From Wavelets to Compressed Sensing

arXiv:0903.3383v1 [astro-ph.IM] 19 Mar 2009 AM,Clfri nttt fTcnlg,MC275,120 217-50, M/C USA. Technology, CA-91125, of PASADENA Institute California (ACM), SRNMCLDT ANALYSIS DATA ASTRONOMICAL eSca,F911GfSrYet ee,France. cedex, Gif-Sur-Yvette F-91191 d’Astr Saclay, Service de SEDI-SAP, IRFU, Diderot, Université Paris mgs omcsrns t) uvlt hudb preferre be should curvelets (planet wavelets. etc), edges strings, containing Hence, cosmic set zero. images, to data close astronomical are specific them for of most coefficients while which few magnitude, i.e. one large representation, the has sparsest is [9 the decomposition to This transform data leads best curvelet objects. the the generally, anisotropic such More constructions analyzing in other wavelets for motived of optimal use from the for motivation key a so been and astrophysics. noise, has non-stationary – Gaussian components, of on combination noise of noise, space Poisson Poisson wavelet – and in physica noise modeling of the kind Gaussian, of in many not study noise often the is since sciences can to the Furthermore, well-adapted image and data. therefore scales, astronomical an different is transform, at transform components wavelet wavelet multisc into the Using decomposed fractals. as be as such hierarch described complex that approaches fact often to the rise structures, to gives cal due generally is This transform data to wavelet astronomical [29]. the through analysis of for sun success Background) holds the broad Microwave of this study (Cosmic and from CMB abstract, domains, their astrophysical in A all “wavelet” System, contain years. keyword papers Data ten 1000 the Astrophysics around last (NASA that the shows ADS adswww.harvard.edu) during with search analysis quick data astronomical in fr Sensing, image dat an Compressed the reconstructing or collecting of measurements. earth for impact incomplete the astronomy to the in them theory, is transferring sampling what new anal data the discuss astronomical also for o proposed based We been methods of have range that a micr sparsity paper cosmic this the in or review in suc We detec strings background. detection the representations cosmic for such proposed galaxy sparse features been anisotropic also recent and have curvelets More or star data ridgelets removal. from to ray ranging deconvolution, purposes, cosmic many and for astronomy filtering in now years etrto,cmrse sensing compressed restoration, .Bbni ihtedprmn fApidadComputational and Applied of department the with is Bobin J. .L trki ihteLbrtieAM(M 18,CEA/DSM 7158), (UMR AIM Laboratoire the with is Starck J.-L. fwvlt ersn eliorpcfaue,te r far are they features, isotropic well represent wavelets If used extensively been has (WT) transform wavelet The Terms Index Abstract srnmclDt nlssadSast:from Sparsity: and Analysis Data Astronomical Wvlt aebe sdetnieyfrseveral for extensively used been have —Wavelets Atooia aaaayi,Wvlt Curvelet, Wavelet, analysis, data —Astronomical .I I. a eest opesdSensing Compressed to Wavelets NTRODUCTION enLcSac n eoeBobin, Jerome and Starck Jean-Luc pyiu,Centre ophysique, E.California, 0 Mathematics -CNRS- aea have inof tion owave ysis. to d ary om ale a, i- ]. n h l otenx n sotie sn h ` ru”agrtm[30 algorithm trous” “à the using obtained is one next the to ( clusters. objec galaxy quasi-isotropic or or galaxies stars, isotropic thi as mostly and such wh contain features, processing, pixels data image isotropic of astronomical the of ver for set success detection is its a the transform explains with wavelet to associated This adapted transform. is well multi-scale image the input of the of pixel rdc fto1 filters 1D two of product hg frequencies)). (high where c { hoywihi togyrltdt priy n rsnsi presents compress data and spatial sparsity, for to especially astronomy, related in sensi strongly impacts compressed is cosmic the for which describes application theory V of Section ver example detection. curvelet discriminate an string give and the We detect features. and to faint powerful wavelet structu very anisotropic the are to transforms together, adapted Combined well transf is analysis. curvelet which the introduced, representation, is another applicat IV, restoration section for In used can support multiresolution trke l[0 8 o oeifrain.Tu,tealgor the Thus, information). outputs more for 28] al.[30, et Starck eopssan decomposes neigi uhthat such is indexing pc sn os oeig loigu obidteso- the build wavelet to in us detected allowing be modeling, can We noise called astronomy. interest a in of using algorithm signal space WT the popular how most show the is i (UIWT which Transform first Wavelet Isotropic We Undecimated representations. the sparse troduce on based methods analysis 2997. NGC Galaxy 1. Fig. w I T II. c h 0 w j j 2 h eopsto sahee sn h le bank filter the using achieved is decomposition The ec,w aea have we Hence, h stoi neiae aee rnfr IW)[25] (IUWT) transform wavelet undecimated Isotropic The nti ae,w eiwarneo srnmcldata astronomical of range a review we paper, this In +1 +1 1 D and w , . . . , g , [ [ HE ,l k, ,l k, utrslto support multiresolution c 2 J D w J I = ] = ] j = SOTROPIC sacas rsot eso fteoiia image original the of version smooth or coarse a is 1 + J ersns‘h eal of details ‘the represents δ c , − J c u-adary fsize of arrays sub-band h 0 } n X c [ 2 sasproiino h form the of superposition a as , m ,l k, j D × [ , ,l k, h ˜ X ut-cl ie representation pixel multi-scale U = ] n n 2 ] j NDECIMATED D image − h h c = 1 = J 1 1D c D [ j δ, ,l k, +1 h asg rmoeresolution one from passage The . [ m hnw rsn nIIhwthis how III in present we Then . g ˜ orsod otefietscale finest the to corresponds c 2 [ + ] ] ,l k, D 0 h 1 = noacefiin set coefficient a into D ] X j =1 J , [ δ n W ) ] w where c c AVELET j 0 j n [ tscale at ’ [ k ,l k, × 2 + ] h n , 2 Tepresent (The . j D T ,l m, RANSFORM stetensor the is ..each i.e. , 2 2 + − j W j ions. orm, ithm (see n ion. ere ng ] ts, n- = re , ts y y (1) s 1 ] ) ASTRONOMICAL DATA ANALYSIS 2

