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  1. Home
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  3. The Mathematics of Signal Processing
  • Cited by 38
Publisher:
Cambridge University Press
Online publication date:
June 2012
Print publication year:
2011
Online ISBN:
9781139003896
DOI:
https://doi.org/10.1017/CBO9781139003896
Subjects:
Mathematics, Communications and Signal Processing, Differential and Integral Equations, Dynamical Systems and Control Theory, Engineering, Mathematical Physics, Numerical Analysis and Computational Science
Series:
Cambridge Texts in Applied Mathematics (48)
Information

Book description

Arising from courses taught by the authors, this largely self-contained treatment is ideal for mathematicians who are interested in applications or for students from applied fields who want to understand the mathematics behind their subject. Early chapters cover Fourier analysis, functional analysis, probability and linear algebra, all of which have been chosen to prepare the reader for the applications to come. The book includes rigorous proofs of core results in compressive sensing and wavelet convergence. Fundamental is the treatment of the linear system y=Φx in both finite and infinite dimensions. There are three possibilities: the system is determined, overdetermined or underdetermined, each with different aspects. The authors assume only basic familiarity with advanced calculus, linear algebra and matrix theory and modest familiarity with signal processing, so the book is accessible to students from the advanced undergraduate level. Many exercises are also included.

Reviews

'Damelin and Miller provide a very detailed and thorough treatment of all the important mathematics related to signal processing. This includes the required background information found in elementary mathematics courses, so their book is really self-contained. The style of writing is suitable not only for mathematicians, but also for practitioners from other areas. Indeed, Damelin and Miller managed to write their text in a form that is accessible to nonspecialists, without giving up mathematical rigor.'

Kai Diethelm Source: Computing Reviews

‘In the last 20 years or so, many books on wavelets have been published; most of them deal with wavelets from either the engineering or the mathematics perspective, but few try to connect the two viewpoints. The book under review falls under the last category … Overall, the book is a good addition to the literature on engineering mathematics.’

