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Fusion Frames

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Finite Frames

Part of the book series: Applied and Numerical Harmonic Analysis ((ANHA))

Abstract

Novel technological advances have significantly increased the demand to model applications requiring distributed processing. Frames are, however, too restrictive for such applications, wherefore it was necessary to go beyond classical frame theory. Fusion frames, which can be regarded as frames of subspaces, satisfy exactly those needs. They analyze signals by projecting them onto multidimensional subspaces, in contrast to frames which consider only one-dimensional projections. This chapter serves as an introduction to and a survey about this exciting area of research as well as a reference for the state of the art of this research field.

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References

  1. Balan, R., Bodmann, B.G., Casazza, P.G., Edidin, D.: Painless reconstruction from magnitudes of frame coefficients. J. Fourier Anal. Appl. 15(4), 488–501 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  2. Benedetto, J.J., Fickus, M.: Finite normalized tight frames. Adv. Comput. Math. 18(2–4), 357–385 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  3. 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 

  4. Bodmann, B.G.: Optimal linear transmission by loss-insensitive packet encoding. Appl. Comput. Harmon. Anal. 22(3), 274–285 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bodmann, B.G., Casazza, P.G., Kutyniok, G.: A quantitative notion of redundancy for finite frames. Appl. Comput. Harmon. Anal. 30, 348–362 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bodmann, B.G., Casazza, P.G., Paulsen, V.I., Speegle, D.: Spanning and independence properties of frame partitions. Proc. Am. Math. Soc. 40(7), 2193–2207 (2012)

    Article  MathSciNet  Google Scholar 

  7. Bodmann, B.G., Casazza, P.G., Peterson, J., Smalyanu, I., Tremain, J.C.: Equi-isoclinic fusion frames and mutually unbiased basic sequences, preprint

    Google Scholar 

  8. Bölcskei, H., Hlawatsch, F., Feichtinger, H.G.: Frame-theoretic analysis of oversampled filter banks. IEEE Trans. Signal Process. 46, 3256–3269 (1998)

    Article  Google Scholar 

  9. Boufounos, B., Kutyniok, G., Rauhut, H.: Sparse recovery from combined fusion frame measurements. IEEE Trans. Inf. Theory 57, 3864–3876 (2011)

    Article  MathSciNet  Google Scholar 

  10. Cahill, J., Casazza, P.G., Li, S.: Non-orthogonal fusion frames and the sparsity of fusion frame operators, preprint

    Google Scholar 

  11. Calderbank, R., Casazza, P.G., Heinecke, A., Kutyniok, G., Pezeshki, A.: Sparse fusion frames: existence and construction. Adv. Comput. Math. 35(1), 1–31 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  12. Calderbank, A.R., Hardin, R.H., Rains, E.M., Shore, P.W., Sloane, N.J.A.: A group-theoretic framework for the construction of packings in Grassmannian spaces. J. Algebr. Comb. 9(2), 129–140 (1999)

    Article  MATH  Google Scholar 

  13. Candés, E.J., Romberg, J.K., Tao, T.: Stable signal recovery from incomplete and inaccurate measurements. Commun. Pure Appl. Math. 59(8), 1207–1223 (2006)

    Article  MATH  Google Scholar 

  14. Casazza, P.G., Fickus, M.: Minimizing fusion frame potential. Acta Appl. Math. 107(103), 7–24 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  15. Casazza, P.G., Fickus, M., Heinecke, A., Wang, Y., Zhou, Z.: Spectral tetris fusion frame constructions, preprint

    Google Scholar 

  16. Casazza, P.G., Fickus, M., Kovačević, J., Leon, M., Tremain, J.C.: A physical interpretation of tight frames. In: Heil, C. (ed.) Harmonic Analysis and Applications, pp. 51–76. Birkhäuser, Boston (2006)

    Chapter  Google Scholar 

  17. Casazza, P.G., Fickus, M., Mixon, D., Wang, Y., Zhou, Z.: Constructing tight fusion frames. Appl. Comput. Harmon. Anal. 30(2), 175–187 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  18. Casazza, P.G., Fickus, M., Mixon, D., Peterson, J., Smalyanau, I.: Every Hilbert space frame has a Naimark complement, preprint

    Google Scholar 

  19. Casazza, P.G., Heinecke, A., Krahmer, F., Kutyniok, G.: Optimally sparse frames. IEEE Trans. Inf. Theory 57, 7279–7287 (2011)

    Article  MathSciNet  Google Scholar 

  20. Casazza, P.G., Heinecke, A., Kutyniok, G.: Optimally sparse fusion frames: existence and construction. In: Proc. SampTA’11 (Singapore, 2011)

    Google Scholar 

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

    Chapter  Google Scholar 

  22. 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 

  23. 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 (2007)

    Google Scholar 

  24. Casazza, P.G., Leon, M.: Existence and construction of finite frames with a given frame operator. Int. J. Pure Appl. Math. 63(2), 149–158 (2010)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  26. Chebira, A., Fickus, M., Mixon, D.G.: Filter bank fusion frames. IEEE Trans. Signal Process. 59, 953–963 (2011)

