TWO DISTRIBUTED ALGORITHMS FOR THE DECONVOLUTION OF LARGE RADIO-INTERFEROMETRIC MULTISPECTRAL IMAGES

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1 WO DISRIBUED ALGORIHMS FOR HE DECONVOLUION OF LARGE RADIO-INERFEROMERIC MULISPECRAL IMAGES Céine Meiier, Pasca Bianchi, Waid Hachem LCI, CNRS, eecom Paristech, Université Paris-Sacay, 73, Paris, France ABSRAC We address in this paper the deconvoution issue for radiointerferometric mutispectra images. Whereas this probem has been widey expored in the recent iterature for singe images, a few agorithms are abe to reconstruct mutispectra images three-dimensiona images [], [2]. We propose in this paper two new distributed agorithms based on the optimization methods ADMM and projected gradient PG for the reconstruction of radio-interferometric mutispectra images. We present an origina distributed architecture and a comparison of their performance on a quasi-rea data cube. Index erms ADMM, deconvoution, distributed optimization, projected gradient, radio-interferometry, mutispectra images.. INRODUCION With the advent of new generations of radio interferometers such as the Low Frequency Array LOFAR and the Square Kiometer Array SKA, a arge amount of mutispectra images wi be produced in the next few years. hese new interferometers have a very arge fied of view miions of pixes with a high spectra resoution hundreds of frequency bands in the radio wave domain. hese massive observed data are corrupted by the noise and by the instrument response. A chaenging point is the design of deconvoution agorithms that are abe to dea with the arge size of the observations. A good dea of the recent research is focused on distributed optimization agorithms aiming to sove the deconvoution issue in two cases: Large two-dimensiona monochromatic i.e., with ony one spectra band image deconvoution with different approaches: et us cite the so-caed PURIFY agorithm [3] based on the Simutaneous Direction Method of Mutipiers SDMM, [4] where the authors propose in particuar two new scaabe spitting agorithms for image reconstruction, or [5] where a compressive sensing approach is proposed. he extension to the mutispectra 3D images bearing two spatia dimensions and a spectra one. he third his work was party supported by the Agence Nationae pour a Recherche, France, under the project ANR-4-CE23-4- MAGELLAN. dimension obviousy increases the size of the deconvoution probem. he authors of [2] propose a method based on the Aternating Direction Method of Mutipiers ADMM that is known amongst other things for its abiity to be distributed [6]. his work starts with the reguarized optimization probem described in [2] and recaed in Section 2. In Section 3, a simper version of ADMM than in [2] is proposed by removing one Lagrange mutipier vector. Concurrenty, we aso propose to sove the optimization probem by resorting to the projected gradient method PG in the dua space Section 4. Distributed impementations for both these agorithms are proposed in Section 5. his distributed architecture differs from the cassica master/save one by the fact that none of the nodes stores the entire mutispectra image in cassica architecture, the goba image is stored on the master node. hanks to a minima amount of data exchange between the nodes, the required memory and the computationa cost supported by a singe node are decreased. he performances of the two agorithms are finay compared in Section PROBLEM FORMULAION Assuming that the observations are made on L frequency bands, the 3D data cube can be seen as a coection of L monochromatic images of N pixes each. Simiar to [2], the image observed at the frequency ν {ν,, ν L }, aso caed the dirty image at ν, is given by the equation y = H x + n R N where x R N is the true image vector, n is the noise vector and H is a convoution matrix representing the so-caed Point Spread Function PSF of the radio-interferometer. Stacking the L dirty images in the vector y = [y,, y L ] R M where M = N L, and denoting respectivey as 2 and the Eucidean and the norms, the optimization probem is written min x 2 y Hx µ ɛ 2 x ı R +x + Wx, 2 where the first term is the objective function, the second is a ikhonov reguarization term controed by the parameter