Fig. 2. Wavelet transform of NGC 2997 by the IUWT. The co-addition of these six images reproduces exactly the original image.

where h1D is typically a symmetric low-pass filter such as the wavelet coefficients wj [k,l] to a threshold level tj . tj is B3-Spline filter. generally taken equal to Kσj , and K is chosen between 3 Fig. 2 shows IUWT of the galaxy NGC 2997 displayed in and 5. The value of 3 corresponds to a probability of false Fig. 1 . Five wavelet scales are shown and the final smoothed detection of 0.27%. If wj [k,l] is small, then it is not significant plane (lower right). The original image is given exactly by the and could be due to noise. If wj [k,l] is large, it is significant: sum of these six images. if w [k,l] t then w [k,l] is significant | j | ≥ j j (4) if wj [k,l]

For other kind of noise (correlated noise, non stationary Example noise, etc), other solutions have been proposed to derive the A simulated Hubble Space Telescope image of a distant multiresolution support [29]. In next section, we show how cluster of galaxies is shown in Fig. 4, middle. The simulated the multiresolution support can be used for denoising and data are shown in Fig. 4, left. Wavelet deconvolution solution deconvolution. is shown Fig. 4, right. The method is stable for any kind of point spread function, and any kind of noise modeling can be considered. III. RESTORATION USING THE WAVELET TRANSFORM

