zbMATH — the first resource for mathematics

TV-based reconstruction of periodic functions. (English) Zbl 1458.94087
94A12 Signal theory (characterization, reconstruction, filtering, etc.)
41A15 Spline approximation
90C25 Convex programming
GitHub; PDCO; Pyff; pyFFS
Full Text: DOI
[1] Unser M, Fageot J and Ward J P 2017 Splines are universal solutions of linear inverse problems with generalized TV regularization SIAM Rev.59 769-93 · Zbl 1382.41011
[2] Tikhonov A 1963 On the solution of ill-posed problems and the regularization method Sov. Math.—Dokl.4 1035-8 · Zbl 0141.11001
[3] Hoerl A E 1962 Application of the ridge analysis to regression problems Chem. Eng. Prog.58 54-9
[4] Tibshirani R 1996 Regression shrinkage and selection via the lasso J. R. Stat. Soc. B 58 267-88 · Zbl 0850.62538
[5] Chen S S, Donoho D L and Saunders M A 2001 Atomic decomposition by basis pursuit SIAM Rev.43 129-59 · Zbl 0979.94010
[6] Donoho D L 2006 Compressed sensing IEEE Trans. Inf. Theory52 1289-306 · Zbl 1288.94016
[7] Candès E J, Romberg J and Tao T 2006 Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information IEEE Trans. Inf. Theory52 489-509 · Zbl 1231.94017
[8] Chandrasekaran V, Recht B, Parrilo P A and Willsky A S 2012 The convex geometry of linear inverse problems Found. Comput. Math.12 805-49 · Zbl 1280.52008
[9] Eldar Y and Kutyniok G 2012 Compressed Sensing: Theory and Applications (Cambridge: Cambridge University Press)
[10] Foucart S and Rauhut H 2013 A Mathematical Introduction to Compressive Sensing vol 1 (Basel: Birkhäuser) · Zbl 1315.94002
[11] Adcock B and Hansen A 2015 Generalized sampling and infinite-dimensional compressed sensing Found. Comput. Math.16 1263 · Zbl 1379.94026
[12] Adcock B, Hansen A, Poon C and Roman B 2017 Breaking the coherence barrier: a new theory for compressed sensing Forum of Mathematics, Sigma vol 5 (Cambridge: Cambridge University Press) · Zbl 1410.94030
[13] Eldar Y 2008 Compressed sensing of analog signals in shift-invariant spaces (arXiv:0806.3332)
[14] Unser M, Fageot J and Gupta H 2016 Representer theorems for sparsity-promoting IEEE Trans. Inf. Theory62 5167-80 · Zbl 1359.94185
[15] Traonmilin Y, Puy G, Gribonval R and Davies M 2017 Compressed sensing in Hilbert spaces Compressed Sensing and Its Applications (Berlin: Springer) pp 359-84
[16] Bodmann B, Flinth A and Kutyniok G 2018 Compressed sensing for analog signals (arXiv:1803.04218)
[17] März M, Boyer C, Kahn J and Weiss P 2020 Sampling rates for ℓ1-synthesis (arXiv:2004.07175)
[18] Vetterli M, Marziliano P and Blu T 2002 Sampling signals with finite rate of innovation IEEE Trans. Signal Process.50 1417-28 · Zbl 1369.94309
[19] Maravic I and Vetterli M 2005 Sampling and reconstruction of signals with finite rate of innovation in the presence of noise IEEE Trans. Signal Process.53 2788-805 · Zbl 1370.94398
[20] Candès E and Fernandez-Granda C 2014 Towards a mathematical theory of super-resolution Commun. Pure Appl. Math.67 906-56 · Zbl 1350.94011
[21] Candès E and Fernandez-Granda C 2013 Super-resolution from noisy data J. Fourier Anal. Appl.19 1229-54 · Zbl 1312.94015
[22] Zukhovitskii S 1962 On approximation of real functions in the sense of P.L. Chebyshev AMS Translations of Mathematical Monographs19 221-52 · Zbl 0119.05603
[23] Fisher S and Jerome J 1975 Spline solutions to ℓ1 extremal problems in one and several variables J. Approx. Theory13 73-83 · Zbl 0295.49002
