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References

Published online by Cambridge University Press:  16 July 2021

Ben Adcock  and
Anders C. Hansen
Ben Adcock
Affiliation:
Simon Fraser University, British Columbia
Anders C. Hansen
Affiliation:
University of Cambridge
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Compressive Imaging: Structure, Sampling, Learning , pp. 568 - 595
Publisher: Cambridge University Press
Print publication year: 2021

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References

The USC-SIPI Image Database.http://sipi.usc.edu/database/.Google Scholar
Adcock, B.. Infinite-dimensional compressed sensing and function interpolation. Found. Comput. Math., 18(3):661–701, 2018.Google Scholar
Adcock, B., Antun, V., and Hansen, A. C.. Uniform recovery in infinite-dimensional compressed sensing and applications to structured binary sampling. arXiv:1905.00126, 2019.Google Scholar
Adcock, B., Bao, A., and Brugiapaglia, S.. Correcting for unknown errors in sparse high-dimensional function approximation. Numer. Math., 142(3):667–711, 2019.CrossRefGoogle Scholar
Adcock, B., Bao, A., Jakeman, J. D., and Narayan, A.. Compressed sensing with sparse corruptions: Fault-tolerant sparse collocation approximations. SIAM/ASA J. Uncertain. Quantif., 6(4):1424–1453, 2018.Google Scholar
Adcock, B., Boyer, C., and Brugiapaglia, S.. On oracle-type local recovery guarantees in compressed sensing. Inf. Inference (in press), 2020.CrossRefGoogle Scholar
Adcock, B., Brugiapaglia, S., and King-Roskamp, M.. Do log factors matter? On optimal wavelet approximation and the foundations of compressed sensing. Found. Comput. Math. (in press), 2021.CrossRefGoogle Scholar
Adcock, B., Brugiapaglia, S., and King-Roskamp, M.. Iterative and greedy algorithms for the sparsity in levels model in compressed sensing. In Ville, D. V. D., Papadakis, M., and Lu, Y. M., editors, Wavelets and Sparsity XVIII, volume 11138, pages 76–89. International Society for Optics and Photonics, SPIE, 2019.Google Scholar
Adcock, B., Brugiapaglia, S., and King-Roskamp, M.. The benefits of acting locally: reconstruction algorithms for sparse in levels signals with stable and robust recovery guarantees. arXiv:2006.1338, 2020.Google Scholar
Adcock, B., Brugiapaglia, S., and Webster, C. G.. Compressed sensing approaches for polynomial approximation of high-dimensional functions. In Boche, H., Caire, G., Calderbank, R., März, M., Kutyniok, G., and Mathar, R., editors, Compressed Sensing and its Applications: Second International MATHEON Conference 2015, Applied and Numerical Harmonic Analysis, pages 93–124. Birkhäuser, Cham, 2017.Google Scholar
Adcock, B. and Dexter, N.. The gap between theory and practice in function approximation with deep neural networks. SIAM J. Math. Data Sci. (in press), 2021.Google Scholar
Adcock, B., Dexter, N., and Xu, Q.. Improved recovery guarantees and sampling strategies for tv minimization in compressive imaging. arXiv:2009.08555, 2020.Google Scholar
Adcock, B., Gataric, M., and Hansen, A. C.. On stable reconstructions from nonuniform Fourier measurements. SIAM J. Imaging Sci., 7(3):1690–1723, 2014.Google Scholar
Adcock, B., Gataric, M., and Hansen, A. C.. Weighted frames of exponentials and stable recovery of multidimensional functions from nonuniform fourier samples. Appl. Comput. Harmon. Anal., 42(3):508–535, 2017.Google Scholar
Adcock, B. and Hansen, A. C.. A generalized sampling theorem for stable reconstructions in arbitrary bases. J. Fourier Anal. Appl., 18(4):685–716, 2012.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):357–388, 2012.Google Scholar
Adcock, B. and Hansen, A. C.. Generalized sampling and infinite-dimensional compressed sensing. Found. Comput. Math., 16(5):1263–1323, 2016.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):3114–3131, 2013.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):387–415, 2014.Google Scholar
Adcock, B., Hansen, A. C., Poon, C., and Roman, B.. Breaking the coherence barrier: a new theory for compressed sensing. Forum Math. Sigma, 5, 2017.CrossRefGoogle Scholar
Adcock, B., Hansen, A. C., and Roman, B.. The quest for optimal sampling: computationally efficient, structure-exploiting measurements for compressed sensing. In Boche, H., Calderbank, R., Kutyniok, G., and Vybíral, J., editors, Compressed Sensing and its Applications: MATHEON Workshop 2013, Applied and Numerical Harmonic Analysis, pages 143–167. Birkhäuser, Cham, 2015.CrossRefGoogle Scholar
Adcock, B., Hansen, A. C., and Roman, B.. A note on compressed sensing of structured sparse wavelet coefficients from subsampled Fourier measurements. IEEE Signal Process. Lett., 23(5):732–736, 2016.Google Scholar
Adcock, B., Hansen, A. C., and Shadrin, A.. A stability barrier for reconstructions from Fourier samples. SIAM J. Numer. Anal., 52(1):125–139, 2014.CrossRefGoogle Scholar
Adcock, B. and Sui, Y.. Compressive Hermite interpolation: sparse, high-dimensional approximation from gradient-augmented measurements. Constr. Approx., 50(1):167–207, 2019.Google Scholar
Adler, J. and Öktem, O.. Solving ill-posed inverse problems using iterative deep neural networks. Inverse Problems, 33(12):124007, 2017.Google Scholar
Adler, J. and Öktem, O.. Learned primal-dual reconstruction. IEEE Trans. Med. Imag., 37(6):1322–1332, 2018.Google Scholar
Ahmad, R., Bouman, C. A., Buzzard, G. T., Chan, S., Liu, S., Reehorst, E. T., and Schniter, P.. Plug-and-play methods for magnetic resonance imaging: using denoisers for image recovery. IEEE Signal Process. Mag., 37(1):105–116, 2020.CrossRefGoogle ScholarPubMed
Ahmad, R. and Schniter, P.. Iteratively reweighted ℓ 1 approaches to sparse composite regularization. IEEE Trans. Comput. Imag., 1(4):220–235, 2015.CrossRefGoogle Scholar
Ahsen, M. E. and Vidyasagar, M.. Error bounds for compressed sensing algorithms with group sparsity: a unified approach. Appl. Comput. Harmon. Anal., 43(2):212–232, 2017.CrossRefGoogle Scholar
Akhtar, N. and Mian, A.. Threat of adversarial attacks on deep learning in computer vision: a survey. IEEE Access, 6:14410–14430, 2018.Google Scholar
Alberti, G. S. and Santacesaria, M.. Infinite dimensional compressed sensing from anisotropic measurements and applications to inverse problems in PDE. Appl. Comput. Harmon. Anal., 50:105–146, 2021.CrossRefGoogle Scholar
Ambrosio, L., Fusco, N., and Pallara, D.. Functions of Bounded Variation and Free Discontinuity Problems. Oxford Mathematical Monographs. Clarendon Press, Oxford, 2000.CrossRefGoogle Scholar
Andersson, J. and Strömberg, J.-O.. On the theorem of uniform recovery of random sampling matrices. IEEE Trans. Inf. Theory, 60(3):1700–1710, 2014.Google Scholar
Antun, V.. Coherence estimates between Hadamard matrices and Daubechies wavelets. Master’s thesis, University of Oslo, 2016.Google Scholar
Antun, V., Colbrook, M. J., and Hansen, A. C.. Can stable and accurate neural networks be computed? – On barriers of deep learning and Smale’s 18th problem. arXiv:2101.08286, 2021.Google Scholar
Antun, V., Renna, F., Poon, C., Adcock, B., and Hansen, A. C.. On instabilities of deep learning in image reconstruction and the potential costs of AI. Proc. Natl. Acad. Sci. USA, 117(48):30088–30095, 2020.Google Scholar
Antun, V. and Ryan, Ø.. On the unification of schemes and software for wavelets on the interval. Acta Appl. Math. (in press), 2021.CrossRefGoogle Scholar
Arce, G. R., Brady, D. J., Carin, L., Arguello, H., and Kittle, D.. Compressive coded aperture spectral imaging: an introduction. IEEE Signal Process. Mag., 31(1):105–115, 2014.CrossRefGoogle Scholar
Arjovsky, M., Chintala, S., and Bottou, L.. Wasserstein generative adversarial networks. In Precup, D. and Teh, Y. W., editors, Proceedings of the 34th International Conference on Machine Learning, volume 70 of Proceedings of Machine Learning Research, pages 214–223. PMLR, 2017.Google Scholar
Arjovsky, M., Shah, A., and Bengio, Y.. Unitary evolution recurrent neural networks. In Balcan, M. F. and Weinberger, K. Q., editors, Proceedings of The 33rd International Conference on Machine Learning, volume 48, pages 1120–1128. Proceedings of Machine Learning Research, 2016.Google Scholar
Arnab, A., Miksik, O., and Torr, P. H.. On the robustness of semantic segmentation models to adversarial attacks. In 2018 IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 888–897, 2018.Google Scholar
Arridge, S., Maass, P., Öktem, O., and Schönlieb, C.-B.. Solving inverse problems using data-driven models. Acta Numer., 28:1–174, 2019.Google Scholar
Asif, M. S., Ayremlou, A., Sankaranarayanan, A. C., Veeraraghavan, A., and Baraniuk, R. G.. Flatcam: thin, bare-sensor cameras using coded aperture and computation. IEEE Trans. Comput. Imag. (in press), 2021.Google Scholar
Athalye, A., Carlini, N., and Wagner, D.. Obfuscated gradients give a false sense of security: circumventing defenses to adversarial examples. In Dy, J. and Krause, A., editors, Proceedings of the 35th International Conference on Machine Learning, volume 80 of Proceedings of Machine Learning Research, pages 274–283. PMLR, 2018.Google Scholar
Babu, P. and Stoica, P.. Connection between SPICE and Square-Root LASSO for sparse parameter estimation. Signal Process., 95:10–14, 2014.Google Scholar
Bach, F.. Breaking the curse of dimensionality with convex neural networks. J. Mach. Learn. Res., 18(19):1–53, 2017.Google Scholar
Baldassarre, L., Li, Y.-H., Scarlett, J., Gözcü, B., Bogunovic, I., and Cevher, V.. Learning-based compressive subsampling. IEEE J. Sel. Topics Signal Process., 10(4):809–822, 2016.CrossRefGoogle Scholar
Baraniuk, R. G., Cevher, V., Duarte, M. F., and Hedge, C.. Model-based compressive sensing. IEEE Trans. Inf. Theory, 56(4):1982–2001, 2010.Google Scholar
Baraniuk, R. G., Goldstein, T., Sankaranarayanan, A. C., Studer, C., Veeraraghavan, A., and Wakin, M. B.. Compressive video sensing: algorithms, architectures, and applications. IEEE Signal Process. Mag., 34(1):52–66, 2017.CrossRefGoogle Scholar
Barrett, H. H. and Myers, K. J.. Foundations of Image Science. Wiley–Interscience, Hoboken, NJ, 2004.Google Scholar
Bastounis, A. and Hansen, A. C.. On the absence of uniform recovery in many real-world applications of compressed sensing and the restricted isometry property and nullspace property in levels. SIAM J. Imaging Sci., 10(1):335–371, 2017.CrossRefGoogle Scholar
Bastounis, A., Hansen, A. C., and Vlačić, V.. The extended Smale’s 9th problem – on computational barriers and paradoxes in estimation, regularisation, computer-assisted proofs, and learning. Preprint, 2021.Google Scholar
Bauschke, H. H. and Combettes, P. L.. Convex Analysis and Monotone Operator Theory in Hilbert Spaces. CMS Books in Mathematics. Springer, New York, NY, 2011.Google Scholar
Beauchamp, K. G.. Walsh Functions and their Applications. Academic Press, London, 1975.Google Scholar
Beck, A.. Introduction to Nonlinear Optimization: Theory, Algorithms, and Applications with MATLAB. MOS-SIAM Series on Optimization. Society for Industrial and Applied Mathematics, Philadelphia, PA, 2014.Google Scholar
Beck, A.. First-Order Methods in Optimization. MOS-SIAM Series on Optimization. Society for Industrial and Applied Mathematics, Philadelphia, PA, 2017.CrossRefGoogle Scholar
Beck, A. and Teboulle, M.. A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM J. Imaging Sci., 2(1):183–202, 2009.CrossRefGoogle Scholar
Beck, A. and Teboulle, M.. Gradient-based algorithms with applications to signal-recovery problems. In Palomar, D. P. and Eldar, Y. C., editors, Convex Optimization in Signal Processing and Communications, pages 42–88. Cambridge University Press, Cambridge, 2009.Google Scholar
Beck, A. and Teboulle, M.. Smoothing and first order methods: a unified framework. SIAM J. Optim., 22(2):557–580, 2012.Google Scholar
Becker, S., Bobin, J., and Candès, E. J.. NESTA: A fast and accurate first-order method for sparse recovery. SIAM J. Imaging Sci., 4(1):1–39, 2011.Google Scholar
