Sparse Astronomical Data Analysis. Jean-Luc Starck. Collaborators: J. Bobin., F. Sureau. CEA Saclay

Transcription

1 Sparse Astronomical Data Analysis Jean-Luc Starck Collaborators: J. Bobin., F. Sureau CEA Saclay

2 What is a good representation for data? A signal s (n samples) can be represented as sum of weighted elements of a given dictionary Dictionary (basis, frame) Atoms Ex: Haar wavelet coefficients Few large coefficients Many small coefficients Sorted index k Fast calculation of the coefficients Analyze the signal through the statistical properties of the coefficients Approximation theory uses the sparsity of the coefficients 2-2

3 Sparse Model: we consider a dictionary which has a fast transform/reconstruction operator: Local DCT Stationary textures Locally oscillatory Wavelet transform Piecewise smooth Isotropic structures Curvelet transform Piecewise smooth, edge

4 NGC2997

5 The STARLET Transform Isotropic Undecimated Wavelet Transform (a trous algorithm) " = B 3 # spline, 1 2 $(x 2 ) = 1 2 "( x 2 ) #"(x) h = [1,4,6,4,1]/16, g = % - h, h = g = % I(k,l) = c J,k,l + " J w j=1 j,k,l

6 Inverse Problem Regularization using Sparsity Y = AX + N Between all possible solutions, we want the one which has the sparsest representation in the dictionary Φ. It leads to the following optimization problem: 1 min α 1,,α T 2σ 2 Y AΦα2 + λ T α i p p, 0 p<2. i=1 X = Φα A sparse model can be interpreted in a Bayesian framework Assuming the coefficients α of the solution in the dictionary Φ follow a leptokurtic PDF with heavy tails such as the generalized Gaussian distribution form: T pdf α (α 1,...,α T ) exp λ α i p p 0 p<2. i=1

7 Denoising using a sparsity model Denoising using a sparsity prior on the solution: X is sparse in Φ, i.e.x = Φα where most of α are negligible. α arg min α 1 2 Y Φα 2 +t α p p, 0 p 1.

8 α arg min α p=0 1 2 Y Φα 2 + t2 2 α 0 ==> Solution via Iterative Hard Thresholding α (t+1) =HardThresh µt ( α (t) + µφ T (Y Φ α (t) )),µ=1/ Φ 2. 1st iteration solution: X = Φ HardThresh t (Φ T Y )= Φ,t (Y ) Exact for Φ orthonormal. Soft prior Noise Modeling

9 p=1 ==> Solution via iterative Soft Thresholding α (t+1) =SoftThresh µt ( α (t) + µφ T (Y Φ α (t) )),µ (0, 2/ Φ 2 ). 1st iteration solution: Exact for Φ orthonormal. X = Φ SoftThresh t (Φ T Y )= Φ,t (Y )

10 Sparsity and FERMI data S. Digel, Stanford I. Grenier, CEA J.-L. Starck, M.J. Fadili, S. Digel, B. Zhang and J. Chiang, "Source Detection Using a 3D Sparse Representation: Application to the Fermi Gamma-ray Space Telescope ", Astronomy and Astrophysics, 504, 2, pp , J. Schmitt, J.L. Starck, J.M. Casandjian, M.J. Fadili, I. Grenier, "Poisson Denoising on the Sphere: Application to the Fermi Gamma Ray Space Telescope", Astronomy and Astrophysics, 717, A26, 2010.

11 X E LAT Data Y underlying sources counts + src positions band: 16th band: 25th very low source intensity (~10-1 or even lower) multi-spectral images FERMI dedicated wavelets: 2D-1D Wavelets ψ(x, y, z) =ψ (xy) (x, y)ψ (z) (z) w j1,j 2 [k x,k y,k z ]=D ψ (xy) j 1 ψ (z) j 2 (x, y, z)

12 Results Multi-Spectral Image Restoration (MS-VST + [2D+1D] Wavelet) J.-L. Starck, M.J. Fadili, S. Digel, B. Zhang and J. Chiang, "Source Detection Using a 3D Sparse Representation: Application to the Fermi Gamma-ray Space Telescope ", Astronomy and Astrophysics, 504, 2, pp , A 35-source grid: 5 lines x 7 columns Thresh. Level = 4σ 16th band MS-VST 25th band MS-VST

13 Inverse Problems and Iterative Thresholding Minimizing Algorithm 1 T min α 1,,α T 2σ 2 Y AΦα2 + λ α i p p, 0 p<2. i=1 Iterative thresholding with a varying threshold was proposed in (Starck et al, 2004; Elad et al, 2005) for sparse signal decomposition in order to accelerate the convergence. The idea consists in using a different threshold at each iteration. For IST: For IHT: More Refs: Vonesch et al, 2007; Elad et al 2008; Wright et al., 2008; Nesterov, 2008 and Beck-Teboulle, 2009; Blumensath, 2008; Maleki et Donoho, 2009 ; etc.