[24] Castro Y D and Gamboa F 2012 Exact reconstruction using Beurling minimal extrapolation J. Math. Anal. Appl.395 336-54 · Zbl 1302.94019
[25] Bredies K and Pikkarainen H 2013 Inverse problems in spaces of measures ESAIM: Control, Optim. Calc. Var.19 190-218 · Zbl 1266.65083
[26] Duval V and Peyré G 2015 Exact support recovery for sparse spikes deconvolution Found. Comput. Math.15 1315-55 · Zbl 1327.65104
[27] Azais J, Castro Y D and Gamboa F 2015 Spike detection from inaccurate samplings Appl. Comput. Harmon. Anal.38 177 · Zbl 1308.94046
[28] Fernandez-Granda C 2016 Super-resolution of point sources via convex programming Inf. Inference5 251 · Zbl 1386.94027
[29] Chambolle A, Duval V, Peyré G and Poon C 2016 Geometric properties of solutions to the total variation denoising problem (arXiv:1602.00087)
[30] Duval V and Peyré G 2017 Sparse regularization on thin grids I: the Lasso Inverse Problems33 055008 · Zbl 1373.65039
[31] Duval V and Peyré G 2017 Sparse spikes super-resolution on thin grids II: the continuous basis pursuit Inverse Problems33 095008 · Zbl 1392.35332
[32] Denoyelle Q, Duval V and Peyré G 2017 Support recovery for sparse super-resolution of positive measures J. Fourier Anal. Appl.23 1153-94 · Zbl 1417.65223
[33] Flinth A, de Gournay F and Weiss P 2020 On the linear convergence rates of exchange and continuous methods for total variation minimization Math. Program. 1-37
[34] Chi Y and Costa M F D 2020 Harnessing sparsity over the continuum: atomic norm minimization for superresolution IEEE Signal Process. Mag.37 39-57
[35] García H, Hernández C, Junca M and Velasco M 2020 Approximate super-resolution of positive measures in all dimensions Appl. Comput. Harmon. Anal.
[36] Denoyelle Q, Duval V, Peyré G and Soubies E 2019 The sliding Frank-Wolfe algorithm and its application to super-resolution microscopy Inverse Problems36 014001 · Zbl 1434.65082
[37] Courbot J-B, Duval V and Legras B 2019 Sparse analysis for mesoscale convective systems tracking Signal Process., Image Commun.85 115854
[38] Schoenberg I 1973 Cardinal Spline Interpolation (Philadelphia, PA: SIAM) · Zbl 0264.41003
[39] Gupta H, Fageot J and Unser M 2018 Continuous-domain solutions of linear inverse problems with Tikhonov vs generalized TV regularization IEEE Trans. Signal Process.66 4670-84 · Zbl 1415.94119
[40] Flinth A and Weiss P 2019 Exact solutions of infinite dimensional total-variation regularized problems Inf. Inference8 407-43
[41] Debarre T, Fageot J, Gupta H and Unser M 2019 B-spline-based exact discretization of continuous-domain inverse problems with generalized TV regularization IEEE Trans. Inf. Theory65 4457-70 · Zbl 1432.94022
[42] Debarre T, Denoyelle Q, Unser M and Fageot J 2020 Sparsest continuous piecewise-linear representation of data (arXiv:2003.10112)
[43] Simeoni M 2020 Functional penalised basis pursuit on spheres (arXiv: 2006.05761)
[44] Simeoni M 2020 Functional inverse problems on spheres: theory, algorithms and applications EPFL Technical Report https://doi.org/10.5075/epfl-thesis-7174
[45] Debarre T, Aziznejad S and Unser M 2019 Hybrid-spline dictionaries for continuous-domain inverse problems IEEE Trans. Signal Process.67 5824-36 · Zbl 07123517
[46] Aziznejad S and Unser M 2019 Multi-kernel regression with sparsity constraint (arXiv:1811.00836)