Becker, S. R.. Practical Compressed Sensing: Modern Data Acquisition and Signal Processing. PhD thesis, Stanford University, 2011.Google Scholar
Belloni, A., Chernozhukov, V., and Wang, L.. Square-root lasso: pivotal recovery of sparse signals via conic programming. Biometrika, 98(4):791–806, 2011.Google Scholar
Belloni, A., Chernozhukov, V., and Wang, L.. Pivotal estimation via square-root Lasso in nonparametric regression. Ann. Statist., 42(2):757–788, 2014.Google Scholar
Ben-Tal, A. and Nemirovski, A.. Interior Point Polynomial Time Methods in Convex Programming. Available online at www2.isye.gatech.edu/~nemirovs/, 1996.Google Scholar
Ben-Tal, A. and Nemirovski, A.. Lectures on Modern Convex Optimization: Analysis, Algorithms, and Engineering Applications. Available online at www2.isye .gatech.edu/∼nemirovs/, 2000.Google Scholar
Ben-Tal, A. and Nemirovski, A.. Lectures on Modern Convex Optimization: Analysis, Algorithms, and Engineering Applications. MOS-SIAM Series on Optimization. Society for Industrial and Applied Mathematics, Philadelphia, PA, 2001.CrossRefGoogle Scholar
Berk, A., Plan, Y., and Yilmaz, O.. Sensitivity of ℓ 1 minimization to parameter choice. Inf. Inference (in press), 2020.Google Scholar
Berner, J., Grohs, P., and Jentzen, A.. Analysis of the generalization error: empirical risk minimization over deep artificial neural networks overcomes the curse of dimensionality in the numerical approximation of Black–Scholes partial differential equations. SIAM J. Math. Data Sci., 2(3):631–657, 2020.Google Scholar
Bickel, P. J., Ritov, Y., and Tsybakov, A. B.. Simultaneous analysis of Lasso and Dantzig selector. Ann. Statist., 37(4):1705–1732, 2009.Google Scholar
Bigdeli, S. A., Zwicker, M., Favaro, P., and Jin, M.. Deep mean-shift priors for image restoration. In Advances in Neural Information Processing Systems, pages 763–772, 2017.Google Scholar
Bigot, J., Boyer, C., and Weiss, P.. An analysis of block sampling strategies in compressed sensing. IEEE Trans. Inf. Theory, 62(4):2125–2139, 2016.Google Scholar
Birgin, E. G., Martínez, J. M., and Raydan, M.. Nonmonotone spectral projected gradient methods on convex sets. SIAM J. Optim., 10(4):1196–1211, 2000.CrossRefGoogle Scholar
Birkes, D. and Dodge, Y.. Alternative Methods of Regression. Wiley-Interscience, New York, 1993.Google Scholar
Blaimer, M., Breuer, F., Mueller, M., Heidemann, R., Griswold, M., and Jakob, P.. SMASH, SENSE, PILS, GRAPPA: how to choose the optimal method. Top. Magn. Reson. Imaging, 15(4):223–236, 2004.CrossRefGoogle ScholarPubMed
Block, K. T., Uecker, M., and Frahm, J.. Suppression of MRI truncation artifacts using total variation constrained data extrapolation. Int. J. Biomed. Imaging, 2008:184123, 2008.Google Scholar
Blum, L., Cucker, F., Shub, M., and Smale, S.. Complexity and Real Computation. Springer, New York, NY, 1998.Google Scholar
Blum, L., Shub, M., and Smale, S.. On a theory of computation and complexity over the real numbers: NP-completeness, recursive functions and universal machines. Bull. Amer. Math. Soc., 21(1):1–46, 1989.Google Scholar
Blumensath, T. and Davies, M. E.. Iterative thresholding for sparse approximations. J. Fourier Anal. Appl., 14(5):629–654, 2008.Google Scholar
Blumensath, T. and Davies, M. E.. Iterative hard thresholding for compressed sensing. Appl. Comput. Harmon. Anal., 27(3):265–274, 2009.Google Scholar
Bodmann, B. G., Flinth, A., and Kutyniok, G.. Compressed sensing for analog signals. arXiv:1803.04218, 2018.Google Scholar
Boominathan, V., Adams, J. K., Asif, M. S., Avants, B. W., Robinson, J. T., Baraniuk, R. G., Sankaranarayanan, A. C., and Veeraraghavan, A.. Lensless imaging: a computational renaissance. IEEE Signal Process. Mag., 33(5):23–35, 2016.Google Scholar
Bora, A., Jalal, A., Price, E., and Dimakis, A. G.. Compressed sensing using generative models. In International Conference on Machine Learning, pages 537–546, 2017.Google Scholar
Böttcher, A.. Infinite matrices and projection methods. In Lectures on Operator Theory and its Applications, volume 3 of Fields Institute Monographs, pages 1–72. American Mathematical Society, Providence, RI, 1996.Google Scholar
Bottou, L., Curtis, F. E., and Nocedal, J.. Optimization methods for large-scale machine learning. SIAM Rev., 60(2):223–311, 2018.Google Scholar
Boucheron, S., Lugosi, G., and Massart, P.. Concentration Inequalities: A Nonasymptotic Theory of Independence. Oxford University Press, Oxford, 2013.Google Scholar
Bourgain, J.. An improved estimate in the restricted isometry problem. In Klartag, B. and Milman, E., editors, Geometric Aspects of Functional Analysis, volume 2116 of Lecture Notes in Mathematics, pages 65–70. Springer, Cham, 2014.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. Inf. Theory, 60(12):7928–7946, 2014.Google Scholar
Boyd, S. and Parikh, N.. Proximal algorithms. Foundations and Trends in Optimization, 1(3):123–231, 2013.Google Scholar
Boyd, S., Parikh, N., Chu, E., Peleato, B., and Eckstein, J.. Distributed optimization and statistical learning via the alternating direction method of multipliers. Foundations and Trends in Machine Learning, 3(1):1–122, 2011.Google Scholar
Boyd, S. and Vandenberghe, L.. Convex Optimization. Cambridge University Press, Cambridge, 2004.Google Scholar
Boyer, C., Bigot, J., and Weiss, P.. Compressed sensing with structured sparsity and structured acquisition. Appl. Comput. Harmon. Anal., 46(2):312–350, 2019.Google Scholar
Boyer, C., Chauffert, N., Ciuciu, P., Kahn, J., and Weiss, P.. On the generation of sampling schemes for magnetic resonance imaging. SIAM J. Imaging Sci., 9(4):2039–2072, 2016.Google Scholar
Boyer, C., Weiss, P., and Bigot, J.. An algorithm for variable density sampling with block-constrained acquisition. SIAM J. Imaging Sci., 7(2):1080–1107, 2014.CrossRefGoogle Scholar
Brady, D. J.. Optical Imaging and Spectroscopy. Wiley, Hoboken, NJ, 2009.Google Scholar
Brady, D. J., Choi, K., Marks, D. L., Horisaki, R., and Lim, S.. Compressive holography. Opt. Express, 17(15):13040–13049, 2009.CrossRefGoogle ScholarPubMed
Bredies, K., Kunisch, K., and Pock, T.. Total generalized variation. SIAM J. Imaging Sci., 3(3):492–526, 2010.Google Scholar
Brugiapaglia, S. and Adcock, B.. Robustness to unknown error in sparse regularization. IEEE Trans. Inf. Theory, 64(10):6638–6661, 2018.Google Scholar
Bubba, T. A., Kutyniok, G., Lassas, M., März, M., Samek, W., Siltanen, S., and Srini-vasan, V.. Learning the invisible: a hybrid deep learning-shearlet framework for limited angle computed tomography. Inverse Problems, 35(6):064002, 2019.Google Scholar
Bubeck, S.. Convex optimization: algorithms and complexity. Foundations and Trends in Machine Learning, 8(3–4):231–358, 2015.Google Scholar
Bunea, F., Lederer, J., and She, Y.. The group square-root Lasso: theoretical properties and fast algorithms. IEEE Trans. Inf. Theory, 60(2):1313–1325, 2014.Google Scholar
Burger, M., Möller, M., Benning, M., and Osher, S.. An adaptive inverse scale space method for compressed sensing. Math. Comp., 82(281):269–299, 2013.Google Scholar
Burger, M., Sawatzky, A., and Steidl, G.. First order algorithms in variational image processing. In Glowinski, R., Osher, S. J., and Yin, W., editors, Splitting Methods in Communication, Imaging, Science, and Engineering, pages 345–407. Springer, Berlin, 2016.Google Scholar
Bürgisser, P. and Cucker, F.. Condition: The Geometry of Numerical Algorithms. Grundlehren der mathematischen Wissenschaften. Springer, Berlin, 2013.Google Scholar
Cahill, J. and Mixon, D. G.. Robust width: a characterization of uniformly stable and robust compressed sensing. arXiv:1408.4409, 2018.Google Scholar
Cai, J.-F. and Xu, W.. Guarantees of total variation minimization for signal recovery. Inf. Inference, 4(4):328–353, 2015.Google Scholar
Cai, T. and Zhang, A.. Sparse representation of a polytope and recovery of sparse signals and low-rank matrices. IEEE Trans. Inf. Theory, 60(1):122–132, 2014.Google Scholar
Calderbank, R., Hansen, A., Roman, B., and Thesing, L.. On reconstructing functions from binary measurements. In Boche, H., Caire, G., Calderbank, R., Kutyniok, G., Mathar, R., and Petersen, P., editors, Compressed Sensing and Its Applications: Third International MATHEON Conference 2017, Applied and Numerical Harmonic Analysis, pages 97–128. Birkhäuser, Cham, 2019.Google Scholar
Candès, E. J.. Compressive sampling. In Sanz-Solé, M., Soria, J., Varona, J. L., and Verdera, J., editors, Proceedings of the International Congress of Mathematicians, Madrid 2006, pages 1433–1452. American Mathematical Society, Providence, RI, 2006.Google Scholar
Candès, E. J.. The restricted isometry property and its implications for compressed sensing. C. R. Math. Acad. Sci. Paris, 346(9–10):589–592, 2008.Google Scholar
Candès, E. J. and Donoho, D. L.. Ridgelets: a key to high dimensional intermittency? Philos. Trans. Roy. Soc. A, 357(1760):2495–2509, 1999.Google Scholar
Candès, E. J. and Donoho, D. L.. Curvelets—a surprisingly effective nonadaptive representation for objects with edges. In Rabut, C., Cohen, A., and Schumaker, L. L., editors, Curves and Surfaces, pages 105–120. Vanderbilt University Press, Nashville, TN, 2000.Google Scholar
Candès, E. J. and Donoho, D. L.. Recovering edges in ill-posed inverse problems: optimality of curvelet frames. Ann. Statist., 30(3):784–842, 2002.Google Scholar
Candès, E. J. and Donoho, D. L.. New tight frames of curvelets and optimal representations of objects with piecewise C 2 singularities. Comm. Pure Appl. Math., 57(2):219–266, 2004.CrossRefGoogle Scholar
Candès, E. J., Eldar, Y. C., Needell, D., and Randall, P.. Compressed sensing with coherent and redundant dictionaries. Appl. Comput. Harmon. Anal., 31(1):59–73, 2010.Google Scholar
Candès, E. J., Eldar, Y. C., Strohmer, T., and Voroninski, V.. Phase retrieval via matrix completion. SIAM J. Imaging Sci., 6(1):199–225, 2013.Google Scholar
Candès, E. J. and Fernandez-Granda, C.. Towards a mathematical theory of super-resolution. Comm. Pure Appl. Math., 67(6):906–956, 2014.Google Scholar
Candès, E. J., Li, X., and Soltanolkotabi, M.. Phase retrieval via Wirtinger flow: theory and algorithms. IEEE Trans. Inf. Theory, 61(4):1985–2007, 2015.Google Scholar
Candès, E. J. and Plan, Y.. A probabilistic and RIPless theory of compressed sensing. IEEE Trans. Inf. Theory, 57(11):7235–7254, 2011.Google Scholar
Candès, E. J. and Romberg, J.. l1-magic.https://statweb.stanford.edu/~candes/ software/l1magic/.Google Scholar
Candès, E. J. and Romberg, J.. Sparsity and incoherence in compressive sampling. Inverse Problems, 23(3):969–985, 2007.Google Scholar
Candès, E. J., Romberg, J., and Tao, T.. Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information. IEEE Trans. Inf. Theory, 52(2):489–509, 2006.Google Scholar
Candès, E. J., Strohmer, T., and Voroninski, V.. Phaselift: exact and stable signal recovery from magnitude measurements via convex programming. Comm. Pure Appl. Math., 66(8):1241– 1274, 2013.Google Scholar
Candès, E. J. and Tao, T.. Near optimal signal recovery from random projections: universal encoding strategies? IEEE Trans. Inf. Theory, 52(12):5406–5425, 2006.Google Scholar
Candès, E. J. and Tao, T.. The Dantzig selector: statistical estimation when p is much larger than n. Ann. Statist., 35(6):2313–2351, 2007.Google Scholar
Candès, E. J. and Wakin, M. B.. An introduction to compressive sampling. IEEE Signal Process. Mag., 25(2):21–30, 2008.Google Scholar
Candès, E. J., Wakin, M. B., and Boyd, S. P.. Enhancing sparsity by reweighted ℓ 1 minimization. J. Fourier Anal. Appl., 14(5):877–905, 2008.Google Scholar
Casazza, P. G. and Christensen, O.. Approximation of the inverse frame operator and applications to Gabor frames. J. Approx. Theory, 103(2):338–356, 2000.Google Scholar
Chakraborty, A., Alam, M., Dey, V., Chattopadhyay, A., and Mukhopadhyay, D.. Adversarial attacks and defences: a survey. arXiv:1810.00069, 2018.Google Scholar
