Imaging with Information Field Theory Philipp Arras November 28, 2017 Max-Planck Institute for Astrophysics, Garching, Germany Technical University of Munich Introduction Radio Aperture Synthesis dirty image 1011 reconstructed sky × 0.0012 5 4 0.0010 3 0.0008 2 flux 0.0006 1 intensity 0.0004 0 0.0002 1 − Figure 1: Dirty image Figure 2: Reconstructed sky model Supernova remnant 3C391, data from CASA tutorial on NRAO website, VLA, 128 MHz bandwidth, two spectral windows at 4.6 and 7.5 GHz. Reconstructed with RESOLVE + noise estimation, only Stokes I. 1 Inverse Problems True sky model Comes with an infinite number of degrees of freedom. 3 Fields (e.g. ρ : R R≥0) −! ! 2 Inverse Problems Data True sky model Always finite number of degrees of Comes with an infinite number of freedom. degrees of freedom. 3 Data arrays Fields (e.g. ρ : R R≥0) −! −! ! (e.g. vis.shape = (4132094,)) 2 Figure 4: Reconstructed image (! s) (s d) P j Inverse Problems (even worse) U-V-coverage 0.3 0.2 0.1 0.0 v 0.1 − 0.2 − 0.3 − 0.4 0.2 0.0 0.2 0.4 − − u Figure 3: UV coverage (! d) 3 (s d) P j Inverse Problems (even worse) U-V-coverage 0.3 0.2 0.1 0.0 v 0.1 − 0.2 − 0.3 − 0.4 0.2 0.0 0.2 0.4 − − u Figure 3: UV coverage (! d) Figure 4: Reconstructed image (! s) 3 Inverse Problems (even worse) U-V-coverage 0.3 0.2 0.1 0.0 v 0.1 − 0.2 − 0.3 − 0.4 0.2 0.0 0.2 0.4 − − u Figure 3: UV coverage (! d) Figure 4: Reconstructed image (! s) (s d) P j 3 (d s) (s) (s d) = P j(dP) P j P 3 Bayesian Inference Bayesian inference Likelihood Probability to obtain a data set given the signal is known. Definitions s := physical signal, d := data Product Rule of Probabilities aka Bayes' theorem (d s) (s) (s d)= P j P P j (d) P 4 Bayesian inference Prior Available information on the signal, prior to the measurement. Definitions s := physical signal, d := data Product Rule of Probabilities aka Bayes' theorem (d s) (s) (s d)= P j P P j (d) P 4 Bayesian inference Posterior Probability for a signal realization given the measured data. Definitions s := physical signal, d := data Product Rule of Probabilities aka Bayes' theorem (d s) (s) (s d)= P j P P j (d) P 4 Bayesian inference Evidence For inferences on s, normalization factor can be ignored in most cases. Definitions s := physical signal, d := data Product Rule of Probabilities aka Bayes' theorem (d s) (s) (s d)= P j P P j (d) P 4 This is a lot! Theoretically, infinite-dimensional. Practically, e.g. 2 5 Npix = 256 ; Nvalues = 10 655 Mio. dimensions. ! Probability Distributions Over All Possible Images ·10−51 4 ) d j s ( P 2 0 Npix Nvalues dimensions · 5 Probability Distributions Over All Possible Images ·10−51 4 ) d j s ( P 2 0 Npix Nvalues dimensions · This is a lot! Theoretically, infinite-dimensional. Practically, e.g. 2 5 5 Npix = 256 ; Nvalues = 10 655 Mio. dimensions. ! Need to extract information about (s d) P j where s is a field. 5 Information Field Theory Provides dictionary: • Statistical mechanics & Bayesian inference Field theory Define (s; d) := log (s; d). Then: • H − P (s; d) e−H(s;d) (s d) = P = : P j (d) R s e−H(s;d) P D Information Field Theory Information Field Theory := Information theory with fields. • 6 Define (s; d) := log (s; d). Then: • H − P (s; d) e−H(s;d) (s d) = P = : P j (d) R s e−H(s;d) P D Information Field Theory Information Field Theory := Information theory with fields. • Provides dictionary: • Statistical mechanics & Bayesian inference Field theory 6 Information Field Theory Information Field Theory := Information theory with fields. • Provides dictionary: • Statistical mechanics & Bayesian inference Field theory Define (s; d) := log (s; d). Then: • H − P (s; d) e−H(s;d) (s d) = P = : P j (d) R s e−H(s;d) P D 6 Information Field Theory Theory on continuous spaces, actual calculations discretized • 7 Information Field Theory Theory on continuous spaces, actual calculations discretized • 7 Wiener filter demo See Jupyter Notebook on google drive. −! 8 Wiener filter demo See Jupyter Notebook on google drive. −! 8 Wiener filter demo See Jupyter Notebook on google drive. −! 8 Wiener filter demo See Jupyter Notebook on google drive. −! 