Imaging Supermassive Black Holes with the Event Horizon Telescope

Model (Original) Model (Convolved) EHT 2017

Kazunori Akiyama, NEROC Symposium: Radio Science and Related Topics, MIT Haystack Observatory, 11/04/2016

Imaging Supermassive Black Holes Forward Jet Model Forward Jet Model Forward Forward Jet Model Forward with the Event Horizon Telescope

Model Model (Convolved) EHT 2017 Fractional Polarization Fractional Polarization Fractional Polarization Counter Jet Model Counter Jet Model Counter Counter Jet Model Counter Kazu Akiyama (MIT Haystack Observatory / JSPS Fellow) On Behalf of the EHT Imaging Working Group: Michael Johnson, Andrew Chael, Ramesh Narayan (CfA/SAO), Katie Bouman (MIT CSAIL), Mareki Honma (NAOJ) et al. Kazunori Akiyama, NEROC Symposium: Radio Science and Related Topics, MIT Haystack Observatory, 11/04/2016 Radio Interferometry: Sampling Fourier Components of the Images Kazunori Akiyama, NEROC Symposium: Radio Science and Related Topics, MIT Haystack Observatory, 11/04/2016 Radio Interferometry: Sampling Fourier Components of the Images

Image Kazunori Akiyama, NEROC Symposium: Radio Science and Related Topics, MIT Haystack Observatory, 11/04/2016 Radio Interferometry: Sampling Fourier Components of the Images

Image Fourier Domain (Visibility) Kazunori Akiyama, NEROC Symposium: Radio Science and Related Topics, MIT Haystack Observatory, 11/04/2016 Radio Interferometry: Sampling Fourier Components of the Images

Image Fourier Domain Sampling Process (Visibility) (Projected Baseline = Spatial Frequency)

(Images: adapted from Akiyama et al. 2015, ApJ ; Movie: Laura Vertatschitsch) Kazunori Akiyama, NEROC Symposium: Radio Science and Related Topics, MIT Haystack Observatory, 11/04/2016 Radio Interferometry: Sampling Fourier Components of the Images

Image Fourier Domain Sampling Process (Visibility) (Projected Baseline = Spatial Frequency)

(Images: adapted from Akiyama et al. 2015, ApJ ; Movie: Laura Vertatschitsch)

Sampling is NOT perfect Kazunori Akiyama, NEROC Symposium: Radio Science and Related Topics, MIT Haystack Observatory, 11/04/2016 Interferometry Imaging: Observational equation is ill-posed Y = A X (Data) (Fourier Matrix) (Image)

exp(i2πu1x1) exp(i2πu1x2) … y1 x1 exp(i2πu1xN) y2 exp(i2πu2x1) exp(i2πu2x2) … x2 y3 = exp(i2πu2xN) : : x3 exp(i2πu3x1) exp(i2πu3x2) … yM exp(i2πu3xN) : :

xN Kazunori Akiyama, NEROC Symposium: Radio Science and Related Topics, MIT Haystack Observatory, 11/04/2016 Interferometry Imaging: Observational equation is ill-posed Y = A X (Data) (Fourier Matrix) (Image)

exp(i2πu1x1) exp(i2πu1x2) … y1 x1 exp(i2πu1xN) y2 exp(i2πu2x1) exp(i2πu2x2) … x2 y3 = exp(i2πu2xN) : : x3 exp(i2πu3x1) exp(i2πu3x2) … yM exp(i2πu3xN) : :

- Sampling is NOT perfect xN Number of data M < Number of image pixels N Kazunori Akiyama, NEROC Symposium: Radio Science and Related Topics, MIT Haystack Observatory, 11/04/2016 Interferometry Imaging: Observational equation is ill-posed Y = A X (Data) (Fourier Matrix) (Image)

exp(i2πu1x1) exp(i2πu1x2) … y1 x1 exp(i2πu1xN) y2 exp(i2πu2x1) exp(i2πu2x2) … x2 y3 = exp(i2πu2xN) : : x3 exp(i2πu3x1) exp(i2πu3x2) … yM exp(i2πu3xN) : :