Chambolle, A., Duval, V., Peyré, G., and Poon, C.. Geometric properties of solutions to the total variation denoising problem. Inverse Problems, 33(1):015002, 2016.Google Scholar
Chambolle, A., Novaga, M., Cremers, D., and Pock, T.. An introduction to total variation for image analysis. In Fornasier, M., editor, Theoretical Foundations and Numerical Methods for Sparse Recovery, volume 9 of Radon Series in Computational and Applied Mathematics, pages 263–340. de Gruyter, Berlin, 2010.CrossRefGoogle Scholar
Chambolle, A. and Pock, T.. A first-order primal-dual algorithm for convex problems with applications to imaging. J. Math. Imaging Vision, 40(1):120–145, 2011.Google Scholar
Chambolle, A. and Pock, T.. An introduction to continuous optimization for imaging. Acta Numer., 25:161–319, 2016.Google Scholar
Chambolle, A. and Pock, T.. On the ergodic convergence rates of a first-order primal-dual algorithm. Math. Program., 159(1–2):253–287, 2016.Google Scholar
Chang, J. H. R., Li, C., Póczos, B., Vijaya Kumar, B. V. K., and Sankaranarayanan, A. C.. One network to solve them all – solving linear inverse problems using deep projection models. In 2017 IEEE International Conference on Computer Vision (ICCV), pages 5888–5897, 2017.CrossRefGoogle Scholar
Chartrand, R.. Fast algorithms for nonconvex compressive sensing: MRI reconstruction from very few data. In 2009 IEEE International Symposium on Biomedical Imaging: From Nano to Macro, pages 262–265, 2009.Google Scholar
Chauffert, N., Ciuciu, P., Kahn, J., and Weiss, P.. Variable density sampling with continuous trajectories. SIAM J. Imaging Sci., 7(4):1962–1992, 2014.Google Scholar
Chávez, C. E., Alonzo-Atienza, F., and Álvarez, D.. Avoiding the inverse crime in the Inverse Problem of electrocardiography: estimating the shape and location of cardiac ischemia. In Computing in Cardiology 2013, volume 687–690, 2013.Google Scholar
Chen, G.-H., Tang, J., and Leng, S.. Prior image constrained compressed sensing (PICCS): a method to accurately reconstruct dynamic CT images from highly undersampled projection data sets. Med. Phys., 35(2):660–663, 2008.Google Scholar
Chen, J. and Huo, X.. Theoretical results on sparse representations of multiple-measurement vectors. IEEE Trans. Signal Process., 54(12):4634–4643, 2006.Google Scholar
Chen, S. S., Donoho, D. L., and Saunders, M. A.. Atomic decomposition by basis pursuit. SIAM J. Sci. Comput., 20(1):33–61, 1998.Google Scholar
Cheraghchi, M., Guruswami, V., and Velingker, A.. Restricted isometry of Fourier matrices and list decodability of random linear codes. SIAM J. Comput., 42(5):1888–1914, 2013.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):2182–2195, 2011.CrossRefGoogle Scholar
Chkifa, A., Dexter, N., Tran, H., and Webster, C. G.. Polynomial approximation via compressed sensing of high-dimensional functions on lower sets. Math. Comp., 87(311):1415–1450, 2018.Google Scholar
Choi, H. and Baraniuk, R. G.. Wavelet statistical models and Besov spaces. In Unser, M. A., Aldroubi, A., and Laine, A. F., editors, Wavelet Applications in Signal and Image Processing VII, volume 3813, pages 489–501. International Society for Optics and Photonics, SPIE, 1999.Google Scholar
Christensen, O. and Strohmer, T.. The finite section method and problems in frame theory. J. Approx. Theory, 133(2):221–237, 2005.Google Scholar
Chun, I.-Y. and Adcock, B.. Compressed sensing and parallel acquisition. IEEE Trans. Inf. Theory, 63(8):4860–4882, 2017.Google Scholar
Chun, I.-Y., Adcock, B., and Talavage, T.. Efficient compressed sensing SENSE pMRI reconstruction with joint sparsity promotion. IEEE Trans. Med. Imag., 31(1):354–368, 2016.Google Scholar
Chun, I.-Y. and Fessler, J. A.. Convolutional dictionary learning: acceleration and convergence. IEEE Trans. Image Process., 27(4):1697–1712, 2018.Google Scholar
Chun, I.-Y. and Fessler, J. A.. Deep BCD-net using identical encoding-decoding CNN structures for iterative image recovery. In 2018 IEEE 13th Image, Video, and Multidimensional Signal Processing Workshop (IVMSP), pages 1–5, 2018.Google Scholar
Cohen, A., Dahmen, W., Daubechies, I., and DeVore, R.. Harmonic analysis of the space BV. Rev. Mat. Iberoam., 19(1):235–263, 2003.Google Scholar
Cohen, A., Dahmen, W., and DeVore, R. A.. Compressed sensing and best k-term approximation. J. Amer. Math. Soc., 22(1):211–231, 2009.Google Scholar
Cohen, A., Daubechies, I., and Feauveau, J.-C.. Biorthogonal bases of compactly supported wavelets. Comm. Pure Appl. Math., 45(5):485–560, 1992.Google Scholar
Cohen, A., Daubechies, I., and Vial, P.. Wavelet bases on the interval and fast algorithms. Appl. Comput. Harmon. Anal., 1(1):54–81, 1993.CrossRefGoogle Scholar
Cohen, A., DeVore, R., Petrushev, P., and Xu, H.. Nonlinear approximation and the space BV (R 2 ). Amer. J. Math., 121(3):587–628, 1999.Google Scholar
Cohen, J., Rosenfeld, E., and Kolter, J. Z.. Certified adversarial robustness via randomized smoothing. In International Conference on Machine Learning, pages 1310–1320, 2019.Google Scholar
Coifman, R. R., Geshwind, F., and Meyer, Y.. Noiselets. Appl. Comput. Harmon. Anal., 10(1):27–44, 2001.Google Scholar
Colton, D. and Kress, R.. Integral Equation Methods in Scattering Theory. Classics in Applied Mathematics. Society for Industrial and Applied Mathematics, Philadelphia, PA, 2013.Google Scholar
Combettes, P. L. and Pesquet, J.-C.. Proximal splitting methods in signal processing. In Bauschke, H. H., Burachik, R. S., Combettes, P. L., Elser, V., Luke, D. R., and Wolkowicz, H., editors, Fixed-Point Algorithms for Inverse Problems in Science and Engineering, volume 49 of Springer Optimization and Its Applications, pages 185–212. Springer, New York, NY, 2011.Google Scholar
Condat, L.. A primal-dual splitting method for convex optimization involving lipschitzian, proximable and linear composite terms. J. Optim. Theory App., 158(2):460–479, 2013.Google Scholar
Crouse, M. S., Nowak, R. D., and Baraniuk, R. G.. Wavelet-based statistical signal processing using hidden Markov models. IEEE Trans. Signal Process., 46(4):886–902, 1998.Google Scholar
Cybenko, G.. Approximation by superposition of a sigmoidal function. Math. Control, Signals, and Systems, 2(4):303–314, 1989.Google Scholar
Dabov, K., Foi, A., Katkovnik, V., and Egiazarian, K.. Image denoising by sparse 3-D transform-domain collaborative filtering. IEEE Trans. Image Process., 16(8):2080–2095, 2007.CrossRefGoogle ScholarPubMed
Dassios, I., Fountoulakis, K., and Gondzio, J.. A preconditioner for a primal-dual Newton conjugate gradient method for compressed sensing problems. SIAM J. Sci. Comput., 37(6):A2783–A2812, 2015.Google Scholar
Daubechies, I.. Ten Lectures on Wavelets. CBMS-NSF Regional Conference Series in Applied Mathematics. Society for Industrial and Applied Mathematics, Philadelphia, PA, 1992.Google Scholar
Daubechies, I., Defrise, M., and Mol, C.. An iterative thresholding algorithm for linear inverse problems with a sparsity constraint. Comm. Pure Appl. Math., 57(11):1413–1457, 2004.Google Scholar
Daubechies, I., DeVore, R., Fornasier, M., and Güntürk, C. S.. Iteratively re-weighted least squares minimization for sparse recovery. Comm. Pure Appl. Math., 63(1):1–38, 2010.Google Scholar
Davenport, M. A., Duarte, M. F., Eldar, Y. C., and Kutyniok, G.. Introduction to compressed sensing. In Eldar, Y. C. and Kutyniok, G., editors, Compressed Sensing: Theory and Applications, pages 1–64. Cambridge University Press, Cambridge, 2012.Google Scholar
Davenport, M. A. and Romberg, J.. An overview of low-rank matrix recovery from incomplete observations. IEEE J. Sel. Topics Signal Process., 10(4):602–622, 2016.Google Scholar
Davenport, M. A. and Wakin, M. B.. Compressive sensing of analog signals using discrete prolate spheroidal sequences. Appl. Comput. Harmon. Anal., 33(3):438–472, 2012.Google Scholar
Davies, M. E. and Eldar, Y. C.. Rank awareness in joint sparse recovery. IEEE Trans. Inf. Theory, 58(2):1135–1146, 2012.Google Scholar
Delaney, A. H. and Bresler, Y.. Globally convergent edge-preserving regularized reconstruction: an application to limited-angle tomography. IEEE Trans. Image Process., 7(2):202– 221, 1998.Google Scholar
Deniţiu, A., Petra, S., Schnörr, C., and Schnörr, C.. Phase transitions and cosparse tomographic recovery of compound solid bodies from few projections. Fundam. Inform., 135(1–2):73– 102, 2014.Google Scholar
DeVore, R., Petrova, G., and Wojtaszczyk, P.. Instance-optimality in probability with an ℓ 1 -minimization decoder. Appl. Comput. Harmon. Anal., 27(3):275–288, 2009.Google Scholar
DeVore, R. A.. Nonlinear approximation. Acta Numer., 7:51–150, 1998.Google Scholar
Diamond, S., Sitzmann, V., Heide, F., and Wetzstein, G.. Unrolled optimization with deep priors. arXiv:1705.08041, 2017.Google Scholar
Do, M. N. and Vetterli, M.. The contourlet transform: an efficient directional multiresolution image representation. IEEE Trans. Image Process., 14(12):2091–2106, 2005.Google Scholar
Do, T. T., Gan, L., Nguyen, N. H., and Tran, T. D.. Fast and efficient compressive sensing using structurally random matrices. IEEE Trans. Signal Process., 60(1):139–154, 2011.Google Scholar
Doneva, M.. Mathematical models for magnetic resonance imaging reconstruction: an overview of the approaches, problems, and future research areas. IEEE Signal Process. Mag., 37(1):24–32, 2020.Google Scholar
Donoho, D. L.. Compressed sensing. IEEE Trans. Inf. Theory, 52(4):1289–1306, 2006.Google Scholar
Donoho, D. L. and Elad, M.. Optimally sparse representation in general (non-orthogonal) dictionaries via ℓ 1 minimization. Proc. Natl. Acad. Sci. USA, 100(5):2197–2002, 2003.Google Scholar
Donoho, D. L. and Huo, X.. Uncertainty principles and ideal atomic decomposition. IEEE Trans. Inf. Theory, 47(7):2845–2862, 2001.Google Scholar
Donoho, D. L., Maleki, A., and Montanari, A.. Message-passing algorithms for compressed sensing. Proc. Natl. Acad. Sci. USA, 106(45):18914–18919, 2009.Google Scholar
Donoho, D. L. and Tanner, J.. Observed universality of phase transitions in high-dimensional geometry, with implications for modern data analysis and signal processing. Philos. Trans. Roy. Soc. A, 367(1906):4273–4293, 2009.Google Scholar
Dorsch, D. and Rauhut, H.. Refined analysis of sparse MIMO radar. J. Fourier Anal. Appl., 23(3):485–529, 2017.CrossRefGoogle Scholar
Douglas, J. and Rachford, H.. On the numerical solution of heat conduction problems in two or three space variables. Trans. Amer. Math. Soc., 82(2):421–439, 1956.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):83–91, 2008.Google Scholar
Duarte, M. F. and Eldar, Y. C.. Structured compressed sensing: from theory to applications. IEEE Trans. Signal Process., 59(9):4053–4085, 2011.Google Scholar
Dumitrescu, B. and Irofti, P.. Dictionary Learning: Algorithms and Applications. Springer, Cham, 2018.Google Scholar
Duval, V. and Peyré, G.. Exact support recovery for sparse spikes deconvolution. Found. Comput. Math., 15(5):1315–1355, 2015.Google Scholar
Efron, B., Hastie, T., Johnstone, I., and Tibshirani, R.. Least angle regression. Ann. Statist., 32(2):407–499, 2004.Google Scholar
Ekeland, I. and Témam, R.. Convex Analysis and Variational Problems. Classics in Applied Mathematics. Society for Industrial and Applied Mathematics, Philadelphia, PA, 1999.Google Scholar
Elad, M.. Sparse and Redundant Representations: From Theory to Applications in Signal and Image Processing. Springer, New York, NY, 2010.Google Scholar
Elad, M. and Bruckstein, A. M.. A generalized uncertainty principle and sparse representation in pairs of bases. IEEE Trans. Inf. Theory, 48(9):2558–2567, 2002.Google Scholar
Elad, M., Milanfar, P., and Rubinstein, R.. Analysis versus synthesis in signal priors. Inverse Problems, 23(3):947, 2007.Google Scholar
Eldar, Y. C. and Kutyniok, G., editors. Compressed Sensing: Theory and Applications. Cambridge University Press, Cambridge, 2012.Google Scholar
Eldar, Y. C. and Mishali, M.. Robust recovery of signals from a structured union of subspaces. IEEE Trans. Inf. Theory, 55(11):5302–5316, 2009.Google Scholar
