MEM_GE: a new maximum entropy method for image reconstruction from solar X-ray visibilities
Paolo Massa, Richard Schwartz, A Kim Tolbert, Anna Maria Massone, Brian R Dennis, Michele Piana, Federico Benvenuto
DDraft version February 20, 2020
Typeset using L A TEX preprint style in AASTeX63
MEM GE: a new maximum entropy method for image reconstructionfrom solar X-ray visibilities
Paolo Massa, Richard Schwartz, A Kim Tolbert, Anna Maria Massone,
1, 3
Brian R Dennis, Michele Piana,
1, 3 and Federico Benvenuto Dipartimento di Matematica, Universit`a di Genova, via Dodecaneso 35 16146 Genova, Italy NASA Goddard Space Flight Center, Greenbelt (MD), USA CNR - SPIN Genova, via Dodecaneso 33 16146 Genova, Italy
ABSTRACTMaximum Entropy is an image reconstruction method conceived to image a sparselyoccupied field of view and therefore particularly appropriate to achieve super-resolutioneffects. Although widely used in image deconvolution, this method has been formulatedin radio astronomy for the analysis of observations in the spatial frequency domain,and an Interactive Data Language (IDL) code has been implemented for image recon-struction from solar X-ray Fourier data. However, this code relies on a non-convexformulation of the constrained optimization problem addressed by the Maximum En-tropy approach and this sometimes results in unreliable reconstructions characterizedby unphysical shrinking effects.This paper introduces a new approach to Maximum Entropy based on the constrainedminimization of a convex functional. In the case of observations recorded by the
ReuvenRamaty High Energy Solar Spectroscopic Imager (RHESSI) , the resulting code providesthe same super-resolution effects of the previous algorithm, while working properlyalso when that code produces unphysical reconstructions. Results are also providedof testing the algorithm with synthetic data simulating observations of the
Spectrome-ter/Telescope for Imaging X-rays (STIX) in Solar Orbiter . The new code is availablein the
HESSI folder of the
Solar SoftWare (SSW) tree. [email protected]@[email protected]@[email protected]@[email protected] a r X i v : . [ a s t r o - ph . S R ] F e b Keywords:
Sun: flares, Sun: X-rays, γ -rays; techniques: image processing; methods:numerical INTRODUCTIONSolar SoftWare (SSW) contains a large collection of computational procedures for the reconstructionof images from the X-ray data recorded by the
Reuven Ramaty High Energy Solar SpectroscopicImager (RHESSI) (Lin et al. 2002) in the time interval between February 2002 to August 2018 - seeDennis & Tolbert (2019) for a recent evaluation of the performance of the different available methods.Some of these methods apply directly to
RHESSI counts while others have been conceived to process
RHESSI visibilities, i.e. calibrated samples of the Fourier transform of the incoming photon flux,generated via a data stacking process. Among count-based methods, SSW includes Back Projection(Hurford et al. 2002), Clean (H¨ogbom 1974), Forward Fit (Aschwanden et al. 2002), Pixon (Metcalfet al. 1996), and Expectation Maximization (Benvenuto et al. 2013); among visibility-based methods,SSW includes MEM NJIT (Bong et al. 2006; Schmahl et al. 2007), a Maximum Entropy method;VIS FWDFIT (Schmahl et al. 2007), which selects pre-defined source shapes based on their bestfitting of visibilities; uv smooth (Massone et al. 2009), an interpolation/extrapolation method in theFourier domain; VIS CS (Felix et al. 2017), a catalogue-based compressed sensing algorithm; andVIS WV (Duval-Poo et al. 2018), a wavelet-based compressed sensing algorithm. Although each oneof these algorithms combines specific values with applicability limitations and specific flaws, a criticalcomparison of the maps of a given flaring event obtained by the application of all (or most) of thesealgorithms provides a good picture of what a reliable image of the event could be.In particular, MEM NJIT provides reconstructions characterized by a notable degree of reliability.The capability of fitting the experimental observations is generally satisfactory both in terms ofcomparison between the predicted and experimental visibility profiles with respect to each
RHESSI detector; and in terms of the reduced χ values computed considering either all detectors or justthe detectors providing the observations. Further, the algorithm is robust with respect to the levelof noise affecting the observations. Moreover, the computational time is among the smallest in theSSW scenario and allows reconstruction within a few seconds. Finally, MEM NJIT is characterizedby super-resolution properties (Cornwell & Evans 1985).However, MEM NJIT sometimes produces images with multiple unrealistically small sources. Theorigin of this problem is not totally clear but is related to the minimization technique and thesetting of the convergence criteria. MEM NJIT addresses a constrained maximization of the entropyfunction, which turns into an optimization problem with two penalty terms, the chi-squared functionrelating the measured and predicted visibilities and a term that ensures the conservation of the overallflux. The optimization of these terms is computationally difficult especially since the optimizationfunctional is not convex, which implies that the numerical schemes may suffer convergence issues.Therefore, the optimization procedure may continue on past the physical meaningful solution tofind a solution involving multiple bright points (see Figure 1, where the reconstructed images havedimension 101 × . .
