Abstract
We consider the problem of spatiotemporal sampling in which an initial state \(f\) of an evolution process \(f_t=A_tf\) is to be recovered from a combined set of coarse spatial samples from varying time levels \(\{t_1,\dots ,t_N\}\). This new way of sampling, which we call dynamical sampling, differs from standard sampling since at any fixed time \(t_i\) there are not enough samples to recover the function \(f\) or the state \(f_{t_i}\). Although dynamical sampling is an inverse problem, it differs from the typical inverse problems in which \(f\) is to be recovered from \(A_Tf\) for a single time \(T\). In this paper, we consider signals that are modeled by \(\ell ^2({\mathbb Z})\) or a shift invariant space \(V\subset L^2({\mathbb R})\), and are evolving under the action of a spatial convolution operator \(A\), so that \(f_{n}=A^nf\). We provide sufficient conditions for the spatiotemporal sampling problem to be solvable. In special cases, we provide error analysis based on the spectral properties of the operator \(A\).
Similar content being viewed by others
References
Aceska, R., Tang, S.: Dynamical sampling in hybrid shift invariant spaces. In: Furst, V., Kornelsen, K., Weber, E. (eds.) Operator Methods in Wavelets, Tilings, and Frame, vol. 626. Contemporary Mathematics, American Mathematics Society, Providence (2014). (To appear)
Aceska, R. Aldroubi, A., Davis, J., Petrosyan, A.: Dynamical sampling in shift invariant spaces. In: Mayeli, A., Iosevich, A., Jorgensen, P.E.T., Ólafsson, G. (eds.) Commutative and Noncommutative Harmonic Analysis and Applications. Vol. 603 of Contemporary Mathematics, American Mathematical Society, Providence, pp. 139–148, (2013)
Akyildiz, I., Su, W., Sankarasubramaniam, Y., Cayirci, E.: A survey on sensor networks. Commun. Mag. IEEE 40, 102–114 (2002)
Aldroubi, A., Gröchenig, K.: Nonuniform sampling and reconstruction in shift-invariant spaces. SIAM Rev. 43, 585–620 (2001). (electronic)
Aldroubi A., Krishtal, I.: Krylov subspace methods in dynamical sampling. Preprint
Aldroubi, A., Krishtal, I., Weber, E.: Finite dimensional dynamical sampling: an overview. In: Balan, R., Begue, M., Benedetto, J. J., Czaja, W., Okodujou, K. Excursions in harmonic analysis. Volume 3, Appl. Numer. Harmon. Anal., Birkhäuser/Springer, New York, 2015. (To appear)
Aldroubi, A., Davis, J., Krishtal, I.: Dynamical sampling: time-space trade-off. Appl. Comput. Harmon. Anal. 34, 495–503 (2013)
Aldroubi, A., Cabrelli, C., Molter, U.: Dynamical sampling trade-off in finite dimensions. Preprint.
Berenstein, C.A., Patrick, E.: Exact deconvolution for multiple convolution operators-an overview, plus performance characterizations for imaging sensors. Proc. IEEE 78, 723–734 (1990)
Bölcskei, H., Hlawatsch, F.: Oversampled modulated filter banks. In: Feichtinger, H.G., Srohmer, T. (eds.) Gabor Analysis and Algorithms, Appl. Numer. Harmon. Anal. Birkhäuser, Boston (1998)
Casazza, P.G., Kutyniok, G., Li, S.: Fusion frames and distributed processing. Appl. Comput. Harmon. Anal. 25, 114–132 (2008)
Colonna, F., Easley, G.R.: The multichannel deconvolution problem: a discrete analysis. J. Fourier Anal. Appl. 10, 351–376 (2004)
Cvetkovic, Z., Vetterli, M.: Oversampled filter banks. IEEE Trans. Signal Process. 46, 1245–1255 (1998)
Davis, J.: Dynamical sampling with a forcing term. In: Furst, V., Kornelsen, K., Weber, E. (eds.) Operator Methods in Wavelets, Tilings, and Frame vol. 626. Contemporary Mathematics, American Mathematician Society, Providence, 2014. (To appear)
Fan, K., Pall, G.: Imbedding conditions for Hermitian and normal matrices. Can. J. Math. 9, 298–304 (1957)