Eldar, Y. C. and Rauhut, H.. Average case analysis of multichannel sparse recovery using convex relaxation. IEEE Trans. Inf. Theory, 56(1):505–519, 2010.Google Scholar
Epstein, C. L.. Introduction to the Mathematics of Medical Imaging. Other Titles in Applied Mathematics. Society for Industrial and Applied Mathematics, 2nd edition, 2007.Google Scholar
Esser, E.. Applications of Lagrangian-based alternating direction methods and connections to split Bregman. Preprint, 2009.Google Scholar
Esser, E., Zhang, X., and Chan, T. F.. A general framework for a class of first order primal-dual algorithms for convex optimization in imaging science. SIAM J. Imaging Sci., 3(4):1015– 1046, 2010.Google Scholar
Evtimov, I., Eykholt, K., Fernandes, E., Kohno, T., Li, B., Prakash, A., Rahmati, A., and Song, D.. Robust physical-world attacks on machine learning models. In 2018 IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 1625–1634, 2018.Google Scholar
Fan, Q., Witzel, T., Nummenmaa, A., Van Dijk, K. R. A., Van Horn, J. D., Drews, M. K., Somerville, L. H., Sheridan, M. A., Santillana, R. M., Snyder, J., Hedden, T., Shaw, E. E., Hollinshead, M. O., Renvall, V., Zanzonico, R., Keil, B., Cauley, S., Polimeni, J. R., Tisdall, D., Buckner, R. L., Wedeen, V. J., Wald, L. L., Toga, A. W., and Rosen, B. R.. MGH–USC human connectome project datasets with ultra-high b-value diffusion MRI. Neuroimage, 124(Pt B):1108–1114, 2016.Google Scholar
Fawzi, A., Moosavi-Dezfooli, S., and Frossard, P.. The robustness of deep networks: a geometrical perspective. IEEE Signal Process. Mag., 34(6):50–62, 2017.Google Scholar
Feeman, T. G.. The Mathematics of Medical Imaging: A Beginner’s Guide. Springer Undergraduate Texts in Mathematics and Technology. Springer, Cham, 2015.Google Scholar
Fessler, J. A.. Model-based image reconstruction for MRI. IEEE Signal Process. Mag., 27(4):81–89, 2010.Google Scholar
Fessler, J. A.. Optimization methods for magnetic resonance image reconstruction: key models and optimization algorithms. IEEE Signal Process. Mag., 37(1):33–40, 2020.Google Scholar
Fessler, J. A. and Sutton, B. P.. Nonuniform fast Fourier transforms using min-max interpolation. IEEE Trans. Signal Process., 51(2):560–574, 2003.Google Scholar
Figueiredo, M. A. T. and Nowak, R. D.. An EM algorithm for wavelet-based image restoration. IEEE Trans. Image Process., 12(8):906–916, 2003.Google Scholar
Figueiredo, M. A. T., Nowak, R. D., and Wright, S. J.. Gradient projection for sparse reconstruction: application to compressed sensing and other inverse problems. IEEE J. Sel. Topics Signal Process., 1(4):586–597, 2007.Google Scholar
Finlayson, S. G., Bowers, J. D., Ito, J., Zittrain, J. L., Beam, A. L., and Kohane, I. S.. Adversarial attacks on medical machine learning. Science, 363(6433):1287–1289, 2019.Google Scholar
Floyd, C. E.. An artificial neural network for SPECT image reconstruction. IEEE Trans. Med. Imag., 10(3):485–487, 1991.Google Scholar
Fornasier, M. and Rauhut, H.. Compressive sensing. In Scherzer, O., editor, Handbook of Mathematical Methods in Imaging, pages 205–256. Springer, New York, NY, 2nd edition, 2015.Google Scholar
Foucart, S.. Stability and robustness of ℓ 1 -minimizations with Weibull matrices and redundant dictionaries. Linear Algebra Appl., 441:4–21, 2014.Google Scholar
Foucart, S. and Rauhut, H.. A Mathematical Introduction to Compressive Sensing. Applied and Numerical Harmonic Analysis. Birkhäuser, New York, NY, 2013.Google Scholar
Fountoulakis, K., Gondzio, J., and Zhlobich, P.. Matrix-free interior point method for compressed sensing problems. Math. Program. Comput., 6(1):1–31, 2014.Google Scholar
Friedlander, M., Mansour, H., Saab, R., and Yilmaz, I.. Recovering compressively sampled signals using partial support information. IEEE Trans. Inf. Theory, 58(2):1122–1134, 2012.Google Scholar
Frikel, J.. Sparse regularization in limited angle tomography. Appl. Comput. Harmon. Anal., 34(1):117–141, 2013.Google Scholar
Fu, L., Lee, T.-C., Kim, S. M., Alessio, A. M., Kinahan, P. E., Chang, Z., Sauer, K., Kalra, M. K., and Man, B.. Comparison between pre-log and post-log statistical models in ultra-low-dose CT reconstruction. IEEE Trans. Med. Imag., 36(3):707–720, 2017.Google Scholar
Fuchs, J. J.. On sparse representations in arbitrary redundant bases. IEEE Trans. Inf. Theory, 50(6):1341–1344, 2004.Google Scholar
Gács, P. and Lovász, L.. Khachiyan’s algorithm for linear programming. In Mathematical Programming at Oberwolfach, pages 61–68. Springer, 1981.Google Scholar
Gao, X.. Penalized Methods for High-Dimensional Least Absolute Deviations Regression. PhD thesis, The University of Iowa, 2008.Google Scholar
Gao, X. and Huang, J.. Asymptotic analysis of high-dimensional LAD regression with LASSO. Statist. Sinica, 20(4):1485–1506, 2010.Google Scholar
Gataric, M. and Poon, C.. A practical guide to the recovery of wavelet coefficients from Fourier measurements. SIAM J. Sci. Comput., 38(2):A1075–A1099, 2016.Google Scholar
Gauss, E.. Walsh Funktionen für Ingenieure und Naturwissenschaftler. Springer, Wiesbaden, 1994.Google Scholar
Gehm, M. E. and Brady, D. J.. Compressive sensing in the EO/IR. Appl. Opt., 54(8):C14– C22, 2015.Google Scholar
Genzel, M., Kutyniok, G., and März, M.. ℓ 1 -analysis minimization and generalized (co-)sparsity: when does recovery succeed? Appl. Comput. Harmon. Anal. (in press), 2020.Google Scholar
Genzel, M., März, M., and Seidel, R.. Compressed sensing with 1D total variation: breaking sample complexity barriers via non-uniform recovery. arXiv:2001.09952, 2020.Google Scholar
Gilton, D., Ongie, G., and Willett, R.. Neumann networks for linear inverse problems in imaging. IEEE Trans. Comput. Imag., 6:328–343, 2020.Google Scholar
Goldstein, T. and Osher, S.. The split Bregman method for L1-regularized problems. SIAM J. Imaging Sci., 2(2):323–343, 2009.Google Scholar
Goldstein, T. and Studer, C.. Phasemax: convex phase retrieval via basis pursuit. IEEE Trans. Inf. Theory, 64(4):2675–2689, 2018.Google Scholar
Goldstein, T., Xu, L., Kelly, K. F., and Baraniuk, R. G.. The STOne transform: multiresolution image enhancement and real-time compressive video. IEEE Trans. Image Process., 24(12):5581–5593, 2015.Google Scholar
Golubov, B., Efimov, A., and Skvortsov, V.. Walsh Series and Transforms: Theory and Applications, volume 64 of Mathematics and its Applications. Springer, Netherlands, 1991.Google Scholar
Goodfellow, I., Bengio, Y., and Courville, A.. Deep Learning. The MIT Press, Cambridge, MA, 2016.Google Scholar
Goodfellow, I., Pouget-Abadie, J., Mirza, M., Xu, B., Warde-Farley, D., Ozair, S., Courville, A., and Bengio, Y.. Generative adversarial nets. In Advances in Neural Information Processing Systems, pages 2672–2680, 2014.Google Scholar
Goodfellow, I., Shlens, J., and Szegedy, C.. Explaining and harnessing adversarial examples. arXiv:1412.6572, 2014.Google Scholar
Gottschling, N. M., Antun, V., Adcock, B., and Hansen, A. C.. The troublesome kernel: why deep learning for inverse problems is typically unstable. arXiv:2001.01258, 2020.Google Scholar
Gözcü, B., Mahabadi, R. J., Li, Y.-H., Ilcak, E., Çukur, T., Scarlett, J., and Cevher, V.. Learning-based compressive MRI. IEEE Trans. Med. Imag., 37(6):1394–1406, 2018.Google Scholar
Graff, C. G. and Sidky, E. Y.. Compressive sensing in medical imaging. Appl. Opt., 54(8):C23–C44, 2015.Google Scholar
Gregor, K. and LeCun, Y.. Learning fast approximations of sparse coding. In International Conference on Machine Learning, pages 399–406, 2010.Google Scholar
Gribonval, R., Rauhut, H., Schnass, K., and Vandergheynst, P.. Atoms of all channels, unite! Average case analysis of multi-channel sparse recovery using greedy algorithms. J. Fourier Anal. Appl., 13(5):655–687, 2008.Google Scholar
Gröchenig, K., Rzeszotnik, Z., and Strohmer, T.. Quantitative estimates for the finite section method and Banach algebras of matrices. Integral Equations and Operator Theory, 67(2):183–202, 2011.Google Scholar
Grohs, P., Hornung, F., Jentzen, A., and Von Wurstemberger, P.. A proof that artificial neural networks overcome the curse of dimensionality in the numerical approximation of Black-Scholes partial differential equations. arXiv:1809.02362, 2018.Google Scholar
Gross, D.. Recovering low-rank matrices from few coefficients in any basis. IEEE Trans. Inf. Theory, 57(3):1548–1566, 2011.Google Scholar
Grötschel, M., Lovász, L., and Schrijver, A.. Geometric Algorithms and Combinatorial Optimization, volume 2 of Algorithms and Combinatorics. Springer, Berlin, 2nd edition, 1993.Google Scholar
Guerquin-Kern, M., Lejeune, L., Pruessmann, K. P., and Unser, M.. Realistic analytical phantoms for parallel magnetic resonance imaging. IEEE Trans. Med. Imag., 31(3):626– 636, 2012.Google Scholar
Gühring, I., Kutyniok, G., and Petersen, P.. Error bounds for approximations with deep ReLU neural networks in W s, p norms. Anal. Appl., 18(5):803–859, 2020.Google Scholar
Guo, K., Kutyniok, G., and Labate, D.. Sparse multidimensional representations using anisotropic dilation and shear operators. In Chen, G. and Lai, M.-J., editors, Wavelets and Splines: Athens 2005, pages 189–201. Nashboro Press, Brentwood, TN, 2006.Google Scholar
Guo, K. and Labate, D.. Optimally sparse multidimensional representation using shearlets. SIAM J. Math. Anal., 39(1):298–318, 2007.Google Scholar
Gupta, H., Jin, K. H., Nguyen, H. Q., McCann, M. T., and Unser, M.. CNN-based projected gradient descent for consistent CT image reconstruction. IEEE Trans. Med. Imag., 37(6):1440–1453, 2018.Google Scholar
Hagen, R., Roch, S., and Silbermann, B.. C ∗ -Algebras and Numerical Analysis, volume 236 of Pure and Applied Mathematics. CRC Press, New York, 2001.Google Scholar
Haldar, J., Hernando, D., and Liang, Z.. Compressed-sensing MRI with random encoding. IEEE Trans. Med. Imag., 30(4):893–903, 2011.Google Scholar
Hale, E. T., Yin, W., and Zhang, Y.. Fixed-point continuation for ℓ 1 -minimization: methodology and convergence. SIAM J. Optim., 19(3):1107–1130, 2008.Google Scholar
Hammernik, K., Klatzer, T., Kobler, E., Recht, M. P., Sodickson, D. K., Pock, T., and Knoll, F.. Learning a variational network for reconstruction of accelerated MRI data. Magn. Reson. Med., 79(6):3055–3071, 2018.Google Scholar
Hansen, A. C.. On the approximation of spectra of linear operators on Hilbert spaces. J. Funct. Anal., 254(8):2092–2126, 2008.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):81–124, 2011.Google Scholar
Hansen, A. C. and Roman, B.. On structure and optimisation in computational harmonic analysis - the key aspects in sparse regularisation. In Harmonic Analysis and Applications (in press). Springer, Berlin, 2020.Google Scholar
Hansen, A. C. and Terhaar, L.. On the stable sampling rate for binary measurements and wavelet reconstruction. Appl. Comput. Harmon. Anal., 48(2):630–654, 2020.Google Scholar
Hastie, T., Tibshirani, R., and Wainwright, M.. Statistical Learning with Sparsity: The Lasso and Generalizations. Number 143 in Monographs on Statistics and Applied Probability. CRC Press, Boca Raton, FL, 2015.Google Scholar
Haviv, I. and Regev, O.. The restricted isometry property of subsampled Fourier matrices. In Proceedings of the 2016 Annual ACM-SIAM Symposium on Discrete Algorithms, pages 288–297. SIAM, Philadelphia, PA, 2016.Google Scholar
He, B., You, Y., and Yuan, X.. On the convergence of primal-dual hybrid gradient algorithm. SIAM J. Imaging Sci., 7(4):2526–2537, 2014.Google Scholar
He, K., Zhang, X., Ren, S., and Sun, J.. Deep residual learning for image recognition. In 2016 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pages 770–778, 2016.Google Scholar
He, L. and Carin, L.. Exploiting structure in wavelet-based Bayesian compressive sensing. IEEE Trans. Signal Process., 57(9):3488–3497, 2009.Google Scholar
He, L., Chen, H., and Carin, L.. Tree-structured compressive sensing with variational Bayesian analysis. IEEE Signal Process. Lett., 17(3):233–236, 2010.Google Scholar
