Lucy Method Research Articles

Joint Deconvolution of Astronomical Images in the Presence of Poisson Noise

We present a new framework for joint likelihood deconvolution (Jolideco) of a set of astronomical observations of the same sky region in the presence of Poisson noise. The observations may be obtained from different instruments with different resolution, and different point-spread functions (PSFs). Jolideco reconstructs a single flux image by optimizing the posterior distribution based on the joint Poisson likelihood of all observations under one of several prior distributions. Most notably, we employ a patch-based image prior that is parameterized via a Gaussian mixture model, which we train on high-signal-to-noise astronomical images, including data from the JWST and the GLEAM radio survey. This prior favors correlation structures among the reconstructed pixel intensities that are characteristic of those observed in the training images. It is, however, not informative for the mean or scale of the reconstruction. By applying the method to simulated data, we show that the combination of multiple observations and the patch-based prior leads to much improved reconstruction quality in many different source scenarios and signal-to-noise regimes. We demonstrate that with the patch prior Jolideco yields superior reconstruction quality relative to alternative standard methods such as the Richardson–Lucy method. We illustrate the results of Jolideco applied to example data from the Chandra X-ray Observatory and the Fermi Gamma-ray Space Telescope. By comparing the measured width of a counts-based and the corresponding Jolideco flux profile of an X-ray filament in SNR 1E 0102.2–7219, we find the deconvolved width of 0.″58 ± 0.″02 to be consistent with the theoretical expectation derived from the known width of the PSF.

Hitomi HXT deconvolution imaging of the Crab Nebula dazzled by the Crab pulsar

Abstract We develop a new deconvolution method to recover the precise Crab Nebula image taken by the Hitomi HXT, suppressing the artifact due to the bright Crab pulsar. We extend the Richardson–Lucy method, introducing two components corresponding to the nebula and pulsar with regularization for smoothness and flux, respectively, and performing simultaneous deconvolution of multi-pulse-phase images. The structures, including the torus and jets, seen in the deconvolved nebula image at the lowest energy band of 3.6–15 keV appear consistent with those identified in the high-resolution Chandra X-ray image. Above 15 keV, we confirm NuSTAR’s findings that the nebula size decreases in higher energy bands. We find that the north-east side of the nebula is fainter in higher energy bands. Our deconvolution method is applicable for any telescope images of faint diffuse objects containing a bright point source.

Joint cluster reconstructions

Context. Galaxy clusters provide a multitude of observational data across wavelengths, and their structure and morphology are of considerable interest in cosmology as well as astrophysics. Aims. We develop a framework that allows the combination of lensing and non-lensing observations in a free-form and mesh-free approach to infer the projected mass distribution of individual galaxy clusters. This method can be used to test common assumptions on the morphology of clusters in parametric models. Methods. We make use of the lensing reconstruction code SAWLENS2, and expand its capabilities by incorporating an estimate of the projected gravitational potential based on X-ray data that are deprojected using the local Richardson–Lucy method and used to infer the Newtonian potential of the cluster. We discuss how potentially arising numerical artefacts can be treated. Results. We demonstrate the feasibility of our method on a simplified mock Navarro–Frenk–White (NFW) halo and on a cluster from a realistic hydrodynamical simulation. We show how the combination of X-ray and weak lensing data can affect a free-form reconstruction, improving the accuracy in the central region in some cases by a factor of two.