- Sampling is NOT perfect xN Number of data M < Number of image pixels N - Equation is ill-posed: inﬁnite numbers of solutions Kazunori Akiyama, NEROC Symposium: Radio Science and Related Topics, MIT Haystack Observatory, 11/04/2016 Interferometry Imaging: Observational equation is ill-posed Y = A X (Data) (Fourier Matrix) (Image)

exp(i2πu1x1) exp(i2πu1x2) … y1 x1 exp(i2πu1xN) y2 exp(i2πu2x1) exp(i2πu2x2) … x2 y3 = exp(i2πu2xN) : : x3 exp(i2πu3x1) exp(i2πu3x2) … yM exp(i2πu3xN) : :

- Sampling is NOT perfect xN Number of data M < Number of image pixels N - Equation is ill-posed: inﬁnite numbers of solutions - Interferometric Imaging: Picking a reasonable solution based on a prior assumption Kazunori Akiyama, NEROC Symposium: Radio Science and Related Topics, MIT Haystack Observatory, 11/04/2016 Approach 1: Sparse Reconstruction Kazunori Akiyama, NEROC Symposium: Radio Science and Related Topics, MIT Haystack Observatory, 11/04/2016 Approach 1: Sparse Reconstruction Philosophy: Reconstructing images with the smallest number of point sources within a given residual error Kazunori Akiyama, NEROC Symposium: Radio Science and Related Topics, MIT Haystack Observatory, 11/04/2016 Approach 1: Sparse Reconstruction Philosophy: Reconstructing images with the smallest number of point sources within a given residual error Kazunori Akiyama, NEROC Symposium: Radio Science and Related Topics, MIT Haystack Observatory, 11/04/2016 Approach 1: Sparse Reconstruction Philosophy: Reconstructing images with the smallest number of point sources within a given residual error

Lp-norm: (p>0)

||x||0 = number of non-zero pixels in the image Kazunori Akiyama, NEROC Symposium: Radio Science and Related Topics, MIT Haystack Observatory, 11/04/2016 Approach 1: Sparse Reconstruction Philosophy: Reconstructing images with the smallest number of point sources within a given residual error

Number of non-zero pixels (point sources)

Lp-norm: (p>0)

Obs. Number Data Image of non-zero pixels Matrix (point sources) Chi-square: Consistency between data and the image

Lp-norm: (p>0)

Obs. Number Data Image of non-zero pixels Matrix (point sources) Chi-square: Consistency Computationally verybetween expensive!! data and the image

Lp-norm:(It can be solved for N < ~100) (p>0) - L0 norm is not continuous, nondifferentiable - Combinational Optimization ||x||0 = number of non-zero pixels in the image Kazunori Akiyama, NEROC Symposium: Radio Science and Related Topics, MIT Haystack Observatory, 11/04/2016 Approach 1: Sparse Reconstruction CLEAN (Hobgom 1974) = Matching Pursuit (Mallet & Zhang 1993) Computationally very cheap, but highly affected by the Point Spread Function

Dirty map: Point Spread Function: Solution: FT of zero-ﬁlled Dirty map Point sources Visibility for the point source + Residual Map

(3C 273, VLBA-MOJAVE data at 15 GHz) Kazunori Akiyama, NEROC Symposium: Radio Science and Related Topics, MIT Haystack Observatory, 11/04/2016 Approach 1: Sparse Reconstruction CLEAN (Hobgom 1974) = Matching Pursuit (Mallet & Zhang 1993) Computationally very cheap, but highly affected by the Point Spread Function

Dirty map: Point Spread Function: Solution: FT of zero-ﬁlled Dirty map Point sources Visibility for the point source + Residual Map

