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BREAKING THE COHERENCE BARRIER: A NEW THEORY FOR COMPRESSED SENSING

Part of: Nontrigonometric harmonic analysis Communication, information Integral equations, integral transforms Physiological, cellular and medical topics

Published online by Cambridge University Press:  15 February 2017

BEN ADCOCK ,
ANDERS C. HANSEN ,
CLARICE POON  and
BOGDAN ROMAN
BEN ADCOCK
Affiliation:
Department of Mathematics, Simon Fraser University, Burnaby BC, V5A 1S6, Canada; [email protected]
ANDERS C. HANSEN
Affiliation:
DAMTP, University of Cambridge, Cambridge CB3 0WA, UK; [email protected], [email protected], [email protected] Department of Mathematics, University of Oslo, 0316 OSLO, Norway
CLARICE POON
Affiliation:
DAMTP, University of Cambridge, Cambridge CB3 0WA, UK; [email protected], [email protected], [email protected]
BOGDAN ROMAN
Affiliation:
DAMTP, University of Cambridge, Cambridge CB3 0WA, UK; [email protected], [email protected], [email protected]

Abstract

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This paper presents a framework for compressed sensing that bridges a gap between existing theory and the current use of compressed sensing in many real-world applications. In doing so, it also introduces a new sampling method that yields substantially improved recovery over existing techniques. In many applications of compressed sensing, including medical imaging, the standard principles of incoherence and sparsity are lacking. Whilst compressed sensing is often used successfully in such applications, it is done largely without mathematical explanation. The framework introduced in this paper provides such a justification. It does so by replacing these standard principles with three more general concepts: asymptotic sparsity, asymptotic incoherence and multilevel random subsampling. Moreover, not only does this work provide such a theoretical justification, it explains several key phenomena witnessed in practice. In particular, and unlike the standard theory, this work demonstrates the dependence of optimal sampling strategies on both the incoherence structure of the sampling operator and on the structure of the signal to be recovered. Another key consequence of this framework is the introduction of a new structured sampling method that exploits these phenomena to achieve significant improvements over current state-of-the-art techniques.

MSC classification

Primary: 94A20: Sampling theory 94A08: Image processing (compression, reconstruction, etc.)
Secondary: 42C40: Wavelets and other special systems 65R32: Inverse problems 92C55: Biomedical imaging and signal processing
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 5 , 2017 , e4
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2017

