Radio astronomical imaging using aperture synthesis telescopes requires deconvolution of the point spread function as well as calibration of the instrumental characteristics (primary beam) and foreground (ionospheric/atmospheric) effects. These effects vary in time and also across the field of view, resulting in directionally-dependent (DD), time-varying gains. The primary beam will deviate from the theoretical estimate in real cases at levels that will limit the dynamic range of images if left uncorrected. Ionospheric electron density variations cause time and position variable refraction of sources. At low frequencies and sufficiently high dynamic range this will also defocus the images producing error patterns that vary with position and also with frequency due to the chromatic aberration of synthesis telescopes. Superposition of such residual sidelobes can lead to spurious spectral signals. Field-based ionospheric calibration as well as "peeling" calibration of strong sources leads to images with higher dynamic range and lower spurious signals but will be limited by sensitivity on the necessary short-time scales. The results are improved images although some artifacts remain.
We conducted radio interferometric observations of six pulsars at 610 MHz using the Giant Metrewave Radio Telescope (GMRT). All these objects were claimed or suspected to be the gigahertz-peaked spectra (GPS) pulsars. For a half of the sources in our sample the interferometric imaging provides the only means to estimate their flux at 610 MHz due to a strong pulse scatter-broadening. In our case, these pulsars have very high dispersion measure values and we present their spectra containing for the first time low-frequency measurements. The remaining three pulsars were observed at low frequencie…
Interferometric measurements of the radio emission of extensive air showers allow reconstructing cosmic-ray properties. A recent simulation study with an idealised detector promised measurements of the depth of the shower maximum $X_\mathrm{max}$ with an accuracy better than 10$\,$g$\,$cm$^{-2}$. In this contribution, we evaluate the potential of interferometric $X_\mathrm{max}$ measurements of (simulated) inclined air showers with realistically dimensioned, sparse antenna arrays. We account for imperfect time synchronisation between individual antennas and study its inter-dependency with the …
Radio interferometry is a powerful technique for astronomical imaging. The theory of Compressed Sensing (CS) has been applied recently to the ill-posed inverse problem of recovering images from the measurements taken by radio interferometric telescopes. We review novel CS radio interferometric imaging techniques, both at the level of acquisition and reconstruction, and discuss their superior performance relative to traditional approaches. In order to remain as close to the theory of CS as possible, these techniques necessarily consider idealised interferometric configurations. To realise the e…
We present an interferometric technique for the reconstruction of ultra-wide band impulsive signals from point sources. This highly sensitive method was developed for the search for ultra-high energy neutrinos with the ANITA experiment but is fully generalizable to any antenna array detecting radio impulsive events. Applications of the interferometric method include event reconstruction, thermal noise and anthropogenic background rejection, and solar imaging for calibrations. We illustrate this technique with applications from the analysis of the ANITA-I and ANITA-II data in the 200-1200 MHz b…
For the next generation of radio interferometric telescopes it is of paramount importance to incorporate wide field-of-view (WFOV) considerations in interferometric imaging, otherwise the fidelity of reconstructed images will suffer greatly. We extend compressed sensing techniques for interferometric imaging to a WFOV and recover images in the spherical coordinate space in which they naturally live, eliminating any distorting projection. The effectiveness of the spread spectrum phenomenon, highlighted recently by one of the authors, is enhanced when going to a WFOV, while sparsity is promoted …
Radio astronomy is transitioning to a big-data era due to the emerging generation of radio interferometric (RI) telescopes, such as the Square Kilometre Array (SKA), which will acquire massive volumes of data. In this article we review methods proposed recently to resolve the ill-posed inverse problem of imaging the raw visibilities acquired by RI telescopes in the big-data scenario. We focus on the recently proposed online reconstruction method [4] and the considerable savings in data storage requirements and computational cost that it yields.
The celebrated CLEAN algorithm has been the cornerstone of deconvolution algorithms in radio interferometry almost since its conception in the 1970s. For all its faults, CLEAN is remarkably fast, robust to calibration artefacts and in its ability to model point sources. We demonstrate how the same assumptions that afford CLEAN its speed can be used to accelerate more sophisticated deconvolution algorithms.