(3C 273, VLBA-MOJAVE data at 15 GHz) 9 9 MEM algorithm with full visibility phase information, directly minimizing Eq 4 with the 2 term in Eq. 5. This choice, 2 while infeasibleMEM in practice algorithm due with to phase full visibility errors, allowed phase information, us to directly directly compare minimizing to CLEAN Eq 4 without with the introducing term in Eq. the 5. This choice, need for self-calibration.while infeasible in practice due to phase errors, allowed us to directly compare to CLEAN without introducing the need for self-calibration. After obtaining MEM and CLEAN reconstructions from the same data, we convolved the reconstructed images with a sequence of GaussianAfter obtaining beams scaled MEM from and the CLEAN elliptical reconstructions Gaussian fitted from to the the Fourier same data, transform we convolved of the u, the v coverage reconstructed (the images with a “clean beam”).sequence We then of computed Gaussian the beams normalized scaled from root-mean-square the elliptical Gaussian error (NRMSE) fitted to of the each Fourier restored transform image: of the u, v coverage (the “clean beam”). We then computed the normalized root-mean-square error (NRMSE) of each restored image: 2 n 2 I0 Ii 2 i=1 i n 2 NRMSE = 2| | , I I (17) v n 2 i=1 i0 i NRMSEIi = 2| | , (17) uP i=1 | | v n 2 u u i=1 Ii t uP | | where I0 is the final restored image and KazunoriI is the Akiyama, true NEROC image. Symposium: ForP Radio the Science CLEANt and Related reconstructions, Topics, MIT Haystack Observatory, we chose 11/04/2016not to add the dirty image residualswhere backI0 is the to the final convolved restored image model, and as theI is residuals the true image. are a sensible ForP the quantity CLEAN only reconstructions, for the full restoring we chose not to add the beam. To minimizedirty the image e↵ect residuals of this back choice to onApproach the the convolved CLEAN model,1: reconstruction, Sparse as the Reconstruction residuals we chose are a compact a sensible model quantity image only with for no the full restoring di↵use structure.beam. After To tuning minimize our the CLEAN e↵ect reconstruction of this choice on for the this CLEAN image, the reconstruction, total flux left we in chose the residuals a compact was model less image with no di↵use structure. AfterCLEAN tuning (Hobgom our CLEAN 1974) reconstruction = Matching Pursuit for this image,(Mallet & the Zhang total 1993) flux left in the residuals was less than 2% of the total image flux. InComputationally performing the CLEAN very cheap, reconstruction, but highly we affected used Briggs by the weighting Point Spread and a loop gain of 0.025, with thethan rest 2% of of the the parameters total image set flux. to the In performing default in the the algorithm’s CLEAN reconstruction, CASA implementation we used Briggs7. weighting and a loop gain of 0.025, with the restFunction ofCLEANCLEAN the parameters vsvs regularizedregularized set maximum tomaximum the default likelihoodlikelihood in the (BSMEM)(BSMEM) algorithm’s CASA implementation 7. CLEANCLEAN vsvs regularizedregularized maximummaximumCLEAN likelihoodlikelihood is problematic (BSMEM)(BSMEM) for the black hole shadows?

UV coverage Model SMT,CARMA,LMT, ALMA,MaunaKea (Broderick) UV coverage GroundModel SMT,CARMA,LMT, (Broderick) ALMA,MaunaKea Truth Fabien Baron – Image reconstruction for the EHT the for reconstruction Image – Baron Fabien

CLEAN BSMEM Fabien Baron – Image reconstruction for the EHT the for reconstruction Image – Baron Fabien CLEAN

CLEAN BSMEM

Fabian EventBaron+ Horizon Telescope ConferenceChael+2016 – 20 January 2012 ApJ Akiyama+2016b,c,

