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Sparse fusion frames: existence and construction

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Abstract

Fusion frame theory is an emerging mathematical theory that provides a natural framework for performing hierarchical data processing. A fusion frame can be regarded as a frame-like collection of subspaces in a Hilbert space, and thereby generalizes the concept of a frame for signal representation. However, when the signal and/or subspace dimensions are large, the decomposition of the signal into its fusion frame measurements through subspace projections typically requires a large number of additions and multiplications, and this makes the decomposition intractable in applications with limited computing budget. To address this problem, in this paper, we introduce the notion of a sparse fusion frame, that is, a fusion frame whose subspaces are generated by orthonormal basis vectors that are sparse in a ‘uniform basis’ over all subspaces, thereby enabling low-complexity fusion frame decompositions. We study the existence and construction of sparse fusion frames, but our focus is on developing simple algorithmic constructions that can easily be adopted in practice to produce sparse fusion frames with desired (given) operators. By a desired (or given) operator we simply mean one that has a desired (or given) set of eigenvalues for the fusion frame operator. We start by presenting a complete characterization of Parseval fusion frames in terms of the existence of special isometries defined on an encompassing Hilbert space. We then introduce two general methodologies to generate new fusion frames from existing ones, namely the Spatial Complement Method and the Naimark Complement Method, and analyze the relationship between the parameters of the original and the new fusion frame. We proceed by establishing existence conditions for 2-sparse fusion frames for any given fusion frame operator, for which the eigenvalues are greater than or equal to two. We then provide an easily implementable algorithm for computing such 2-sparse fusion frames.

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References

  1. Balan, R., Casazza, P.G., Edidin, D.: On signal reconstruction without noisy phase. Appl. Comput. Harmon. Anal. 20, 345–356 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  2. Balan, R., Daubechies, I., Vaishampayan, V.: The analysis and design of windowed Fourier frame based multiple description source coding schemes. IEEE Trans. Inf. Theory 46, 2491–2536 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  3. Benedetto J.J., Fickus, M.: Finite normalized tight frames. Adv. Comput. Math. 18, 357–385 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bjørstad, P.J., Mandel, J.: On the spectra of sums of orthogonal projections with applications to parallel computing. BIT 1, 76–88 (1991)

    Article  Google Scholar 

  5. Bodmann, B.G.: Optimal linear transmission by loss-insensitive packet encoding. Appl. Comput. Harmon. Anal. 22, 274–285 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bodmann, B.G., Casazza, P.G., Kutyniok, G.: A quantative notion of redundancy for finite frames. arXiv:0910.5904v2

  7. Bodmann, B.G., Casazza, P.G., Kutyniok, G., Li, S., Pezeshki, A., Rozell, C.J.: www.fusionframe.org

  8. Bodmann, B.G., Paulsen, V.I.: Frame paths and error bounds for sigma-delta quantization. Appl. Comput. Harmon. Anal. 22, 176–197 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  9. Boufounos, P., Kutyniok, G., Rauhut, H.: Sparse recovery from combined fusion frame measurements. arXiv:0912.4988v1

  10. Bruckstein, A.M., Donoho, D.L., Elad, M.: From sparse solutions of systems of equations to sparse modeling of signals and images. SIAM Rev. 51, 34–81 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  11. Candès, E.J., Donoho, D.L.: New tight frames of curvelets and optimal representations of objects with piecewise C 2 singularities. Commun. Pure Appl. Math. 57, 219–266 (2004)

    Article  MATH  Google Scholar 

  12. Candès, E.J., Romberg, J., Tao T.: Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information. IEEE Trans. Inf. Theory 52, 489–509 (2006)

    Article  Google Scholar 

  13. Casazza, P.G.: The art of frame theory. Taiwan. J. Math 4, 129–201 (2000)

    MathSciNet  MATH  Google Scholar 

  14. Casazza, P.G.: Custom building finite frames. In: Wavelets, Frames and Operator Theory: Contemp. Math., vol. 345, pp. 61–86. Amer. Math. Soc., Providence, RI (2004)

    Google Scholar 

  15. Casazza, P.G., Fickus, M.: Minimizing fusion frame potential. http://www.framerc.org/ (2010). Accessed 5 June 2010

  16. Casazza, P.G., Fickus, M., Mixon, D.G., Wang, Y., Zhou, Z.: Constructing tight fusion frames. http://www.framerc.org/ (2010). Accessed 5 June 2010

