Imaging Supermassive Black Holes with the Event Horizon Telescope

Transcription

1 Counter Jet Model Forward Jet Kazunori Akiyama, NEROC Symposium: Radio Science and Related Topics, MIT Haystack Observatory, 11/04/2016 Imaging Supermassive Black Holes with the Event Horizon Telescope Model Model (Convolved) EHT 2017 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.

2 Radio Interferometry: Sampling Fourier Components of the Images

3 Radio Interferometry: Sampling Fourier Components of the Images Image

4 Radio Interferometry: Sampling Fourier Components of the Images Image Fourier Domain (Visibility)

5 Radio Interferometry: Sampling Fourier Components of the Images Image Fourier Domain (Visibility) Sampling Process (Projected Baseline = Spatial Frequenc (Images: adapted from Akiyama et al. 2015, ApJ ; Movie: Laura Vertatschitsch)

6 Radio Interferometry: Sampling Fourier Components of the Images Image Fourier Domain (Visibility) Sampling Process (Projected Baseline = Spatial Frequenc (Images: adapted from Akiyama et al. 2015, ApJ ; Movie: Laura Vertatschitsch)

7 Radio Interferometry: Sampling Fourier Components of the Images Image Fourier Domain (Visibility) Sampling Process (Projected Baseline = Spatial Frequenc (Images: adapted from Akiyama et al. 2015, ApJ ; Movie: Laura Vertatschitsch) Sampling is NOT perfect

8 Interferometry Imaging: Observational equation is ill-posed Y = A X (Data) (Fourier Matrix) (Image) y1 y2 y3 : : = exp(i2πu1x1) exp(i2πu1x2) exp(i2πu1xn) exp(i2πu2x1) exp(i2πu2x2) exp(i2πu2xn) exp(i2πu3x1) exp(i2πu3x2) x1 x2 x3 ym exp(i2πu3xn) : : xn

9 Interferometry Imaging: Observational equation is ill-posed Y = A X (Data) (Fourier Matrix) (Image) y1 y2 y3 : : = exp(i2πu1x1) exp(i2πu1x2) exp(i2πu1xn) exp(i2πu2x1) exp(i2πu2x2) exp(i2πu2xn) exp(i2πu3x1) exp(i2πu3x2) x1 x2 x3 ym exp(i2πu3xn) : : - Sampling is NOT perfect Number of data M < Number of image pixels N xn

10 Interferometry Imaging: Observational equation is ill-posed Y = A X (Data) (Fourier Matrix) (Image) y1 y2 y3 : : = exp(i2πu1x1) exp(i2πu1x2) exp(i2πu1xn) exp(i2πu2x1) exp(i2πu2x2) exp(i2πu2xn) exp(i2πu3x1) exp(i2πu3x2) x1 x2 x3 ym exp(i2πu3xn) : : - Sampling is NOT perfect Number of data M < Number of image pixels N xn - Equation is ill-posed: infinite numbers of solutions

11 Interferometry Imaging: Observational equation is ill-posed Y = A X (Data) (Fourier Matrix) (Image) y1 y2 y3 : : = exp(i2πu1x1) exp(i2πu1x2) exp(i2πu1xn) exp(i2πu2x1) exp(i2πu2x2) exp(i2πu2xn) exp(i2πu3x1) exp(i2πu3x2) x1 x2 x3 ym exp(i2πu3xn) : : - Sampling is NOT perfect Number of data M < Number of image pixels N xn - Equation is ill-posed: infinite numbers of solutions - Interferometric Imaging: Picking a reasonable solution based on a prior assumption

12 Approach 1: Sparse Reconstruction

13 Approach 1: Sparse Reconstruction Philosophy: Reconstructing images with the smallest number of point sources within a given residual error

14 Approach 1: Sparse Reconstruction Philosophy: Reconstructing images with the smallest number of point sources within a given residual error

15 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

16 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) x 0 = number of non-zero pixels in the image

17 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) Obs. Data Image Matrix Chi-square: Consistency between data and the image Lp-norm: (p>0) x 0 = number of non-zero pixels in the image

18 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: Obs. Data Image Matrix Chi-square: Consistency Computationally very expensive!! between data and the image (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

19 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: FT of zero-filled Visibility Point Spread Function: Dirty map for the point source Solution: Point sources + Residual Map (3C 273, VLBA-MOJAVE data at 15 GHz)

20 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: FT of zero-filled Visibility Point Spread Function: Dirty map for the point source Solution: Point sources + Residual Map (3C 273, VLBA-MOJAVE data at 15 GHz)