References

Adcock, B. and Hansen, A. C., ‘A generalized sampling theorem for stable reconstructions in arbitrary bases’, J. Fourier Anal. Appl. 18(4) (2012), 685–716.Google Scholar
Adcock, B. and Hansen, A. C., ‘Stable reconstructions in Hilbert spaces and the resolution of the Gibbs phenomenon’, Appl. Comput. Harmon. Anal. 32(3) (2012), 357–388.CrossRefGoogle Scholar
Adcock, B. and Hansen, A. C., ‘Generalized sampling and infinite-dimensional compressed sensing’, Found. Comput. Math. 16(5) (2016), 1263–1323.CrossRefGoogle Scholar
Adcock, B., Hansen, A. C., Herrholz, E. and Teschke, G., ‘Generalized sampling: extension to frames and inverse and ill-posed problems’, Inverse Problems 29(1) (2013), 015008.Google Scholar
Adcock, B., Hansen, A. C. and Poon, C., ‘Beyond consistent reconstructions: optimality and sharp bounds for generalized sampling, and application to the uniform resampling problem’, SIAM J. Math. Anal. 45(5) (2013), 3114–3131.CrossRefGoogle Scholar
Adcock, B., Hansen, A. C. and Poon, C., ‘On optimal wavelet reconstructions from Fourier samples: linearity and universality of the stable sampling rate’, Appl. Comput. Harmon. Anal. 36(3) (2014), 387–415.Google Scholar
Adcock, B., Hansen, A. C., Roman, B. and Teschke, G., ‘Generalized sampling: stable reconstructions, inverse problems and compressed sensing over the continuum’, Adv. Imaging Electron Phys. 182 (2014), 187–279.Google Scholar
Baraniuk, R. G., Cevher, V., Duarte, M. F. and Hedge, C., ‘Model-based compressive sensing’, IEEE Trans. Inform. Theory 56(4) (2010), 1982–2001.CrossRefGoogle Scholar
Bastounis, A. and Hansen, A. C., ‘On the absence of uniform recovery in many real-world applications of compressed sensing and the RIP & nullspace property in levels’, SIAM J. Imaging Sci. (to appear).Google Scholar
Bigot, J., Boyer, C. and Weiss, P., ‘An analysis of block sampling strategies in compressed sensing’, IEEE Trans. Inform. Theory 62(4) (2016), 2125–2139.Google Scholar
Bourrier, A., Davies, M. E., Peleg, T., Pérez, P. and Gribonval, R., ‘Fundamental performance limits for ideal decoders in high-dimensional linear inverse problems’, IEEE Trans. Inform. Theory 60(12) (2014), 7928–7946.CrossRefGoogle Scholar
Boyer, C., Bigot, J. and Weiss, P., ‘Compressed sensing with structured sparsity and structured acquisition’, Preprint, 2015, arXiv:1505.01619.Google Scholar
Candès, E. and Donoho, D. L., ‘Recovering edges in ill-posed inverse problems: optimality of curvelet frames’, Ann. Statist. 30(3) (2002), 784–842.CrossRefGoogle Scholar
Candès, E. J., ‘An introduction to compressive sensing’, IEEE Signal Process. Mag. 25(2) (2008), 21–30.CrossRefGoogle Scholar
Candès, E. J. and Donoho, D., ‘New tight frames of curvelets and optimal representations of objects with piecewise C 2 singularities’, Comm. Pure Appl. Math. 57(2) (2004), 219–266.Google Scholar
Candès, E. J. and Plan, Y., ‘A probabilistic and RIPless theory of compressed sensing’, IEEE Trans. Inform. Theory 57(11) (2011), 7235–7254.Google Scholar
Candès, E. J. and Romberg, J., ‘Sparsity and incoherence in compressive sampling’, Inverse Problems 23(3) (2007), 969–985.Google Scholar
Candès, E. J., Romberg, J. and Tao, T., ‘Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information’, IEEE Trans. Inform. Theory 52(2) (2006), 489–509.Google Scholar
Carson, W. R., Chen, M., Rodrigues, M. R. D., Calderbank, R. and Carin, L., ‘Communications-inspired projection design with application to compressive sensing’, SIAM J. Imaging Sci. 5(4) (2012), 1185–1212.Google Scholar
Chauffert, N., Ciuciu, P., Kahn, J. and Weiss, P., ‘Variable density sampling with continuous trajectories’, SIAM J. Imaging Sci. 7(4) (2014), 1962–1992.Google Scholar