03) to data-dependent higher values, but this typically results in a worsefitting of the observations and a less accurate estimate of the emission flux and, in general, impedesthe use of MEM NJIT in an automatic pipeline for data processing.The present paper illustrates a new algorithm for the constrained maximization of the image entropywhich, differently than MEM NJIT, relies on the optimization of a convex functional. Indeed, thisapproach aims at the minimization of χ under the constraint of maximum entropy, which leads tothe formulation of a convex functional characterized by a single penalty term and, therefore a singleregularization parameter that is fixed a priori . The minimization of this functional is performediteratively and in an alternate fashion (Combettes & Pesquet 2011): for each iteration, a gradientstep minimizes the χ functional and then a proximal step minimizes the negative entropy andprojects the result of the first step onto the hyper-surface of vectors with positive components andconstant flux.The method is implemented in Solar SoftWare (SSW) and can be reached via the HESSI GUIwith the name MEM GE. We point out that the IDL code is implemented in a way that is relativelyindependent of the instrument providing the experimental visibilities and the related uncertainties. Inparticular, when using RHESSI visibilities, MEM GE images are very similar to MEM NJIT imageswhen this latter approach works but MEM GE also provides meaningful images for those cases wherethe MEM NJIT images, made with the standard value of the tolerance parameter, are unphysical.The plan of the paper is as follows. Section 2 provides some details about the formulation ofMEM GE and the optimization algorithm. Section 3 illustrates the results of its application againstboth experimental visibilities recorded by
RHESSI and synthetic visibilities generated within thesimulation software of the
Spectrometer Telescope for Imaging X-rays (STIX) on-board
Solar Orbiter .Comments on these results are contained in Section 4. Our conclusions will be offered in Section 5. THE OPTIMIZATION PROBLEMBoth
RHESSI and
STIX provide observations, named visibilities, which are calibrated Fouriertransforms of the incoming photon flux picked up at specific sampling points ξ , . . . , ξ N v in the spatialfrequency plane ( u, v ) (for RHESSI , the number N v of recorded visibilities is variable and depends onthe observation; for STIX , it is fixed at 30). Therefore image reconstruction for
RHESSI and
STIX involves the solution of the inverse Fourier transform problem with limited data v = Fx , (1)where v ∈ C N v is the vector containing the observed visibilities; the photon flux N × N image toreconstruct is transformed into the M -dimension vector x ∈ R M with M = N , by means of thestandard column-wise concatenation procedure; F ∈ C N v × M is the matrix computing the DiscreteFourier Transform (DFT) of x at the spatial frequencies ξ , . . . , ξ N v sampled by either RHESSI or STIX .The mathematical basis of MEM NJIT is the constrained maximization of the entropy H = − M (cid:88) j =1 x j log x j me , (2)where x j is the signal content of pixel j , m = F (cid:48) /M , F (cid:48) is the total flux in the image and e is theEuler’s number ( e = 2 . χ = 0 , (3) Figure 1.