[47] Bredies K and Carioni M 2018 Sparsity of solutions for variational inverse problems with finite-dimensional data (arXiv:1809.05045)
[48] Boyer C, Chambolle A, Castro Y D, Duval V, de Gournay F and Weiss P 2019 On representer theorems and convex regularization SIAM J. Optim.29 1260-81 · Zbl 1423.49036
[49] Unser M and Fageot J 2019 Native Banach spaces for splines and variational inverse problems (arXiv:1904.10818)
[50] Unser M 2019 A representer theorem for deep neural networks J. Mach. Learn. Res.20 1-30 · Zbl 1434.68526
[51] Aziznejad S, Gupta H, Campos J and Unser M 2020 Deep neural networks with trainable activations and controlled Lipschitz constant (arXiv:2001.06263)
[52] Novosadová M and Rajmic P 2018 Image edges resolved well when using an overcomplete piecewise-polynomial model 2018 12th Int. Conf. on Signal Processing and Communication Systems (ICSPCS) 1-10
[53] Simeoni M, Besson A, Hurley P and Vetterli M 2020 CPGD: Cadzow plug-and-play gradient descent for generalised FRI (arXiv: 2006.06374)
[54] Delgado-Gonzalo R, Thévenaz P, Seelamantula C and Unser M 2012 Snakes with an ellipse-reproducing property IEEE Trans. Image Process.21 1258-71 · Zbl 1372.94066
[55] Uhlmann V, Fageot J and Unser M 2016 Hermite snakes with control of tangents IEEE Trans. Image Process.25 2803-16 · Zbl 1408.94654
[56] Light W and Cheney E 1992 Interpolation by periodic radial basis functions J. Math. Anal. Appl.168 111-30 · Zbl 0794.41002
[57] Jacob M, Blu T and Unser M 2002 Sampling of periodic signals: a quantitative error analysis IEEE Trans. Signal Process.50 1153-9 · Zbl 1369.94422
[58] Simeoni M, Kashani S, Hurley P and Vetterli M 2019 Deepwave a recurrent neural-network for real-time acoustic imaging Advances in Neural Information Processing Systems 15274-86
[59] Pan H, Scheibler R, Bezzam E, Dokmanić I and Vetterli M 2017 FRIDA FRI-based DOA estimation for arbitrary array layouts 2017 IEEE Int. Conf. on Acoustics, Speech and Signal Processing (ICASSP) 3186-90
[60] Krim H and Viberg M 1996 Two decades of array signal processing research: the parametric approach IEEE Signal Process. Mag.13 67-94
[61] Hurley P and Simeoni M 2016 Flexibeam: analytic spatial filtering by beamforming 2016 IEEE Int. Conf. on Acoustics, Speech and Signal Processing (ICASSP) 2877-80
[62] Hurley P and Simeoni M 2017 Flexarray random phased array layouts for analytical spatial filtering 2017 IEEE Int. Conf. on Acoustics, Speech and Signal Processing (ICASSP) 3380-4
[63] Ghysels E, Osborn D R and Sargent T J 2001 The Econometric Analysis of Seasonal Time Series (Cambridge: Cambridge University Press)
[64] Zhang G P and Qi M 2005 Neural network forecasting for seasonal and trend time series Eur. J. Oper. Res.160 501-14 · Zbl 1066.62094
[65] Unser M 2019 A representer theorem for deep neural networks J. Mach. Learn. Res.20 1-30 · Zbl 1434.68526
[66] Sitzmann V, Martel J N, Bergman A W, Lindell D B and Wetzstein G 2020 Implicit neural representations with periodic activation functions (arXiv:2006.09661)
[67] Badoual A, Fageot J and Unser M 2018 Periodic splines and Gaussian processes for the resolution of linear inverse problems IEEE Trans. Signal Process.66 6047-61 · Zbl 1415.94039
[68] Simeoni M 2020 Periodispline https://github.com/matthieumeo/periodispline
[69] Trèves F 1967 Topological Vector Spaces, Distributions and Kernels (New York: Academic) · Zbl 0171.10402