Chauffert, N., Weiss, P., Kahn, J. and Ciuciu, P., ‘Gradient waveform design for variable density sampling in magnetic resonance imaging’, Preprint, 2014, arXiv:1412.4621.Google Scholar
Chi, Y., Scharf, L. L., Pezeshki, A. and Calderbank, R., ‘Sensitivity to basis mismatch in compressed sensing’, IEEE Trans. Signal Process. 59(5) (2011), 2182–2195.Google Scholar
Cohen, A., Dahmen, W. and DeVore, R., ‘Compressed sensing and best k-term approximation’, J. Amer. Math. Soc. 22(1) (2009), 211–231.Google Scholar
Cormen, T. H., Stein, C., Rivest, R. L. and Leiserson, C. E., Introduction to Algorithms, 2nd edn (MIT Press, Cambridge, MA; McGraw-Hill Book Co., Boston, MA, 2001).Google Scholar
Dahlke, S., Kutyniok, G., Maass, P., Sagiv, C., Stark, H.-G. and Teschke, G., ‘The uncertainty principle associated with the continuous shearlet transform’, Int. J. Wavelets Multiresolut. Inf. Process. 6(2) (2008), 157–181.Google Scholar
Dahlke, S., Kutyniok, G., Steidl, G. and Teschke, G., ‘Shearlet coorbit spaces and associated Banach frames’, Appl. Comput. Harmon. Anal. 27(2) (2009), 195–214.Google Scholar
Daubechies, I., ‘Orthonormal bases of compactly supported wavelets’, Comm. Pure Appl. Math. 41(7) (1988), 909–996.Google Scholar
Daubechies, I., Ten Lectures on Wavelets, CBMS-NSF Regional Conference Series in Applied Mathematics (Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992).Google Scholar
Davenport, M. A., Duarte, M. F., Eldar, Y. C. and Kutyniok, G., ‘Introduction to compressed sensing’, inCompressed Sensing: Theory and Applications (Cambridge University Press, Cambridge, 2011).Google Scholar
DeVore, R. A., ‘Nonlinear approximation’, Acta Numer. 7 (1998), 51–150.Google Scholar
Do, M. N. and Vetterli, M., ‘The contourlet transform: an efficient directional multiresolution image representation’, IEEE Trans. Image Process. 14(12) (2005), 2091–2106.Google Scholar
Donoho, D. L., ‘Compressed sensing’, IEEE Trans. Inform. Theory 52(4) (2006), 1289–1306.Google Scholar
Donoho, D. L. and Huo, X., ‘Uncertainty principles and ideal atomic decomposition’, IEEE Trans. Inform. Theory 47 (2001), 2845–2862.Google Scholar
Donoho, D. L. and Kutyniok, G., ‘Microlocal analysis of the geometric separation problem’, Comm. Pure Appl. Math. 66(1) (2013), 1–47.Google Scholar
Donoho, D. L. and Tanner, J., ‘Neighborliness of randomly-projected simplices in high dimensions’, Proc. Natl Acad. Sci. USA 102(27) (2005), 9452–9457.CrossRefGoogle Scholar
Donoho, D. L. and Tanner, J., ‘Counting faces of randomly-projected polytopes when the projection radically lowers dimension’, J. Amer. Math. Soc. 22(1) (2009), 1–53.Google Scholar
Duarte, M. F., Davenport, M. A., Takhar, D., Laska, J., Kelly, K. and Baraniuk, R. G., ‘Single-pixel imaging via compressive sampling’, IEEE Signal Process. Mag. 25(2) (2008), 83–91.Google Scholar
Duarte, M. F. and Eldar, Y. C., ‘Structured compressed sensing: from theory to applications’, IEEE Trans. Signal Process. 59(9) (2011), 4053–4085.CrossRefGoogle Scholar
Eldar, Y. C. and Kutyniok, G. (Eds.), Compressed Sensing: Theory and Applications (Cambridge University Press, Cambridge, 2012).CrossRefGoogle Scholar
Fornasier, M. and Rauhut, H., ‘Compressive sensing’, inHandbook of Mathematical Methods in Imaging (Springer, New York, NY, 2011), 187–228.Google Scholar
Foucart, S. and Rauhut, H., A Mathematical Introduction to Compressive Sensing (Birkhäuser/Springer, New York, NY, 2013).CrossRefGoogle Scholar
Gröchenig, K., Rzeszotnik, Z. and Strohmer, T., ‘Convergence analysis of the finite section method and banach algebras of matrices’, Integr. Equat. Oper. Th. 67(2) (2010), 183–202.Google Scholar
Gross, D., ‘Recovering low-rank matrices from few coefficients in any basis’, IEEE Trans. Inform. Theory 57(3) (2011), 1548–1566.CrossRefGoogle Scholar