Three examples of MEM NJIT misbehaviour. First row: the February 20 2002 event; timeinterval: 11:06:05 - 11:07:42; energy range: 25 −
50 keV; detectors: 3 through 9. Second row: the May1 2002 event; time interval: 19:21:29 - 19:22:29; energy range: 3 − −
12 keV; detectors:3 through 9. Left column: MEM NJIT reconstructions. Right column: comparison between predicted andmeasured visibilities with χ := N v (cid:88) i =1 | ( Fx ) i − v i | σ i − N v , (4)where σ is the vector of the experimental uncertainties. A second constraint involves the flux and isgiven by F = 0 , (5)with F = M (cid:88) j =1 x j − F (cid:48) . (6)Finally, a third constraint requires that all components of x must be non-negative. The algorithmimplemented in the MEM NJIT IDL code addresses the constrained maximum problemarg max x ≥ { H − αχ − βF } , (7)where α and β are the Lagrange multipliers associated to constraints (3) and (5); these two parametersare not estimated a priori but are updated during the maximization process. The first main drawbackof this approach is that the maximization problem (7) involves a functional which is not convex andtherefore numerical schemes may lead to unstable solutions. Further, the functional is characterizedby two Lagrange multipliers, whose updating process is sometimes non optimal. When one of theseconditions occurs, MEM NJIT produces unphysical reconstructions like the ones shown in Figure 1.MEM GE addresses these two issues by providing a different formulation of the maximum entropyoptimization problem. The first idea is to replace the maximization problem (7) with the minimizationproblem arg min x ≥ { χ − λH } , (8)under the flux constraint (5) and where λ is the regularization parameter. The main advantage ofthis approach is that now the optimization problem is convex and therefore it can be addressed byrelying on several numerical methods whose convergence properties are well-established. In particu-lar, in MEM GE we adopted the following iterative scheme whereby, at each iteration we compute(Combettes & Pesquet 2011)1. A gradient step to minimize χ ;2. A proximal step to maximize the entropy subjected to the positivity and flux constraints.After this second step, the algorithm is accelerated by computing a linear combination with theapproximation corresponding to the previous iteration and a monotonicity check is also performed(Beck & Teboulle 2009).Two main technical aspects are involved by the implementation of the algorithm. First, the reg-ularization parameter λ is a priori determined relying on the observation that an over-regularizingvalue λ of this parameter implies that the corresponding regularized solution must be related to theaverage flux by (cid:12)(cid:12)(cid:12)(cid:12) ( x λ ) j − mm (cid:12)(cid:12)(cid:12)(cid:12) ≤ j . Simple numerical approximations show that (9) implies λ ≥ max j | ∂ j χ ( x λ ) | (10)where ∂ j indicates the partial derivative along the j -th direction. In order to determine λ weapproximate each component of x λ with m . This results in a over-estimate of the regularizationparameter; therefore the optimal value for the regularization parameter is chosen as a rate of theestimated λ , where this rate is determined as a function of the visibility signal-to-noise ratio usinga heuristic look-up table.Second, the realization of the flux constraint in the second step relies on the solution of the not-regularized constrained minimum problem arg min x ≥ χ , (11)which is performed by applying the projected Landweber method (Piana & Bertero 1997): theestimate of the flux is computed by summing up the pixel content of the solution of problem (11). APPLICATION TO X-RAY VISIBILITIESThis section validates the reliability of MEM GE in the case of both observations provided by
RHESSI and synthetic visibilies simulated according to the
STIX imaging concept setup.3.1.