Monte-Carlo methods for NLTE spectral synthesis of supernovae

We present JEKYLL, a new code for modelling of supernova (SN) spectra and lightcurves based on Monte-Carlo (MC) techniques for the radiative transfer. The code assumes spherical symmetry, homologous expansion and steady state for the matter, but is otherwise capable of solving the time-dependent radiative transfer problem in non-local-thermodynamic-equilibrium (NLTE). The method used was introduced in a series of papers by Lucy, but the full time-dependent NLTE capabilities of it have never been tested. Here, we have extended the method to include non-thermal excitation and ionization as well as charge-transfer and two-photon processes. Based on earlier work, the non-thermal rates are calculated by solving the Spencer-Fano equation. Using a method previously developed for the SUMO code, macroscopic mixing of the material is taken into account in a statistical sense. To save computational power a diffusion solver is used in the inner region, where the radiation field may be assumed to be thermalized. In addition, a statistical Markov-chain model is used to sample the emission frequency more efficiently, and we introduce a method to control the sampling of the radiation field, which is used to reduce the noise in the radiation field estimators. Except for a description of JEKYLL, we provide comparisons with the ARTIS, SUMO and CMFGEN codes, which show good agreement in the calculated spectra as well as the state of the gas. In particular, the comparison with CMFGEN, which is similar in terms of physics but uses a different technique, shows that the Lucy method does indeed converge in the time-dependent NLTE case. Finally, as an example of the time-dependent NLTE capabilities of JEKYLL, we present a model of a Type IIb SN, taken from a set of models presented and discussed in detail in an accompanying paper. Based on this model we investigate the effects of NLTE, in particular those arising from non-thermal excitation and ionization, and find strong effects even on the bolometric lightcurve. This highlights the need for full NLTE calculations when simulating the spectra and lightcurves of SNe.

Blind deconvolution with principal components analysis for wide-field and small-aperture telescopes

Telescopes with a wide field of view (greater than 1°) and small apertures (less than 2 m) are workhorses for observations such as sky surveys and fast-moving object detection, and play an important role in time-domain astronomy. However, images captured by these telescopes are contaminated by optical system aberrations, atmospheric turbulence, tracking errors and wind shear. To increase the quality of images and maximize their scientific output, we propose a new blind deconvolution algorithm based on statistical properties of the point spread functions (PSFs) of these telescopes. In this new algorithm, we first construct the PSF feature space through principal component analysis, and then classify PSFs from a different position and time using a self-organizing map. According to the classification results, we divide images of the same PSF types and select these PSFs to construct a prior PSF. The prior PSF is then used to restore these images. To investigate the improvement that this algorithm provides for data reduction, we process images of space debris captured by our small-aperture wide-field telescopes. Comparing the reduced results of the original images and the images processed with the standard Richardson–Lucy method, our method shows a promising improvement in astrometry accuracy.

Iterative Methods of Richardson-Lucy-Type for Image Deblurring

Image deconvolution problems with a symmetric point-spread function arise in many areas of science and engineering. These problems often are solved by the Richardson-Lucy method, a nonlinear iterative method. We first show a convergence result for the Richardson-Lucy method. The proof sheds light on why the method may converge slowly. Subsequently, we describe an iterative active set method that imposes the same constraints on the computed solution as the Richardson-Lucy method. Com- puted examples show the latter method to yield better restorations than the Richardson- Lucy method and typically require less computational effort. AMS subject classifications: 65R20, 65R32, 65K05

A fast iterative technique for restoring scanning electron microscope images

This paper proposes a fast new technique for restoring scanning electron microscope images to improve their sharpness. The images with our approach are sharpened by deconvolution with the point spread function modeled as the intensity distribution of the electron beam at the specimen׳s surface. We propose an iterative technique that employs a modified cost function based on the Richardson–Lucy method to achieve faster processing. The empirical results indicate significant improvements in image quality. The proposed approach speeds up deconvolution by about 10–50 times faster than that with the conventional Richardson–Lucy method.

An improved deconvolution method for X-ray coded imaging in inertial confinement fusion

In inertial confinement fusion (ICF), X-ray coded imaging is considered as the most potential means to diagnose the compressed core. The traditional Richardson—Lucy (RL) method has a strong ability to deblur the image where the noise follows the Poisson distribution. However, it always suffers from over-fitting and noise amplification, especially when the signal-to-noise ratio of image is relatively low. In this paper, we propose an improved deconvolution method for X-ray coded imaging. We model the image data as a set of independent Gaussian distributions and derive the iterative solution with a maximum-likelihood scheme. The experimental results on X-ray coded imaging data demonstrate that this method is superior to the RL method in terms of anti-overfitting and noise suppression.

