Optimization and Engineering

, Volume 19, Issue 1, pp 39–70 | Cite as

Sparse \(\ell _{1}\) regularisation of matrix valued models for acoustic source characterisation

  • Laurent Hoeltgen
  • Michael Breuß
  • Gert Herold
  • Ennes Sarradj


We present a strategy for the recovery of a sparse solution of a common problem in acoustic engineering, which is the reconstruction of sound source levels and locations applying microphone array measurements. The considered task bears similarities to the basis pursuit formalism but also relies on additional model assumptions that are challenging from a mathematical point of view. Our approach reformulates the original task as a convex optimisation model. The sought solution shall be a matrix with a certain desired structure. We enforce this structure through additional constraints. By combining popular splitting algorithms and matrix differential theory in a novel framework we obtain a numerically efficient strategy. Besides a thorough theoretical consideration we also provide an experimental setup that certifies the usability of our strategy. Finally, we also address practical issues, such as the handling of inaccuracies in the measurement and corruption of the given data. We provide a post processing step that is capable of yielding an almost perfect solution in such circumstances.


Convex optimisation Sparse recovery Split Bregman Matrix differentiation Acoustic source characterisation Microphone array 

Mathematics Subject Classification

MSC 65K10 MSC 65Z05 MSC47A99 



This work has partially been funded by the German Research Foundation (DFG) within the Grant No. SA 1502/5-1. This funding is thankfully acknowledged.