8 IFT actually works. 8 Imaging the Radio Sky: RESOLVE Challenges to be addressed Extended sources (exclude point sources for now). • Uncertainty maps. • Low Signal-to-Noise performance. • Reproducibility. • 9 Data Prior power spectrum, Posterior power spectrum 2 10 power smoothness 100 2 10− correlation 4 10− 6 prior, flat Inv- 10− 100 101 102 harmonic mode Gamma prior Data model d = Res + n RESOLVE (without noise estimation) U-V-coverage 0.3 0.2 0.1 0.0 v 0.1 − 0.2 − 0.3 − 0.4 0.2 0.0 0.2 0.4 − − u Inference algorithm 10 Prior power spectrum, Posterior power spectrum 2 10 power smoothness 100 2 10− correlation 4 10− 6 prior, flat Inv- 10− 100 101 102 harmonic mode Gamma prior Data model d = Res + n RESOLVE (without noise estimation) U-V-coverage 0.3 Data 0.2 0.1 0.0 v 0.1 − 0.2 − 0.3 − 0.4 0.2 0.0 0.2 0.4 − − u Inference algorithm 10 Prior power spectrum, Posterior power spectrum 2 10 power smoothness 100 2 10− correlation 4 10− 6 prior, flat Inv- 10− 100 101 102 harmonic mode Gamma prior RESOLVE (without noise estimation) U-V-coverage 0.3 Data 0.2 0.1 0.0 v 0.1 − 0.2 − 0.3 − 0.4 0.2 0.0 0.2 0.4 − − u Inference algorithm Data model d = Res + n 10 Posterior power spectrum 2 10 power 100 2 10− correlation 4 10− 6 10− 100 101 102 harmonic mode RESOLVE (without noise estimation) U-V-coverage 0.3 Data 0.2 0.1 0.0 v 0.1 − 0.2 − 0.3 − Prior 0.4 0.2 0.0 0.2 0.4 power− spectrum,− u Inference smoothness algorithm prior, flat Inv- Gamma prior Data model d = Res + n 10 Information Hamiltonian (without noise estimation) (d; s; τ ) = log (d; s; τ ) H − P 1 s y −1 s 1 = 2 (d Re ) N (d Re ) + 2 log N | − {z− j }j Likelihood 1 y −1 1 + 2 s S s + 2 log S | {z j }j Prior / regularization y y −τ 1 y + (α 1) τ + q e + 2 τ T τ | − {z } hyper-prior with τ being the power spectrum of signal covariance S correlation structure: X τk S = e Sk k 11 Inference Algorithm (without noise estimation) Approximate posterior: • (s d) = (s m; D) δ(τ τ ∗) P j G − · − Solve for m and τ ∗ (with NIFTy's1 help): • d Map m Power spectrum τ ∗ 1https://gitlab.mpcdf.mpg.de/ift/nifty 12 Posterior power spectrum 2 10 power 100 2 10− correlation 4 10− 6 10− 100 101 102 harmonic mode RESOLVE (without noise estimation) U-V-coverage 0.3 Data 0.2 0.1 0.0 v 0.1 − 0.2 − 0.3 − Prior 0.4 0.2 0.0 0.2 0.4 power− spectrum,− u Inference smoothness algorithm prior, flat Inv- Gamma prior Data model d = Res + n 13 power spectrum 2 10 power 100 2 10− correlation 4 10− 6 10− 100 101 102 harmonic mode RESOLVE (without noise estimation) U-V-coverage 0.3 Data 0.2 0.1 0.0 v 0.1 − 0.2 − 0.3 Prior − Posterior 0.4 0.2 0.0 0.2 0.4 power− spectrum,− u Inference smoothness algorithm prior, flat Inv- Gamma prior Data model d = Res + n 13 3C391 again. dirty image 1011 × 5 4 3 2 flux 1 0 1 − 14 3C391 again. dirty image 1011 × 5 4 3 2 flux 1 0 1 − 14 3C391 again. reconstructed sky 0.0012 0.0010 0.0008 0.0006 intensity 0.0004 0.0002 14 3C391 again. 14 power spectrum 2 10 power 100 2 10− correlation 4 10− 6 10− 100 101 102 harmonic mode 3C391 again. relative error 4.0 3.5 3.0 2.5 2.0 1.5 relative error 1.0 0.5 14 power spectrum 2 10 power 100 2 10− correlation 4 10− 6 10− 100 101 102 harmonic mode 3C391 again. relative error 4.0 3.5 3.0 2.5 2.0 1.5 relative error 1.0 0.5 14 Imaging the Gamma-Ray Sky: D3PO D3PO D3PO: Denoise, Deconvolve and Decompose Photon Observations. • Assumptions: Like RESOLVE but with Poisson statistics and add • point sources. Paper and codes: www.mpa-garching.mpg.de/ift/d3po. • 15 D3PO in action: 6.5 years all sky data Data... Selig, Vacca, Oppermann, Enlin (2015) 16 D3PO in action: 6.5 years all sky data Data...Log-data... Selig, Vacca, Oppermann, Enlin (2015) 16 D3PO in action: 6.5 years all sky data Data...Log-data...Denoised... Selig, Vacca, Oppermann, Enlin (2015) 16 D3PO in action: 6.5 years all sky data Data...Log-data...Denoised...Deconvolved... Selig, Vacca, Oppermann, Enlin (2015) 16 D3PO in action: 6.5 years all sky data Data...Log-data...Denoised...Deconvolved...Decomposed. Selig, Vacca, Oppermann, Enlin (2015) 16 D3PO in action: 6.5 years all sky data Data...Log-data...Denoised...Deconvolved...Decomposed.