A. Denoising C. Inpainting The most used filtering method is the hard thresholding, Missing data are a standard problem in astronomy. They can which consists of setting to 0 all wavelet coefficients of Y be due to bad pixels, or image area we consider as problematic which have an absolute value lower than a threshold tj due to calibration or observational problems. These masked area lead to many difficulties for post-processing, especially wj [k,l] if wj [k,l] >tj w˜j [k,l]= | | (8) to estimate statistical information such the power spectrum or 0 otherwise the bispectrum. The inpainting technique consists in filling the More generally, for a given sparse representation (wavelet, gaps. The classical image inpainting problem can be defined curvelet, etc) with its associate fast transform and fast as follows. Let X be the ideal complete image, Y the observed Tw reconstruction w, we can derive a hard thresholding de- incomplete image and L the binary mask (i.e. L[k,l]=1 if noising solutionRX from the data Y , by first estimating the we have information at pixel (k,l), L[k,l]=0 otherwise). In multiresolution support M using a given noise model, and short, we have: Y = LX. Inpainting consists in recovering X then calculating: knowing Y and L. X = wM wY. (9) Noting z 0 the l0 pseudo-norm, i.e. the number of non- R T || || 2 zero entries in z and z the classical l2 norm (i.e. z = We transform the data, multiply the coefficients by the support 2 || || || || (zk) ), we thus want to minimize: and reconstruct the solution. k P T The solution can however be improved considering the min Φ X 0 subject to Y LX ℓ2 σ, (13) X k k k − k ≤ following optimization problem min M( Y X) 2 X w w 2 where σ stands for the noise standard deviation in the noisy where M is the multiresolution supportk of YT. A solution− T cank case. It has also been shown that if is sparse enough, be obtained using the Landweber iterative scheme [22, 30]: X the l0 pseudo-norm can also be replaced by the convex l1 n+1 n n X = X + wM [ wY wX ] (10) norm (i.e. z 1 = k zk ) [14]. The solution of such R T − T an optimization|| || task can| be| obtained through an iterative If the solution is known to be positive, the positivity constraint P thresholding algorithm called MCA [15, 16] : can be introduced using the following equation: n+1 n n X = ∆ n (X + Y LX ) (14) Xn+1 = P (Xn + M [ Y Xn]) (11) Φ,λ − + Rw Tw − Tw where the nonlinear operator ∆Φ,λ(Z) consists in: where P+ is the projection on the cone of non-negative images. • decomposing the signal on the dictionary to derive This algorithm allows us to constraint the residual to have Z Φ the coefficients T . a zero value inside the multiresolution support [30]. For α =Φ Z • threshold the coefficients: , where the thresh- astronomical image filtering, iterating improves significantly α˜ = ρ(α, λ) olding operator can either be a hard thresholding (i.e. the results, especially for the photometry (i.e. the integrated ρ if and otherwise) or a soft number of photons in a given object). ρ(αi, λ) = αi αi > λ 0 thresholding (i.e. ρ|(α|, λ) = sign(α )max(0, α λ)). i i | i|− The hard thresholding corresponds to the l0 optimization problem while the soft-threshold solves that for l . B. Deconvolution 1 • reconstruct Z˜ from the thresholds coefficients α˜.

In a deconvolution problem, Y = HX+N, when the sensor The threshold parameter λn decreases with the iteration num- is linear, H is the block Toeplitz matrix. Similarly to the ber and it plays a role similar to the cooling parameter of denoising problem, the solution can be obtained minimizing the simulated annealing techniques, i.e. it allows the solution 2 minX M w(Y HX) under a positivity constraint, to escape from local minima. More details relative to this k T − k2 leading to the Landweber iterative scheme [22, 30]: optimization problem can be found in [12, 16]. For many dictionaries such as wavelets or Fourier, fast operators exist Xn+1 = P Xn + Ht M [Y HXn] (12) + Rw Tw − to decompose the signal so that the iteration of eq. 14 is very Only coefficients that belong to the multiresolution support are fast. It requires only to perform at each iteration a forward kept, while the others are set to zero [22]. At each iteration, the transform, a thresholding of the coefficients and an inverse multiresolution support M can be updated by selecting new transform. coefficients in the wavelet transform of the residual which have Example: The experiment was conducted on a simulated an absolute value larger than a given threshold. weak lensing mass map masked by a typical mask patterns ASTRONOMICAL DATA ANALYSIS 4

Fig. 4. Simulated Hubble Space Telescope image of a distant cluster of galaxies. Left: original, unaberrated and noise-free. middle: input, aberrated, noise added. Right, wavelet restoration wavelet.