We study the impact of the spread spectrum effect in radio interferometry on the quality of image reconstruction. This spread spectrum effect will be induced by the wide field-of-view of forthcoming radio interferometric telescopes. The resulting chirp modulation improves the quality of reconstructed interferometric images by increasing the incoherence of the measurement and sparsity dictionaries. We extend previous studies of this effect to consider the more realistic setting where the chirp modulation varies for each visibility measurement made by the telescope. In these first preliminary re…
Next-generation radio interferometers, such as the Square Kilometre Array (SKA), will revolutionise our understanding of the universe through their unprecedented sensitivity and resolution. However, standard methods in radio interferometry produce reconstructed interferometric images that are limited in quality and they are not scalable for big data. In this work we apply and evaluate alternative interferometric reconstruction methods that make use of state-of-the-art sparse image reconstruction algorithms motivated by compressive sensing, which have been implemented in the PURIFY software pac…
The main objective of this work is to accelerate the Maximum-Likelihood (ML) estimation procedure in radio interferometric calibration. We introduce the OS-LS and the OS-SAGE radio interferometric calibration methods, as a combination of the Ordered-Subsets (OS) method with the Least-Squares (LS) and Space Alternating Generalized Expectation maximization (SAGE) calibration techniques, respectively. The OS algorithm speeds up the ML estimation and achieves nearly the same level of accuracy of solutions as the one obtained by the non-OS methods. We apply the OS-LS and OS-SAGE calibration methods…
We describe a flagging and calibration pipeline intended for making quick look images from GMRT data. The package identifies and flags corrupted visibilities, computes calibration solutions and interpolates these onto the target source. These flagged calibrated visibilities can be directly imaged using any standard imaging package. The pipeline is written in "C" with the most compute intensive algorithms being parallelized using OpenMP.
Object detection in astronomical images, generically referred to as source finding, is often performed before the object characterisation stage in astrophysical processing work flows. In radio astronomy, source finding has historically been performed by bespoke off-line systems; however, modern data acquisition systems as well as those proposed for upcoming observatories such as the Square Kilometre Array (SKA), will make this approach unfeasible. One area where a change of approach is particularly necessary is in the design of fast imaging systems for transient studies. This paper presents a …
Radio interferometric gain calibration can be biased by incomplete sky models and radio frequency interference, resulting in calibration artefacts that can restrict the dynamic range of the resulting images. It has been suggested that calibration algorithms employing heavy-tailed likelihood functions are less susceptible to this due to their robustness against outliers in the data. We present an algorithm based on a Student's t-distribution which leverages the framework of complex optimisation and Wirtinger calculus for efficient and robust interferometric gain calibration. We integrate this a…
The spatial-frequency coverage of a radio interferometer is increased by combining samples acquired at different times and observing frequencies. However, astrophysical sources often contain complicated spatial structure that varies within the time-range of an observation, or the bandwidth of the receiver being used, or both. Image reconstruction algorithms can been designed to model time and frequency variability in addition to the average intensity distribution, and provide an improvement over traditional methods that ignore all variability. This paper describes an algorithm designed for suc…
Solution intervals are often used to improve the signal-to-noise ratio during radio interferometric gain calibration. This work investigates how factors such as the noise level, intrinsic gain variability, degree of model incompleteness, and the presence of radio frequency interference impact the selection of solution intervals for calibration. We perform different interferometric simulations to demonstrate how these factors, in combination with the choice of solution intervals, affect calibration and imaging outputs and discuss practical guidelines for choosing optimal solution intervals. Fur…
The concept of a recently proposed small-scale interferometric optical imaging device, an instrument known as the Segmented Planar Imaging Detector for Electro-optical Reconnaissance (SPIDER), is of great interest for its possible applications in astronomy and space science. Due to low weight, low power consumption, and high resolution, the SPIDER telescope could replace the large space telescopes that exist today. Unlike traditional optical interferometry the SPIDER accurately retrieves both phase and amplitude information, making the measurement process analogous to a radio interferometer. S…