Figure 4. (Left) Normalized root-mean-squareEvent Horizon Telescope Conference error – 20 (NRMSE, January 2012 Eq. 17) of MEM and CLEAN reconstructedsubmitted Stokes toI ApJimages as a function of the fractional restoringFigure 4. beam(Left) size. Normalized For comparison, root-mean-square the NRMSE error of (NRMSE, the model Eq. image 17) is of also MEM plotted. and CLEAN The reconstructed reconstructed images Stokes wereI images as a function produced using simulatedof the fractional data from restoring the EHT beam array; size. for straightforward For comparison, comparison the NRMSE with of CLEAN, the model realistic image thermal is also plotted. noise was The added reconstructed to the images were simulated visibilitiesproduced but gain using calibration simulated errors, data random from the atmospheric EHT array; phases, for straightforward and blurring comparison due to interstellar with CLEAN, scattering realistic were allthermal neglected. noise was added to the The images weresimulated convolved visibilities with scaled but versions gain calibration of the fitted errors, clean random beam. atmospheric The minimum phases, for and each blurring NRMSE due curve to interstellar indicates the scattering optimal were all neglected. restoring beam, whichThe isimages significantly were convolved smaller for with MEM scaled (25% versions of nominal) of the than fitted for clean CLEAN beam. (0.78% The of minimum nominal). for each NRMSE curve indicates the optimal (Right) Example reconstructionsrestoring beam, restored which is with significantly scaled beams smaller from for curves MEM in (25% the left of nominal) panel. The than center-left for CLEAN panels (0.78% are the of MEM nominal). and CLEAN reconstructions restored(Right) at Example the nominal reconstructions resolution, restored with the with fitted scaled clean beams beam. from The curves center-right in the left panels panel. show The the center-left reconstructions panels are restored the MEM and CLEAN with the optimal beamreconstructions for the CLEAN restored reconstruction at the nominal and resolution, the far right with panels the fitted show both clean reconstructions beam. The center-right restored with panels the show optimal the reconstructions MEM restored beam. The CLEANwith reconstructions the optimal beam consist for ofthe only CLEAN the CLEAN reconstruction components and the convolved far right with panels the show restoring both reconstructions beam and do not restored include with the the optimal MEM dirty image residuals,beam. as discussed The CLEAN at the reconstructions end of Section consist 3. of only the CLEAN components convolved with the restoring beam and do not include the dirty image residuals, as discussed at the end of Section 3. The results are displayed in Fig. 4. In the left panel, we see that the MEM curve has a minimum in NRMSE at a significantly smallerThe results beam size are than displayed the CLEAN in Fig. reconstruction, 4. In the left panel, demonstrating we see that MEM’s the superior MEM curve ability has to a superresolve minimum in NRMSE at a source structuresignificantly over CLEAN. smaller Furthermore, beam size the than value the ofCLEAN NRMSE reconstruction, from the MEM demonstrating reconstruction MEM’s is consistently superior ability lower to superresolve than from CLEANsource for structure all values over of restoring CLEAN. beam Furthermore, size. Most the importantly, value of NRMSE while the from CLEAN the MEM curve reconstruction NRMSE increases is consistently lower rapidly for restoringthan from beams CLEAN smaller for than all values the optimal of restoring resolution, beam size. the Most MEM importantly, image fidelity while is the relatively CLEAN una curve↵ected NRMSE increases by choosing a restoringrapidly for beam restoring that is beams too small. smaller Choosing than the a optimal restoring