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Equichordal tight fusion frames. (English) Zbl 1456.42043

Summary: A Grassmannian fusion frame is an optimal configuration of subspaces of a given vector space, that are useful in some applications related to representing data in signal processing. Grassmannian fusion frames are robust against noise and erasures when the signal is reconstructed. In this paper, we present an approach to construct optimal Grassmannian fusion frames based on a given Grassmannian frame. We also analyse an algorithm for sparse fusion frames which was introduced by R. Calderbank et al. [Adv. Comput. Math. 35, No. 1, 1–31 (2011; Zbl 1264.94042)] and present necessary and sufficient conditions for the output of that algorithm to be an optimal Grassmannian fusion frame.

MSC:

42C15 General harmonic expansions, frames
41A58 Series expansions (e.g., Taylor, Lidstone series, but not Fourier series)
94A12 Signal theory (characterization, reconstruction, filtering, etc.)

Citations:

Zbl 1264.94042
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Full Text: DOI

References:

[1] Bachoc, C.; Ehler, M., Tight p-fusion frames, Appl. Comput. Harmon. Anal., 35, 1, 1-15 (2013) · Zbl 1293.42031
[2] Balan, R.; Casazza, P. G.; Edidin, D., On signal reconstruction without noisy phase, Appl. Comp. Harm. Anal., 20, 3, 345-356 (2006) · Zbl 1090.94006
[3] Balan, R. V.; Daubechies, I.; Vaishampayan, V., The analysis and design of windowed Fourier frame based multiple description source coding schemes, IEEE Trans. Inf. Theory, 46, 7, 2491-2536 (2000) · Zbl 0998.94011
[4] Bodmann, B. G.; Paulsen, V. I., Frame paths and error bounds for sigma-delta quantization, Appl. Comput. Harmon. Anal., 22, 2, 176-197 (2007) · Zbl 1133.94013
[5] Boufounos, B.; Kutyniok, G.; Rauhut, H., Sparse recovery from combined fusion frame measuements, IEEE Trans. Inform. Theory., 57, 6, 3864-3876 (2011) · Zbl 1365.94066
[6] Calderbank, R.; Casazza, P. G.; Heinecke, A.; Kutyniok, G.; Pezeshki, A., Sparse fusion frame: Existence and construction, Appl. Comput. Harmon. Anal., 22, 2, 176-197 (2007) · Zbl 1133.94013
[7] Candes, E. J.; Donoho, D. L., New tight frames of curvelets and optimal representations of objects with piecewise C^2 singularities, Comm. Pure and Appl. Math., 57, 2, 219-266 (2004) · Zbl 1038.94502
[8] P. G. Casazza, A. Heinecke, and G. Kutyniok, Optimally sparse fusion frames: Existence and construction, Sampta’11 Singapore, Proc., (2011). · Zbl 1264.94042
[9] P. G. Casazza and G. Kutyniok, Frames of subspaces, In: Wavelets, frames and operator theory, College Park, MD, 2003, In: Contemp. Math., 345, Amer. Math. Soc., Providence, RI, (2004), 87-113. · Zbl 1058.42019
[10] Casazza, P. G.; Kutyniok, G., Fusion frames and distributed processing, Appl. Comput. Harmon. Anal., 25, 1, 114-132 (2008) · Zbl 1258.42029
[11] Conway, J. H.; Hardin, R. H.; Sloane, N. J A., Packing lines, planes, etc.: Packings in Grassmannian spaces, Experiment. Math., 5, 2, 139-159 (1996) · Zbl 0864.51012
[12] Daubechies, I.; Han, B.; Ron, A.; Shen, Z., Framelets: MRA-based constructions of wavelet frames, Appl. Comp. Harm. Anal., 14, 1, 1-46 (2003) · Zbl 1035.42031
[13] Eldar, Y. C., Sampling and reconstruction in arbitrary spaces and oblique dual frame vectors, J. Fourier Analys. Appl., 1, 9, 77-96 (2003) · Zbl 1028.94020
[14] Goyal, V. K.; Kovacevic, J., Quantized frame expansions with erasures, J. Fourier Analys. Appl., 1, 9, 77-96 (2003)
[15] Grochenig, K., Fundation of Time-frequency analysis (1998), Boston, MA: Birkhauser, Boston, MA
[16] Emily J. King, Grassmannian fusion frames, arXiv:1004.1086, (2010).
[17] Kutyniok, G.; Pezeshki, A.; Calderbank, R.; Liu, T., Robust dimension reduction, fusion frames, and Grassmannian packings, Appl. Comput. Harmon. Anal., 26, 1, 64-76 (2009) · Zbl 1283.42046
[18] Osgooei, E.; Arefi-Jamaal, A. A., Compare and contrast duals of fusion and discrete frames, Sahand Commun. Math. Anal., 8, 1, 83-96 (2016)
[19] Rahimi, A.; Darvishi, Z.; Daraby, B., On the duality of c-fusion frames in Hilbert spaces, Anal. Math. Phys., 7, 4, 335-348 (2016) · Zbl 1432.42024
[20] Strohmer, T.; Heath, R. W Jr., Grassmannian frames with applications to coding and communication, Appl. Comput. Harmon. Anal., 14, 3, 257-275 (2003) · Zbl 1028.42020
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