Gross, D., Krahmer, F. and Kueng, R., ‘A partial derandomization of phaselift using spherical designs’, J. Fourier Anal. Appl. 21(2) (2015), 229–266.Google Scholar
Guerquin-Kern, M., Häberlin, M., Pruessmann, K. and Unser, M., ‘A fast wavelet-based reconstruction method for magnetic resonance imaging’, IEEE Transactions on Medical Imaging 30(9) (2011), 1649–1660.Google Scholar
Guerquin-Kern, M., Lejeune, L., Pruessmann, K. P. and Unser, M., ‘Realistic analytical phantoms for parallel Magnetic Resonance Imaging’, IEEE Trans. Med. Imaging 31(3) (2012), 626–636.CrossRefGoogle ScholarPubMed
Hansen, A. C., ‘On the approximation of spectra of linear operators on hilbert spaces’, J. Funct. Anal. 254(8) (2008), 2092–2126.Google Scholar
Hansen, A. C., ‘On the solvability complexity index, the n-pseudospectrum and approximations of spectra of operators’, J. Amer. Math. Soc. 24(1) (2011), 81–124.CrossRefGoogle Scholar
Herman, M. A., ‘Compressive sensing with partial-complete, multiscale Hadamard waveforms’, inImaging and Applied Optics (Optical Society of America, Arlington, VA, 2013), CM4C.3.Google Scholar
Herman, M. A., Weston, T., McMackin, L., Li, Y., Chen, J. and Kelly, K. F., ‘Recent results in single-pixel compressive imaging using selective measurement strategies’, inProc. SPIE 9484, Compressive Sensing IV 94840A (SPIE, Baltimore, MD, 2015).Google Scholar
Hernández, E. and Weiss, G., A First Course on Wavelets, Studies in Advanced Mathematics (CRC Press, Boca Raton, FL, 1996).Google Scholar
Hrycak, T. and Gröchenig, K., ‘Pseudospectral Fourier reconstruction with the modified inverse polynomial reconstruction method’, J. Comput. Phys. 229(3) (2010), 933–946.Google Scholar
Jones, A., Tamtögl, A., Calvo-Almazán, I. and Hansen, A., ‘Continuous compressed sensing for surface dynamical processes with helium atom scattering’, Sci. Rep. 6 (2016), 27776 EP –, 06.Google Scholar
Jones, A. D., Adcock, B. and Hansen, A. C., ‘On asymptotic incoherence and its implications for compressed sensing of inverse problems’, IEEE Trans. Inform. Theory 62(2) (2016), 1020–1037.Google Scholar
Krahmer, F. and Ward, R., ‘Stable and robust sampling strategies for compressive imaging’, IEEE Trans. Image Process. 23(2) (2014), 612–622.CrossRefGoogle ScholarPubMed
Kutyniok, G., Lemvig, J. and Lim, W.-Q., ‘Compactly supported shearlets’, inApproximation Theory XIII: San Antonio 2010, (eds. Neamtu, M. and Schumaker, L.) Springer Proceedings in Mathematics, 13 (Springer, New York, 2012), 163–186.Google Scholar
Kutyniok, G. and Lim, W.-Q., ‘Optimal compressive imaging of Fourier data’, Preprint, 2015, arXiv:1510.05029.Google Scholar
Larson, P. E. Z., Hu, S., Lustig, M., Kerr, A. B., Nelson, S. J., Kurhanewicz, J., Pauly, J. M. and Vigneron, D. B., ‘Fast dynamic 3D MR spectroscopic imaging with compressed sensing and multiband excitation pulses for hyperpolarized 13c studies’, Magn. Reson. Med. 65(3) (2011), 610–619.CrossRefGoogle ScholarPubMed
Ledoux, M., The Concentration of Measure Phenomenon, Mathematical Surveys and Monographs, 89 (American Mathematical Society, Providence, RI, 2001).Google Scholar
Li, C. and Adcock, B., ‘Compressed sensing with local structure: uniform recovery guarantees for the sparsity in levels class’, Preprint, 2016, arXiv:1601.01988.Google Scholar
Lustig, M., ‘Sparse MRI’, PhD Thesis, Stanford University, 2008.Google Scholar
Lustig, M., Donoho, D. L. and Pauly, J. M., ‘Sparse MRI: the application of compressed sensing for rapid MRI imaging’, Magn. Reson. Imaging 58(6) (2007), 1182–1195.Google Scholar
Lustig, M., Donoho, D. L., Santos, J. M. and Pauly, J. M., ‘Compressed Sensing MRI’, IEEE Signal Process. Mag. 25(2) (2008), 72–82.Google Scholar