resolution, beam that the is MEM too large image produces fidelity an is image relatively una↵ected with the sameby fidelity choosing as the a restoringmodel blurred beam to that that is resolution. too small. The Choosing right panel a restoring of Fig. beam 4 shows that the is too model large image, produces an image the interferometerwith “clean” the same beam, fidelity and as the the reconstructions model blurred blurred to that with resolution. the clean The beam right (nominal) panel of and Fig. the 4 measured shows the model image, optimal fractionalthe beams. interferometer In addition “clean” to lower beam, resolution and the reconstructions and fidelity, the blurred CLEAN with reconstructions the clean beam show (nominal) prominent and the measured striping featuresoptimal from isolated fractional components beams. In being addition restored to lower with the resolution restoring and beam. fidelity, the CLEAN reconstructions show prominent striping features from isolated components being restored with the restoring beam. While Fig. 4 demonstrates that in this case the MEM reconstruction has superior resolution and fidelity to the CLEAN reconstruction,While Fig. the 4 optimal demonstrates restoring that beam in this size case for the the CLEAN MEM reconstruction reconstruction has is still superior less than resolution unity. This and fidelity to the result was observedCLEAN in several reconstruction, similar reconstructions, the optimal restoring suggesting beam that size shrinking for the CLEAN the restoring reconstruction beam used is in still CLEAN less than unity. This reconstructionsresult to 75% was of observed the nominal in several fitted beam similar can reconstructions, enhance resolution suggesting without that introducing shrinking the imaging restoring artifacts, beam at used in CLEAN least on imagesreconstructions of compact objects to 75% similar of the to those nominal used fitted in these beam tests. can enhance resolution without introducing imaging artifacts, at least on images of compact objects similar to those used in these tests. Repeating the exercise of Fig. 4 with observations taken with increased or decreased signal-to-noise ratio resulted in NRMSE curvesRepeating that are the only exercise slightly of higher Fig. 4 and with lower observations than the taken curves with in Fig. increased 4, but or shared decreased the same signal-to-noise form - in ratio resulted particular, thein location NRMSE of the curves minimum that are NRMSE only slightly values higher was barely and lower shifted. than This the insensitivity curves in Fig. to additional4, but shared noise the is same form - in particular, the location of the minimum NRMSE values was barely shifted. This insensitivity to additional noise is 7 http://casa.nrao.edu/docs/TaskRef/clean-task.html 7 http://casa.nrao.edu/docs/TaskRef/clean-task.html Kazunori Akiyama, NEROC Symposium: Radio Science and Related Topics, MIT Haystack Observatory, 11/04/2016 Approach 1: Sparse Reconstruction L1 regularization (LASSO, Tibishirani 1996) Kazunori Akiyama, NEROC Symposium: Radio Science and Related Topics, MIT Haystack Observatory, 11/04/2016 Approach 1: Sparse Reconstruction L1 regularization (LASSO, Tibishirani 1996) Convex Relaxation: Relaxing L0-norm to a convex, continuous, and differentiable function Kazunori Akiyama, NEROC Symposium: Radio Science and Related Topics, MIT Haystack Observatory, 11/04/2016 Approach 1: Sparse Reconstruction L1 regularization (LASSO, Tibishirani 1996) Convex Relaxation: Relaxing L0-norm to a convex, continuous, and differentiable function Kazunori Akiyama, NEROC Symposium: Radio Science and Related Topics, MIT Haystack Observatory, 11/04/2016 Approach 1: Sparse Reconstruction L1 regularization (LASSO, Tibishirani 1996) Convex Relaxation: Relaxing L0-norm to a convex, continuous, and differentiable function