2 µ ɛ >, the third is a positivity constraint since we are recovering sky brightnesses where ı R +x = if one of the eements of x at east is negative, and otherwise, and the ast term is a sparsity term. Here, the sparsity is induced in the mutiresoution waveet domain for each monochromatic image, and in the Discrete Cosine ransform DC domain at each mutifrequency pixe. Specificay, µ s W s W = µ ν W ν where µ s, µ ν > are reguarization parameters, and W s W s =... R msm M W s is a bock-diagona matrix where each bock W s R msn N, acting on a monochromatic image, is the concatenation of m s orthogona waveet bases. Simiar to [2] and [3], we identify W s with a dictionary consisting in the concatenation of the first eight Daubechies waveet bases m s = 8. Finay, W ν = W ν... W ν P R M M where P is the permutation matrix that rearranges the eements of the vector x pixe by pixe, each of these pixes being represented by a vector of L frequencies, and where each bock W ν is a L L matrix representing the DC. Note that W s Ws = m s I M and W ν W ν = I M. It is moreover obvious that Wx = µ s W s x + µ ν W ν x. 3. ADMM DESCRIPION In order to sove Probem 2, we reformuate it as foows: where min x fx + gz subject to: Ax + Bz = fx = /2 y Hx µ ɛ /2 x 2 2, gz = ı R +p + µ s t + µ ν v where z = p, t, v R M R msm R M, I A = M W s, and B = I M I msm. W ν I M he associated augmented Lagrangian for ρ > is L ρ x, z, γ = fx + gz + γ p x p + γ t W s x t + γ v W ν x v + ρ 2 x p ρ 2 W s x t ρ 2 W ν x v where γ = [γ p, γ t, γ v ] R 2+msM is the vector of Lagrange mutipiers, decomposed in accordance with the right hand side of 3. Note that the dimension of this vector is smaer than in [2], where four sets of Lagrange mutipiers were used instead of three here. As it is we known, ADMM consists of the foowing iterations: x k+ = argmin L ρ x, z k, γ k 4 x z k+ = [p k+, t k+, v k+ ] = argmin L ρ x k+, z, γ k 5 z γ k+ = [γ k+ p, γ k+, γ k+ ] t v = γ k + ρax k+ + Bz k+. 6 We now write x k = [x k,..., x k L ] where each bock is of size N and thus corresponds to a monochromatic image. We do the same decomposition for p k and γ k p. Simiar decompositions are aso done for t k and γ k t where the dimensions of the bocks are this time equa to m s N. Let us consider the minimization 4. Soving the equation shows that the minimization w.r.t. x is separabe with respect to the frequencies thanks to the bock diagona structure of Ws and to the orthogonaity of Wν. After a straightforward computation, we obtain that x k+ = Q b k for each {,..., L} where Q = H H + µ ɛ m s ρi N and b k = H y γ k p, Ws γ k t, ρt k + ρ p k + P v k. 7 P W ν γ k v Here denotes the th size-n bock of a vector. he computations eading to the expression of Q make use of the isometric nature of W s. Observe that the Q can be computed once at the beginning of the agorithm whie the vectors b k need to be computed at every iteration. Finay, since H is a convoution operator, the equation Q b k can be practicay approximated by using the Fast Fourier ransform operator. Minimizations w.r.t. the vectors p and t in 5 are structuray separabe with respect to the frequencies. Writing p k+ = ρ γ k p, + xk+, we get p k+ = arg min ı R +u + γ k u R M p, x k+ u + ρ 2 xk+ u 2 2 = max, p k+ where max is taken eementwise.

3 Writing t k+ = W s x k+ + ρ γ k t,, we aso have t k+ = arg min µ s u + ρ u R msm 2 u t k+ 2 2 = t k+ max, ρ µ s t k+ where is the eementwise product. We recognize here the usua soft threshoding operator. he update of the vector v k is done at each mutifrequency pixe. Write this time v k = [v k,..., v k N ] and γ k v = [γ k v,,..., γ k v, ] where the bocks within these vectors have the size L. Let ṽ k+ i = W ν x k+ i + ρ γ k v,i where i is a size-l bock, we get v k+ i = arg min µ ν u + ρ u R L 2 = ṽ k+ i max, ρ µ ν ṽ k+ i u ṽk+ i 2 2 Finay, the inspection of Equation 6 shows that γ k p and γ k t are updated at the eve of the monochormatic images whie γ k v is updated at the mutifrequency pixe eve.. 4. PROJECED GRADIEN ON HE DUAL PROBLEM 4.. Prima and dua probems In this section, we repace Probem 2 with the probem min fx + hwx, 8 x R M where we reca that fx = /2 y Hx 2 2 +µ ɛ /2 x 2 2, and where we set h : R ms+m R +, u u. Note that the positivity assumption is now absent. In order to sove this probem, we start by writing its dua probem min f W λ + h λ 9 λ R ms+m where f is the Legendre-Fenche transform of f, defined as f φ = sup x φ, x fx. After a standard cacuation, we get f φ = 2 φ φ+φ H y+ 2 y H H Iy where = H H + µ ɛ I is a bock diagona matrix of size M M with oepitz bocks, and h λ = ı B λ where B = {λ R ms+m : λ } is the unit ba for the norm. he inspection of f and h shows that the quaification conditions for the duaity gap to be zero hod. Moreover, the sadde point x, λ satisfies x = f W λ where f is the gradient of f, given by f φ = φ + Hy Soving the dua probem using PG he dua probem 9 can be reformuated as: min f W λ. λ: λ Since f is smooth, this probem can be soved with the hep of PG see e.g. [7]. In our context, this agorithm reads: λ k+ = P λ k + ρw f W λ k = P λ k ρw W λ k Hy where ρ > and where P is the projection operator on B, being the proximity operator of h. At the ast iteration k max, the 3D cube x kmax is recovered according to the equation x kmax = f W λ kmax = H y W λ kmax. In order to provide a distributed impementation of this agorithm, we write λ k = [λ k s, λ k ν ] where λ k s R msm is processed at the eve of the monochromatic images and λ k ν R M is processed at the eve of the mutifrequency pixes. Detais are provided in the next section. 