This paper introduces an amendment to radio interferometric calibration of sources below the noise level. The main idea is to employ the information of the stronger sources' measured signals as a plug-in criterion to solve for the weaker ones. For this purpose, we construct a number of source clusters, with centroids mainly near the strongest sources, assuming that the signals of the sources belonging to a single cluster are corrupted by almost the same errors. Due to this characteristic of clusters, each cluster is calibrated as a single source, using all the coherencies of its sources simult…
Interferometric measurements with arrays of radio antennas are a powerful and widely used technique in astronomy. Recently, this technique has been revisited for the reconstruction of extensive air showers [1]. This radio-interferometric technique exploits the coherence in the radio emission emitted by billions of secondary shower particles to reconstruct the shower parameters, in particular the shower axis and depth of the shower maximum $X_\mathrm{max}$. The accuracy previously demonstrated on simulations with an idealized detector is very promising. In this article we evaluate the potential…
For a ground-based radio interferometer observing at low frequencies, the ionosphere causes propagation delays and refraction of cosmic radio waves which result in phase errors in the received signal. These phase errors can be corrected using a calibration method that assumes a two-dimensional phase screen at a fixed altitude above the surface of the Earth, known as the thin-layer model. Here we investigate the validity of the thin-layer model and provide a simple equation with which users can check when this approximation can be applied to observations for varying time of day, zenith angle, i…
CLEAN algorithms are a class of deconvolution solvers which are widely used to remove the effect of the telescope Point Spread Function (PSF). Loop gain is one important parameter in CLEAN algorithms. Currently the parameter is fixed during deconvolution, which restricts the performance of CLEAN algorithms. In this paper, we propose a new deconvolution algorithm with an adaptive loop gain scheme, which is referred to as the adaptive-loop-gain adaptivescale CLEAN (Algas-Clean) algorithm. The test results show that the new algorithm can give a more accurate model with faster convergence.
In recent works, sparse models and convex optimization techniques have been applied to radio-interferometric (RI) imaging showing the potential to outperform state-of-the-art imaging algorithms in the field. In this talk, I will review our latest contributions in RI imaging, which leverage the versatility of convex optimization to both handle realistic continuous visibilities and offer a highly parallelizable structure paving the way to high-dimensional data scalability. Firstly, I will review our recently proposed average sparsity approach, SARA, which relies on the observation that natural i…
In recent works, compressed sensing (CS) and convex optimization techniques have been applied to radio-interferometric imaging showing the potential to outperform state-of-the-art imaging algorithms in the field. We review our latest contributions, which leverage the versatility of convex optimization to both handle realistic continuous visibilities and offer a highly parallelizable structure paving the way to significant acceleration of the reconstruction and high-dimensional data scalability. The new algorithmic structure promoted in a new software PURIFY (beta version) relies on the simulta…
Radio interferometers observe the Fourier space of the sky, at locations determined by the array geometry. Before a real space image is constructed by a Fourier transform, the data is weighted to improve the quality of reconstruction. Two criteria for calculation of weights are maximizing sensitivity and minimizing point spread function (PSF) sidelobe levels. In this paper, we propose a novel weighting scheme suitable for ultra deep imaging experiments. The proposed weighting scheme is used to maximize sensitivity while minimizing PSF sidelobe variation across frequency and multiple epochs. We…
Next generation radio telescopes, like the Square Kilometre Array, will acquire an unprecedented amount of data for radio astronomy. The development of fast, parallelisable or distributed algorithms for handling such large-scale data sets is of prime importance. Motivated by this, we investigate herein a convex optimisation algorithmic structure, based on primal-dual forward-backward iterations, for solving the radio interferometric imaging problem. It can encompass any convex prior of interest. It allows for the distributed processing of the measured data and introduces further flexibility by…
We propose a novel algorithm for image reconstruction in radio interferometry. The ill-posed inverse problem associated with the incomplete Fourier sampling identified by the visibility measurements is regularized by the assumption of average signal sparsity over representations in multiple wavelet bases. The algorithm, defined in the versatile framework of convex optimization, is dubbed Sparsity Averaging Reweighted Analysis (SARA). We show through simulations that the proposed approach outperforms state-of-the-art imaging methods in the field, which are based on the assumption of signal spar…
argosim: a Python package for radio interferometric simulations