equivalent Chi-square Regularization on sparsity Kazunori Akiyama, NEROC Symposium: Radio Science and Related Topics, MIT Haystack Observatory, 11/04/2016 Approach 1: Sparse Reconstruction L1 regularization (LASSO, Tibishirani 1996) Convex Relaxation: Relaxing L0-norm to a convex, continuous, and differentiable function

equivalent Chi-square Regularization on sparsity

- Reconstruction purely in the visibility domain: Not affected by de-convolution beam (point spread function)

- Many applications after appearance of Compressed Sensing (Donoho, Candes+) Kazunori Akiyama, NEROC Symposium: Radio Science and Related Topics, MIT Haystack Observatory, 11/04/2016 Approach 1: Sparse Reconstruction Application of LASSO (Honma et al. 2014)

(Honma, Akiyama, Uemura & Ikeda 2014, PASJ) Kazunori Akiyama, NEROC Symposium: Radio Science and Related Topics, MIT Haystack Observatory, 11/04/2016 Approach 1: Sparse Reconstruction For Smoother Image: sparsity on gradient domain Total Variation: Sparse regularizer of the image in its gradient domain

L1 + TV regularization (Akiyama et al. 2016b,c, Kuramochi+ in prep.)

(Sgr A*; Kuramochi, Akiyama, et al. in prep.) Kazunori Akiyama, NEROC Symposium: Radio Science and Related Topics, MIT Haystack Observatory, 11/04/2016 Approach 1: Sparse Reconstruction For Smoother Image: sparsity on gradient domain

Total Variation: SparseModel (Original) regularizerModel of (Convolved) the image inEHT its 2017gradient domain Forward Jet Model Forward

L1 + TV regularization (Akiyama et al. 2016b,c, Kuramochi+ in prep.) Fractional Polarization Counter Jet Model Counter

(Akiyama et al. 2016c subm. to ApJ)

(Sgr A*; Kuramochi, Akiyama, et al. in prep.) Kazunori Akiyama, NEROC Symposium: Radio Science and Related Topics, MIT Haystack Observatory, 11/04/2016 Approach 2: Maximize the Information Entropy Maximum Entropy Methods (MEM; Frieden 1972; Gull & Daniell 1978) Kazunori Akiyama, NEROC Symposium: Radio Science and Related Topics, MIT Haystack Observatory, 11/04/2016 Approach 2: Maximize the Information Entropy Maximum Entropy Methods (MEM; Frieden 1972; Gull & Daniell 1978) 9 Kazunori Akiyama, NEROC Symposium: Radio Science and Related Topics, MIT Haystack Observatory, 11/04/2016 MEM algorithm with full visibility phase information, directly minimizing Eq 4 with the 2 term in Eq. 5. This choice, while infeasible in practice due to phase errors, allowed us to directly compare to CLEAN without introducing the need for self-calibration.Approach 2: Maximize the Information Entropy AfterMaximum obtaining MEM Entropy and CLEAN Methods reconstructions (MEM; from the Frieden same data, we1972; convolved Gull the reconstructed& Daniell images 1978) with a sequence of Gaussian beams scaled from the elliptical Gaussian ﬁtted to the Fourier transform of the u, v coverage (the “clean beam”). We then computed the normalized root-mean-square error (NRMSE) of each restored image:

2 n 2 i=1 Ii0 Ii NRMSE = 2| | , (17) v n 2 u i=1 Ii uP | | t where I0 is the final restored image and I is the true image. ForP the CLEAN reconstructions, we chose not to add the dirty image residuals back to the convolved model, as the residuals are a sensible quantity only for the full restoring beam. To minimize the e↵ect of this choice on the CLEAN reconstruction, we chose a compact model image with no di↵use- Compared structure. After tuning with our CLEAN: CLEAN reconstruction for this image, the total flux left in the residuals was less than 2% of the total image flux. In performing the CLEAN reconstruction, we used Briggs weighting and a loop gain of 0.025,(1) with Better the rest fidelity of the parameters for Smooth set to the Structure default in the (2) algorithm’s Better CASA optimal implementation resolution7.

Figure 4. (Left) Normalized root-mean-square error (NRMSE, Eq. 17) of MEM and CLEAN reconstructed Stokes I images as a function of the fractional restoring beam size. For comparison, the NRMSE of the model image is(Chael also plotted. et The al. reconstructed 2016, imagesApJ) were produced using simulated data from the EHT array; for straightforward comparison with CLEAN, realistic thermal noise was added to the simulated visibilities but gain calibration errors, random atmospheric phases, and blurring due to interstellar scattering were all neglected. The images were convolved with scaled versions of the fitted clean beam. The minimum for each NRMSE curve indicates the optimal restoring beam, which is significantly smaller for MEM (25% of nominal) than for CLEAN (0.78% of nominal). (Right) Example reconstructions restored with scaled beams from curves in the left panel. The center-left panels are the MEM and CLEAN reconstructions restored at the nominal resolution, with the fitted clean beam. The center-right panels show the reconstructions restored with the optimal beam for the CLEAN reconstruction and the far right panels show both reconstructions restored with the optimal MEM beam. The CLEAN reconstructions consist of only the CLEAN components convolved with the restoring beam and do not include the dirty image residuals, as discussed at the end of Section 3.