RHESSI
In order to illustrate the behavior of the new algorithm when applied to
RHESSI observations, weconsider two sets of events. The first set is made of the same cases considered in Figure 1 and weverified whether MEM GE is able to provide reconstructions that do not suffer the same pathologicalbehavior characterizing MEM NJIT, while guaranteeing the same data fidelity. Specifically, Figure2 refers to the same events considered in Figure 1, but this time the reconstruction method em-ployed is MEM GE. Then, Figures 3 through 5 compare the reconstructions provided by MEM GEand MEM NJIT with the ones of VIS CS, EM, Clean and uv smooth for these same three datasets.Finally, Figures 6 through 8 show the reconstructions provided by all six algorithms for three eventswhereby MEM NJIT works properly (for the reconstructions in Figures 3 through 8 we do not re-port the comparisons between the measured and predicted visibilities since they show very similarbehaviors among the reconstruction methods, with the only exception of uv smooth, which is charac-terized by fitting performances systematically slightly worse). Tables 1 and 2 illustrate a quantitativecomparison of performances from all codes and for all events considered in this sub-section.3.2.
STIX
We simulated the following four configurations with an overall incident flux of 10 photons cm − s − (see Figure 9 and Table 3 for all parameters): • a double footpoint flare in which one of the sources is more extended and has double the fluxof the other source (Configuration 1); • a double footpoint flare in which the sources have the same size and the same flux (Configuration2); • a loop flare with small curvature (Configuration 3); • a loop flare with large curvature (Configuration 4). Figure 2.
Left: MEM GE reconstructions (left) and corresponding comparisons between predicted andobserved visibilities (right) for the same cases as in Figure 1.
For each one of these configurations the
STIX software utilized a Monte Carlo approach to producea set of synthetic visibilities with associated the corresponding standard deviations. These visibilitysets have been processed by MEM GE and the results are compared in Table 2 with the ones providedby the two other methods currently implemented in the
STIX software tree, i.e. visibility-based Cleanand count-based Expectation Maximization (EM) (Massa et al. 2019). DISCUSSION OF RESULTS
MEM GE MEM NJIT VIS CSEM Clean uv smooth
Figure 3.
Reconstructions of the May 1 2002 event provided by six imaging methods available from the
HESSI
GUI. The observations and conditions are the same as in Figure 1 for this same event. For sakeof comparison, contour levels of MEM GE corresponding to 10% and 50% of the maximum intensity aresuperimposed to the reconstructions.
One of the nice aspects of MEM GE (see Figures 1 and 2) is that it provides reliable reconstructionsfor those events where MEM NJIT, with its default tolerance parameter set to 0 .
03, produces multipleunrealistically small sources, while predicting the experimental visibilities with a statistical fidelityclose to the MEM NJIT one. On the other hand, it behaves similarly to MEM NJIT for thoseflaring events where MEM NJIT reconstructions are reliable (see Figures 6, 7, and 8). As far as themorphological properties are concerned, MEM GE systematically introduces high-resolution effects.This is particularly visibile in the case of the reconstruction of the February 13 2002 event (Figure6), where a convex loop-shape is reproduced by MEM GE and MEM NJIT (and also EM), while thisconvexity is much smeared out in the reconstructions provided by Clean and uv smooth (however,we point out that the performances of Clean depend on what Regression Combined Method is usedfor the final Clean beam image and what Beam Width Factor is used). In this case, VIS CS failsto produce the correct orientation of the loop and reproduces a concave shape. Analogously, thespatial resolution achieved by the two MEM codes (and by EM and, partly, by uv smooth) is asfine as can be expected in the reconstruction of the July 23 2002 X Class flare, given the angularresolution of the modulation collimators used in the analysis: all four sources characterizing this
MEM GE MEM NJIT VIS CSEM Clean uv smooth
Figure 4.