Mallat, S. G., A Wavelet Tour of Signal Processing: The Sparse Way, 3rd edn (Elsevier/Academic Press, Amsterdam, 2009).Google Scholar
McDiarmid, C., ‘Concentration’, inProbabilistic Methods for Algorithmic Discrete Mathematics, Algorithms and Combinatorics, 16 (Springer, Berlin, 1998), 195–248.Google Scholar
Po, D. D.-Y. and Do, M. N., ‘Directional multiscale modeling of images using the contourlet transform’, IEEE Trans. Image Process. 15(6) (2006), 1610–1620.Google Scholar
Poon, C., ‘A consistent, and stable approach to generalized sampling’, J. Fourier Anal. Appl. 20(5) (2014), 985–1019.CrossRefGoogle Scholar
Poon, C., ‘On the role of total variation in compressed sensing’, SIAM J. Imaging Sci. 8(1) (2015), 682–720.Google Scholar
Poon, C., ‘Structure dependent sampling in compressed sensing: theoretical guarantees for tight frames’, Appl. Comput. Harmon. Anal. (2015), (to appear).Google Scholar
Puy, G., Marques, J. P., Gruetter, R., Thiran, J., Van De Ville, D., Vandergheynst, P. and Wiaux, Y., ‘Spread spectrum Magnetic Resonance Imaging’, IEEE Trans. Med. Imaging 31(3) (2012), 586–598.Google Scholar
Puy, G., Vandergheynst, P. and Wiaux, Y., ‘On variable density compressive sampling’, IEEE Signal Process. Letters 18 (2011), 595–598.Google Scholar
Rauhut, H. and Ward, R., ‘Interpolation via weighted ℓ 1 minimization’, Appl. Comput. Harmon. Anal. 40(2) (2016), 321–351.CrossRefGoogle Scholar
Roman, B., Adcock, B. and Hansen, A. C., ‘On asymptotic structure in compressed sensing’, Preprint, 2014, arXiv:1406.4178.Google Scholar
Romberg, J., ‘Imaging via compressive sampling’, IEEE Signal Process. Mag. 25(2) (2008), 14–20.CrossRefGoogle Scholar
Rudelson, M., ‘Random vectors in the isotropic position’, J. Funct. Anal. 164(1) (1999), 60–72.CrossRefGoogle Scholar
Strohmer, T., ‘Measure what should be measured: progress and challenges in compressive sensing’, IEEE Signal Process. Letters 19(12) (2012), 887–893.Google Scholar
Studer, V., Bobin, J., Chahid, M., Mousavi, H. S., Candès, E. and Dahan, M., ‘Compressive fluorescence microscopy for biological and hyperspectral imaging’, Proc. Natl. Acad. Sci. USA 109(26) (2012), E1679–E1687.Google Scholar
Takhar, D., Laska, J. N., Wakin, M. B., Duarte, M. F., Baron, D., Sarvotham, S., Kelly, K. F. and Baraniuk, R. G., ‘A new compressive imaging camera architecture using optical-domain compression’, inProc. of Computational Imaging IV at SPIE Electronic Imaging (2006), 43–52.Google Scholar
Talagrand, M., ‘New concentration inequalities in product spaces’, Invent. Math. 126(3) (1996), 505–563.Google Scholar
Traonmilin, Y. and Gribonval, R., ‘Stable recovery of low-dimensional cones in Hilbert spaces: one RIP to rule them all’, Appl. Comput. Harmon. Anal. (2016), (to appear).Google Scholar
Tropp, J. A., ‘On the conditioning of random subdictionaries’, Appl. Comput. Harmon. Anal. 25(1) (2008), 1–24.Google Scholar
Tsaig, Y. and Donoho, D. L., ‘Extensions of compressed sensing’, Signal Process. 86(3) (2006), 549–571.Google Scholar
Wang, L., Carlson, D., Rodrigues, M. R. D., Wilcox, D., Calderbank, R. and Carin, L., ‘Designed measurements for vector count data’, inAdvances in Neural Information Processing Systems (2013), 1142–1150.Google Scholar
Wang, Q., Zenge, M., Cetingul, H. E., Mueller, E. and Nadar, M. S., ‘Novel sampling strategies for sparse mr image reconstruction’, Proc. Int. Soc. Mag. Res. in Med. (22) (2014).Google Scholar
Wang, Z. and Arce, G. R., ‘Variable density compressed image sampling’, IEEE Trans. Image Process. 19(1) (2010), 264–270.Google Scholar
Zomet, A. and Nayar, S. K., ‘Lensless imaging with a controllable aperture’, in2006 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (IEEE, 2006), 339–346.Google Scholar