[70] Schwartz L 1966 Théorie des distributions (Paris: Hermann)
[71] Unser M and Tafti P D 2014 An Introduction to Sparse Stochastic Processes (Cambridge: Cambridge University Press) · Zbl 1329.60002
[72] Campbell S and Meyer C 2009 Generalized Inverses of Linear Transformations (Philadelphia, PA: SIAM)
[73] Ben-Israel A and Greville T 2003 Generalized Inverses: Theory and Applications vol 15 (Berlin: Springer)
[74] Freeden W and Schreiner M 2008 Spherical Functions of Mathematical Geosciences: A Scalar, Vectorial, and Tensorial Setup (Berlin: Springer)
[75] Unser M 1999 Splines: a perfect fit for signal and image processing IEEE Signal Process. Mag.16 22-38
[76] Gray J 1984 The shaping of the Riesz representation theorem: a chapter in the history of analysis Arch. Hist. Exact. Sci.31 127-87 · Zbl 0549.01010
[77] Rudin W 2006 Real and Complex Analysis (New Delhi: Tata McGraw-Hill)
[78] Simon B 2003 Distributions and their Hermite expansions J. Math. Phys.12 140-8 · Zbl 0205.12901
[79] Tibshirani R 2013 The lasso problem and uniqueness Electron. J. Stat.7 1456-90 · Zbl 1337.62173
[80] Rudin W 1991 Functional Analysis(International Series in Pure and Applied Mathematics) (New York: McGraw-Hill)
[81] Kashani S 2020 pyFFS: a fast Fourier series library for Python 3 https://github.com/imagingofthings/pyFFS
[82] Samko S, Kilbas A and Marichev O 1993 Fractional Integrals and Derivatives (London: Gordon and Breach)
[83] Zygmund A 2002 Trigonometric Series vol 1 (Cambridge: Cambridge University Press) · Zbl 1084.42003
[84] Gia Q L, Sloan I and Wendland H 2012 Multiscale approximation for functions in arbitrary Sobolev spaces by scaled radial basis functions on the unit sphere Appl. Comput. Harmon. Anal.32 401-12 · Zbl 1238.41020
[85] Rasmussen C 2003 Gaussian processes in machine learning Summer School on Machine Learning (Berlin: Springer) pp 63-71 · Zbl 1120.68436
[86] Abramowitz M and Stegun I 1948 Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables vol 55 (Washington, DC: US Government Printing Office)
[87] Wendland H 2004 Scattered Data Approximation vol 17 (Cambridge: Cambridge University Press)
[88] Zhu S 2012 Compactly supported radial basis functions: how and why? OCCAM Preprint Number 12/57
[89] Hubbert S 2012 Closed form representations for a class of compactly supported radial basis functions Adv. Comput. Math.36 115-36 · Zbl 1251.41007
[90] Madych W and Nelson S 1990 Polyharmonic cardinal splines J. Approx. Theory60 141-56 · Zbl 0702.41020
[91] Rabut C 1992 Elementary m-harmonic cardinal B-splines Numer. Algorithms2 39-61 · Zbl 0851.41010
[92] Van De Ville D, Blu T and Unser M 2005 Isotropic polyharmonic B-splines: scaling functions and wavelets IEEE Trans. Image Process.14 1798-813
[93] Boas R 1966 Fourier series with positive coefficients Bull. Am. Math. Soc72 863-5 · Zbl 0163.07603
[94] Shapiro V 2011 Fourier Series in Several Variables with Applications to Partial Differential Equations (Boca Raton, FL: CRC Press)
[95] Lu Y M and Do M 2008 A theory for sampling signals from a union of subspaces IEEE Trans. Signal Process.56 2334-45 · Zbl 1390.94656
[96] Eldar Y and Mishali M 2009 Robust recovery of signals from a structured union of subspaces IEEE Trans. Inf. Theory55 5302-16 · Zbl 1367.94087
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.