The results are displayed in Fig. 4. In the left panel, we see that the MEM curve has a minimum in NRMSE at a significantly smaller beam size than the CLEAN reconstruction, demonstrating MEM’s superior ability to superresolve source structure over CLEAN. Furthermore, the value of NRMSE from the MEM reconstruction is consistently lower than from CLEAN for all values of restoring beam size. Most importantly, while the CLEAN curve NRMSE increases rapidly for restoring beams smaller than the optimal resolution, the MEM image fidelity is relatively una↵ected by choosing a restoring beam that is too small. Choosing a restoring beam that is too large produces an image with the same fidelity as the model blurred to that resolution. The right panel of Fig. 4 shows the model image, the interferometer “clean” beam, and the reconstructions blurred with the clean beam (nominal) and the measured optimal fractional beams. In addition to lower resolution and fidelity, the CLEAN reconstructions show prominent striping features from isolated components being restored with the restoring beam. While Fig. 4 demonstrates that in this case the MEM reconstruction has superior resolution and fidelity to the CLEAN reconstruction, the optimal restoring beam size for the CLEAN reconstruction is still less than unity. This result was observed in several similar reconstructions, suggesting that shrinking the restoring beam used in CLEAN reconstructions to 75% of the nominal fitted beam can enhance resolution without introducing imaging artifacts, at least on images of compact objects similar to those used in these tests. Repeating the exercise of Fig. 4 with observations taken with increased or decreased signal-to-noise ratio resulted in NRMSE curves that are only slightly higher and lower than the curves in Fig. 4, but shared the same form - in particular, the location of the minimum NRMSE values was barely shifted. This insensitivity to additional noise is

7 http://casa.nrao.edu/docs/TaskRef/clean-task.html Kazunori Akiyama, NEROC Symposium: Radio Science and Related Topics, MIT Haystack Observatory, 11/04/2016 Approach 2: Maximize the Information Entropy Maximum Entropy Methods (MEM; Frieden 1972; Gull & Daniell 1978)

- PolMEM: Extension of MEM to full-polarimetric Imaging 14 (Chael+16) Chael et al.

Figure 9. (Top) 1.3-mm MEM reconstructions of a magnetically arrested disk simulation of the Sgr A* accretion flow, courtesy of Jason Dexter (Dexter 2014). Color indicates Stokes I flux and ticks marking the direction of linear(Chael polarization et are al. plotted 2016, in regions ApJ) with I greater than 4 its RMS value and P greater than 2 its RMS value. After blurring the image with the Sgr A* scattering kernel at 1.3 mm, data were⇥ simulated with realistic| | thermal noise,⇥ amplitude calibration errors, and random atmospheric phases. The center right panel shows a reconstruction with data simulated on EHT baselines expected in 2016 and the rightmost panel shows the reconstruction with the full array expected in 2017. Each reconstruction was restored with a Gaussian beam 1/2 the size of the fitted clean beam (93 32 µas FWHM in 2016 ; 27 14 µas FWHM in 2017). For comparison, the center left panel shows the model smoothed to the same resolution⇥ as the 2017 image. (Bottom)⇥ 1.3-mm MEM reconstructions of a simulation of the jet in M87, courtesy of Avery Broderick (Broderick & Loeb 2009; Lu et al. 2014b). Data were simulated on 2016 and 2017 EHT baselines as in the top panel, but without the contributions from interstellar scattering that are significant for Sgr A⇤.BothreconstructionswererestoredwithaGaussianbeam1/2thesizeofthefitted clean beam (72 36 µas FWHM in 2016 ; 28 20 µas FWHM in 2017). ⇥ ⇥ restoring beam, the I and P NRMSE values drop to 24.0% and 59.0% for the 2016 reconstruction and 19.8% and 61.9% for the 2017 image. The polarization position angle weighted error drops to 20.0 and 21.6 for the 2016 and 2017 images, respectively. Even with minimal baseline coverage, MEM is able to reconstruct a reasonably accurate image when compared to the true image viewed at the same resolution. The 2016 image of an M87 jet model (Fig. 9, bottom panel) gave NRMSE values of 55.61% for Stokes I and 77.34% for Stokes P , with a weighted angular error of 23.5. In 2017, the NRMSE values were 36.71% for Stokes I and 54.40% for P , with an angular error of 17.9. When we instead compare the reconstructions to the model image smoothed to the same resolution as the restoring beam, the I and P NRMSE values drop to 21.3% and 34.5% for the 2016 image and 18.3% and 27.7% for the 2017 image, while the polarization position angle weighted error drops to 21.6 and 14.8 for the 2016 and 2017 images, respectively.