14 DECONVOLUTION SIMULATION LUCY PIXON Wavelet

15 DECONVOLUTION - E. Pantin, J.-L. Starck, and F. Murtagh, "Deconvolution and Blind Deconvolution in Astronomy", in Blind image deconvolution: theory and applications, pp , J.-L. Starck, F. Murtagh, and M. Bertero, "The Starlet Transform in Astronomical Data Processing: Application to Source Detection and Image Deconvolution", Springer, Handbook of Mathematical Methods in Imaging, in press, 2011.

16 The Curse of Missing Data - Period detection in temporal series COROT: HD Bad pixels, cosmic rays, point sources in 2D images,...

17 Inp inting M. Elad, J.-L. Starck, D.L. Donoho, P. Querre, Simultaneous Cartoon and Texture Image Inpainting using Morphological Component Analysis (MCA)", ACHA, Vol. 19, pp , M.J. Fadili, J.-L. Starck and F. Murtagh, "Inpainting and Zooming using Sparse Representations", The Computer Journal, 52, 1, pp 64-79, Where M is the mask: M(i,j) = 0 ==> missing data M(i,j) = 1 ==> good data Iterative Hard Thresholding with a decreasing threshold. MCAlab available at:

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19 Interpolation of Missing Data: Sparse Inpainting M. Elad, J.-L. Starck, D.L. Donoho, P. Querre, Simultaneous Cartoon and Texture Image Inpainting using Morphological Component Analysis (MCA)", ACHA, Vol. 19, pp , P. Abrial, Y. Moudden, J.L. Starck, M.J. Fadili, J. Delabrouille, and M. Nguyen, "CMB Data Analysis and Sparsity", Statistical Methodology, Vol 5, No 4, pp , Where M is the mask: M(i,j) = 0 ==> missing data M(i,j) = 1 ==> good data Sparse-Inpainting preserves the ISW and the Weak Lensing signal. L. Perotto, J. Bobin, S. Plaszczynski, J.-L. Starck, and A. Lavabre, "Reconstruction of the CMB lensing for Planck", A&A, F.-X. Dupe, A. Rassat, J.-L. Starck, M. J. Fadili, Measuring the Integrated Sachs-Wolfe Effect, in press, A&A, arxiv: Theoretical justification through the sampling theory of Compressed Sensing? Rauhut and Ward, Sparse Legendre expansion via l1 minimization, Constructive Approximation journal, 2010.

20 PLANCK and Sparsity Successor of WMAP (better resolution, better sensitivity, more channels) Launched on May 14, 2009 Two instruments LFI and HFI Nine Temperature maps at 30,44,70,100,143,217,353,545,857 GHz + Polarization Angular resolutions: 33, 24, 14, 10, 7.1, 5, 5, 5, 5 ==> DATA Released in 2013

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22 The CMB exhibits Fluctuations WMAP The Cosmic Microwave Background (CMB) is a relic radiation (with a temperature equals to Kelvin) emitted 13 billion years ago when the Universe was about years old.

23 PLANCK SIMULATED POLARIZED DATA: Magnitude Orientation 2-23

24 CMB simulated map The importance of Source Separation Extra foregrounds are superimposed with the CMB!!! Point sources, galactic foregrounds,... etc IWCS - June, 1st 2011

25 30 GHz 44 GHz 70 GHz 100 GHz 143 GHz 857 GHz 545 GHz 353 GHz 217 GHz

26 Is Sparse models adapted to PLANCK Data? Components are sparse in a wavelet dictionary Spatial Domain Wavelet Domain Dust Dust CMB CMB

27 Isotropic Undecimated Wavelet on the Sphere Wavelets, Ridgelets and Curvelets on the Sphere, Astronomy & Astrophysics, 446, , Undecimated Wavelet Transform j=1 j=2 j=3 j=4 2-27

28 Curvelet functions on the sphere

29 Dictionaries Spherical Harmonics Wavelets Curvelets 29

30 Polarized Dictionary: E/B Polarized Wavelet J.-L. Starck, Y. Moudden and J. Bobin, "Polarized Wavelets and Curvelets on the Sphere", Astronomy and Astrophysics, 497, 3, pp , j=4 2-30

31 Curvelet and E/B Mode Decomposition 2-31

32 Polarized Data Denoising Where Hard thresholding corresponds to the following non linear operation:

33 Polarized Data Denoising

34 MRS Version V2.0 available since June 2010 Wavelet, Ridgelet and Curvelet on the Sphere : Software available at: J.-L. Starck, P. Abrial, Y. Moudden and M. Nguyen, Wavelets, Ridgelets and Curvelets on the Sphere, Astronomy & Astrophysics, 446, , Wavelet transforms Continuous Wavelet Transform (Mexican Hat) Orthogonal Wavelets Undecimated isotropic wavelet transform (Spline, Meyer and Needlet filters). Pyramidal wavelet transform 2. Ridgelet and Curvelet Transforms 3. Denoising using Wavelets and Curvelets 4. Gaussianity tests: Skewness, Kurtosis, Moment of order 5 and 6, Max, Higher Criticism 5. Astrophysical Component Separation (ICA on the Sphere): JADE, Fast ICA, GMCA. 6. Sparse Inpainting. Polarized Spherical Wavelets and Curvelets: SparsePol/Version 1.0 Software available at: J.-L. Starck, Y. Moudden and J. Bobin, "Polarized Wavelets and Curvelets on the Sphere", Astronomy and Astrophysics, 497, 3, pp , 2009.