Université Paris-Saclay, Université Paris Cité, CEA, CNRS, AIM（巴黎萨克雷大学，巴黎西岱大学，法国原子能和替代能源委员会，法国国家科学研究中心，天体物理学与仪器方法实验室）；Institutes of Computer Science and Astrophysics, Foundation for Research and Technology Hellas (FORTH)（计算机科学与天体物理研究所，希腊研究与技术基金会）；Max-Planck-Institut für Radioastronomie（马克斯·普朗克射电天文研究所）
Compressed sensing theory is slowly making its way to solve more and more astronomical inverse problems. We address here the application of sparse representations, convex optimization and proximal theory to radio interferometric imaging. First, we expose the theory behind interferometric imaging, sparse representations and convex optimization, and second, we illustrate their application with numerical tests with SASIR, an implementation of the FISTA, a Forward-Backward splitting algorithm hosted in a LOFAR imager. Various tests have been conducted in Garsden et al., 2015. The main results are:…
Imaging by aperture synthesis from interferometric data is a well-known, but is a strong ill-posed inverse problem. Strong and faint radio sources can be imaged unambiguously using time and frequency integration to gather more Fourier samples of the sky. However, these imagers assumes a steady sky and the complexity of the problem increases when transients radio sources are also present in the data. Hopefully, in the context of transient imaging, the spatial and temporal information are separable which enable extension of an imager fit for a steady sky. We introduce independent spatial and tem…
Residual calibration errors are difficult to predict in interferometric radio polarimetry because they depend on the employed observational calibration strategy, encompassing the Stokes vector of the calibrator and parallactic angle coverage. This work presents analytic derivations and simulations that enable examination of residual on-axis instrumental leakage and position angle errors for a suite of calibration strategies. The focus is on arrays comprising alt-az antennas with common feeds over which parallactic angle is approximately uniform. The results indicate that calibration schemes re…
The data reduction procedure for radio interferometers can be viewed as a combined calibration and imaging problem. We present an algorithm that unifies cross-calibration, self-calibration, and imaging. Being a Bayesian method, that algorithm does not only calculate an estimate of the sky brightness distribution, but also provides an estimate of the joint uncertainty which entails both the uncertainty of the calibration and the one of the actual observation. The algorithm is formulated in the language of information field theory and uses Metric Gaussian Variational Inference (MGVI) as the unde…
Radio interferometric imaging aims to estimate an unknown sky intensity image from degraded observations, acquired through an antenna array. In the theoretical case of a perfectly calibrated array, it has been shown that solving the corresponding imaging problem by iterative algorithms based on convex optimization and compressive sensing theory can be competitive with classical algorithms such as CLEAN. However, in practice, antenna-based gains are unknown and have to be calibrated. Future radio telescopes, such as the SKA, aim at improving imaging resolution and sensitivity by orders of magni…
We describe and demonstrate a technique for correcting direction dependent artifacts due to asymmetries in antenna patterns and differences among antennas used in radio interoferometric imaging. The technique can correct images in all Stokes parameters I, Q, U and V and is shown with simulated data to reduce the level of artifacts to near the level of those from the basic imaging technique. The demonstrations use simulations of a mixed array of 13.5 and 15 m antennas with asymmetric patterns. The flux densities and spectral indices of the sources in a high dynamic range realistic simulated sky…
Recent developments in optimization theory have extended some traditional algorithms for least-squares optimization of real-valued functions (Gauss-Newton, Levenberg-Marquardt, etc.) into the domain of complex functions of a complex variable. This employs a formalism called the Wirtinger derivative, and derives a full-complex Jacobian counterpart to the conventional real Jacobian. We apply these developments to the problem of radio interferometric gain calibration, and show how the general complex Jacobian formalism, when combined with conventional optimization approaches, yields a whole new f…
(arXiv abridged abstract) The current years are seeing huge developments of radio telescopes and a tremendous increase of their capabilities. Such systems make mandatory the design of more sophisticated techniques not only for transporting, storing and processing this new generation of radio interferometric data, but also for restoring the astrophysical information contained in such data. In this paper we present a new radio deconvolution algorithm named MORESANE and its application to fully realistic simulated data of MeerKAT, one of the SKA precursors. This method has been designed for the d…
In order to cope with the increased data volumes generated by modern radio interferometers such as LOFAR (Low Frequency Array) or SKA (Square Kilometre Array), fast and efficient calibration algorithms are essential. Traditional radio interferometric calibration is performed using nonlinear optimization techniques such as the Levenberg-Marquardt algorithm in Euclidean space. In this paper, we reformulate radio interferometric calibration as a nonlinear optimization problem on a Riemannian manifold. The reformulated calibration problem is solved using the Riemannian trust-region method. We show…