21 age and Kazunori I is the trueneroc image. For Radio the Science CLEAN reconstructions, we chose11/04/2016 not to Akiyama, Symposium: and Related Topics, MIT Haystack Observatory, nal restored image I is residuals the true image. For thequantity CLEAN only reconstructions, e convolved model,and as the are a sensible for the full uals the convolved as the Reconstruction residuals a sensiblemodel quantity on this back choicetoon the CLEAN model, reconstruction, we choseare a compact image Approach 1: Sparse ize the e ectreconstruction of this choice on reconstruction, weinchose a compa our CLEAN for the thisclean image, the total flux left the residuals CLEAN (Hobgom 1974)reconstruction = Matching Pursuit (Mallet & Zhang 1993) After tuning our CLEAN for this image, the total flux left i InComputationally performing the CLEAN reconstruction, we used Briggs weighting and a l very cheap, but highlyreconstruction, affected by the Point Spread 7 otal image flux. In performing the CLEAN we used Briggs we rameters set to the default in the algorithm s CASA implementation. Function vs regularized likelihood (BSMEM) e rest of CLEAN the parameters set maximum to the default in the algorithm s CASA impleme Fabien Baron Image reconstruction for the EHT CLEAN is problematic for the black hole shadows? zed maximum likelihood (BSMEM) UV coverage SMT,CARMA,LMT, Ground Model (Broderick) Truth ALMA,MaunaKea CLEAN CLEAN Model (Broderick) BSMEM BSMEM Horizon Telescope Conference 20 January 2012 Fabian Event Baron+ Chael+2016 ApJ Akiyama+2016b,c, ean-square error (NRMSE, Eq. 17) of MEM and CLEAN reconstructed StokestoI ApJ images as submitted Horizon Telescope Conference 20 January 2012 For comparison, the NRMSE of (NRMSE, the model Eq. image plotted. The reconstructed ormalized root-mean-square error 17) is of also MEM and CLEAN reconstructedim S

22 Approach 1: Sparse Reconstruction L1 regularization (LASSO, Tibishirani 1996)

23 Approach 1: Sparse Reconstruction L1 regularization (LASSO, Tibishirani 1996) Convex Relaxation: Relaxing L0-norm to a convex, continuous, and differentiable function

24 Approach 1: Sparse Reconstruction L1 regularization (LASSO, Tibishirani 1996) Convex Relaxation: Relaxing L0-norm to a convex, continuous, and differentiable function

25 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

26 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+)

27 Approach 1: Sparse Reconstruction Application of LASSO (Honma et al. 2014) (Honma, Akiyama, Uemura & Ikeda 2014, PASJ)

28 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.)

29 Approach 1: Sparse Reconstruction For Smoother Image: sparsity on gradient domain Total Variation: Sparse Model (Original) regularizer Model of (Convolved) the image in EHT its 2017 gradient domain Forward Jet Model Counter Jet Model L1 + TV regularization (Akiyama et al. 2016b,c, Kuramochi+ in prep.) Fractional Polarization (Akiyama et al. 2016c subm. to ApJ) (Sgr A*; Kuramochi, Akiyama, et al. in prep.)

30 Approach 2: Maximize the Information Entropy Maximum Entropy Methods (MEM; Frieden 1972; Gull & Daniell 1978)

31 Approach 2: Maximize the Information Entropy Maximum Entropy Methods (MEM; Frieden 1972; Gull & Daniell 1978)

32 Approach 2: Maximize the Information Entropy Maximum Entropy Methods (MEM; Frieden 1972; Gull & Daniell 1978) v u P t P - Compared with CLEAN: (1) Better fidelity for Smooth Structure (2) Better optimal resolution (Chael et al. 2016, ApJ)

33 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) 14 PolMEM: Extension of MEM to full-polarimetric Imaging Chael et al. (Chael+16) (Chael et al. 2016, ApJ) 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 polarization are plotted in regions with I

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

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

36 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)

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

38 Approach 3: Machine-learn Distribution of Image Patches A patch prior (CHIRP; Bouman et al. 2015) HIRP: Continuous High Image Resolution using Patch priors Reconstruct the image so that it maximizes consistency with a machine-leaned patch prior distribution GROUND TRUTH BLURRED GROUND TRUTH CHIRP Courtesy of Kazunori Akiyama Jason Dexter, MonikaMoscibrodzka, Avery Brodrick, Hotaka Shiokawa (courtesy of Katie Bouman)

39 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)