  17. Casazza, P.G., Han, D., Larson, D.R.: Frames for Banach spaces. In: The Functional and Harmonic Analysis of Wavelets and Frames: San Antonio 1999, Contemp. Math., vol. 247, pp. 149–181. Amer. Math. Soc., Providence, RI (2000)

    Google Scholar 

  18. Casazza, P.G., Kutyniok, G.: Frames of subspaces. In: Wavelets, Frames and Operator Theory: College Park 2003, Contemp. Math., vol. 345, pp. 87–113. Amer. Math. Soc., Providence, RI (2004)

    Google Scholar 

  19. Casazza, P.G., Kutyniok, G.: A generalization of Gram-Schmidt orthogonalization generating all Parseval frames. Adv. Comput. Math. 27, 65–78 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  20. Casazza, P.G., Kutyniok, G.: Robustness of fusion frames under erasures of subspaces and of local frame vectors. In: Radon Transforms, Geometry and Wavelets: New Orleans 2006, Contemp. Math., vol. 464, pp. 149–160. Amer. Math. Soc., Providence, RI (2008)

    Google Scholar 

  21. Casazza, P.G., Kutyniok, G., Li, S.: Fusion frames and distributed processing. Appl. Comput. Harmon. Anal. 25, 114–132 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  22. Casazza, P.G., Kutyniok, G., Li, S., Rozell, C.J.: Modeling sensor networks with fusion frames. In: Wavelets XII: San Diego 2007, SPIE Proc., vol. 6701, pp. 67011M-1–67011M-11. SPIE, Bellingham, WA (2007)

    Google Scholar 

  23. Casazza, P.G., Leon, M.T.: Constructing frames with a given frame operator. http://www.framerc.org/ (2010). Accessed 5 June 2010

  24. Casazza, P.G., Leon, M.T.: Existence and construction of finite tight frames. J. Concr. Appl. Math. 4, 277–289 (2006)

    MathSciNet  MATH  Google Scholar 

  25. Casazza, P.G., Tremain, J.C.: The Kadison-Singer Problem in mathematics and engineering. Proc. Natl. Acad. Sci. USA 103, 2032–2039 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  26. Channappayya, S.S., Lee, J., Heath R.W. Jr., Bovik, A.C.: Frame based multiple description image coding in the wavelet domain. In: Proc. IEEE Int. Conf. Image Process.: Genova 2005, Proc., vol. 3, pp. 920–923. IEEE, Los Alamitos, CA (2005)

    Google Scholar 

  27. Chebira A., Kovačević, J.: Frames in bioimaging. In: Proc. 42nd Annual Conference on Information Sciences and Systems (CISS), pp. 727–732. Princeton University, Princeton, NJ (2008)

    Chapter  Google Scholar 

  28. Christensen, O.: An Introduction to Frames and Riesz Bases. Birkhauser, Boston, MA (2003)

    MATH  Google Scholar 

  29. Donoho, D.L.: Compressed sensing. IEEE Trans. Inf. Theory 52, 1289–1306 (2006)

    Article  MathSciNet  Google Scholar 

  30. Daubechies, I., Han, B.: The canonical dual frame of a wavelet frame. Appl. Comput. Harmon. Anal. 12, 269–285 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  31. Daubechies, I., Han, B., Ron A., Shen, Z.: Framelets: MRA-based constructions of wavelet frames. Appl. Comput. Harmon. Anal. 14, 1–46 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  32. Dykema, K., Freeman, D., Kornelson, K., Larson, D.R., Ordower, M., Weber, E.: Ellipsoidal tight frames and projection decompositions of operators. Ill. J. Math. 48, 477–489 (2004)

    MathSciNet  MATH  Google Scholar 

  33. Eldar, Y.C.: Sampling and reconstruction in arbitrary spaces and oblique dual frame vectors. J. Fourier Anal. Appl. 1, 77–96 (2003)

    Article  MathSciNet  Google Scholar 

  34. Eldar, Y.C., Forney, G.D. Jr.: Optimal tight frames and quantum measurement. IEEE Trans. Inf. Theory 48, 599–610 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  35. Heath, R.W. Jr., Bölcskei, H., Paulraj, A.J.: Space-time signaling and frame theory. In: Proc. IEEE Int. Conf. Acoust., Speech, and Signal Process.: Salt Lake City 2001, Proc., vol. 4, pp. 2445–2448. IEEE, Los Alamitos, CA (2001)