Hernández, E. and Weiss, G.. A First Course on Wavelets. Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1996.Google Scholar
Higham, C. F. and Higham, D. J.. Deep learning: an introduction for applied mathematicians. SIAM Rev., 61(4):860–891, 2019.Google Scholar
Hogan, J. A. and Lakey, J. D.. Duration and Bandwidth Limiting: Prolate Functions, Sampling, and Applications. Applied and Numerical Harmonic Analysis. Birkhäuser, Basel, 2012.Google Scholar
Holland, D. J., Bostock, M. J., Gladden, L. F., and Nietlispach, D.. Fast multidimensional NMR spectroscopy using compressed sensing. Angew. Chem. Int. Ed., 50(29):6548–6551, 2011.Google Scholar
Hornik, K.. Approximation capabilities of multilayer feedforward networks. Neural Networks, 4(2):251–257, 1991.Google Scholar
Huang, G., Jiang, H., Matthews, K., and Wilford, P.. Lensless imaging by compressive sensing. In 20th IEEE International Conference on Image Processing, 2013.Google Scholar
Huang, J. and Zhang, T.. The benefit of group sparsity. Ann. Statist., 38(4):1978–2004, 2010.Google Scholar
Huang, Y., Würfl, T., Breininger, K., Liu, L., Lauritsch, G., and Maier, A.. Some investigations on robustness of deep learning in limited angle tomography. In International Conference on Medical Image Computing and Computer-Assisted Intervention, pages 145–153, 2018.Google Scholar
Hütter, J.-C. and Rigollet, P.. Optimal rates for total variation denoising. In Feldman, V., Rakhlin, A., and Shamir, O., editors, 29th Annual Conference on Learning Theory, volume 49 of Proceedings of Machine Learning Research, pages 1115–1146. PMLR, 2016.Google Scholar
Huynh-Thu, Q. and Ghanbari, M.. Scope of validity of PSNR in image/video quality assessment. Electron. Lett., 44(13):800–801, 2008.Google Scholar
Jaganathan, K., Eldar, Y. C., and Hassibi, B.. Phase retrieval: an overview of recent developments. In Stern, A., editor, Optical Compressive Imaging, Series in Optics and Optoelec-tronics, pages 263–296. CRC Press, Boca Raton, FL, 2017.Google Scholar
Jang, Y., Zhao, T., Hong, S., and Lee, H.. Adversarial defense via learning to generate diverse attacks. In Proceedings of the IEEE International Conference on Computer Vision, pages 2740–2749, 2019.Google Scholar
Jaspan, O. R., Fleysher, R., and Lipton, M. L.. Compressed sensing MRI: a review of the clinical literature. Brit. J. Radiol., 88(1056):20150487, 2015.Google Scholar
Jiang, H., Chen, Z., Shi, Y., Dai, B., and Zhao, T.. Learning to defense by learning to attack. arXiv:1811.01213, 2018.Google Scholar
Jin, K. H., McCann, M. T., Froustey, E., and Unser, M.. Deep convolutional neural network for inverse problems in imaging. IEEE Trans. Image Process., 26(9):4509–4522, 2017.Google Scholar
Jones, A., Tamtögl, A., Calvo-Almazán, I., and Hansen, A. C.. Continuous compressed sensing for surface dynamical processes with helium atom scattering. Sci. Rep., 6(1):27776, 2016.Google Scholar
Jørgensen, J. S., Coban, S. B., Lionheart, W. R. B., McDonald, S. A., and Withers, P. J.. SparseBeads data: benchmarking sparsity-regularized computed tomography. Meas. Sci. Technol., 28(12):124005, 2017.Google Scholar
Jørgensen, J. S., Kruschel, C., and Lorenz, D. A.. Testable uniqueness conditions for empirical assessment of undersampling levels in total variation-regularized X-ray CT. Inverse Probl. Sci. Eng., 23(8):1283–1305, 2015.Google Scholar
Jørgensen, J. S. and Sidky, E. Y.. How little data is enough? Phase-diagram analysis of sparsity-regularized X-ray CT. Philos. Trans. Roy. Soc. A, 373(2043):20140387, 2015.Google Scholar
Jørgensen, J. S., Sidky, E. Y., and Pan, X.. Quantifying admissible undersampling for sparsity-exploiting iterative image reconstruction in X-ray CT. IEEE Trans. Med. Imag., 32(2):460– 473, 2013.Google Scholar
Kabanava, M. and Rauhut, H.. Analysis ℓ 1 -recovery with frames and Gaussian measurements. Acta Appl. Math., 140(1):173–195, 2015.Google Scholar
Kabanava, M. and Rauhut, H.. Cosparsity in compressed sensing. In Boche, H., Calderbank, R., Kutyniok, G., and Vybíral, J., editors, Compressed Sensing and its Applications: MATHEON Workshop 2013, Applied and Numerical Harmonic Analysis, pages 315–339. Birkhäuser, Cham, 2015.Google Scholar
Karimi, H., Nutini, J., and Schmidt, M.. Linear convergence of gradient and proximal-gradient methods under the Polyak-Łojasiewicz condition. In Frasconi, P., Landwehr, N., Manco, G., and Vreeken, J., editors, Machine Learning and Knowledge Discovery in Databases. ECML PKDD 2016, volume 9851 of Lecture Notes in Computer Science, pages 795–811. Springer, Cham, 2016.Google Scholar
Karmarkar, N.. A new polynomial-time algorithm for linear programming. Combinatorica, 4(4):373–395, 1984.Google Scholar
Katz, O., Bromberg, Y., and Silberberg, Y.. Compressive ghost imaging. Appl. Phys. Lett., 95(13):131110, 2009.Google Scholar
Kazimierczuk, K. and Orekhov, V. Y.. Accelerated NMR spectroscopy by using compressed sensing. Angew. Chem. Int. Ed., 50(24):5556–5559, 2011.Google Scholar
Khachiyan, L. G.. Polynomial algorithms in linear programming. Zh. Vychisl. Mat. Mat. Fiz., 20(1):51–68, 1980.Google Scholar
Kim, S. J., Koh, K., Lustig, M., Boyd, S., and Gorinevsky, D.. An interior-point method for large-scale ℓ 1 -regularized least squares. IEEE J. Sel. Topics Signal Process., 1(4):606–617, 2007.Google Scholar
Knoll, F., Bredies, K., Pock, T., and Stollberger, R.. Second order total generalized variation (TGV) for MRI. Magn. Reson. Med., 65(2):480–491, 2011.Google Scholar
Knoll, F., Clason, C., Diwoky, C., and Stollberger, R.. Adapted random sampling patterns for accelerated MRI. Magn. Reson. Mater. Phy., 24(1):43–50, 2011.Google Scholar
Knoll, F., Hammernik, K., Zhang, C., Moeller, S., Pock, T., Sodickson, D. K., and Akçakaya, M.. Deep-learning methods for parallel magnetic resonance imaging reconstruction: a survey of the current approaches, trends, and issues. IEEE Signal Process. Mag., 37(1):128–140, 2020.Google Scholar
Knoll, F., Murrell, T., Sriram, A., Yakubova, N., Zbontar, J., Rabbat, M., Defazio, A., Muckley, M. J., Sodickson, D. K., Zitnick, C. L., and Recht, M. P.. Advancing machine learning for MR image reconstruction with an open competition: overview of the 2019 fastMRI challenge. Magn. Reson. Med., 84(6):3054–3070, 2020.Google Scholar
Knopp, T., Kunis, S., and Potts, D.. A note on the iterative MRI reconstruction from nonuniform k-space data. Int. J. Biomed. Imaging, 2007:024727, 2007.Google Scholar
Komodakis, N. and Pesquet, J.-C.. Playing with duality: an overview of recent primal-dual approaches for solving large-scale optimization problems. IEEE Signal Process. Mag., 32(6):31–54, 2015.Google Scholar
Krahmer, F., Kruschel, C., and Sandbichler, M.. Total variation minimization in compressed sensing. In Boche, H., Caire, G., Calderbank, R., März, M., Kutyniok, G., and Mathar, R., editors, Compressed Sensing and its Applications: Second International MATHEON Conference 2015, Applied and Numerical Harmonic Analysis, pages 333–358. Birkhäuser, Cham, 2017.Google Scholar
Krahmer, F., Mendelson, S., and Rauhut, H.. Suprema of chaos processes and the restricted isometry property. Comm. Pure Appl. Math., 67(11):1877–1904, 2014.Google Scholar
Krahmer, F., Needell, D., and Ward, R.. Compressive sensing with redundant dictionaries and structured measurements. SIAM J. Math. Anal., 47(6):4606–4629, 2015.Google Scholar
Krahmer, F., Rauhut, H., and Ward, R.. Local coherence sampling in compressed sensing. In Proceedings of the 10th International Conference on Sampling Theory and Applications, pages 476–480, 2013.Google Scholar
Krahmer, F. and Ward, R.. Stable and robust sampling strategies for compressive imaging. IEEE Trans. Image Process., 23(2):612–622, 2013.Google Scholar
Krizhevsky, A., Sutskever, I., and Hinton, G. E.. ImageNet classification with deep convolutional neural networks. Advances in Neural Information Processing Systems, pages 1097–1105, 2012.Google Scholar
Krogh, A. and Hertz, J. A.. A simple weight decay can improve generalization. In Advances in Neural Information Processing Systems, pages 950–957, 1992.Google Scholar
Kuchment, P.. The Radon Transform and Medical Imaging. CBMS-NSF Regional Conference Series in Applied Mathematics. Society for Industrial and Applied Mathematics, Philadelphia, PA, 2013.Google Scholar
Kueng, R. and Gross, D.. RIPless compressed sensing from anisotropic measurements. Linear Algebra Appl., 441:110–123, 2014.Google Scholar
Kurakin, A., Goodfellow, I. J., and Bengio, S.. Adversarial machine learning at scale. arXiv:1611.01236, 2016.Google Scholar
Kutyniok, G. and Labate, D., editors. Shearlets: Multiscale Analysis for Multivariate Data. Applied and Numerical Harmonic Analysis. Birkhäuser, Basel, 2012.Google Scholar
Kutyniok, G. and Lim, W.-Q.. Compactly supported shearlets are optimally sparse. J. Approx. Theory, 163(11):1564–1589, 2011.Google Scholar
Kutyniok, G. and Lim, W.-Q.. Optimal compressive imaging of Fourier data. SIAM J. Imaging Sci., 11(1):507–546, 2018.Google Scholar
Labate, D., Lim, W.-Q., Kutyniok, G., and Weiss, G.. Sparse multidimensional representation using shearlets. In Papadakis, M., Laine, A. F., and Unser, M. A., editors, Wavelets XI, volume 5914, pages 254–262. International Society for Optics and Photonics, SPIE, 2005.Google Scholar
Lai, M.-J. and Liu, Y.. The null space property for sparse recovery from multiple measurement vectors. Appl. Comput. Harmon. Anal., 30(3):402–406, 2011.Google Scholar
Larkman, D. J. and Nunes, R. G.. Parallel magnetic resonance imaging. Phys. Med. Biol., 52(7):R15, 2007.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):610– 619, 2011.Google Scholar
Laska, J. N., Davenport, M. A., and Baraniuk, R. G.. Exact signal recovery from sparsely corrupted measurements through the pursuit of justice. In 2009 Conference Record of the Forty-Third Asilomar Conference on Signals, Systems and Computers, pages 1556–1560, 2009.Google Scholar
Lawler, E. L.. The great mathematical Sputnik of 1979. Math. Intelligencer, 2(4):191–198, 1980.Google Scholar
Cun, Y. Y. Bengio, , and Hinton, G.. Deep learning. Nature, 521(7553):436–444, 2015.Google Scholar
Pennec, E. and Mallat, S.. Sparse geometric image representations with bandelets. IEEE Trans. Image Process., 14(4):423–438, 2005.Google Scholar
Leary, R., Saghi, Z., Midgley, P. A., and Holland, D. J.. Compressed sensing electron tomography. Ultramicroscopy, 131:70–91, 2013.Google Scholar
Ledoux, M.. The Concentration of Measure Phenomenon. Number 89 in Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2001.Google Scholar
Lee, K., Bresler, Y., and Junge, M.. Oblique pursuits for compressed sensing. IEEE Trans. Inf. Theory, 59(9):6111–6141, 2013.Google Scholar
Li, C. and Adcock, B.. Compressed sensing with local structure: uniform recovery guarantees for the sparsity in levels class. Appl. Comput. Harmon. Anal., 46(3):453–477, 2019.Google Scholar
Li, X.. Compressed sensing and matrix completion with a constant proportion of corruptions. Constr. Approx., 37(1):73–99, 2013.Google Scholar
Liang, D., Cheng, J., Ke, Z., and Ying, L.. Deep magnetic resonance image reconstruction: inverse problems meet neural networks. IEEE Signal Process. Mag., 37(1):141–151, 2020.Google Scholar
Liang, S. and Srikant, R.. Why deep neural networks for function approximation? arXiv:1610.04161, 2016.Google Scholar
Liang, Z.-P. and Lauterbur, P. C.. Principles of Magnetic Resonance Imaging: A Signal Processing Perspective. IEEE Press Series on Biomedical Engineering. Wiley–IEEE Press, New York, 2000.Google Scholar
Lin, D. J., Johnson, P. M., Knoll, F., and Lui, Y. W.. Artificial intelligence for MR image reconstruction: an overview for clinicians. J. Magn. Reson. Imaging (in press), 2020.Google Scholar
Lindner, M.. Infinite Matrices and their Finite Sections: An Introduction to the Limit Operator Method. Frontiers in Mathematics. Birkhäuser, Basel, 2006.Google Scholar
Ling, S. and Strohmer, T.. Self-calibration and biconvex compressive sensing. Inverse Problems, 31(11):115002, 2015.Google Scholar