  1. Bauschke HH, Borwein JM (1997) Legendre functions and the method of random Bregman projections. J Convex Anal 4(1):27–67MathSciNetMATHGoogle Scholar
  2. Blacodon D, Elias G (2004) Level estimation of extended acoustic sources using a parametric method. J Aircr 41(6):1360–1369CrossRefGoogle Scholar
  3. Bregman LM (1967) The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming. USSR Comput Math Math Phys 7(3):200–217MathSciNetCrossRefMATHGoogle Scholar
  4. Brune C, Sawatzky A, Burger M (2011) Primal and dual Bregman methods with application to optical nanoscopy. Int J Comput Vis 92:211–229MathSciNetCrossRefMATHGoogle Scholar
  5. Burger M, Osher S, Xu J, Gilboa G (2005) Nonlinear inverse scale space methods for image restoration. In: Paragios N, Faugeras OD, Chan T, Schnörr C (eds) Variational, geometric, and level set methods in computer vision (VLSM). Third international workshop, Lecture Notes in Computer Science, vol. 3752. Springer, Beijing, pp 25–36Google Scholar
  6. Burger M, Gilboa G, Osher S, Xu J (2006) Nonlinear inverse scale space methods. Commun Math Sci 4(1):175–208MathSciNetMATHGoogle Scholar
  7. Burger M, Resmerita E, He L (2007) Error estimation for Bregman iterations and inverse scale space methods. Computing 81(2–3):109–135MathSciNetCrossRefMATHGoogle Scholar
  8. Cai JF, Osher S, Shen Z (2009a) Linearized Bregman iterations for compressed sensing. Math Comput 78(267):1515–1536MathSciNetCrossRefMATHGoogle Scholar
  9. Cai JF, Osher S, Shen Z (2009b) Split Bregman methods and frame based image restoration. Multiscale Model Simul 8(2):337–369MathSciNetCrossRefMATHGoogle Scholar
  10. Chambolle A, Pock T (2011) A first-order primal-dual algorithm for convex problems with applications to imaging. J Math Imaging Vis 40:120–145MathSciNetCrossRefMATHGoogle Scholar
  11. Geiger C, Kanzow C (2002) Theorie und Numerik restringierter Optimierungsaufgaben. Springer-Lehrbuch Masterclass. Springer, BerlinCrossRefMATHGoogle Scholar
  12. Ghadimi E, Teixeira A, Shames I, Johansson M (2015) Optimal parameter selection for the alternating direction method of multipliers (ADMM): quadratic problems. IEEE Trans Automat Contr 60(3):644–658MathSciNetCrossRefMATHGoogle Scholar
  13. Goldstein T, Osher S (2009) The split Bregman method for \(\ell _1\) regularized problems. SIAM J Imaging Sci 2(2):323–343MathSciNetCrossRefMATHGoogle Scholar
  14. Goldstein T, O’Donoghue B, Setzer S, Baraniuk R (2014) Fast alternating direction optimization methods. SIAM J Imaging Sci 7(3):1588–1623MathSciNetCrossRefMATHGoogle Scholar
  15. Goldstein T, Taylor G, Barabin K, Sayre K (2015) Unwrapping ADMM: efficient distributed computing via transpose reduction. Arxiv Preprint 1504.02147, University of Maryland, College Park and United States Naval Academy, AnnapolisGoogle Scholar
  16. He L, Chang TC, Osher S, Fang T, Speier P (2006) MR image reconstruction by using the iterative refinement method and nonlinear inverse scale space methods. UCLA CAM report 06-35, University of California, Los AngelesGoogle Scholar
  17. Herold G, Sarradj E (2014) Preliminary benchmarking of microphone array methods. In: Proceedings, Berlin beamforming conference, 2014. Gesellschaft zur Förderung angewandter InformatikGoogle Scholar
  18. Herold G, Sarradj E (2015) An approach to estimate the reliability of microphone array methods. In: 21st AIAA/CEAS aeroacoustics conferenceGoogle Scholar
  19. Herold G, Sarradj E, Geyer T (2013) Covariance matrix fitting for aeroacoustic application. Fortschritte der Akustik-AIA-DAGA 2013:1926–1928Google Scholar
  20. Horn RA, Johnson CR (1990) Matrix analysis. Cambridge University Press, CambridgeMATHGoogle Scholar
  21. Horn RA, Johnson CR (1994) Topics in matrix analysis. Cambridge University Press, CambridgeMATHGoogle Scholar
  22. Högbom JA (1974) Aperture synthesis with a non-regular distribution of interferometer baselines. Astron Astrophys Suppl 15:417–426Google Scholar
  23. Kadkhodaie M, Christakopoulou K, Sanjabi M, Banerjee A (2015) Accelerated alternating direction method of multipliers. In: KDD ’15 proceedings of the 21th ACM SIGKDD international conference on knowledge discovery and data mining. ACM, New York, pp 497–506Google Scholar
  24. Lin CJ (2007) Projected gradient methods for nonnegative matrix factorization. Neural Comput 19(10):2756–2779MathSciNetCrossRefMATHGoogle Scholar
  25. Magnus JR, Neudecker H (2007) Matrix differential calculus with applications in statistics and econometrics., Wiley series in probability and statisticsWiley, LondonMATHGoogle Scholar