A quantitative evaluation of various deconvolution methods and their applications in the deconvolution of plasma spectra

A quantitative evaluation of various deconvolution methods and their applications in processing plasma emitted spectra was performed. The iterative deconvolution algorithms evaluated here include Jansson's method, Richardson–Lucy's method, the maximum a posteriori method and Gold's method. The evaluation criteria include minimization of the sum of squared errors and the sum of squared relative error of parameters, and their rate of convergence. After comparing deconvolved results using these methods, it was concluded that Jansson's and Gold's methods were able to provide good profiles that are visually close to the original spectra. Additionally, Gold's method generally gives the best results when considering all the criteria above. The applications to the actual plasma spectra obtained from the EAST tokamak with these methods are also presented in this paper. The deconvolution results with Gold's and Jansson's methods show that the effects of instruments can be satisfactorily eliminated and clear spectra are recovered.

Comparison of the Richardson–Lucy method and a classical approach for spectrometer bandpass correction

Bandpass correction in spectrometer measurements using monochromators is often necessary in order to obtain accurate measurement results. The classical approach of spectrometer bandpass correction is based on local polynomial approximations and the use of finite differences. Here we compare this approach with an extension of the Richardson–Lucy method, which is well known in image processing, but has not been applied to spectrum bandpass correction yet. Using an extensive simulation study and a practical example, we demonstrate the potential of the Richardson–Lucy method. In contrast to the classical approach, it is robust with respect to wavelength step size and measurement noise. In almost all cases the Richardson–Lucy method turns out to be superior to the classical approach both in terms of spectrum estimate and its associated uncertainties.

Unfolding the response of a zero-degree magnetic spectrometer from measurements of the [formula omitted] resonance

The magnetic spectrometer FRagment Separator at GSI has been used to investigate the in-medium Δ-resonance excitation in peripheral heavy-ion reactions. The resolving power of this spectrometer makes it possible to disentangle the longitudinal-momentum loss induced by the excitation of the Δ resonance in the projectile residues produced in isobaric charge-exchange collisions. However, beam emittance, electromagnetic interactions of projectile and residual nuclei in the target, and the accuracy of the tracking detectors limit the final resolution. The characterization of the Δ resonance requires then to unfold the measured longitudinal-momentum distribution from the response of the spectrometer. In this work, we use an unfolding procedure based on the Richardson–Lucy method with a regularization technique to optimize the stability of the solution against statistical fluctuations. The method is validated using measurements of isobaric charge-changing collisions with a 136Xe beam at 500AMeV.

Blind deconvolution subject to sparse representation for fluorescence microscopy

Blind deconvolution is an effective fluorescence microscopic image processing technique to improve the quality of degraded digital images resulting from photon counting noise and out-of-focus blur. For solving the severely ill-posed problem of deconvolution, in this paper we propose an alternate minimized blind deconvolution method which considers sparse representation as constraint condition to confine the solution space of traditional Richardson–Lucy method and Gaussian model as initial value of point spread function (PSF). We assume that Poisson noise is dominating during the course of imaging. The maximum-likelihood estimation on a fluorescence image and corresponding point spread function (PSF) is developed. By solving the Euler–Lagrange equation of the total cost function, including the data term obtained by the hypothetical Poisson noise distribution model and the regularized term corresponding to the sparse representation constraint, and using gradient descent method we can get the iterative equations of the original fluorescence image and PSF respectively. Compared with the related blind deconvolution methods, our model shows superior performance in terms of both objective criteria and subjective human vision via processing simulated and real fluorescence microscopic degraded images.

Hybrid simulation of scatter intensity in industrial cone-beam computed tomography

A cone-beam computed tomography (CT) system using a 450 kV X-ray tube has been developed to challenge the three-dimensional imaging of parts of the automotive industry in short acquisition time. Because the probability of detecting scattered photons is high regarding the energy range and the area of detection, a scattering correction becomes mandatory for generating reliable images with enhanced contrast detectability. In this paper, we present a hybrid simulator for the fast and accurate calculation of the scattering intensity distribution. The full acquisition chain, from the generation of a polyenergetic photon beam, its interaction with the scanned object and the energy deposit in the detector is simulated. Object phantoms can be spatially described in form of voxels, mathematical primitives or CAD models. Uncollided radiation is treated with a ray-tracing method and scattered radiation is split into single and multiple scattering. The single scattering is calculated with a deterministic approach accelerated with a forced detection method. The residual noisy signal is subsequently deconvoluted with the iterative Richardson–Lucy method. Finally the multiple scattering is addressed with a coarse Monte Carlo (MC) simulation. The proposed hybrid method has been validated on aluminium phantoms with varying size and object-to-detector distance, and found in good agreement with the MC code Geant4. The acceleration achieved by the hybrid method over the standard MC on a single projection is approximately of three orders of magnitude.