Liu, B., Zou, Y. M., and Ying, L.. SparseSENSE: application of compressed sensing in parallel MRI. In 2008 International Conference on Information Technology and Applications in Biomedicine, pages 127–130, 2008.Google Scholar
Long, J., Shelhamer, E., and Darrell, T.. Fully convolutional networks for semantic segmentation. In 2015 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pages 3431–3440, 2015.Google Scholar
Lou, Y., Zeng, T., Osher, S., and Xin, J.. A weighted difference of anisotropic and isotropic total variation model for image processing. SIAM J. Imaging Sci., 8(3):1798–1823, 2015.Google Scholar
Lucas, A., Iliadis, M., Molina, R., and Katsaggelos, A. K.. Using deep neural networks for inverse problems in imaging: beyond analytical methods. IEEE Signal Process. Mag., 35(1):20–36, 2018.Google Scholar
Lundervold, A. and Lundervold, A.. An overview of deep learning in medical imaging focusing on MRI. Z. Med. Phys., 29(2):102–127, 2019.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. Med., 58(6):1182–1195, 2007.Google Scholar
Lustig, M., Donoho, D. L., Santos, J. M., and Pauly, J. M.. Compressed sensing MRI. IEEE Signal Process. Mag., 25(2):72–82, 2008.Google Scholar
Ma, J. and März, M.. A multilevel based reweighting algorithm with joint regularizers for sparse recovery. arXiv:1604.06941, 2016.Google Scholar
Ma, J. and Plonka, G.. The curvelet transform. IEEE Signal Process. Mag., 27(2):118–133, 2010.Google Scholar
Macovski, A.. Noise in MRI. Magn. Reson. Imaging, 36(3):494–497, 1996.Google Scholar
Madry, A., Makelov, A., Schmidt, L., Tsipras, D., and Vladu, A.. Towards deep learning models resistant to adversarial attacks. arXiv:1706.06083, 2017.Google Scholar
Majumdar, A.. Compressed Sensing for Magnetic Resonance Image Reconstruction. Cambridge University Press, Cambridge, 2015.Google Scholar
Majumdar, A. and Ward, R. K.. Calibration-less multi-coil MR image reconstruction. Magn. Reson. Imaging, 30(7):1032–1045, 2012.Google Scholar
Mallat, S.. A Wavelet Tour of Signal Processing: The Sparse Way. Academic Press, Amsterdam, 3rd edition, 2009.Google Scholar
Mallat, S.. Understanding deep convolutional networks. Philos. Trans. Roy. Soc. A, 374(2065):20150203, 2016.Google Scholar
Mallat, S. and Peyré, G.. A review of bandlet methods for geometrical image representation. Numer. Algorithms, 44(3):205–234, 2007.Google Scholar
Marcia, R. F., Harmany, Z. T., and Willett, R. M.. Compressive coded aperture imaging. In Bouman, C. A., Miller, E. L., and Pollak, I., editors, Computational Imaging VII, volume 7246, pages 106–118. International Society for Optics and Photonics, SPIE, 2009.Google Scholar
Marcia, R. F., Willett, R. M., and Harmany, Z. T.. Compressive optical imaging: architectures and algorithms. In Cristobal, G., Schelken, P., and Thienpont, H., editors, Optical and Digital Image Processing: Fundamentals and Applications, pages 485–505. Wiley, New York, 2011.Google Scholar
Mardani, M., Sun, Q., Vasawanala, S., Papyan, V., Monajemi, H., Pauly, J., and Donoho, D.. Neural proximal gradient descent for compressive imaging. In Advances in Neural Information Processing Systems, pages 9596–9606, 2018.Google Scholar
Markoff, J.. Scientists see promise in deep-learning programs.www.nytimes .com/2012/11/24/science/scientists-see-advances-in-deep-learning-a-part-of-artificial-intelligence.html, November 2012.Google Scholar
Marwah, K., Wetzstein, G., Bando, Y., and Raskar, R.. Compressive light field photography using overcomplete dictionaries and optimized projections. ACM Trans. Graph., 32(46), 2013.Google Scholar
Matiyasevich, Y. V.. Hilbert’s Tenth Problem. The MIT Press, Cambridge, MA, 1993.Google Scholar
McCann, M. T., Jin, K. H., and Unser, M.. Convolutional neural networks for inverse problems in imaging: a review. IEEE Signal Process. Mag., 34(6):85–95, 2017.Google Scholar
McRobbie, D. W., Moore, E. A., Graves, M. J., and Prince, M. R.. MRI: From Picture to Proton. Cambridge University Press, Cambridge, 2nd edition, 2006.Google Scholar
Metzler, C. A., Mousavi, A., and Baraniuk, R. G.. Learned D-AMP: principled neural network based compressive image recovery. In Advances in Neural Information Processing Systems, pages 1770–1781, 2017.Google Scholar
Mishali, M. and Eldar, Y. C.. Xampling: analog to digital at sub-Nyquist rates. IET Circuits, Devices, & Systems, 5(1):8–20, 2011.Google Scholar
Mishali, M., Eldar, Y. C., and Elron, A. J.. Xampling: signal acquisition and processing in union of subspaces. IEEE Trans. Signal Process., 59(10):4719–4734, 2011.Google Scholar
Monga, V., Li, Y., and Eldar, Y. C.. Algorithm unrolling: interpretable, efficient deep learning for signal and image processing. arXiv:1912.10557, 2019.Google Scholar
Moosavi-Dezfooli, S., Fawzi, A., Fawzi, O., and Frossard, P.. Universal adversarial perturbations. In 2017 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pages 86–94, 2017.Google Scholar
Moosavi-Dezfooli, S. M., Fawzi, A., and Frossard, P.. DeepFool: a simple and accurate method to fool deep neural networks. In 2016 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pages 2574–2582, 2016.Google Scholar
Moshtaghpour, A.. Computational Interferometry for Hyperspectral Imaging. PhD thesis, Université catholique de Louvain, 2019.Google Scholar
Moshtaghpour, A., Dias, J. B., and Jacques, L.. Close encounters of the binary kind: signal reconstruction guarantees for compressive Hadamard sampling with Haar wavelet basis. IEEE Trans. Inf. Theory, 66(11):7253–7273, 2020.Google Scholar
Mota, J. F. C., Deligiannis, N., and Rodrigues, M. R. D.. Compressed sensing with prior information: strategies, geometry, and bounds. IEEE Trans. Inf. Theory, 63(7):4472–4496, 2017.Google Scholar
Motwani, R. and Raghavan, P.. Randomized Algorithms. Cambridge University Press, Cambridge, 1995.Google Scholar
Muckley, M. J., Riemenschneider, B., Radmanesh, A., Kim, S., Jeong, G., Ko, J., Jun, Y., Shin, H., Hwang, D., Mostapha, M., Arberet, S., Nickel, D., Ramzi, Z., Ciuciu, P., Starck, J.-L., Teuwen, J., Karkalousos, D., Zhang, C., Sriram, A., Huang, Z., Yakubova, N., Lui, Y., and Knoll, F.. State-of-the-art machine learning MRI reconstruction in 2020: results of the second fastMRI challenge. arXiv:2012.06318, 2020.Google Scholar
Mueller, J. L. and Siltanen, S.. Linear and Nonlinear Inverse Problems with Practical Applications. Computational Science & Engineering. Society for Industrial and Applied Mathematics, Philadelphia, PA, 2012.Google Scholar
Murphy, M., Alley, M., Demmel, J., Keutzer, K., Vasanawala, S., and Lustig, M.. Fast l 1 -SPIRiT compressed sensing parallel imaging MRI: scalable parallel implementation and clinically feasible runtime. IEEE Trans. Med. Imag., 31(6):1250–1262, Jun. 2012.Google Scholar
Nam, S., Davies, M. E., Elad, M., and Gribonval, R.. The cosparse analysis model and algorithms. Appl. Comput. Harmon. Anal., 34(1):30–56, 2013.Google Scholar
Natarajan, B. K.. Sparse approximate solutions to linear systems. SIAM J. Comput., 24(2):227–234, 1995.Google Scholar
Natterer, F. and Wübbeling, F.. Mathematical Methods in Image Reconstruction. Mathematical Modeling and Computation. Society for Industrial and Applied Mathematics, Philadelphia, PA, 2001.Google Scholar
Needell, D. and Tropp, J. A.. CoSaMP: Iterative signal recovery from incomplete and inaccurate samples. Appl. Comput. Harmon. Anal., 26(3):301–321, 2008.Google Scholar
Needell, D. and Ward, R.. Near-optimal compressed sensing guarantees for total variation minimization. IEEE Trans. Image Process., 22(10):3941–3949, 2013.Google Scholar
Needell, D. and Ward, R.. Stable image reconstruction using total variation minimization. SIAM J. Imaging Sci., 6(2):1035–1058, 2013.Google Scholar
Nemirovski, A.. Polynomial time methods in convex programming. In Renegar, J., Shub, M., and Smale, S., editors, The Mathematics of Numerical Analysis, volume 32, pages 543–589. AMS-SIAM Summer Seminar on Applied Mathematics, 1995.Google Scholar
Nemirovski, A. S. and Yudin, D. B.. Problem Complexity and Method Efficiency in Optimization. Wiley-Interscience, New York, 1983.Google Scholar
Nesterov, Y.. A method for solving the convex programming problem with convergence rate O(1/k 2 ). Soviet Math. Dokl., 27(2):372–376, 1983.Google Scholar
Nesterov, Y.. Smooth minimization of non-smooth functions. Math. Program., 103(1):127– 152, 2005.Google Scholar
Nesterov, Y.. Lectures on Convex Optimization, volume 137 of Springer Optimization and Its Applications. Springer, Cham, 2nd edition, 2018.Google Scholar
Nesterov, Y. and Nemirovskii, A.. Interior-Point Polynomial Algorithms in Convex Programming. Studies in Applied and Numerical Mathematics. Society for Industrial and Applied Mathematics, Philadelphia, PA, 1994.Google Scholar
Nguyen, A., Yosinki, J., and Clune, J.. Deep neural networks are easily fooled: high confidence predictions for unrecognizable images. In 2015 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pages 427–436, 2015.Google Scholar
Nguyen, T. and Tran, T. D.. Exact recoverability from dense corrupted observations via ℓ 1 -minimization. IEEE Trans. Inf. Theory, 59(4):2017–2035, 2013.Google Scholar
Nishimura, D. G.. Principles of Magnetic Resonance Imaging. Lulu Press, Raleigh, NC, 2010.Google Scholar
Nocedal, J. and Wright, S. J.. Numerical Optimization. Springer Series in Operations Research and Financial Engineering. Springer, New York, NY, 2nd edition, 2006.Google Scholar
Ongie, G., Jalal, A., Metzler, C. A., Baraniuk, R. G., Dimakis, A. G., and Willett, R.. Deep learning techniques for inverse problems in imaging. IEEE J. Sel. Areas Inf. Theory, 1(1):39–56, 2020.Google Scholar
Osborne, M., Presnell, B., and Turlach, B.. A new approach to variable selection in least squares problems. IMA J. Numer. Anal., 20(3):389–403, 2000.Google Scholar
Pan, X., Sidky, E. Y., and Vannier, M.. Why do commercial CT scanners still employ traditional, filtered back-projection for image reconstruction? Inverse Problems, 25(12):1230009, 2009.Google Scholar
Petersen, P. and Voigtlaender, F.. Optimal approximation of piecewise smooth functions using deep ReLU neural networks. Neural Networks, 108:296–330, 2018.Google Scholar
Petra, S. and Schnörr, C.. Average case recovery analysis of tomographic compressive sensing. Linear Algebra Appl., 441:168–198, 2014.Google Scholar
Peyré, G., Fadili, J., and Starck, J.-L.. Learning the morphological diversity. SIAM J. Imaging Sci., 3(3):646–669, 2010.Google Scholar
Pinkus, A.. Approximation theory of the MLP model in neural networks. Acta Numer., 8:143–195, 1999.Google Scholar
Plan, Y.. Compressed Sensing, Sparse Approximation, and Low-Rank Matrix Estimation. PhD thesis, California Institute of Technology, 2011.Google Scholar
Pock, T., Cremers, D., Bischof, H., and Chambolle, A.. An algorithm for minimizing the Mumford–Shah functional. In 2009 IEEE 12th International Conference on Computer Vision, pages 1133–1140, 2009.Google Scholar
Poon, C.. On the role of total variation in compressed sensing. SIAM J. Imaging Sci., 8(1):682–720, 2015.Google Scholar
Poon, C.. Structure dependent sampling in compressed sensing: theoretical guarantees for tight frames. Appl. Comput. Harmon. Anal., 42(3):402–451, 2017.Google Scholar
Poon, C., Keriven, N., and Peyré, G.. The geometry of off-the-grid compressed sensing. arXiv:1802.08464, 2018.Google Scholar
Potts, D., Steidl, G., and Tasche, M.. Fast Fourier Transforms for nonequispaced data: a tutorial. In Benedetto, J. J. and Ferreira, P. J. S. G., editors, Modern Sampling Theory, Applied and Numerical Harmonic Analysis, pages 247–270. Birkhäuser, Boston, MA, 2001.Google Scholar
Pruessmann, K. P., Weiger, M., Scheidigger, M. B., and Boesiger, P.. SENSE: Sensitivity encoding for fast MRI. Magn. Reson. Med., 42(5):952–962, 1999.Google Scholar
Puy, G., Davies, M. E., and Gribonval, R.. Recipes for stable linear embeddings from Hilbert spaces to R m . IEEE Trans. Inf. Theory, 63(4):2171–2187, 2017.Google Scholar