Fig. 5. Left panel, simulated weak lensing mass map, middle panel, simulated mass map with the mask pattern of CFHTLS data, right panels, inpainted mass map. The region shown is 1◦ x 1◦.

[1]. They may also be simply due to foreground emission, or Fig. 6. A few first generation curvelets. to non-Gaussian instrumental noise and systematics . All these sources of non-Gaussian signatures might have (see Fig. 5). The left panel shows the simulated mass map different origins and thus different statistical and morpholog- and the middle panel show the masked map. The result of the ical characteristics. It is therefore not surprising that a large inpainting method is shown in the right panel. We note that number of studies have recently been devoted to the subject the gaps are undistinguishable by eye. More interesting, it has of the detection of non-Gaussian signatures. In [2, 21], it was been shown that, using the inpainted map, we can reach an shown that the wavelet transform was a very powerful tool to accuracy of about 1% for the power spectrum and 3% for the detect the non-Gaussian signatures. Indeed, the excess kurtosis bispectrum [19]. (4th moment) of the wavelet coefficients outperformed all the other methods (when the signal is characterized by a non-zero 4th moment). IV. FROM WAVELET TO CURVELET Finally, a major issue of the non-Gaussian studies in CMB The 2D curvelet transform [9] was developed in an attempt remains our ability to disentangle all the sources of non- to overcome some limitations inherent in former multiscale Gaussianity from one another. It has been shown it was methods e.g. the 2D wavelet, when handling smooth images possible to separate the non-Gaussian signatures associated with edges i.e. singularities along smooth curves. Basically, with topological defects (cosmic strings) from those due to the the curvelet dictionary is a multiscale pyramid of localized Doppler effect of moving clusters of galaxies (i.e. the kinetic directional functions with anisotropic support obeying a spe- Sunyaev-Zel’dovich effect), both dominated by a Gaussian −j cific parabolic scaling such that at scale 2 , its length is CMB field, by combining the excess kurtosis derived from −j/2 −j 2 and its width is 2 . This is motivated by the parabolic both the wavelet and the curvelet transforms [21]. scaling property of smooth curves. Other properties of the curvelet transform as well as decisive optimality results in The wavelet transform is suited to spherical-like sources approximation theory are reported in [8]. Notably, curvelets of non-Gaussianity, and a curvelet transform is suited to provide optimally sparse representations of manifolds which structures representing sharp and elongated structures such as are smooth away from edge singularities along smooth curves. cosmic strings. The combination of these transforms highlights Several digital curvelet transforms [23, 7] have been pro- the presence of the cosmic strings in a mixture CMB+SZ+CS. posed which attempt to preserve the essential properties of Such a combination gives information about the nature of the the continuous curvelet transform and several papers report non-Gaussian signals. The sensitivity of each transform to a on their successful application in astrophysical experiments particular shape makes it a very strong discriminating tool [24, 21, 26]. [21, 17]. Fig. 6 shows a few curvelets at different scales, orientations and locations. Fig. 7. Top, primary Cosmic Microwave Background anisotropies (left) and kinetic Sunyaev-Zel’dovich fluctuations (right). Bottom, cosmic string Application to the detection of cosmic strings simulated map (left) and simulated observation containing the previous three components (right). The wavelet function is overplotted on the Sunyaev- Some applications require the use of sophisticated statistical Zel’dovich map and the curvelet function is overplotted on the cosmic string tools in order to detect a very faint signals, embedded in noise. map. An interesting case is the detection of non-Gaussian signatures in Cosmic Microwave Background (CMB). which is of great interest for cosmologists. Indeed, the non-Gaussian signatures In order to illustrate this, we show in Fig. 7 a set of in the CMB can be related to very fundamental questions simulated maps. Primary CMB, kinetic SZ and cosmic string such as the global topology of the universe [20], superstring maps are shown respectively in Fig. 7 top left, top right theory, topological defects such as cosmic strings [6], and and bottom left. The “simulated observed map”, containing multi-field inflation [4]. The non-Gaussian signatures can, the three previous components, is displayed in Fig. 7 bottom however, have a different but still cosmological origin. They right. The primary CMB anisotropies dominate all the signals can be associated with the Sunyaev-Zel’dovich (SZ) effect [31] except at very high multipoles (very small angular scales). (inverse Compton effect) of the hot and ionized intra-cluster The wavelet function is overplotted on the kinetic Sunyaev- gas of galaxy clusters [1], with the gravitational lensing by Zel’dovich map and the curvelet function is overplotted on large scale structures, or with the reionization of the universe cosmic string map. ASTRONOMICAL DATA ANALYSIS 5