5. DISRIBUED ARCHIECURE Impementation of deconvoution agorithms on a distributed architecture is needed for memory charge reasons; SKA mutispectra data cubes are expected to be 8 tera bytes. 5.. Structure of the custer From the 2D + D structure of the mutispectra data and the spatia and spectra sparsity constraints of the minimization probems 2 and 8, the variabes x, p, t, γ p, γ t, λ s resp. v, γ v, λ ν can be evauated ony on monochromatic images resp. on pixes. A of these cacuations can be done with parae programming w.r.t. the frequencies resp. the pixes. We use a custer of machines that we divide into two groups: one for the cacuations w.r.t. the frequencies group A, the other for the cacuations w.r.t. the pixes group B. Figure iustrates the two groups of nodes architecture and exchanges between nodes. For the sake of simpicity, in figure, we assume there are as many nodes in group A resp. in group B as frequency bands resp. pixes in the mutispectra image. Note that in rea impementation, each node is in charge of severa frequencies or pixes, depending on the capacity of the custer and the image size Distributed impementation of ADMM and PG Agorithm summarizes the distribution of the aternated updating steps of the prima and dua variabes described in section 3 for the ADMM agorithm. A the cacuations are distributed on the two groups of nodes architecture introduced in the previous paragraph w.r.t. the frequencies resp. w.r.t.

4 N A A 2 A L B B 2 B N Fig. : Iustration of the distributed architecture. Nodes of group A orange bocks do parae cacuation w.r.t. the frequencies, whie nodes of group B bue bocks do parae cacuation w.r.t. the pixes. the pixes. Agorithm 2 summarizes the PG approach where the cacuation of the gradient defined in equation is distributed on the two groups of nodes architecture. he parae programming of both the agorithms on the distributed architecture figure is done using message passing interface MPI that aows the nodes of the custer to exchange data. B N 6. RESULS AND CONCLUSION he two agorithms are tested on a quasi-rea mutispectra image of size pixes. We use the radio emission image from an HII region of the M3 gaaxy figure 2 on the eft as the reference image. We buid a mutispectra image by appying a sine wave spectrum to each pixe of the image. he dirty image figure 2 on the right is obtained by convoution with a 2D Gaussian PSF and corrupted by a white Gaussian noise whose signa-to-noise ratio SNR is from 5dB to 25dB according to the frequencies L Fig. 2: Left: reference image of the M3 gaaxy. Right: Dirty image at a given frequency SNR = 25dB. We present in figure 3 the deconvoution resuts produced by ADMM and PG approaches. he reguarization parameters for ADMM and PG on the dua are set to the foowing vaues: µ ɛ =, µ s =. and µ ν =.5 that provide the best reconstruction on synthetic mutispectra images. he Agorithm Distributed ADMM agorithm Initiaize x, p, t, v, γ p, γ t, γ v whie δx k 5 do Evauate b k+ Sove Q x k+ = b k+ Send x k+ to nodes of group B p k+ max, p k+ t k+ t k+ max γ k+ p, γ k+, ρ µ s t k+ γ k p, + ρxk+ p k+ t, = γ k t, + ρw sx k+ t k+ v k+ i ṽ k+ i max, ρ µ ν ṽ k+ i γ k+ v,i γ k v,i + ρ W ν x k+ i v k+ Send v k+ i and γ k+ v k = k+ end whie return x k i to nodes of group A ADMM parameter ρ is set to, whie ρ = /K for PG where K = µ2 s ms+µ2 ν µ ɛ is the Lipschitz constant of the gradient [f W ]. he custer contains 4 nodes 2 for group A and 2 for group B. Since the two probems 2 and 8 are not stricty equivaent, we propose to use the reative variation δ x for the reconstructed image for the convergence criterion and the mean square error δ x = xk+ x k 2 x k+ 2 and MSE = E [ x k x 2], where x is the theoretica deconvoved image. As expected, using PG on the dua requires ess iterations to converge to a soution than using ADMM agorithm: the variation of the reconstructed image is ower than for ADMM case. Evauation of the reconstruction quaity is not the purpose of this paper, but MSE curves are coherent according to images reconstructed with ADMM and PG. In both case, the soution is sighty biased it can be seen on reconstructed spectrum due to the ikhonov reguarization. In this work we have proposed a preiminary study of two distributed optimization agorithm to sove the reguarized optimization probem described in [2]. Detais of both the agorithms wi be reported in an extended paper where the distributed architecture wi be more precisey deveoped. In this paper we appied the agorithms on a image.