Reconstructions of the February 20 2002 event provided by six imaging methods available fromthe
HESSI
GUI. The observations and conditions are the same as in Figure 1 for this same event. For sakeof comparison, contour levels of MEM GE corresponding to 10% and 50% of the maximum intensity aresuperimposed to the reconstructions. emission topography are clearly visibile in the reconstructions. On the contrary, neither VIS CS norClean are able to distinguish all four emitting regions. Tables 1 and 2 show that, when MEM NJITworks properly, the two maximum entropy methods provide similar estimates of the quantitativeparameters, although MEM GE may reproduce the data with significantly smaller χ values (as inthe case of the December 13 2007 and July 23 2002 events) but may require a higher computationaltime (as in the case of the February 13 2002 and July 23 2002 events). We finally point out that thetolerance parameter in MEM NJIT can be manually tuned to higher values in order to obtain moreregularized reconstructions. However, Figure 10 shows that increasing the tolerance value results intomorphologies closer to the ones corresponding to MEM GE reconstructions but with a less accurateability of the method to both fit the observations and conserve the flux value given as input to thealgorithm. Even more importantly, the figure shows that this tuning procedure is dependent on theexperimental dataset considered.Interestingly, the super-resolution properties of Maximum Entropy are confirmed by the analysisof synthetic STIX data illustrated in Table 3, where the performances of MEM GE are comparedto the ones of visibility-based Clean and count-based EM (there is no MEM NJIT code adapted to0
MEM GE MEM NJIT VIS CSEM Clean uv smooth
Figure 5.
Reconstructions of the December 13 2007 event provided by six imaging methods available fromthe
HESSI
GUI. The observations and conditions are the same as in Figure 1 for this same event. For sakeof comparison, contour levels of MEM GE corresponding to 10% and 50% of the maximum intensity aresuperimposed to the reconstructions. the
STIX framework). In fact, in the reconstructions of the foot-points, the ability of MEM GE torecover the ground-truth full width at half maximum (FWHM) outperforms the effectiveness of theother visibility-based algorithm (EM works systematically better, probably because the signal-to-noise ratio associated to counts is higher than the ones associated to the real and imaginary parts ofthe visibilities, as shown by Massa et al. (2019)).Also, similarly to EM, MEM GE behaves better than Clean in both reproducing the exact totalflux and best-fitting the synthetic measurements. EM and Clean seem more accurate in separatelyreproducing the flux of each one of the two circular sources in Configuration 1 and Configuration 2. CONCLUSIONSThis paper introduces a novel algorithm, named MEM GE, implementing a Maximum Entropyapproach to image reconstruction from X-ray visibilities in solar astronomy. The motivation of thiseffort relies on the fact that the only existing algorithm (MEM NJIT) realizing this approach forthis kind of data provides reliable and super-resolved reconstructions for datasets associated to mostevents, but sometimes suffers numerical instabilities and lack of convergence to meaningful solutions.Differently than the old algorithm, the new implementation relies on the constrained minimization1
MEM GE MEM NJIT VIS CSEM Clean uv smooth
Figure 6.
Reconstructions of the February 13 2002 event provided by the same six imaging methodsconsidered in Figures 3 through 5. Time interval: 12:29:40 - 12:31:22; energy range: 6 −
12 keV; detectors:3 through 9. For sake of comparison, contour levels of MEM GE corresponding to 10% and 50% of themaximum intensity are superimposed to the reconstructions. of a convex functional, which is realized by alternating gradient descent and proximal steps. Onthe one hand, the resulting code maintains the good imaging properties of the previous one, whileguaranteeing, on the other hand, convergence to reliable reconstructions when MEM NJIT givesunphysical sources (unless the tolerance parameter is tuned in a data-dependent fashion).MEM GE is available in the Solar SoftWare tree and can be most easily accessed through theHESSI GUI. It is one of the algorithms that is currently used for the population of the
RHESSI image archive, and can be used also for the processing of
STIX synthetic visibilities.The nice imaging properties of this algorithm, together with its systematic reliability, make it par-ticularly appropriate for the accurate estimation of morphological properties like the ones consideredin the analysis of coronal hard X-ray sources (Xu et al. 2008; Guo et al. 2012b,a; Dennis et al. 2018).MEM GE is an appropriate algorithm to realize a statistical study of these kinds of events.REFERENCES
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Figure 7.