Garcia, A.G., Kim, J.M., Kwon, K.H., Yoon, G.J.: Multi-channel sampling on shift-invariant spaces with frame generators. Int. J. Wavelets Multiresolut. Inf. Process. 10, 1250003 (2012)
Gautschi, W.: On inverses of Vandermonde and confluent Vandermonde matrices. Numer. Math. 4, 117–123 (1962)
Halmos, P.R.: Introduction to Hilbert Space and the Theory of Spectral Multiplicity. AMS Chelsea Publishing, Providence (1998). (Reprint of the second (1957) edition)
Hormati, A., Roy, O., Lu, Y., Vetterli, M.: Distributed sampling of signals linked by sparse filtering: theory and applications. IEEE Trans. Signal Process. 58, 1095–1109 (2010)
Lu, Y., Vetterli, M., Spatial super-resolution of a diffusion field by temporal oversampling in sensor networks. In: Acoustics, Speech and Signal Processing, 2009. IEEE International Conference on ICASSP 2009, April 2009, pp. 2249–2252
Lu, Y., Dragotti, P.-L., Vetterli, M.: Localization of diffusive sources using spatiotemporal measurements. In: 49th Annual Allerton Conference on Communication, Control, and Computing (Allerton), 2011, Sept 2011, pp. 1072–1076
Mallat, S.: A Wavelet Tour of Signal Processing. Academic Press Inc., San Diego (1998)
Papoulis, A.: Generalized sampling expansion. IEEE Trans. Circ. Syst. CAS–24, 652–654 (1977)
Ranieri, J., Chebira, A., Lu, Y.M., Vetterli, M.: Sampling and reconstructing diffusion fields with localized sources. In: IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 2011, May 2011, pp. 4016–4019
Reise, G., Matz, G.: Clustered wireless sensor networks for robust distributed field reconstruction based on hybrid shift-invariant spaces. In: IEEE 10th Workshop on Signal Processing Advances in Wireless Communications, 2009. SPAWC ’09. 2009, pp. 66–70
Reise, G., Matz, G.: Distributed sampling and reconstruction of non-bandlimited fields in sensor networks based on shift-invariant spaces. In: IEEE International Conference on Acoustics, Speech and Signal Processing, ICASSP 2009. 2061–2064 (2009)
Reise, G., Matz, G.: Reconstruction of time-varying fields in wireless sensor networks using shift-invariant spaces: Iterative algorithms and impact of sensor localization errors. In: 2010 IEEE Eleventh International Workshop on Signal Processing Advances in Wireless Communications (SPAWC). pp. 1–5 (2010)
Reise, G., Matz, G., Grochenig, K.: Distributed field reconstruction in wireless sensor networks based on hybrid shift-invariant spaces. IEEE Trans. Signal Process. 60, 5426–5439 (2012)
Shannon, C.: Communication in the presence of noise. Proc. IEEE 86, 447–457 (1998)
Strang, G., Nguyen, T.: Wavelets and Filter Banks. Wellesley-Cambridge Press, Wellesley (1996)
Strohmer, T.: Finite- and infinite-dimensional models for oversampled filter banks. In: Modern Sampling Theory. Birkhäuser, Boston (2001)
Vaidyanathan, P.P., Liu, V.C.: Classical sampling theorems in the context of multirate and polyphase digital filter bank structures. IEEE Trans. Acoust. Speech Signal Process. 36, 1480–1495 (1988)
Acknowledgments
We would like to thank Rosie the cat for engaging Penelope the toddler in many games of chase, thereby leaving us time to write this manuscript. This work is supported by NSF Grant DMS-1322127 and DMS-1322099.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Karlheinz Gröchenig.
Rights and permissions
About this article
Cite this article
Aldroubi, A., Davis, J. & Krishtal, I. Exact Reconstruction of Signals in Evolutionary Systems Via Spatiotemporal Trade-off. J Fourier Anal Appl 21, 11–31 (2015). https://doi.org/10.1007/s00041-014-9359-9
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00041-014-9359-9
Keywords
- Distributed sampling
- Reconstruction
- Frames
- Wireless sensor networks
- Multichannel deconvolution
- Filter banks