V. COMPRESSED SENSING where δi is an operator that shifts the original image X with a shift T . The term models instrumental noise or model im- A. Compressed Sensing in a nutshell δi ηi perfections. According to the compressed sensing framework, Compressed sensing (CS) [10, 13] is a new sam- each signal is projected onto the subspace ranged by Θ. Each pling/compression theory based on the revelation that one can compressed observation is then obtained as follows : exploit sparsity or compressibility when acquiring signals of i 1, ,N ; Y =Θ X (17) general interest, and that one can design nonadaptive sampling ∀ ∈{ · · · } i iΛi i techniques that condense the information in a compressible where the sets Λi are such that the union of all the signal into a small amount of data. The gist of Compressed { } Rn measurement matrices [ΘΛ1 , , ΘΛ1 ] span . In practice, Sensing (CS) relies on two fundamental properties : · · · the subsets Λi are disjoint and have a cardinality m = n/N . 1) Compressibility of the data : The signal X is said to When there is no shift between consecutive images,⌊ these⌋ be compressible if it exists a dictionary Φ where the conditions guarantee that the signal X can be reconstructed T coefficients α = Φ X, obtained after decomposing X univocally from Yi i=1,··· ,N , up to noise. The decoding step on Φ, are sparsely distributed. amounts to seeking{ } the signal x as follows : 2) Acquiring incoherent measurements : In the Com- N pressed Sensing framework, the signal X is not acquired min α ℓ1 s. t. Yi ΘΛi Φα ℓ2 < √Nǫ (18) directly; one then acquires a signal X by collecting data α k k k − k Xi=1 of the form Y = AX + η : A is an m n (with The solution of this optimization problem can be found via an mbasis and the Dirac basis) N n+1 n T n entails that the information carried by X is diluted in all X = ∆Φ,λn (X + µΘ ΘΛi (Yi ΘΛi X )) (19) − the measurements Y . Combining the incoherence of A Xi=1 and Φ with the sparsity of X in Φ makes the decoding where the nonlinear operator ∆Φ,λ(Z) is defined in Equa- T problem tractable. tion 14 and the step-size µ Λ < 2/ Θ Θ i . Similarly Θ i k Λi Λ k2 In the following, we choose the measurement matrix A to to the MCA algorithm, the thresholdP λn decreases with the be a submatrix of an orthogonal matrix Θ : the resulting iteration number towards the final value : λf ; a typical value measurement matrix is denoted Θ and obtained by picking is λf = 2 3σ. This algorithm has been shown to be very Λ − a set of columns of Θ indexed by Λ; ΘΛ is obtained by efficient for solving the problem in Equation 15 in [5]. subsampling the transformed signal ΘX. In practice, when Θ admits a fast implicit transform (i.e. discrete Fourier transform, Hadamard transform, noiselet transform), the compression step is very fast and made reliable for on-board satellite implementation. Fig. 8. Top left : Original image. Top right : First input noisy map (out of 6). A standard approach in CS attempts to reconstruct X by The PACS data already contains approximately Gaussian noise. Bottom left : Mean of the 6 input images. Bottom right : Reconstruction from noiselet-based solving CS projections. The iterative algorithm has been used with 100 iterations. min α ℓ1 s. t. Y ΘΛΦα ℓ2 <ǫ (15) α k k k − k where ǫ2 is an estimated upper bound on the noise power. a) Illustration: We compare two approaches to solve the Herschel/PACS compression problem : i) transmitting the B. Compressed sensing for the Herschel data average of 6 consecutive images (MO6), ii) compressing 6 The Herschel/PACS mission of the European Space Agency consecutive images of a raster scan and decompressing using (ESA) is facing with a strenuous compression dilemma : it Compressed Sensing. Real Herschel/PACS data are complex : needs a compression rate equal to ρ = 1/N with N = 6. A the original datum X is contaminated with a slowly varying first approach has been proposed which consists in averaging “flat field” component cf . In a short sequence of 6 consecutive N = 6 consecutive images of a raster scan and transmitting images, the flat field component is almost fixed. In this context, the final average image. Nevertheless, doing so with high the data xi i=0,··· ,1 can then be modeled as follows : { } speed raster scanning leads to a dramatic loss in resolution. X = i (X)+ η + c (20) In [5], we emphasized on the redundancy of raster scan i Tδ i f data : 2 consecutive images are almost the same images up If cf is known (which will be the case in the forthcoming (n) (n) to a small shift δ. Then, jointly compressing/decompressing experiments), i X is replaced by i X + c in Tδ Tδ f consecutive images of the same raster scan has been put Equation 19. The data have been designed by adding realistic forward to alleviate the Herchel/PACS compression dilemma. pointwise sources to real calibration measurements performed The problem then consists in recovering a single image X in mid-2007. In the following experiment, the sparsifying from N compressed and shifted noisy versions of X : dictionary is Φ is an undecimated wavelet tight frame and the measurement matrices are submatrices of the noiselet basis