    Article  MathSciNet  Google Scholar 

  27. Conway, J.H., Hardin, R.H., Sloane, N.J.A.: Packing lines, planes, etc.: packings in Grassmannian spaces. Exp. Math. 5(2), 139–159 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  28. Cvetković, Z., Vetterli, M.: Oversampled filter banks. IEEE Trans. Signal Process. 46, 1245–1255 (1998)

    Article  Google Scholar 

  29. Donoho, D.L., Elad, M.: Optimally sparse representation in general (nonorthogonal) dictionaries via l 1 minimization. Proc. Natl. Acad. Sci. USA 100(5), 2197–2202 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  30. Et-Taoui, B., Fruchard, A.: Equi-isoclinic subspaces of Euclidean space. Adv. Geom. 9(4), 471–515 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  31. Et-Taoui, B.: Equi-isoclinic planes in Euclidean even dimensional spaces. Adv. Geom. 7(3), 379–384 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  32. Et-Taoui, B.: Equi-isoclinic planes of Euclidean spaces. Indag. Math. (N. S.) 17(2), 205–219 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  33. Fornasier, M.: Quasi-orthogonal decompositions of structured frames. J. Math. Anal. Appl. 289, 180–199 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  34. Fulton, W.: Young Tableaux. With Applications to Representation Theory and Geometry. London Math. Society Student Texts, vol. 35. Cambridge University Press, Cambridge (1997)

    MATH  Google Scholar 

  35. Godsil, C.D., Hensel, A.D.: Distance regular covers of the complete graph. J. Comb. Theory, Ser. B 56(2), 205–238 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  36. Goyal, V., Kovačević, J., Kelner, J.A.: Quantized frame expansions with erasures. Appl. Comput. Harmon. Anal. 10(3), 203–233 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  37. Hoggar, S.G.: New sets of equi-isoclinic n-planes from old. Proc. Edinb. Math. Soc. 20(4), 287–291 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  38. Kutyniok, G., Pezeshki, A., Calderbank, A.R., Liu, T.: Robust dimension reduction, fusion frames, and Grassmannian packings. Appl. Comput. Harmon. Anal. 26(1), 64–76 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  39. Lemmens, P.W.H., Seidel, J.J.: Equi-isoclinic subspaces of Euclidean spaces. Ned. Akad. Wet. Proc. Ser. A 76, Indag. Math. 35, 98–107 (1973)

    MathSciNet  Google Scholar 

  40. Li, S., Yan, D.: Frame fundamental sensor modeling and stability of one-sided frame perturbation. Acta Appl. Math. 107(1–3), 91–103 (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. (to appear)

    Google Scholar 

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

    Google Scholar 

  43. 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 

  44. Sun, W.: G-frames and G-Riesz bases. J. Math. Anal. Appl. 322, 437–452 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  45. Sun, W.: Stability of G-frames. J. Math. Anal. Appl. 326, 858–868 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  46. Wootters, W.K.: Quantum mechanics without probability amplitudes. Found. Phys. 16(4), 391–405 (1986)

    Article  MathSciNet  Google Scholar 

  47. Wootters, W.K., Fields, B.D.: Optimal state-determination by mutually unbiased measurements. Ann. Phys. 191(2), 363–381 (1989)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The first author is supported by NSF DMS 1008183, NSF ATD 1042701, and AFOSR FA9550-11-1-0245. The second author acknowledges support by the Einstein Foundation Berlin, by Deutsche Forschungsgemeinschaft (DFG) Grant SPP-1324 KU 1446/13 and DFG Grant KU 1446/14, and by the DFG Research Center Matheon “Mathematics for key technologies” in Berlin. The authors are indebted to Andreas Heinecke for his careful reading of this chapter and various useful comments and suggestions.

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Authors and Affiliations

  1. Mathematics Department, University of Missouri, Columbia, MO, 65211, USA

    Peter G. Casazza

  2. Institut für Mathematik, Technische Universität Berlin, 10623, Berlin, Germany

    Gitta Kutyniok

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  1. Peter G. Casazza
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  2. Gitta Kutyniok
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Correspondence to Gitta Kutyniok .

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Editors and Affiliations

  1. , Department of Mathematics, University of Missouri, 202 Mathematical Sciences Building, Columbia, 65211, Missouri, USA

    Peter G. Casazza

  2. , Institut für Mathematik, Technische Universität Berlin, Strasse des 17. Juni 136, Berlin, 10623, Berlin, Germany

    Gitta Kutyniok

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Casazza, P.G., Kutyniok, G. (2013). Fusion Frames. In: Casazza, P., Kutyniok, G. (eds) Finite Frames. Applied and Numerical Harmonic Analysis. Birkhäuser, Boston. https://doi.org/10.1007/978-0-8176-8373-3_13

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