Introduction to linear and nonlinear programming

Image deblurring is an important topic in imaging science. In this review, we consider together fluorescence microscopy and optical/infrared astronomy because of two common features: in both cases the imaging system can be described, with a sufficiently good approximation, by a convolution operator, whose kernel is the so-called point-spread function (PSF); moreover, the data are affected by photon noise, described by a Poisson process. This statistical property of the noise, that is common also to emission tomography, is the basis of maximum likelihood and Bayesian approaches introduced in the mid eighties. From then on, a huge amount of literature has been produced on these topics. This review is a tutorial and a review of a relevant part of this literature, including some of our previous contributions. We discuss the mathematical modeling of the process of image formation and detection, and we introduce the so-called Bayesian paradigm that provides the basis of the statistical treatment of the problem. Next, we describe and discuss the most frequently used algorithms as well as other approaches based on a different description of the Poisson noise. We conclude with a review of other topics related to image deblurring such as boundary effect correction, space-variant PSFs, super-resolution, blind deconvolution and multiple-image deconvolution.

https://iopscience.iop.org/article/10.1088/0266-5611/25/12/123006/pdf

Image deblurring with Poisson data: from cells to galaxies

We present a method for the analysis of time-resolved spectroscopic data following first-order kinetics. The time traces at all the available spectroscopic channels (e.g., wavelength or wavenumber) are inverse Laplace transformed. The transformation is stabilized by the maximum entropy method generalized for solutions without sign-restriction. In this way, time-resolved spectra can be converted to lifetime-resolved spectra, where bands appear at coordinates corresponding to their spectroscopic maxima and time constant of appearance (negative amplitude) or disappearance (positive amplitude). From the lifetime-resolved spectra, the number of exponentially decaying components, their time constants, and their decay-associated spectra are readily available. Moreover, since bands are spread in two dimensions extra band-resolution is possible. We named this method of transforming time-resolved spectra into lifetime-resolved spectra multi-spectroscopic channel maximum entropy inversion of the Laplace transform (M-MaxEnt-iLT). The basis of M-MaxEnt-iLT is presented in detail and its properties and limitations are thoroughly discussed. We also show how the combination of M-MaxEnt-iLT with spectral smoothing or deconvolution can improve the appearance and/or band resolution of the obtained lifetime-resolved spectra.

Transformation of time-resolved spectra to lifetime-resolved spectra by maximum entropy inversion of the laplace transform.

An adaptive algorithm for image restoration using combined penalty functions

This article first provides a historical and detailed introduction to the image formation models for diluted pupils array and their densified versions called hypertelescopes . We propose in particular an original derivation showing that densification using a periscopic setting like in Michelson’s 20 −  foot interferometer, or using inverted Galilean telescopes are fully equivalent. After a review based on previous reference studies (Tallon & Tallon-Bosc 1992; Labeyrie 1996; Aime 2008 and Aime et al. 2012), the introductory part ends with a tutorial section for simulating optical interferometric images produced by cophased arrays. We illustrate in details how the optical image formation model can be used to simulate hypertelescopes images, including sampling issues and their effects on the observed images. In a second part of the article, we address the issue of restoring hypertelescope images and present numerical illustrations obtained for classical (constrained Maximum Likelihood) methods. We also provide a detailed survey of more recent deconvolution methods based on sparse representations and of their spread in interferometric image reconstruction. The last part of the article is dedicated to two original and numerical studies. The first study shows by Monte Carlo simulations that the restoration quality achieved by constrained ML methods applied to photon limited images obtained from a diluted array on a square grid, or from a densified array (without spectral aliasing) on a grid, are essentially equivalent. The second study shows that it is possible to recover in hypertelescopes images quasi point sources that are not only far outside the clean field, but also superimposed on the replicas of other objects. This is true at least for the considered pupil array and in the limit of vanishing noise.

Formation, Simulation and Restoration of Hypertelescopes Images

Ringing reduction in image restoration algorithms using a constraint on the inferior bound of the solution

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Ringing reduction in image restoration algorithms using a constraint on the inferior bound of the solution