Avez-vous entendu parler du Compressed sensing? Atelier Inversion d ISTerre 15 Mai 2012 H-C. Nataf
Compressed Sensing (CS) Candès & Wakin, 2008 15 Mai 2012 Atelier Inversion d'isterre 2
15 Mai 2012 Atelier Inversion d'isterre Candès & Wakin, 2008 3
Sparsity and incoherence Sparsity expresses the idea that the information rate of a continuous time signal may be much smaller than suggested by its bandwidth, or that a discrete-time signal depends on a number of degrees of freedom which is comparably much smaller than its (finite) length. More precisely, CS exploits the fact that many natural signals are sparse or compressible in the sense that they have concise representations when expressed in the proper basis Y. Candès & Wakin, 2008 15 Mai 2012 Atelier Inversion d'isterre 4
Sparsity and incoherence Incoherence extends the duality between time and frequency and expresses the idea that objects having a sparse representation in Y must be spread out in the domain in which they are acquired, just as a Dirac or a spike in the time domain is spread out in the frequency domain. Put differently, incoherence says that unlike the signal of interest, the sampling/sensing waveforms have an extremely dense representation in Y. Candès & Wakin, 2008 15 Mai 2012 Atelier Inversion d'isterre 5
Sparse l1-norm Candès & Wakin, 2008 15 Mai 2012 Atelier Inversion d'isterre 6
Un petit test pour voir -20 40 Reconstruction du potentiel magnétique de la «graine» de notre expérience DTS à partir de mesures de B dans un doigt de gant. 432 data. 102 Gauss coefficients (m 6, l 11). 15 Mai 2012 Atelier Inversion d'isterre 7
Inversion par moindre carré model coefficients of the magnetic potential for the least-square fit (overdetermined ; uniform errors) predicted data (red) from this model compared to the actual measurements (blue) spread = 0.406 l1-spread = 17.3 normalized misfit = 2.875 15 Mai 2012 Atelier Inversion d'isterre 8
Inversion par moindre carré avec a priori sur les paramètres same as above for the classical Bayesian inversion with C pp -1 = l (l+1)m 2 same as above for the classical Bayesian inversion with C pp -1 = l (l+1)m 2 spread = 0.036 l1-spread = 1.3535 normalized misfit = 2.94 15 Mai 2012 Atelier Inversion d'isterre 9
Inversion «compressed sensing» model coefficients of the magnetic potential obtained by l1-norm minimization predicted data (red) from this model compared to the actual measurements (blue spread = 0.0436 l1-spread = 1.559 normalized misfit = 3.026 15 Mai 2012 Atelier Inversion d'isterre 10
Le potentiel magnétique de la graine 15 Mai 2012 Atelier Inversion d'isterre 11
Et un cas où ça ne marche pas Installer un réseau de 30 magnétomètres (B r ) à la surface de BigSister pour mesurer le champ jusqu au degré et ordre 4. 15 Mai 2012 Atelier Inversion d'isterre 12
Stratégie CS: distribution aléatoire des magnétomètres 30 probes at random positions (1 realization) 30 probes at random positions, 10% noise added, aliasing (l synthe =6, l inv =4) normalized model error ~ 10 15 Mai 2012 Atelier Inversion d'isterre 13
distribution «raisonnée» des magnétomètres 31 probes at Simon s positions 31 probes at Simon s positions, 10% noise added, aliasing (l synthe =6, l inv =4) normalized model error = 0.17 15 Mai 2012 Atelier Inversion d'isterre 14
Justin Romberg s l1magic matlab set (2005) "l1eq" = L1 minimization with equality constraints, "l1qc" = L1 minimization with quadratic (L2 norm) constraints, "l1decode" = L1 norm approximation (for channel decoding), "l1dantzig" = L1 minimization with minimal residual correlation (the Dantzig selector). "tveq" = TV minimization with equality constraints, "tvqc" = TV minimization with quadratic constrains, "tvdantzig" = TV minimization with minimal residual correlation. function xp = l1qc_logbarrier(x0, A, At, b, epsilon, lbtol, mu, cgtol, cgmaxiter) % Solve quadratically constrained l1 minimization: % min x 1 s.t. Ax - b 2 epsilon 15 Mai 2012 Atelier Inversion d'isterre 15
Mark Schmidt s LASSO * matlab set (2005) LassoActiveSet.m LassoBlockCoordinate.m LassoConstrained.m LassoGaussSeidel.m LassoGrafting.m LassoIteratedRidge.m LassoIterativeSoftThresholding.m LassoNonNegativeSquared.m LassoPrimalDualLogBarrier.m LassoProjection.m LassoShooting.m LassoSignConstraints.m LassoSubGradient.m LassoUnconstrainedApx.m function [w,wp,iteration] = LassoGaussSeidel(X, y, gamma,varargin) % This function computes the Least Squares parameters % with a penalty on the L1-norm of the parameters * Least Squares with a penalty on the L1-norm of the parameters 15 Mai 2012 Atelier Inversion d'isterre 16
Biblio An introduction to Compressive Sampling, E.J. Candès and M.B. Wakin, IEEE Signal Processing Magazine, 25, 2008. Structured Compressed Sensing: From Theory to Applications, M.F. Duarte and Y.C. Eldar, IEEE Transactions on Signal Processing, 59, 4053-4085, 2011. Compressed Sensing in Astronomy, J. Bobin, J-L. Starck and R. Ottensamer, IEEE J. Selected Topics in Signal Processing, 2, 718-726, 2008. Robust modeling with erratic data, Claerbout J-F. and F. Muir F., Geophys. Mag., 38, 826-844, 1973. 15 Mai 2012 Atelier Inversion d'isterre 17
Web sites Emmanuel Candès & Justin Romberg matlab l1 routines: http://users.ece.gatech.edu/~justin/l1magic/#code Mark Schmidt s matlab LASSO routines: http://www.di.ens.fr/~mschmidt/software/lasso.html Igor Caron s living site: Compressive Sensing: The Big Picture http://sites.google.com/site/igorcarron2/cs#reconstruction 15 Mai 2012 Atelier Inversion d'isterre 18