It has recently been shown that radio interferometric gain calibration can be expressed succinctly in the language of complex optimisation. In addition to providing an elegant framework for further development, it exposes properties of the calibration problem which can be exploited to accelerate traditional non-linear least squares solvers such as Gauss-Newton and Levenberg-Marquardt. We extend existing derivations to chains of Jones terms: products of several gains which model different aberrant effects. In doing so, we find that the useful properties found in the single term case still hold.…
Next-generation radio arrays, including the Square Kilometre Array (SKA) and its pathfinders, will open up new avenues for exciting transient science at radio wavelengths. Their innovative designs, comprising a large number of small elements, pose several challenges in digital processing and optimal observing strategies. The Giant Metrewave Radio Telescope (GMRT) presents an excellent test-bed for developing and validating suitable observing modes and strategies for transient experiments with future arrays. Here we describe the first phase of the ongoing development of a transient detection sy…
We report on a broader evaluation of statistical bootstrap resampling methods as a tool for pixel-level calibration and imaging fidelity assessment in radio interferometry. Pixel-level imaging fidelity assessment is a challenging problem, important for the value it holds in robust scientific interpretation of interferometric images, enhancement of automated pipeline reduction systems needed to broaden the user community for these instruments, and understanding leadingedge direction-dependent calibration and imaging challenges for future telescopes such as the Square Kilometer Array. This new c…
Radio interferometric data are used to estimate the sky brightness distributions in radio frequencies. Here we focus on estimators of the large-scale structure and the power spectrum of the sky brightness distribution inferred from radio interferometric observations and assess their efficacy using simulated observations of the model sky. We find that while the large-scale distribution can be unbiasedly estimated from the reconstructed image from the interferometric data, estimates of the power spectrum of the intensity fluctuations calculated from the image are generally biased. The bias is mo…
Transient radio signals of astrophysical origin present an avenue for studying the dynamic universe. With the next generation of radio interferometers being planned and built, there is great potential for detecting and studying large samples of radio transients. Currently-used image-based techniques for detecting radio sources have not been demonstrated to be optimal, and there is a need for development of more sophisticated algorithms, and methodology for comparing different detection techniques. A visibility-space detector benefits from our good understanding of visibility-space noise proper…
The analysis of astronomical interferometric data is often performed on the images obtained after deconvolution of the interferometer's point spread function (PSF). This strategy can be understood (especially for cases of sparse arrays) as fitting models to models, since the deconvolved images are already non-unique model representations of the actual data (i.e., the visibilities). Indeed, the interferometric images may be affected by visibility gridding, weighting schemes (e.g., natural vs. uniform), and the particulars of the (non-linear) deconvolution algorithms. Fitting models to the direc…
Radio frequency interference (RFI) is the principal factor limiting the sensitivities of radio telescopes, particularly at frequencies below 1 GHz. I present a conceptually new approach to mitigation of RFI in interferometric data. This has been used to develop a software tool (RfiX) to remove RFI from observations using the Giant Metrewave Radio Telescope, India. However, the concept can be used to excise RFI in any interferometer. Briefly, the fringe-stopped correlator output of an interferometer baseline oscillates with the fringe-stop period in the presence of RFI. RfiX works by identifyin…
Fast radio bursts remain one of the most enigmatic astrophysical sources. Observations have significantly progressed over the last few years, thanks to the capabilities of new radio telescopes and the refurbishment of existing ones. Here we describe the upgrade of the Northern Cross radio telescope, operating in the 400-416~MHz frequency band, with the ultimate goal of turning the array into a dedicated instrument to survey the sky for fast radio bursts. We present test observations of the pulsar B0329+54 to characterize the system performance and forecast detectability. Observations with the …
Optimal transport regularized dynamic radio interferometric reconstruction
University of Cambridge（剑桥大学）；Kavli Institute for Cosmology, University of Cambridge（卡弗里宇宙学研究所）；Yale University（耶鲁大学）；National Astronomical Observatory of Japan（日本国立天文台）；Instituto de Astrofísica de Andalucía-CSIC（安达卢西亚天体物理研究所）；Mizusawa VLBI Observatory, National Astronomical Observatory of Japan（水泽VLBI观测站）；The Institute of Statistical Mathematics（数理统计学研究所）；The Graduate University for Advanced Studies (SOKENDAI)（综合研究大学院大学）；The University of Tokyo（东京大学）；Heriot-Watt University（赫瑞瓦特大学）；Massachusetts Institute of Technology Haystack Observatory（麻省理工学院海斯塔克天文台）