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50 keV; detectors: 3 through9. For sake of comparison, contour levels of MEM GE corresponding to 10% and 50% of the maximumintensity are superimposed to the reconstructions.Benvenuto, F., Schwartz, R., Piana, M., &Massone, A. M. 2013, A&A, 555, A61,doi: 10.1051/0004-6361/201321295Bong, S.-C., Lee, J., Gary, D. E., & Yun, H. S.2006, ApJ, 636, 1159, doi: 10.1086/498225Combettes, P. L., & Pesquet, J.-C. 2011, inFixed-Point Algorithms for Inverse Problems inScience and Engineering, ed. Bauschke,H. Burachik, R. Combettes, P. Elser, V. Luke,D. Wolkowicz, & H. (Eds.) (Springer), 185–212.https://hal.inria.fr/hal-00643807Cornwell, T. J., & Evans, K. F. 1985, A&A, 143,77Dennis, B. R., Duval-Poo, M. A., Piana, M., et al.2018, ApJ, 867, 82,doi: 10.3847/1538-4357/aae0f5Dennis, B. R., & Tolbert, A. K. 2019, ApJ, 887,131, doi: 10.3847/1538-4357/ab4f81 Duval-Poo, M. A., Piana, M., & Massone, A. M.2018, A&A, 615, A59,doi: 10.1051/0004-6361/201731765Felix, S., Bolzern, R., & Battaglia, M. 2017, ApJ,849, 10, doi: 10.3847/1538-4357/aa8d1eGuo, J., Emslie, A. G., Kontar, E. P., et al. 2012a,A&A, 543, A53,doi: 10.1051/0004-6361/201219341Guo, J., Emslie, A. G., Massone, A. M., & Piana,M. 2012b, ApJ, 755, 32,doi: 10.1088/0004-637X/755/1/32H¨ogbom, J. A. 1974, A&AS, 15, 417Hurford, G. J., Schmahl, E. J., Schwartz, R. A.,et al. 2002, SoPh, 210, 61,doi: 10.1023/A:1022436213688Lin, R. P., Dennis, B. R., Hurford, G. J., et al.2002, SoPh, 210, 3,doi: 10.1023/A:1022428818870 MEM GE MEM NJIT VIS CSEM Clean uv smooth
Figure 8.
Reconstructions of the August 31 2004 event provided by the same six imaging methods consideredin Figures 3 through 5. Time interval: 05:34:47 - 05:35:47; energy range: 6 −
12 keV; detectors: 3 through9. For sake of comparison, contour levels of MEM GE corresponding to 10% and 50% of the maximumintensity are superimposed to the reconstructions.Massa, P., Piana, M., Massone, A. M., &Benvenuto, F. 2019, A&A, 624, A130,doi: 10.1051/0004-6361/201935323Massone, A. M., Emslie, A. G., Hurford, G. J.,et al. 2009, ApJ, 703, 2004,doi: 10.1088/0004-637X/703/2/2004 Metcalf, T. R., Hudson, H. S., Kosugi, T.,Puetter, R. C., & Pina, R. K. 1996, ApJ, 466,585, doi: 10.1086/177533Piana, M., & Bertero, M. 1997, Inverse problems,13, 441Schmahl, E. J., Pernak, R. L., Hurford, G. J., Lee,J., & Bong, S. 2007, SoPh, 240, 241,doi: 10.1007/s11207-007-0263-1Xu, Y., Emslie, A. G., & Hurford, G. J. 2008,ApJ, 673, 576, doi: 10.1086/524184 reduced χ ( all ) reduced χ (used) flux timeFebruary 20 2002 (20 −
50 keV; 11 : 06 : 05 −
11 : 07 : 42 UT)MEM GE 2.45 2.99 26.2 22MEM NJIT 2.43 2.54 21.0 11VIS CS 2.88 3.52 18.1 3CLEAN 2.74 3.30 39.6 7EM 2.39 2.91 18.4 83UV SMOOTH 5.39 7.21 40.8 1May 1 2002 (3 − −
19 : 22 : 29 UT)MEM GE 3.02 3.93 1.81 × × × × × × −
12 keV; 22 : 11 : 33 −
22 : 12 : 56 UT)MEM GE 1.76 1.61 12.6 × × × × × × Table 1.