6. CONCLUSION

As the EHT opens up new, extreme environments to direct VLBI imaging, a renewed exploration of VLBI imaging strategies is necessary for extracting physical signatures from challenging datasets. In this paper, we have shown the e↵ectiveness of imaging linear polarization from VLBI data using extensions of the Maximum Entropy Method. We explored extensions of MEM using previously proposed polarimetric regularizers like PNN and adaptations of regularizers new to VLBI imaging like total variation. We furthermore adapted standard MEM to operate on robust bispectrum and polarimetric ratio measurements instead of calibrated visibilities. MEM imaging of polarization can provide increased resolution over CLEAN (Fig. 5) and is more adapted to continuous distributions, as are expected for the black hole accretion disks and jets targeted by the Event Horizon Telescope. Furthermore, MEM imaging algorithms can naturally incorporate both physical constraints on flux and polarization fraction as well as constraints from prior information or expected source structure. Extending our code to run on data from connected-element interferometers like ALMA is a logical next step, but it will require new methods to eciently handle large amount of data and image pixels across a wide field of view. Polarimetric MEM is also a promising tool for synthesis imaging of a diversity of other astrophysical systems typically observed with connected element interferometers. For example, the polarized dust emission from protostellar cores frequently exhibits a smooth morphology (Girart et al. 2006; Hull et al. 2013), so MEM may be better-suited to study both the large-scale magnetic-field morphologies and their small Kazunori Akiyama, NEROC Symposium: Radio Science and Related Topics, MIT Haystack Observatory, 11/04/2016

Approach 3: Machine-learn Distribution of Image Patches A patch prior (CHIRP; Bouman et al. 2015)

Simple Example

(courtesy of Katie Bouman) Kazunori Akiyama, NEROC Symposium: Radio Science and Related Topics, MIT Haystack Observatory, 11/04/2016

Approach 3: Machine-learn Distribution of Image Patches A patch prior (CHIRP; Bouman et al. 2015)

Simple Example

(courtesy of Katie Bouman) Kazunori Akiyama, NEROC Symposium: Radio Science and Related Topics, MIT Haystack Observatory, 11/04/2016

Approach 3: Machine-learn Distribution of Image Patches A patch prior (CHIRP; Bouman et al. 2015)

Simple Example Probability Distribution of “Multi-scale Patches”

Can be used as “A Prior Knowledge”

(courtesy of Katie Bouman) Kazunori Akiyama, NEROC Symposium: Radio Science and Related Topics, MIT Haystack Observatory, 11/04/2016

Approach 3: Machine-learn Distribution of Image Patches A patch prior (CHIRP; Bouman et al. 2015) CHIRP: Continuous High Image Resolution using Patch priors Reconstruct the image so that it maximizes consistency with a machine-leaned patch prior distribution Sample

Cluster (courtesy of Katie Bouman) Kazunori Akiyama, NEROC Symposium: Radio Science and Related Topics, MIT Haystack Observatory, 11/04/2016

GROUND TRUTH

BLURRED GROUND TRUTH

CHIRP

Courtesy of Kazunori Akiyama Jason Dexter, MonikaMoscibrodzka, Avery Brodrick, Hotaka Shiokawa

(courtesy of Katie Bouman) Kazunori Akiyama, NEROC Symposium: Radio Science and Related Topics, MIT Haystack Observatory, 11/04/2016

Summary

• All state-of-the-art imaging techniques developed for the EHT have shown much better performance than the traditional CLEAN.

• These techniques can be applied to any existing interferometers

• These techniques would be applicable to similar Fourier-inverse problems (e.g. ) Faraday Tomography (RM Synthesis) Mostly equivalent to linear polarimetric imaging

M87 jets (Application to VLBA data) - Color: CLEAN (3mm) - Lines: Sparse Modeling (7mm)