i 1, ,N ; X = i (X)+ η (16) [11]. ∀ ∈{ · · · } i Tδ i ASTRONOMICAL DATA ANALYSIS 6

The top-left picture of Figure 8 features the original signal X. [8] E. J. Candès and D. L. Donoho. Curvelets – a surprisingly In the top-right panel of Figure 8. The “flat field” component effective nonadaptive representation for objects with edges. overwhelms the useful part of the data so that x has at best In A. Cohen, C. Rabut, and L.L. Schumaker, editors, Curve and Surface Fitting: Saint-Malo 1999, Nashville, TN, 1999. a level that is 30 times lower than the “flat field” component. Vanderbilt University Press. The MO6 solution (resp. the CS-based solution) is shown on [9] E.J. Candès and D. Donoho. Ridgelets: the key to high the left (resp. right) and at the bottom of Figure 8. We showed dimensional intermittency? Philosophical Transactions of the in [5] that Compressed Sensing provides a resolution enhance- Royal Society of London A, 357:2495–2509, 1999. ment that can reach 30% of the FWHM of the instrument’s [10] Emmanuel Candès, Justin Romberg, , and Terence Tao. Robust uncertainty principles: Exact signal reconstruction from highly PSF for a wide range of signal intensities (i.e. flux of X). incomplete frequency information. IEEE Trans. on Information This experiment illustrates the reliability of the CS-based Theory, 52(2):489–509, 2006. compression to deal with real-world data compression. The ef- [11] R. Coifman, F. Geshwind, and Y. Meyer. Noiselets. Appl. ficiency of Compressed Sensing applied to the Herschel/PACS Comput. Harmon. Anal., 10(1):27–44, 2001. data compression relies also on the redundancy of the data : [12] P. L. Combettes and V. R. Wajs. Signal recovery by proxi- mal forward-backward splitting. SIAM Journal on Multiscale consecutive images of a raster scan are fairly shifted versions Modeling and Simulation, 4(4):1168–1200, 2005. of a reference image. The good perfomances of CS is obtained [13] D. Donoho. Compressed sensing. IEEE Trans. on Information by merging the information of consecutive images. The same Theory, 52(4):1289–1306, 2006. data fusion scheme could be used to reconstruct with high [14] D.L. Donoho and X. Huo. Uncertainty principles and ideal accuracy wide sky areas from full raster scans. atomic decomposition. IEEE Transactions on Information Theory, 47:2845–2862, 2001. [15] M. Elad, J.-L. Starck, P. Querre, and D.L. Donoho. Simultane- VI. CONCLUSION ous Cartoon and Texture Image Inpainting using Morphological By establishing a direct link between sampling and sparsity, Component Analysis (MCA). J. on Applied and Computational compressed sensing had a huge impact in many scientific Harmonic Analysis, 19(3):340–358, 2005. [16] M.J. Fadili, J.-L Starck, and F. Murtagh. Inpainting and zooming fields, especially in astronomy. We have seen that CS could using sparse representations. The Computer Journal, 2006. offer an elegant solution to the Herschel data transfer problem. submitted. By emphasing so rigorously the importance of sparsity, com- [17] J. Jin, J.-L. Starck, D.L. Donoho, N. Aghanim, and O. Forni. pressed sensing has also shed light on all work related to sparse Cosmological non-gaussian signatures detection: Comparison of data representation (such as the wavelet transform, curvelet statistical tests. Eurasip Journal, 15:2470–2485, 2005. [18] F. Murtagh, J.