The short-spacing problem describes the inherent inability of radio-interferometric arrays to measure the integrated flux and structure of diffuse emission associated with extended sources. New interferometric arrays, such as SKA, require solutions to efficiently combine interferometer and single-dish data. We present a new and open source approach for merging single-dish and cleaned interferometric data sets requiring a minimum of data manipulation while offering a rigid flux determination and full high angular resolution. Our approach combines single-dish and cleaned interferometric data in …
We present a new method for interferometric imaging that is ideal for the large fields of view and compact arrays common in 21 cm cosmology. We first demonstrate the method with simulations for two very different low frequency interferometers, the Murchison Widefield Array (MWA) and the MIT Epoch of Reionization (MITEoR) Experiment. We then apply the method to the MITEoR data set collected in July 2013 to obtain the first northern sky map from 128 MHz to 175 MHz at about 2 degree resolution, and find an overall spectral index of -2.73+/-0.11. The success of this imaging method bodes well for u…
We introduce a new pipeline for analyzing and mitigating radio frequency interference (RFI), which we call Sky-Subtracted Incoherent Noise Spectra (SSINS). SSINS is designed to identify and remove faint RFI below the single baseline thermal noise by employing a frequency-matched detection algorithm on baseline-averaged amplitudes of time-differenced visibilities. We demonstrate the capabilities of SSINS using the Murchison Widefield Array (MWA) in Western Australia. We successfully image aircraft flying over the array via digital television (DTV) reflection detected using SSINS and summarize a…
Distributed radio interferometric calibration based on consensus optimization has been shown to improve the estimation of systematic errors in radio astronomical observations. The intrinsic continuity of systematic errors across frequency is used by a consensus polynomial to penalize traditional calibration. Consensus is achieved via the use of alternating direction method of multipliers (ADMM) algorithm. In this paper, we extend the existing distributed calibration algorithms to use ADMM with an adaptive penalty parameter update. Compared to a fixed penalty, its adaptive update has been shown…
Calibration of a typical radio interferometric array yields thousands of parameters as solutions. These solutions contain valuable information about the systematic errors in the data (ionosphere and beam shape). This information could be reused in calibration to improve the accuracy and also can be fed into imaging to improve the fidelity. We propose a distributed optimization strategy to construct models for the systematic errors in the data using the calibration solutions. We formulate this as an elastic net regularized distributed optimization problem which we solve using the alternating di…
LOPES was the radio extension of the KASCADE-Grande particle-detector array consisting of up to 30 antennas measuring the radio emission of cosmic-ray air showers between 40 and 80 MHz with an energy threshold of around 100 PeV. Even with the external trigger by the particle detectors, the separation of the air-shower signal from the radio background was difficult in the noisy environment of the Karlsruhe Institute of Technology. For the typical event this was only possible because of the digital, interferometric beamforming technique pioneered by LOPES for cosmic-ray detection. Using this tec…
The new generation of low radio-frequency interferometric arrays have enabled the imaging of the solar corona at high spectro-temporal resolutions and sensitivity. In this article we introduce and implement a formalism to generate flux density and brightness temperature ($T_{\mathrm B}$) maps from such images, using independently obtained disc-integrated solar flux density dynamic spectra. These images collectively generate a 4D data cube, with axes spanning angular coordinates ($θ,ϕ$), frequency ($ν$) and time ($t$). This 4D data cube is the most informative data product which can be generate…
The paper present correlation of radio signal with air shower parameters: shower energy E0 and depth of maximum Xmax. It is shown that from radio emission measurements of air showers one can obtain individual showers parameters and mass composition of cosmic rays. We also derived generalized formula for calculating energy of the air showers.
Uncertainty quantification is a critical missing component in radio interferometric imaging that will only become increasingly important as the big-data era of radio interferometry emerges. Since radio interferometric imaging requires solving a high-dimensional, ill-posed inverse problem, uncertainty quantification is difficult but also critical to the accurate scientific interpretation of radio observations. Statistical sampling approaches to perform Bayesian inference, like Markov Chain Monte Carlo (MCMC) sampling, can in principle recover the full posterior distribution of the image, from w…