Puy, G., Marques, J. P., Gruetter, R., Thiran, J., Ville, D. P. Vandergheynst, , and Wiaux, Y.. Spread spectrum magnetic resonance imaging. IEEE Trans. Med. Imag., 31(3):586–598, 2012.Google Scholar
Puy, G., Vandergheynst, P., and Wiaux, Y.. On variable density compressive sampling. IEEE Signal Process. Lett., 18(10):595–598, 2011.Google Scholar
Qu, X., Chen, Y., Zhuang, X., Yan, Z., Guo, D., and Chen, Z.. Spread spectrum compressed sensing MRI using chirp radio frequency pulses. In 2016 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pages 689–693, 2016.Google Scholar
Raj, A., Bresler, Y., and Li, B.. Improving robustness of deep-learning-based image reconstruction. In Daumé, H. III and Singh, A., editors, Proceedings of the 37th International Conference on Machine Learning, volume 119 of Proceedings of Machine Learning Research, pages 7932–7942. PMLR, 2020.Google Scholar
Ramani, S., Liu, Z., Rosen, J., Nielsen, J.-F., and Fessler, J. A.. Regularization parameter selection for nonlinear iterative image restoration and MRI reconstruction using GCV and SURE-based methods. IEEE Trans. Image Process., 21(8):3659–3672, 2012.Google Scholar
Rauhut, H.. Random sampling of sparse trigonometric polynomials. Appl. Comput. Harmon. Anal., 22(1):16–42, 2007.Google Scholar
Rauhut, H.. Compressive sensing and structured random matrices. In Fornasier, M., editor, Theoretical Foundations and Numerical Methods for Sparse Recovery, volume 9 of Radon Series in Computational and Applied Mathematics, pages 1–92. de Gruyter, Berlin, 2010.Google Scholar
Rauhut, H. and Ward, R.. Sparse recovery for spherical harmonic expansions. In Proceedings of the 9th International Conference on Sampling Theory and Applications, 2011.Google Scholar
Rauhut, H. and Ward, R.. Sparse Legendre expansions via ℓ 1 -minimization. J. Approx. Theory, 164(5):517–533, 2012.Google Scholar
Rauhut, H. and Ward, R.. Interpolation via weighted ℓ 1 minimization. Appl. Comput. Harmon. Anal., 40(2):321–351, 2016.Google Scholar
Ravishankar, S. and Bresler, Y.. Adaptive sampling design for compressed sensing MRI. In 2011 Annual International Conference of the IEEE Engineering in Medicine and Biology Society, pages 3751–3755, 2011.Google Scholar
Ravishankar, S. and Bresler, Y.. MR image reconstruction from highly undersampled k-space data by dictionary learning. IEEE Trans. Med. Imag., 30(5):1028–1041, 2011.Google Scholar
Ravishankar, S., Ye, J. C., and Fessler, J. A.. Image reconstruction: from sparsity to data-adaptive methods and machine learning. Proc. IEEE, 108(1):86–109, 2020.Google Scholar
Recht, B., Fazel, M., and Parrilo, P. A.. Guaranteed minimum-rank solutions of linear matrix equations via nuclear norm minimization. SIAM Rev., 52(3):471–501, 2010.Google Scholar
Reed, R.. Pruning algorithms-a survey. IEEE Trans. Neural Netw., 4(5):740–747, 1993.Google Scholar
Rockafeller, R. T.. Convex Analysis. Princeton Mathematical Series. Princeton University Press, Princeton, NJ, 1970.Google Scholar
Roman, B., Bastounis, A., Adcock, B., and Hansen, A. C.. On fundamentals of models and sampling in compressed sensing. Preprint, 2015.Google Scholar
Roman, B., Hansen, A. C., and Adcock, B.. On asymptotic structure in compressed sensing. arXiv:1406.4178, 2014.Google Scholar
Romano, Y., Elad, M., and Milanfar, P.. The little engine that could: Regularization by Denoising (RED). SIAM J. Imaging Sci., 10(4):1804–1844, 2017.Google Scholar
Romberg, J.. Imaging via compressive sampling. IEEE Signal Process. Mag., 25(2):14–20, 2008.Google Scholar
Romberg, J.. Compressive sensing by random convolution. SIAM J. Imaging Sci., 2(4):1098– 1128, 2009.Google Scholar
Ronneberger, O., Fischer, P., and Brox, T.. U-net: convolutional networks for biomedical image segmentation. In International Conference on Medical Image Computing and Computer Assisted Intervention, pages 234–241, 2015.Google Scholar
Roth, S. and Black, M. J.. Field of experts. Int. J. Comput. Vis., 82(2):205, 2009.Google Scholar
Roulet, V. and Boumal, A., d’Aspremont, N.. Computational complexity versus statistical performance on sparse recovery problems. Inf. Inference, 9(1):1–32, 2020.Google Scholar
Rudelson, M. and Vershynin, R.. On sparse reconstruction from Fourier and Gaussian measurements. Comm. Pure Appl. Math., 61(8):1025–1045, 2008.Google Scholar
Rudin, L. I., Osher, S., and Fatemi, E.. Nonlinear total variation based noise removal algorithms. Physica D, 60(1–4):259–268, 1992.Google Scholar
Russakovsky, O., Deng, J., Su, H., Krause, J., Satheesh, S., Ma, S., Huang, Z., Karpathy, A., Khosla, A., Bernstein, M., Berg, A. C., and Li, F.-F.. Imagenet large scale visual recognition challenge. Int. J. Comput. Vis., 115(3):211–252, 2015.Google Scholar
Sandino, C. M., Cheng, J. Y., Chen, F., Mardani, M., Pauly, J. M., and Vasanawala, S. S.. Compressed sensing: from research to clinical practice with deep neural networks. IEEE Signal Process. Mag., 31(1):117–127, 2020.Google Scholar
Schlemper, J., Caballero, J., Hajnal, J. V., Price, A., and Rueckert, D.. A deep cascade of convolutional neural networks for MR image reconstruction. In Niethammer, M., Styner, M., Aylward, S., Zhu, H., Oguz, I., Yap, P.-T., and Shen, D., editors, Information Processing in Medical Imaging, volume 10265 of Lecture Notes in Computer Science, pages 647–658. Springer, Cham, 2017.Google Scholar
Schmidhuber, J.. Deep learning in neural networks: an overview. Neural Networks, 61:85– 117, 2015.Google Scholar
Schmidt, L., Santurkar, S., Tsipras, D., Talwar, K., and Madry, A.. Adversarially robust generalization requires more data. In Advances in Neural Information Processing Systems, pages 5014–5026, 2018.Google Scholar
Schniter, P. and Rangan, S.. Compressive phase retrieval via generalized approximate message passing. IEEE Trans. Signal Process., 63(4):1043–1055, 2015.Google Scholar
Schwab, C. and Zech, J.. Deep learning in high dimension: neural network expression rates for generalized polynomial chaos expansions in UQ. Anal. Appl., 17(1):19–55, 2019.Google Scholar
Selesnick, I. W., Baraniuk, R. G., and Kingsbury, N. C.. The dual-tree complex wavelet transform. IEEE Signal Process. Mag., 22(6):123–151, 2005.Google Scholar
Selesnick, I. W. and Figueiredo, M. A. T.. Signal restoration with overcomplete wavelet transforms: comparison of analysis and synthesis priors. In Goyal, V. K., Papadakis, M., and Ville, D. V. D., editors, Wavelets XIII, volume 7446, pages 107–121. International Society for Optics and Photonics, SPIE, 2009.Google Scholar
Setzer, S.. Operator splittings, Bregman methods and frame shrinkage in image processing. Int. J. Comput. Vis., 92(3):265–280, 2011.Google Scholar
Sharif, M., Bhagavatula, S., Bauer, L., and Reiter, M. K.. Accessorize to a crime: real and stealthy attacks on state-of-the-art face recognition. In Proceedings of the 2016 ACM SIGSAC Conference on Computer and Communications Security, pages 1528–1540. Association for Computing Machinery, 2016.Google Scholar
Shen, C., Nguyen, D., Zhou, Z., Jiang, S. B., Dong, B., and Jia, X.. An introduction to deep learning in medical physics: advantages, potential, and challenges. Phys. Med. Biol., 65(5):05TR01, 2020.Google Scholar
Shepp, L. A. and Logan, B. F.. The Fourier reconstruction of a head section. IEEE Trans. Nucl. Sci., 21(3):21–43, 1974.Google Scholar
Shin, P. J., Larson, P. E. Z., Ohliger, M. A., Elad, M., Pauly, J. M., Vigneron, D. B., and Lustig, M.. Calibrationless parallel imaging reconstruction based on structured low-rank matrix completion. Magn. Reson. Med., 72(4):959–970, 2014.Google Scholar
Sidky, E. Y., Kao, C. M., and Pan, X.. Accurate image reconstruction from few-views and limited-angle data in divergent-beam CT. J. X-Ray. Sci. Technol., 14(2):119–139, 2006.Google Scholar
Sidky, E. Y. and Pan, X.. Image reconstruction in circular cone-beam computed tomography by constrained, total-variation minimization. Phys. Med. Biol., 53(17):4777–4807, 2008.Google Scholar
Sinha, A., Namkoong, H., and Duchi, J.. Certifying some distributional robustness with principled adversarial training. arXiv:1710.10571, 2017.Google Scholar
Sinha, V.. Iterative reconstruction with ZEISS OptiRecon.www.zeiss.com/ microscopy/int/about-us/press-releases/2018/zeiss-optirecon .html, April 2018.Google Scholar
Slepian, D.. Prolate spheroidal wave functions, Fourier analysis, and uncertainty — V: the discrete case. Bell Syst. Tech. J., 57(5):1371–1430, 1978.Google Scholar
Smale, S.. Complexity theory and numerical analysis. Acta Numer., 6:523–551, 1997.Google Scholar
Smale, S.. Mathematical problems for the next century. Math. Intelligencer, 20(2):7–15, 1998.Google Scholar
Som, S. and Schniter, P.. Compressive imaging using approximate message passing and a Markov-tree prior. IEEE Trans. Signal Process., 60(7):3439–3448, 2012.Google Scholar
Song, G. and Gelb, A.. Approximating the inverse frame operator from localized frames. Appl. Comput. Harmon. Anal., 35(1):94–110, 2013.Google Scholar
Starck, J.-L., Murtagh, F., and Fadili, J.. Sparse Image and Signal Processing: Wavelets and Related Geometric Multiscale Analysis. Cambridge University Press, Cambridge, 2nd edition, 2015.Google Scholar
Strang, G. and Fix, G.. A Fourier analysis of the finite element variational method. In Geymonat, G., editor, Constructive Aspect of Functional Analysis, volume 57 of C.I.M.E. Summer Schools, pages 793–840. Springer, Berlin, 1971.Google Scholar
Strang, G. and Nguyen, T.. Wavelets and Filter Banks. Wellesley–Cambridge Press, Wellesley, MA, 1996.Google Scholar
Strohmer, T.. Measure what should be measured: progress and challenges in compressive sensing. IEEE Signal Process. Lett., 19(12):887–893, 2012.Google Scholar
Studer, C., Kuppinger, P., Pope, G., and Bölcskei, H.. Recovery of sparsely corrupted signals. IEEE Trans. Inf. Theory, 58(5):3115–3130, 2012.Google Scholar
Studer, V., Bobin, J., Chahid, M., Moussavi, H., Candès, E. J., and Dahan, M.. Compressive fluorescence microscopy for biological and hyperspectral imaging. Proc. Natl. Acad. Sci. USA, 109(26):1679–1687, 2011.Google Scholar
Szegedy, C., Zaremba, W., Sutskever, I., Bruna, J., Erhan, D., Goodfellow, I. J., and Fergus, R.. Intriguing properties of neural networks. arXiv:1312.6199, 2013.Google Scholar
Tang, G., Bhaskar, B. N., Shah, P., and Recht, B.. Compressed sensing off the grid. IEEE Trans. Inf. Theory, 59(11):7465–7490, 2013.Google Scholar
Tezcan, K. C., Baumgartner, C. F., Luechinger, R., Pruessmann, K. P., and Konukoglu, E.. MR image reconstruction using deep density priors. IEEE Trans. Med. Imag., 38(7):1633–1642, 2019.Google Scholar
Thesing, L., Antun, V, and Hansen, A. C.. What do AI algorithms actually learn? – on false structures in deep learning. arXiv:1906.01478, 2019.Google Scholar
Thesing, L. and Hansen, A. C.. Non-uniform recovery guarantees for binary measurements and infinite-dimensional compressed sensing. arXiv:1909.01143, 2019.Google Scholar
Thys, S., Van Ranst, W., and Goedemé, T.. Fooling automated surveillance cameras: adversarial patches to attack person detection. In 2019 IEEE/CVF Conference on Computer Vision and Pattern Recognition Workshops (CVPRW), pages 49–55, 2019.Google Scholar
Tibshirani, R.. Regression shrinkage and selection via the lasso. J. R. Statist. Soc. B., 58(1):267–288, 1996.Google Scholar
Tillmann, A. M. and Pfetsch, M. E.. The computational complexity of the restricted isometry property, the nullspace property, and related concepts in compressed sensing. IEEE Trans. Inf. Theory, 60(2):1248–1259, 2014.Google Scholar
Trabelsi, C., Bilaniuk, O., Serdyuk, D., Subramanian, S., Santos, J. F., Mehri, S., Rostamzadeh, N., Bengio, Y., and Pal, C. J.. Deep complex networks. arXiv:1705.09792, 2017.Google Scholar
Tramèr, F., Kurakin, A., Papernot, N., Goodfellow, I., Boneh, D., and McDaniel, P.. Ensemble adversarial training: attacks and defenses. arXiv:1705.07204, 2017.Google Scholar
Tran, H. and Webster, C.. A class of null space conditions for sparse recovery via nonconvex, non-separable minimizations. Results Appl. Math., 3:100011, 2019.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., 45(1):170–205, 2018.Google Scholar