-L. Starck, and A. Bijaoui. Image restoration with transform, etc.). Indeed, a signal is generally not sparse in noise suppression using a multiresolution support. Astronomy direct space (i.e. pixel space), but it can be very sparse after and Astrophysics, Supplement Series, 112:179–189, 1995. being decomposed on a specific set of functions. For inverse [19] S. Pires, J. . Starck, A. Amara, R. Teyssier, A. Refregier, and problems, compressed sensing gives a strong theoretical sup- J. Fadili. FASTLens (FAst STatistics for weak Lensing) : Fast port for methods which seek a sparse solution, since such a method for Weak Lensing Statistics and map making. AA, 2009. in press. solution may be (under appropriate conditions) the exact one. [20] A. Riazuelo, J.-P. Uzan, R. Lehoucq, and J. Weeks. Simulating Similar results are hardly accessible with other regularization cosmic microwave background maps in multi-connected spaces. methods. This explain why wavelets and curvelets are so astro-ph/0212223, 2002. successful for astronomical image denoising, deconvolution [21] J.-L. Starck, N. Aghanim, and O. Forni. Detecting cosmological and inpainting. non-gaussian signatures by multi-scale methods. Astronomy and Astrophysics, 416:9–17, 2004. [22] J.-L. Starck, A. Bijaoui, and F. Murtagh. Multiresolution support ACKNOWLEDGMENT applied to image filtering and deconvolution. CVGIP: Graphical This work was partially supported by the French National Models and Image Processing, 57:420–431, 1995. Agency for Research (ANR -08-EMER-009-01). [23] J.-L. Starck, E. Candès, and D.L. Donoho. The curvelet transform for image denoising. 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Jean-Luc Starck Jean-Luc Starck has a Ph.D from University Nice-Sophia Antipolis and an Habilita- tion from University Paris XI. He was a visitor at the European Southern Observatory (ESO) in 1993, at UCLA in 2004 and at Stanford’s statistics depart- =1.25in,clip,keepaspectratio]starck.epsment in 2000 and 2005. He has been a Researcher at CEA since 1994. His research interests include image processing, statistical methods in astrophysics and cosmology. He is an expert in multiscale methods such wavelets and curvelets, He is leader of the project Multiresolution at CEA and he is a core team member of the PLANCK ESA project. He has published more than 100 papers in different areas in scientiﬁc journals. He is also author of two books entitled Image Processing and Data Analysis: the Multiscale Approach (Cambridge University Press, 1998), and Astronomical Image and Data Analysis (Springer, 2nd edition, 2006).

Jerome Bobin Jrme Bobin graduated from the Ecole Normale Superieure (ENS) de Cachan, France, in 2005 and received the M.Sc. degree in signal and image processing from ENS Cachan and Universit Paris XI, Orsay, France. He received the Agrgation =1.25in,clip,keepaspectratio]jbobin.epsde Physique in 2004. Since 2005, he is pursuing his Ph.D with J-L.Starck at the CEA. His research interests include statistics, information theory, multiscale methods and sparse representations in signal and image processing.