Quantitative parameters corresponding to the reconstructions of the three events presented inFigures 3 through 5. The flux is measured in photon cm − s − , time is measured in s. Figure 9.
Four ground-truth configurations utilized to generate synthetic
STIX visibilities. Frome left toright: two foot-points with different size (Configuration 1); two foot-points with same size (Configuration2);a loop with orientation in the bottom left - top right direction (Configuration 3); a loop with orientation inthe right - left direction (Configuration 4). reduced χ ( all ) reduced χ (used) flux timeFebruary 13 2002 (6 −
12 keV; 12 : 29 : 40 −
12 : 31 : 22 UT)MEM GE 0.96 0.99 4.15 × × × × × × −
50 keV; 00 : 29 : 23 −
00 : 29 : 39 UT)MEM GE 3.42 3.53 2.90 × × × × × × −
12 keV; 05 : 34 : 47 −
05 : 35 : 47 UT)MEM GE 1.24 1.32 1.44 × × × × × × Table 2.
Quantitative parameters corresponding to the reconstructions of the three events presented inFigures 6 through 8. The flux is measured in photon cm − s − , time is measured in s. Figure 10.
Analysis of the outcome of MEM NJIT for different values of the tolerance parameter. First row:the February 20 2002 event. Second row: the July 23 2002 event. First column: MEM NJIT reconstructionsfor tolerance=1, with superimposed the level curves corresponding to MEM GE reconstructions; secondcolumn: reduced χ value vs tolerance computed considering both used and all detectors; third column:total flux of the source obtained from the image (red line) vs tolerance. The blue line corresponds to an apriori estimate used by MEM NJIT as input constraint. Configuration 1First Peak Second Peak Total flux ( × ) C-statisticX Y FWHM Flux ( × ) X Y FWHM Flux ( × )Simulated − . − . . .
53 12 . . . .
33 10 . − . ± . − . ± . . ± . . ± .
12 12 . ± . . ± . . ± . . ± .
11 10 . ± .
30 6 . ± . − . ± . − . ± . . ± . . ± .
13 12 . ± . . ± . . ± . . ± .
10 10 . ± .
04 3 . ± . − . ± . − . ± . . ± . . ± .
10 11 . ± . . ± . . ± . . ± .
09 8 . ± .
12 26 . ± . × ) C-statisticX Y FWHM Flux ( × ) X Y FWHM Flux ( × )Simulated − . . . .
95 15 . − . . .
95 10 . − . ± . . ± . . ± . . ± .
13 14 . ± . − . ± . . ± . . ± .
12 10 . ± .
21 5 . ± . − . ± . . ± . . ± . . ± .
14 14 . ± . − . ± . . ± . . ± .
13 10 . ± .
03 3 . ± . − . ± . . ± . . ± . . ± .
10 13 . ± . − . ± . . ± . . ± .
10 8 . ± .
11 31 . ± . × ) C-statisticX YSimulated − . − . − . ± . − . ± . . ± .
21 6 . ± . − . ± . − . ± . . ± .
04 3 . ± . − . ± . − . ± . . ± .
11 29 . ± . × ) C-statisticX YSimulated 20 . . . ± . . ± . . ± .
17 5 . ± . . ± . . ± . . ± .
03 3 . ± . . ± . . ± . . ± .
11 32 . ± . Table 3.
Reconstruction of four source configurations characterized by an overall incident photon flux of10 photons cm − s − . The morphological and photometric parameters reconstructed by MEM GE arecompared with the ground truth and with the values provided by EM and CLEAN. The positions X, Y andthe full width at half maximum (FWHM) of the sources are given in arcseconds, while the total flux and theflux corresponding to each foot point in Configurations 1 and 2 are given in photons cm − s −1