Traonmilin, Y., Puy, G., Gribonval, R., and Davies, M. E.. Compressed sensing in Hilbert spaces. InH. Boche, G. Caire, R. Calderbank, M. März, G. Kutyniok, andR. Mathar, editors, Compressed Sensing and its Applications: Second International MATHEON Conference 2015, Applied and Numerical Harmonic Analysis, pages 359–384. Birkhäuser, Cham, 2017.Google Scholar
Tropp, J. A.. Recovery of short, complex linear combinations via l 1 minimization. IEEE Trans. Inf. Theory, 51(4):1568–1570, 2005.Google Scholar
Tropp, J. A.. Just relax: convex programming methods for identifying sparse signals in noise. IEEE Trans. Inf. Theory, 52(3):1030–1051, 2006.Google Scholar
Tropp, J. A., Gilbert, A. C., and Strauss, M. J.. Algorithms for simultaneous sparse approximation. Part II: Convex relaxation. Signal Process., 86(3):589–602, 2006.Google Scholar
Tsaig, Y. and Donoho, D. L. Extensions of compressed sensing. Signal Process., 86(3):549– 571, 2006.Google Scholar
Tseng, P.. On accelerated proximal gradient methods for convex-concave optimization. Preprint, 2008.Google Scholar
Turing, A. M.. On computable numbers, with an application to the Entscheidungsproblem. Proc. Lond. Math. Soc., s2–42(1):230–265, 1937.Google Scholar
Uecker, M.. Parallel magnetic resonance imaging. arXiv:1501.06209, 2015.Google Scholar
Unser, M.. A representer theorem for deep neural networks. J. Mach. Learn. Res., 20(110):1– 30, 2019.Google Scholar
van de Geer, S.. Estimation and Testing Under Sparsity: École d’Été de Probabilités de Saint-Flour XLV – 2015, volume 2159 of Lecture Notes in Mathematics. Springer, Cham, 2016.Google Scholar
van de Geer, S. A. and Bühlmann, P.. On the conditions used to prove oracle results for the Lasso. Electron. J. Stat., 3:1360–1392, 2009.Google Scholar
Berg, E. and Friedlander, M. P.. Probing the Pareto frontier for basis pursuit solutions. SIAM J. Sci. Comput., 31(2):890–912, 2009.Google Scholar
Berg, E. and Friedlander, M. P.. Theoretical and empirical results for recovery from multiple measurements. IEEE Trans. Inf. Theory, 56(5):2516–2527, 2010.Google Scholar
Vasanwala, S. S., Murphy, M. J., Alley, M. T., Lai, P., Keutzer, K., Pauly, J. M., and Lustig, M.. Practical parallel imaging compressed sensing MRI: summary of two years of experience in accelerating body MRI of pediatric patients. In 2011 IEEE International Symposium on Biomedical Imaging: From Nano to Macro, pages 1039–1043, 2011.Google Scholar
Velisavljevic, V., Beferull-Lozano, B., Vetterli, M., and Dragotti, P. L.. Directionlets: anisotropic multidirectional representation with separable filtering. IEEE Trans. Image Process., 15(7):1916–1933, 2006.Google Scholar
Venkatakrishnan, S. V., Bouman, C. A., and Wohlberg, B.. Plug-and-play priors for model based reconstruction. In 2013 IEEE Global Conference on Signal and Information Processing, pages 945–948, 2013.Google Scholar
Vidal, R., Bruna, J., Giryes, R., and Soatto, S.. Mathematics of deep learning. arXiv:1712.04721, 2017.Google Scholar
Vidyasagar, M.. An Introduction to Compressed Sensing. Society for Industrial and Applied Mathematics, Philadelphia, PA, 2019.Google Scholar
Waldspurger, I., d’Aspremont, A., and Mallat, S.. Phase recovery, MaxCut and complex semidefinite programming. Math. Program., 149(1):47–81, 2015.Google Scholar
Walsh, J. L.. A closed set of normal orthogonal functions. Amer. J. Math., 45(1):5–24, 1923.Google Scholar
Wang, G., Ye, J. C., Mueller, K., and Fessler, J. A.. Image reconstruction is a new frontier of machine learning. IEEE Trans. Med. Imag., 37(6):1289–1296, 2018.Google Scholar
Wang, H., Li, G., and Jiang, G.. Robust regression shrinkage and consistent variable selection through the LAD-Lasso. J. Bus. Econom. Statist., 25(3):347–355, 2007.Google Scholar
Wang, H. and Yu, C.-N.. A direct approach to robust deep learning using adversarial networks. arXiv:1905.09591, 2019.Google Scholar
Wang, S., Fidler, S., and Urtasun, R.. Proximal deep structured models. In Advances in Neural Information Processing Systems, pages 865–873, 2016.Google Scholar
Wang, Z. and Arce, G. R.. Variable density compressed image sampling. IEEE Trans. Image Process., 19(1):264–270, 2010.Google Scholar
Wang, Z. and Bovik, A. C.. Mean squared error: love it or leave it? A new look at signal fidelity measures. IEEE Signal Process. Mag., 26(1):98–117, 2009.Google Scholar
Weller, D. S., Polimeni, J. R., Grady, L., Wald, L. L., Adalsteinsson, E., and Goyal, V. K.. Sparsity-promoting calibration for GRAPPA accelerated parallel MRI reconstruction. IEEE Trans. Med. Imag., 32(7):1325–1335, 2013.Google Scholar
Wen, B., Ravishankar, S., Pfister, L., and Bresler, Y.. Transform learning for magnetic resonance image reconstruction: from model-based learning to building neural networks. IEEE Signal Process. Mag., 37(1):41–53, 2020.Google Scholar
Wen, Z., Yin, W., Goldfarb, D., and Zhang, Y.. A fast algorithm for sparse reconstruction based on shrinkage, subspace optimization, and continuation. SIAM J. Sci. Comput., 32(4):1832– 1857, 2010.Google Scholar
Wiaux, Y., Jacques, L., Puy, G., Scaife, A. M. M., and Vandergheynst, P.. Compressed sensing imaging techniques for radio interferometry. Mon. Not. R. Astron. Soc., 395(3):1733–1742, 2009.Google Scholar
Willett, R. M.. The dark side of image reconstruction: emerging methods for photon-limited imaging. SIAM News, October 2014.Google Scholar
Willett, R. M., Marcia, R. F., and Nichols, J. M.. Compressed sensing for practical optical imaging systems: a tutorial. Opt. Eng., 50(7):072601, 2011.Google Scholar
Willett, R. M. and Nowak, R. D.. Platelets: a multiscale approach for recovering edges and surfaces in photon-limited medical imaging. IEEE Trans. Med. Imag., 22(3):332–350, 2003.Google Scholar
Wiyatno, R. R., Xu, A., Dia, O., and de Berker, A.. Adversarial examples in modern machine learning: a review. arXiv:1911.05268, 2019.Google Scholar
Wohlberg, B.. Efficient algorithms for convolutional sparse representations. IEEE Trans. Image Process., 25(1):301–315, 2016.Google Scholar
Wojtaszczyk, P.. A Mathematical Introduction to Wavelets. Cambridge University Press, Cambridge, 1997.Google Scholar
Wojtaszczyk, P.. Stability and instance optimality for Gaussian measurements in compressed sensing. Found. Comput. Math., 10(1):1–13, 2010.Google Scholar
Wong, E., Schmidt, F., Metzen, J. H., and Kolter, J. Z.. Scaling provable adversarial defenses. In Advances in Neural Information Processing Systems, pages 8400–8409, 2018.Google Scholar
Woodworth, J. and Chartrand, R.. Compressed sensing recovery via nonconvex shrinkage penalties. Inverse Problems, 32(7):075004, 2016.Google Scholar
Wright, J. and Ma, Y.. Dense correction via ℓ 1 -minimization. IEEE Trans. Inf. Theory, 56(7):3540–3560, 2010.Google Scholar
Wright, S. J., Nowak, R. D., and Figueiredo, M. A. T.. Sparse reconstruction by separable approximation. IEEE Trans. Signal Process., 57(7):2479–2493, 2009.Google Scholar
Wu, B., Millane, R. P., Watts, R., and Bones, P.. Applying compressed sensing in parallel MRI. In Proc. Intl. Soc. Mag. Reson. Med., 2008.Google Scholar
Xiao, C., Li, B., yan Zhu, J., He, W., Liu, M., and Song, D.. Generating adversarial examples with adversarial networks. In Proceedings of the Twenty-Seventh International Joint Conference on Artificial Intelligence, IJCAI-18, pages 3905–3911. International Joint Conferences on Artificial Intelligence Organization, 7 2018.Google Scholar
Xu, J.. Parameter estimation, model selection and inferences in L1-based linear regression. PhD thesis, Columbia University, 2005.Google Scholar
Xu, J. and Ying, Z.. Simultaneous estimation and variable selection in median regression using Lasso-type penalty. Ann. Inst. Statist. Math., 62(3):487–514, 2010.Google Scholar
Xu, W., Evans, D., and Qi, Y.. Feature squeezing: detecting adversarial examples in deep neural networks. arXiv:1704.01155, 2017.Google Scholar
Yang, G., Yu, S., Dong, H., Slabaugh, G., Dragotti, P. L., Ye, X., Liu, F., Arridge, S., Keegan, J., Guo, Y., and Firmin, D.. DAGAN: deep de-aliasing generative adversarial networks for fast compressed sensing MRI reconstruction. IEEE Trans. Med. Imag., 37(6):1310–1321, 2018.Google Scholar
Yang, J. and Zhang, J.. Alternating direction algorithms for ℓ 1 -problems in compressive sensing. SIAM J. Sci. Comput., 33(1):250–278, 2011.Google Scholar
Yang, Y., Sun, J., Li, H., and Xu, Z.. Deep ADMM-Net for compressive sensing MRI. In Advances in Neural Information Processing Systems, pages 10–18, 2016.Google Scholar
Yang, Y., Sun, J., Li, H., and Xu, Z.. ADMM-CSNet: a deep learning approach for image compressive sensing. IEEE Trans. Pattern Anal. Machine Intell., 42(3):521–538, 2020.Google Scholar
Yarotsky, D.. Error bounds for approximations with deep ReLU networks. Neural Networks, 94:103–114, 2017.Google Scholar
Yarotsky, D.. Optimal approximation of continuous functions by very deep ReLU networks. In Bubeck, S., Perchet, V., and Rigollet, P., editors, Proceedings of the 31st Conference On Learning Theory, volume 75 of Proceedings of Machine Learning Research, pages 639–649. PMLR, 2018.Google Scholar
Yin, P., Lou, Y., He, Q., and Xin, J.. Minimization of ℓ 1−2 for compressed sensing. SIAM J. Sci. Comput., 37(1):A536–A563, 2015.Google Scholar
Ying, L. and Sheng, J.. Joint image reconstruction and sensitivity estimation in SENSE (JSENSE). Magn. Reson. Med., 57(6):1196–1202, 2007.Google Scholar
Zhang, H. and Dong, B.. A review on deep learning in medical image reconstruction. J. Oper. Res. Soc. China, 8(2):311–340, 2020.Google Scholar
Zhang, J. and Ghanem, B.. ISTA-Net: interpretable optimization-inspired deep network for image compressive sensing. In 2018 IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 1828–1837, 2018.Google Scholar
Zhang, K., Zuo, W., Chen, Y., Meng, D., and Zhang, L.. Beyond a Gaussian denoiser: residual learning of deep CNN for image denoising. IEEE Trans. Image Process., 26(7):3142–3155, 2017.Google Scholar
Zhang, R. and Li, S.. Optimal D-RIP bounds in compressed sensing. Acta Math. Sin. (Engl. Ser.), 31(5):755–766, 2015.Google Scholar
Zhang, X., Burger, M., and Osher, S.. A unified primal-dual algorithm framework based on Bregman iteration. J. Sci. Comput., 46(1):20–46, 2011.Google Scholar
Zhou, Y. T., Chellappa, R., Vaid, A., and Jenkin, B. K.. Image restoration using a neural network. IEEE Trans. Acoust., Speech, Signal Process., 36(7):1141–1151, 1988.Google Scholar
Zhou, Z. and So, A. M.. A unified approach to error bounds for structured convex optimization problems. Math. Program., 165(2):689–728, 2017.Google Scholar
Zhu, B., Liu, J. Z., Cauley, S. F., Rosen, B. R., and Rosen, M. S.. Image reconstruction by domain-transform manifold learning. Nature, 555(7697):487–492, 2018.Google Scholar
Zhu, L., Zhang, W., Elnatan, D., and Huang, B.. Faster STORM using compressed sensing. Nat. Methods, 9(7):721–723, 2012.Google Scholar

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  • References
  • Ben Adcock, Simon Fraser University, British Columbia, Anders C. Hansen, University of Cambridge
  • Book: Compressive Imaging: Structure, Sampling, Learning
  • Online publication: 16 July 2021
  • Chapter DOI: https://doi.org/10.1017/9781108377447.038
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  • References
  • Ben Adcock, Simon Fraser University, British Columbia, Anders C. Hansen, University of Cambridge
  • Book: Compressive Imaging: Structure, Sampling, Learning
  • Online publication: 16 July 2021
  • Chapter DOI: https://doi.org/10.1017/9781108377447.038
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  • References
  • Ben Adcock, Simon Fraser University, British Columbia, Anders C. Hansen, University of Cambridge
  • Book: Compressive Imaging: Structure, Sampling, Learning
  • Online publication: 16 July 2021
  • Chapter DOI: https://doi.org/10.1017/9781108377447.038
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