  26. McQueen JB (1967) Some methods for classification and analysis of multivariate observations. In: Proceedings of 5th Berkeley symposium on mathematical statistics and probability, vol. 1. University of California Press, pp 281–297Google Scholar
  27. Minka TP (2000) Old and new matrix algebra useful for statistics.
  28. Moreau JJ (1965) Proximité et dualité dans un espace hilbertien. Bulletin de la Société Mathématique de France 93:273–299MathSciNetCrossRefMATHGoogle Scholar
  29. Mueller TJ (ed) (2002) Aeroacoustic measurements. Springer, Berlin. doi: 10.1007/978-3-662-05058-3 Google Scholar
  30. Nocedal J, Wright SJ (2006) Numerical optimization., Springer series in operations research and financial engineeringSpringer, New YorkMATHGoogle Scholar
  31. Osher S, Burger M, Goldfarb D, Xu J, Yin W (2005) An iterative regularization method for total variation-based image restoration. Multiscale Model Simul 4(2):460–489MathSciNetCrossRefMATHGoogle Scholar
  32. Osher S, Mao Y, Dong B, Yin W (2010) Fast linearized Bregman iteration for compressive sensing and sparse denoising. Commun Math Sci 8(1):93–111MathSciNetCrossRefMATHGoogle Scholar
  33. Petersen KB, Pedersen MS (2012) The matrix cookbook.
  34. Pock T, Chambolle A (2011) Diagonal preconditioning for first order primal dual algorithms in convex optimization. In: 2011 IEEE international conference on computer vision (ICCV). IEEE, pp 1762–1769Google Scholar
  35. Pollock DSG (1985) Tensor products and matrix differential calculus. Linear Algebra Appl 67:169–193MathSciNetCrossRefMATHGoogle Scholar
  36. Rousseeuw PJ (1987) Silhouettes: a graphical aid to the interpretation and validation of cluster analysis. Comput Appl Math 20:53–65CrossRefMATHGoogle Scholar
  37. Rudin LI, Osher S, Fatemi E (1992) Nonlinear total variation based noise removal algorithms. Phys D 60(1–4):259–268MathSciNetCrossRefMATHGoogle Scholar
  38. Sarradj E (2010) A fast signal subspace approach for the determination of absolute levels from phased microphone array measurements. J Sound Vib 329:1553–1569CrossRefGoogle Scholar
  39. Setzer S (2009) Split Bregman algorithm, Douglas–Rachford splitting and frame shrinkage. In: Tai XC, Mörken K, Lysaker M, Lie KA (eds) Scale space and variational methods in computer vision. Lecture Notes in Computer Science, vol. 5567. Springer, pp 464–476Google Scholar
  40. Setzer S (2010) Operator splittings, Bregman methods and frame shrinkage in image processing. Int J Comput Vis 92:265–280MathSciNetCrossRefMATHGoogle Scholar
  41. Sijitsma P (2007) CLEAN based on spatial source coherence. Int J Aeroacoust 6(4):357–374CrossRefGoogle Scholar
  42. Tibshirani R, Walther G, Hastie T (2001) Estimating the number of clusters in a data set via the gap statistic. J R Stat Soc B 63(2):411–423MathSciNetCrossRefMATHGoogle Scholar
  43. Welch PD (1967) The use of fast Fourier transform for the estimation of power spectra: a method based on time averaging over short, modified periodograms. IEEE Trans Audio Electroacoust 15(2):70–73CrossRefGoogle Scholar
  44. Xu J, Osher S (2006) Iterative regularization and nonlinear inverse scale space applied to wavelet-based denoising. IEEE Trans Image Process 16(2):534–544MathSciNetCrossRefGoogle Scholar
  45. Xu Y, Yin W (2012) A block coordinate descent method for multi-convex optimization with applications to nonnegative tensor factorization and completion. Rice CAAM technical report TR12-15, Rice UniversityGoogle Scholar
  46. Xu Y, Yin W, Wen Z, Zhang Y (2012) An alternating direction algorithm for matrix completion with nonnegative factors. Front Math China 7(2):365–384MathSciNetCrossRefMATHGoogle Scholar
  47. Yardibi T, Li J, Stoica P, Cattafesta LN (2008) Sparsity constrained deconvolution approaches for acoustic source mapping. J Acoust Soc Am 123(5):2631–2642CrossRefGoogle Scholar
  48. Yin W, Osher S (2013) Error forgetting of Bregman iteration. J Sci Comput 54(2):684–695MathSciNetCrossRefMATHGoogle Scholar
  49. Yin W, Osher S, Goldfarb D, Darbon J (2007) Bregman iterative algorithms for \(\ell 1\)-minimization with applications to compressed sensing. SIAM J Imaging Sci 1(1):143–168MathSciNetCrossRefMATHGoogle Scholar
  50. Zhang X, Burger M, Bresson X, Osher S (2009) Bregmanized nonlocal regularization for deconvolution and sparse reconstruction. UCLA CAM report 09-03, University of California, Los AngelesGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  • Laurent Hoeltgen
    • 1
  • Michael Breuß
    • 1
  • Gert Herold
    • 2
  • Ennes Sarradj
    • 2
  1. 1.Chair for Applied MathematicsBrandenburg University of Technology Cottbus-SenftenbergCottbusGermany
  2. 2.Institut für Strömungsmechanik und Technische AkustikTechnische Universität BerlinBerlinGermany

Personalised recommendations