our approach represents a decisive advance for a wider use of HR microscopy through structured illumination.

Bayesian estimation of regularization and point spread function parameters for Wiener–Hunt deconvolution

This paper tackles the problem of image deconvolution with joint estimation of point spread function (PSF) parameters and hyperparameters. Within a Bayesian framework, the solution is inferred via a global a posteriori law for unknown parameters and object. The estimate is chosen as the posterior mean, numerically calculated by means of a Monte Carlo Markov chain algorithm. The estimates are efficiently computed in the Fourier domain, and the effectiveness of the method is shown on simulated examples. Results show precise estimates for PSF parameters and hyperparameters as well as precise image estimates including restoration of high frequencies and spatial details, within a global and coherent approach.

Sampling High-Dimensional Gaussian Distributions for General Linear Inverse Problems

This paper is devoted to the problem of sampling Gaussian distributions in high dimension. Solutions exist for two specific structures of inverse covariance: sparse and circulant. The proposed algorithm is valid in a more general case especially as it emerges in linear inverse problems as well as in some hierarchical or latent Gaussian models. It relies on a perturbation-optimization principle: adequate stochastic perturbation of a criterion and optimization of the perturbed criterion. It is proved that the criterion optimizer is a sample of the target distribution. The main motivation is in inverse problems related to general (nonconvolutive) linear observation models and their solution in a Bayesian framework implemented through sampling algorithms when existing samplers are infeasible. It finds a direct application in myopic/unsupervised inversion methods as well as in some non-Gaussian inversion methods. An illustration focused on hyperparameter estimation for super-resolution method shows the interest and the feasibility of the proposed algorithm.

Data Inversion for Over-Resolved Spectral Imaging in Astronomy

We present an original method for reconstructing a 3-D object having two spatial dimensions and one spectral dimension from data provided by the infrared slit spectrograph on board the Spitzer Space Telescope. During acquisition, the light flux is deformed by a complex process comprising four main elements (the telescope aperture, the slit, the diffraction grating, and optical distortion) before it reaches the 2-D sensor. The originality of this work lies in the physical modeling, in integral form, of this process of data formation in continuous variables. The inversion is also approached with continuous variables in a semi-parametric format decomposing the object into a family of Gaussian functions. The estimate is built in a deterministic regularization framework as the minimizer of a quadratic criterion. These specificities give our method the power to over-resolve. Its performance is illustrated using real and simulated data. We also present a study of the resolution showing a 1.5-fold improvement relative to conventional methods.

Super-resolution in map-making based on a physical instrument model and regularized inversion - Application to SPIRE/Herschel

We investigate super-resolution methods for image reconstruction from data provided by a family of scanning instruments like the Herschel observatory. To do this, we constructed a model of the instrument that faithfully reflects the physical reality, accurately taking the acquisition process into account to explain the data in a reliable manner. The inversion, i.e. the image reconstruction process, is based on a linear approach resulting from a quadratic regularized criterion and numerical optimization tools. The application concerns the reconstruction of maps for the SPIRE instrument of the Herschel observatory. The numerical evaluation uses simulated and real data to compare the standard tool (coaddition) and the proposed method. The inversion approach is capable to restore spatial frequencies over a bandwidth four times that possible with coaddition and thus to correctly show details invisible on standard maps. The approach is also applied to real data with significant improvement in spatial resolution.

Gradient Scan Gibbs Sampler: An Efficient Algorithm for High-Dimensional Gaussian Distributions

This paper deals with Gibbs samplers that include high dimensional conditional Gaussian distributions. It proposes an efficient algorithm that avoids the high dimensional Gaussian sampling and relies on a random excursion along a small set of directions. The algorithm is proved to converge, i.e., the drawn samples are asymptotically distributed according to the target distribution. Our main motivation is in inverse problems related to general linear observation models and their solution in a hierarchical Bayesian framework implemented through sampling algorithms. It finds direct applications in semi-blind/unsupervised methods as well as in some non-Gaussian methods. The paper provides an illustration focused on the unsupervised estimation for super-resolution methods.

Estimating hyperparameters and instrument parameters in regularized inversion. Illustration for SPIRE/Herschel map making.

We describe regularized methods for image reconstruction and focus on the question of hyperparameter and instrument parameter estimation, i.e. unsupervised and myopic problems. We developed a Bayesian framework that is based on the \post density for all unknown quantities, given the observations. This density is explored by a Markov Chain Monte-Carlo sampling technique based on a Gibbs loop and including a Metropolis-Hastings step. The numerical evaluation relies on the SPIRE instrument of the Herschel observatory. Using simulated and real observations, we show that the hyperparameters and instrument parameters are correctly estimated, which opens up many perspectives for imaging in astrophysics.

Deconvolutionwith gaussian blur parameter and hyperparameters estimation

This paper proposes a Bayesian approach for unsupervised image deconvolution when the parameter of the gaussian PSF is unknown. The parameters of the regularization parameters are also unknown and jointly estimated with the other parameters. The solution is found by inferring on a global a posteriori law for unknown object and parameters. The estimate is chosen in the sense of the posterior mean, numerically calculated by means of a Monte-Carlo Markov chain algorithm. The computation is efficiently done in Fourier space and the practicability of the method is shown on simulated examples. Results show high-frequencies restoration in the estimated image with correct estimation of the hyperparameters and instrument parameters.

Convolution kernels for multi-wavelength imaging

Astrophysical images issued from different instruments and/or spectral bands often require to be processed together, either for fitting or comparison purposes. However each image is affected by an instrumental response, also known as PSF, that depends on the characteristics of the instrument as well as the wavelength and the observing strategy. Given the knowledge of the PSF in each band, a straightforward way of processing images is to homogenise them all to a target PSF using convolution kernels, so that they appear as if they had been acquired by the same instrument. We propose an algorithm that generates such PSF-matching kernels, based on Wiener filtering with a tunable regularisation parameter. This method ensures all anisotropic features in the PSFs to be taken into account. We compare our method to existing procedures using measured Herschel/PACS and SPIRE PSFs and simulated JWST/MIRI PSFs. Significant gains up to two orders of magnitude are obtained with respect to the use of kernels computed assuming Gaussian or circularised PSFs.

Gradient scan Gibbs sampler: An efficient high-dimensional sampler application in inverse problems

The paper deals with Gibbs samplers that include high-dimensional conditional Gaussian distributions. It proposes an efficient algorithm that only requires a scalar Gaussian sampling. The algorithm relies on a random excursion along a random direction. It is proved to converge, i.e. the drawn samples are asymptotically under the target distribution. Our original motivation is in unsupervised inverse problems related to general linear observation models and their solution in a hierarchical Bayesian framework implemented through sampling algorithms. The paper provides an illustration focused on 2-D simulations and on the super-resolution problem.