35 Semi-Blind Source Separation Standard approaches amount to model each observation as a linear mixture of elementary components (i.e. CMB, SZ, Synchrotron, Free-Free, Dust...) : i; x i = j a ij s j + n j Which can be recast as: X = AS + N Blind source separation: The objective is to estimate both A and S simultaneously!! - CMB, SZ and Free-Free emission : their electromagnetic spectrum is well known (i.e. the related columns of A are known and fixed). - Synchrotron emission : rank-1 assumption / its electromagnetic spectrum is a power law with an unknown spectral index. We have nine channels and we search for nine sources: 4 sources are modeled and 5 are not modeled. 35

36 Sparse Component Separation: the GMCA Method ==> J. Bobin, J.-L. Starck, M.J. Fadili, and Y. Moudden, "Sparsity, Morphological Diversity and Blind Source Separation", IEEE Trans. on Image Processing, Vol 16, No 11, pp , J. Bobin, J.-L. Starck, M.J. Fadili, and Y. Moudden, "Blind Source Separation: The Sparsity Revolution", Advances in Imaging and Electron Physics, Vol 152, pp , Source: S =[s 1,s n ] We define a dictionary φ Data: X =[x 1,..., x n ]=AS + N GMCA searches a sparse solution S in the dictionary subject to the constraint that the norm X AS 2 is minimal. φ A and S are estimated alternately and iteratively in two steps : 1) Estimate S assuming A is fixed (iterative thresholding) : {S} = Argmin S j λ j s j W 1 + X AS 2 F,Σ 2) Estimate A assuming S is fixed (a simple least square problem) : {A} = Argmin A X AS 2 F,Σ 36

37 Planck - WG2 - Challenger 2 comparaisons - power spectrum GMCA has the lowest spectral residuals at low l. Plot shows C_l of (reconstructed CMB input CMB) evaluated at high galactic latitudes. Sam Leach (SISSA), June 19, 2008, WG2 meeting, Munich 37

38 Component Separation Planck provides data that do not share the same resolution : GHz 70GHz 44GHz 143GHz 100GHz Planck beam Spherical harmonic

39 More formally: Component Separation: more problems i; x i = b i j Globally: X = H (AS)+N where H a ij s j + n i is the multichannel convolution operator The mixture model no more holds! is singular! Spectral behavior varies spatially for some components (dust, synchroton): H 39

40 Component Separation => Use Wavelets to work at different resolutions: Planck beam H [1 9] 1 H [2 9] 2 H [3 9] 3 H [5 9] Spherical harmonic => Assume the mixing matrix varies smoothly Partitionning of the Wavelet Scales 40

41 LOCAL GMCA Undecimated Isotropic Wavelet Transform + Block Partitioning j=1 j=2 j=3 j=4 12*32 Blocks 12*16 Blocks 12*4 Blocks 12 Blocks Global

42 Channel 1 Channel 2 Channel 9 GMCA Scale

43 Inverse problem WVD y = Kf + n Wavelet-Vaguelette GMCA Decomposition f = j f = j Kf,Ψ j,k ψ j,k with K Ψ j,k = ψ j,k k (y, Ψ j,k )ψ j,k k Multi-channel WVD Y i = j K i(a j X) i + N i β j,k = Y i, Ψ (i) j,k = A j α j,k = X s, ψ j,k X s = j k à + j Y i, Ψ (i) j,k with K i Ψ(i) j,k = ψ j,k ψ j,k s GHz The sparse GMCA solution is obtained by minimizing: min α j,a j j 1 2σ 2 β j A j α j 2 s.t. α is sparse

44 CMB map estimation Input Input CMB Map

45 CMB map estimation lgmca Estimate CMB Map

46 CMB map estimation Residual at 60min Residual map for Global GMCA

47 CMB map estimation Residual at 60min Residual map for L-GMCA

48 CMB map estimation Residual per latitude Residual per band at 60min Standard deviation in mk Angle in degrees

49 Sparsity in Astrophysics Conclusions Sparsity is very efficient for Inverse problems (denoising, deconvolution, etc). Inpainting Component Separation. Perspectives PLANCK component separation. Euclid 49