    Google Scholar 

  36. Han, B., Larson, D.R.: Frames, bases and group representations. Mem. Am. Math Soc. 147 (2000)

  37. Heath, R.W., Jr., Paulraj, A.J.: Linear dispersion codes for MIMO systems based on frame theory. IEEE Trans. Signal Process. 50, 2429–2441 (2002)

    Article  Google Scholar 

  38. Kovačević, J., Chebira, A.: Life beyond bases: the advent of frames (Part I). IEEE Signal Process. Mag. 24, 86–104 (2007)

    Google Scholar 

  39. Kovačević, J., Chebira, A.: Life beyond bases: the advent of frames (Part II). IEEE Signal Process. Mag. 24, 115–125 (2007)

    Google Scholar 

  40. Kutyniok, G., Pezeshki, A., Calderbank, A.R., Liu, T.: Robust dimension reduction, fusion frames, and Grassmannian packings. Appl. Comput. Harmon. Anal. 26, 64–76 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  41. Massey, P.G., Ruiz, M.A., Stojanoff, D.: The structure of minimizers of the frame potential on fusion frames. J. Fourier Anal. Appl. (2009). doi:10.1007/s00041-009-9098-5

    Google Scholar 

  42. Oswald, P.: Frames and Space Splittings in Hilbert Spaces. Lecture Notes, Part 1, Bell Labs, Technical Report, pp. 1–32 (1997)

  43. Kutyniok, G., Pezeshki, A., Calderbank, A.R.: Fusion frames and robust dimension reduction. In: Proc. 42nd Annual Conference on Information Sciences and Systems (CISS), pp. 264–268. Princeton University, Princeton, NJ (2008)

    Google Scholar 

  44. Rozell, C.J., Johnson, D.H.: Analyzing the robustness of redundant population codes in sensory and feature extraction systems. Neurocomputing 69, 1215–1218 (2006)

    Article  Google Scholar 

  45. Strohmer, T.: Approximation of dual Gabor frames, window decay, and wireless communications. Appl. Comput. Harmon. Anal. 11, 243–262 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  46. Strohmer, T., Heath, R.W. Jr.: Grassmannian frames with applications to coding and communication. Appl. Comput. Harmon. Anal. 14, 257–275 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  47. Tremain, J.C.: Concrete constructions of equiangular line sets II. http://www.framerc.org/ (2010). Accessed 5 June 2010

  48. Werther, T., Eldar, Y.C., Subbanna, N.K.: Dual Gabor frames: theory and computational aspects. IEEE Trans. Signal Process. 53, 4147–4158 (2005)

    Article  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Program in Applied and Computational Mathematics, Princeton University, Princeton, NJ, 08544-1000, USA

    Robert Calderbank

  2. Department of Mathematics, University of Missouri, Columbia, MO, 65211-4100, USA

    Peter G. Casazza & Andreas Heinecke

  3. Institute of Mathematics, University of Osnabrück, 49069, Osnabrück, Germany

    Gitta Kutyniok

  4. Department of Electrical and Computer Engineering, Colorado State University, Fort Collins, CO, 80523-1373, USA

    Ali Pezeshki

Authors
  1. Robert Calderbank
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  2. Peter G. Casazza
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  3. Andreas Heinecke
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  4. Gitta Kutyniok
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  5. Ali Pezeshki
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Corresponding author

Correspondence to Gitta Kutyniok.

Additional information

Communicated by Qiyu Sun.

P.G.C. and A.H. were supported by NSF DMS 0704216. G.K. would like to thank the Department of Statistics at Stanford University and the Mathematics Department at Yale University for their hospitality and support during her visits. She was supported by Deutsche Forschungsgemeinschaft (DFG) Heisenberg-Fellowship KU 1446/8-1. R.C. and A.P. were supported in part by NSF under Grant CCF-0916314, by ONR under Grant N00173-06-1- G006, and by AFOSR under Grant FA9550-05-1-0443. The authors would like to thank the American Institute of Mathematics in Palo Alto, CA, for sponsoring the workshop on “Frames for the finite world: Sampling, coding and quantization” in August 2008, which provided an opportunity for the authors to complete a major part of this work. The authors also thank the anonymous referees for their constructive suggestions, for pointing out a mistake in an earlier version of the paper, and for bringing [32] to their attention.

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Calderbank, R., Casazza, P.G., Heinecke, A. et al. Sparse fusion frames: existence and construction. Adv Comput Math 35, 1–31 (2011). https://doi.org/10.1007/s10444-010-9162-3

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