Richardson–Lucy method for decoding x-ray ring code image

The Richardson–Lucy (RL) method of decoding used for x-rays code-aperture imaging (RAM) is proposed in this paper. The Wiener filter method cannot get the ideal result because of a lack of prior knowledge about the signal-noise ratio while the RL method gets a very good decoding result because it is based on the maximal probability estimation of the Poisson noise. We held experiments at the XingGuang II laser facility using an x-ray ring-code-microscope. The Wiener filtering method and the RL method are both used to decode the coded image. Compared with the Wiener filtering method, the RL method gets more ideal results, in good agreement with the pinhole image. This method can also be applied to penumbral imaging and other coded imaging techniques in ICF research.

Iterative methods for the reconstruction of astronomical images with high dynamic range

In most cases astronomical images contain objects with very different intensities such as bright stars combined with faint nebulae. Since the noise is mainly due to photon counting (Poisson noise), the signal-to-noise ratio may be very different in different regions of the image. Moreover, the bright and faint objects have, in general, different angular scales. These features imply that the iterative methods which are most frequently used for the reconstruction of astronomical images, namely the Richardson–Lucy Method (RLM), also known in tomography as Expectation Maximization (EM) method, and the Iterative Space Reconstruction Algorithm (ISRA) do not work well in these cases. Also standard regularization approaches do not provide satisfactory results since a kind of adaptive regularization is required, in the sense that one needs a different regularization for bright and faint objects. In this paper we analyze a number of regularization functionals with this particular kind of adaptivity and we propose a simple modification of RLM and ISRA which takes into account these regularization terms. The preliminary results on a test object are promising.

Power spectrum analysis of the ESO Slice Project galaxy redshift survey

We measure the power spectrum of the galaxy distribution in the ESO Slice Project (ESP) galaxy redshift survey. We develope a technique to describe the survey window function analytically, and then deconvolve it from the measured power spectrum using a variant of the Lucy method. We test the whole deconvolution procedure on ESP mock catalogues drawn from large N–body simulations, and find that it is reliable for recovering the correct amplitude and shape of P(k) at k > 0.065h Mpc 1 . In general, the technique is applicable to any survey composed by a collection of circular fields with arbitrary pattern on the sky, as typical of surveys based on fibre spectrographs. The estimated power spectrum has a well–defined power–law shape k n with n ≃ −2.2 for k ≥ 0.2h Mpc 1 , and a smooth bend to a flatter shape (n ≃ −1.6) for smaller k’s. The smallest wavenumber, where a meaningful reconstruction can be performed (k ∼ 0.06h Mpc 1 ), does not allow us to explore the range of scales where other power spectra seem to show a flattening and hints for a turnover. We also find, by direct comparison of the Fourier transforms, that the estimate of the two–point correlation function �(s) is much less sensitive to the effect of a problematic window function as that of the ESP, than the power spectrum. Comparison to other surveys shows an excellent agreement with estimates from blue–selected surveys. In particular, the ESP power spectrum is virtually indistinguishable from that of the Durham-UKST survey over the common range of k’s, an indirect confirmation of the quality of the deconvolution technique applied.

Observations of Comet P/Faye 1991 XXI with the Planetary Camera of the Hubble Space Telescope

Comet P/Faye 1991 XXI (1991 n) was observed with the planetary camera (PC) of the Hubble Space Telescope (HST) between October 29 and November 21, 1991, when its geocentric distance was in the range 0.62–0.67 AU. The resulting high resolution—a single PC pixel projected to a distance of ∼20 km at the comet—made it possible to clearly discriminate the nucleus and to study the dust coma within ∼100 km from the nucleus. The spherical aberration which affected the early operation of the HST has severely complicated the analysis and image restoration using the Richardson–Lucy method has failed to give satisfactory results. The coma is dominated by the point spread function (PSF) of the nucleus up to a radial distance of ∼100 km and cannot be recovered. Beyond, it is affected by the spherical aberration up to approximately 750 km and was analyzed by comparison with a grid of models convolved with appropriate PSFs. Beyond 750 km, it remains unaffected and can be studied directly. From the outer to the innermost regions, the coma presents an elongated shape which may be explained either by an active source on the nucleus or, more likely, by a projection effect of the dust tail. If this second interpretation is correct, then the temporal evolution of the comet is very slow and smooth, suggesting an extended source of dust on the nucleus. The dust production rate, corrected for the projection effect of the tail, amounts to 125 kg sec−1.