5 Agorithm 2 Distributed PG for dua probem Initiaization Eva. H y Send H y to B Evauate and save µ s W s H y Receive H y from A Evauate and save µ ν Wν H y whie δλ k 5 do Eva. µ s W s λ k s Eva. µ ν W ν λ k ν Send µ ν W ν λ k ν to A Receive µ ν W ν λ k ν from B Eva. µ s W s λ k s + µ ν W ν λ k ν Send µ s W s λ k s + µ ν W ν λ k ν to B Eva. θ s =µ s Ws µ s W s λ k s + µ ν W ν λ k ν λ k+ s P λ k s ρ θ s µ s W s H y for nodes in group B do Receive µ s W s λ k s + µ ν W ν λ k ν from A Eva. θ ν =µ ν W ν µ s W s λ k s + µ ν W ν λ k ν λ k+ ν P λ k ν ρ θ ν µ ν W ν H y k = k+ end whie return λ k his heped to test the agorithms before appying them to rea data of size that shoud be soon avaiabe. We aso woud ike to modify the minimization probem formuation to try smoother way for imposing the positivity on the reconstructed image. 7. ACKNOWLEDGEMEN We thank Maxime Fauder for usefu suggestions regarding the distributed impementation. REFERENCES [] U. Rau and. J. Cornwe, A muti-scae muti-frequency deconvoution agorithm for synthesis imaging in radio interferometry, Astronomy & Astrophysics, vo. 532, pp. A7, 2. spectrum arbitrary unit δx frequency index 3 4 iterations ADMM PG 3 2 MSE ADMM PG iterations Fig. 3: Left coumn: resuts from ADMM agorithm. Right coumn: resuts from PG on the dua probem. From top to bottom: deconvoution resut at a given frequency, error between the reference image and the reconstructed one, estimated spectrum bue vs theoretica spectrum green, ast ine: reative variation of reconstructed image eft and mean square error right for both ADMM orange ines and PG magenta ines. [2] A. Ferrari, J. Deguignet, C. Ferrari, D. Mary, A. Schutz, and O. Smirnov, Muti-frequency image reconstruction for radio interferometry. A reguarized inverse probem approach, in SKA Pathfindes Radio Continuum Surveys SPARCS, 25. [3] R. E. Carrio, J. D. McEwen, and Y. Wiaux, Purify: a new approach to radio-interferometric imaging, Monthy Notices of the Roya Astronomica Society, vo. 439, no. 4, pp , 24. [4] A. Onose, R. E. Carrio, A. Repetti, J. D. McEwen, J.-P. hiran, J.-C. Pesquet, and Y. Wiaux, Scaabe spitting agorithms for big-data interferometric imaging in the SKA era, arxiv preprint arxiv:6.426, 26. [5] H. Garsden, J.N. Girard, J.-L. Starck, S. Corbe, C. asse, A. Woisee, J.-P. Mckean, A. S. Van Amesfoort, J. Anderson, I. M. Avruch, et a., Lofar sparse image reconstruction, Astronomy & astrophysics, vo. 575, pp. A9, 25. [6] S. Boyd, N. Parikh, E. Chu, B. Peeato, and J. Eckstein, Distributed optimization and statistica earning via the aternating direction method of mutipiers, Foundations and rends in Machine Learning, vo. 3, no., pp. 22, 2. [7] I. Daubechies, M. Defrise, and C. De Mo, An iterative threshoding agorithm for inear inverse probems with a sparsity constraint, Communications on pure and appied mathematics, vo. 57, no., pp , 4.