Hubble Space Telescope Medium Deep Survey. 2: Deconvolution of Wide Field Camera field galaxy images in the 13 hour + 43 deg field

We present isophotal profiles of six faint field galaxies from some of the first deep images taken for the Hubble Space Telescope (HST) Medium Deep Survey (MDS). These have redshifts in the range z = 0.126 to 0.402. The images were taken with the Wide Field Camera (WFC) in `parallel mode' and deconvolved with the Lucy method using as the point-spread function nearby stars in the image stack. The WFC deconvolutions have a dynamic range of 16 to 20 dB (4 to 5 mag) and an effective resolution approximately less than 0.2 sec (FWHM). The multiorbit HST images allow us to trace the morphology, light profiles, and color gradients of faint field galaxies down to V approximately equal to 22 to 23 mag at sub-kpc resolution, since the redshift range covered is z = 0.1 to 0.4. The goals of the MDS are to study the sub-kpc scale morphology, light profiles, and color gradients for a large samole of faint field galaxies down to V approximately equal to 23 mag, and to trace the fraction of early to late-type galaxies as function of cosmic time. In this paper we study the brighter MDS galaxies in the 13 hour + 43 deg MDS field in detail, and investigate to what extent model fits with pure exponential disks or a(exp 1/4) bulges are justified at V approximately less than 22 mag. Four of the six field galaxies have light profiles that indicate (small) inner bulges following r(exp 1/4) laws down to 0.2 sec resolution, plus a dominant surrounding exponential disk with little or no color gradients. Two occur in a group at z = 0.401, two are barred spiral galaxies at z = 0.179 and z = 0.302, and two are rather subluminous (and edge-on) disk galaxies at z = 0.126 and z = 0.179. Our deep MDS images can detect galaxies down to V, I approximately less than 25 to 26 mag, and demonstrate the impressive potential of HST--even with its pre-refurbished optics--to resolve morphological details in galaxies at cosmologically significant distances (v approximately less than 23 mag). Since the median redshift of these galaxies is approximately less than 0.4, the HST resolution allows us to study sub kpc size scales at the galaxy, which cannot be done with stable images over wide fields from the best ground-based sites.

Image Deconvolution of Extended Objects: A Comparison of the Inverse Fourier and the Lucy Techniques

We have investigated the relative effectiveness of two deconvolution techniques in restoring spherically aberrated Hubble Space Telescope (HST) images. These images were restricted to extended objects (such as planets) of high signal-to-noise. Our reason for pursuing these studies was that one of the deconvolution methods, the Lucy iterative technique, is generally thought to be far superior to the inverse Fourier method. The application of the Lucy algorithm takes considerably more computing time than the Fourier method and is therefore less desirable if computing resources are limited. The principle deficiency of the Lucy method is that it lacks a general stopping mechanism for the iterations—a condition that the Fourier method is not prone to. For the class of objects described above we find that when properly applied the Fourier technique is completely adequate and equivalent to results from the application of the Lucy algorithm. We tested these methods on the basis of many different starting criteria and found that the final restored image was independent of most of the test parameters. Similar tests applied to a Voyager image of Jupiter (that had been blurred to HST resolution and then had Gaussian and readout noise added to it) provided further verification of these results.

Deprojection of Planetary Nebula Images

We present first results of the deprojection of some optical and radio images of planetary nebulae using an iterative technique based on the Lucy method, as discussed in Leahy (1991). This approach allows a greater range of possible geometries to be investigated and is less sensitive to the effects of noise compared with the direct matrix-inversion method which can be used for the spherical case. We assume that the emissivity function can be represented as