Ahmed I. Zayed Source: Mathematical Reviews

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Contents

Contents
References
References
References
[1] D., Achlioptas, Database-friendly random projections: Johnson-Lindenstrauss with binary coins, J. Comp. Sys. Sci., 66 (4), (2003), 671–687.
[2] A., Akansu and R., Haddad, Multiresolution Signal Decomposition, Academic Press, 1993.
[3] G. E., Andrews, R., Askey and R., Roy, Special Functions, Encyclopedia of Mathematics and its Applications, Cambridge University Press, 1999.
[4] T. M., Apostol, Calculus, Blaisdell Publishing Co., 1967–1969.
[5] L. J., Bain and M., Engelhardt, Introduction to Probability and Mathe-matical Statistics, 2nd edn., PWS-Kent, 1992.
[6] R., Baraniuk, M., Davenport, R., DeVore and M., Wakin, A simple proof of the restricted isometry property for random matrices, Const. Approx., 28, (2008), 253–263.
[7] R., Baraniuk and M. B., Wakin, Random projections on smooth manifolds, Found. Comput. Math., 9, (2009), 51–77.
[8] M., Belkin and P., Niyogi, Laplacian eigenmaps for dimensionality reduction and data representation, Neural Comput., 15 (6), (2003), 1373–1396.
[9] A., Berlinet and C. T., Agnan, Reproducing Kernel Hilbert Spaces in Probability and Statistics, Kluwer, 2004.
[10] J. J, Benedetto and M. W., Frazier, Wavelets: Mathematics and Applications, CRC Press, 1994.
[11] J. J., Benedetto and A. I., Zayed, Sampling, Wavelets and Tomography, Birkhauser, 2004.
[12] P., Binev, A., Cohen, W., Dahmen and V., Temlyakov, Universal algorithms for learning theory part I: piecewise constant functions, J. Mach. Learn. Res., 6, (2005), 1297–1321.
[13] M. M., Bronstein and I., Kokkinos, Scale Invariance in Local Heat Kernel Descriptors without Scale Selection and Normalization, INRIA Research Report 7161, 2009.
[14] A. M., Bruckstein, D. L., Donoho and M., Elad, From sparse solutions of systems of equations to sparse modeling of signals and images, SIAM Review, 51, February 2009, 34–81.
[15] A., Boggess and F. J., Narcowich, A First Course in Wavelets with Fourier Analysis, Prentice-Hall, 2001.
[16] S., Boyd and L., Vandenberghe, Convex Optimization, Cambridge University Press, 2004.
[17] C. S., Burrus, R. A., Gopinath and H., Guo, Introduction to Wavelets and Wavelet Transforms. A Primer, Prentice-Hall, 1998.
[18] A., Bultheel, Learning to swim in the sea of wavelets, Bull. Belg. Math. Soc., 2, (1995), 1–44.
[19] E. J., Candes, Compressive sampling, Proceedings of the International Congress of Mathematicians, Madrid, Spain, (2006).
[20] E. J., Candes, Ridgelets and their derivatives: Representation of images with edges, in Curves and Surfaces, ed. L. L., Schumaker et al., Vanderbilt University Press, 2000, pp. 1–10.
[21] E. J., Candes, The restricted isometry property and its implications for compressed sensing, C. R. Acad, Sci. Paris Ser. I Math., 346, (2008), 589–592.
[22] E. J., Candes and D. L., Donoho, Continuous curvelet transform: I. Resolution of the wavefront set, Appl. Comput. Harmon. Anal., 19, (2005), 162–197.
[23] E. J., Candes and E. L., Donoho, Continuous curvelet transform: II. Discretization and frames, Appl. Comput. Harmon. Anal., 19, (2005), 198–222.
[24] E. J., Candes and E. L., Donoho, New tight frames of curvelets and optimal representations of objects with piecewise-C2 singularities, Comm. Pure Appl. Math, 57, (2006), 219–266.
[25] E. J., Candes, X., Li, Y., Ma and J., Wright, Robust principal component analysis?, J. ACM, 58 (1), 1–37.
[26] E. J., Candes, J., Romberg and T., Tao, Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information, IEEE Trans. Inform. Theory, 52, (2006), 489–509.
[27] E. J., Candes, J., Romberg and T., Tao, Stable signal recovery from incomplete and inaccurate measurements, Comm. Pure and Appl. Math., 59, (2006), 1207–1223.
[28] E. J., Candes and T., Tao, Decoding by linear programing, IEEE Trans. Inform. Theory, 51, (2005), 4203–4215.
[29] E. J., Candes and T., Tao, Near optimal signal recovery from random projections: universal encoding strategies, IEEE Trans. Inform. Theory, 52, 5406–5425.
[30] E. J., Candes and T., Tao, The Dantzig selector: Statistical estimation when p is much larger than n, Ann. Stat., 35, (2007), 2313–2351.
[31] K., Carter, R., Raich and A., Hero, On local dimension estimation and its applications, IEEE Trans. Signal Process., 58 (2), (2010), 762–780.
[32] K., Carter, R., Raich, W.G.|Finn and A. O., Hero, FINE: Fisher information non-parametric embedding, IEEE Trans. Pattern Analysis and Machine Intelligence (PAMI), 31 (3), (2009), 2093–2098.
[33] K., Causwe, M, Sears, A., Robin, Using random matrix theory to determine the number of endmembers in a hyperspectral image, Whispers Proceedings, 3, (2010), 32–40.
[34] T. F., Chan and J., Shen, Image Processing and Analysis, SIAM, 2005.
[35] C., Chui, An Introduction to Wavelets, Academic Press, 1992.
[36] C., Chui and H. N., Mhaskar, MRA contextual-recovery extension of smooth functions on manifolds, App. Comput. Harmon. Anal., 28, (2010), 104–113.
[37] A., Cohen, W., Dahmen, I., Daubechies and R., DeVore, Tree approximation and encoding, Appl. Comput. Harmon. Anal., 11, (2001), 192–226.
[38] A., Cohen, W., Dahmen and R., DeVore, Compressed sensing and k-term approximation, Journal of the AMS, 22, (2009), 211–231.
[39] A., Cohen, I., Daubechies and J. C., Feauveau, Biorthogonal bases of compactly supported wavelets, Comm. Pure Appl. Math., 45, (1992), 485–560.
[40] A., Cohen and R. D, Ryan, Wavelets and Multiscale Signal Processing, Chapman and Hall, 1995.
[41] A., Cohen, Wavelets and digital signal processing, in Wavelets and their Applications, ed. M. B., Ruskai et al., Jones and Bartlett, 1992.
[42] A., Cohen and J., Froment, Image compression and multiscale approximation, in Wavelets and Applications (Marseille, Juin 89), ed. Y., Meyer, Masson, 1992.
[43] A., Cohen, Wavelets, approximation theory and subdivision schemes, in Approximation Theory VII (Austin, 1992), ed. E. W., Cheney et al., Academic Press, 1993.
[44] R. R., Coifman and S., Lafon, Diffusion maps, Appl. Comp. Harmon. Anal., 21, (2006), 5–30.
[45] R. R., Coifman and M., Maggioni, Diffusion wavelets, Appl. Comput. Harmon. Anal., 21 (1), (2006), 53–94.
[46] J. A., Costa and A. O., Hero, Geodesic entropic graphs for dimension and entropy estimation in manifold learning, IEEE Trans. Signal Process., 52 (8), (2004), 2210–2221.
[47] F., Cucker and S., Smale, On the mathematical foundations of learning theory, Bull. Amer. Math. Soc., 39, (2002), 1–49.
[48] S. B., Damelin, A walk through energy, discrepancy, numerical integration and group invariant measures on measurable subsets of euclidean space, Numer. Algorithms, 48 (1), (2008), 213–235.
[49] S. B., Damelin, On bounds for diffusion, discrepancy and fill distance metrics, in Principal Manfolds, Springer Lecture Notes in Computational Science and Engineering, vol. 58, 2008, pp. 32–42.
[50] S. B., Damelin and A. J., Devaney, Local Paley Wiener theorems for analytic functions on the unit sphere, Inverse Probl., 23, (2007), 1–12.
[51] S. B., Damelin, F., Hickernell, D., Ragozin and X., Zeng, On energy, discrepancy and G invariant measures on measurable subsets of Euclidean space, J. Fourier Anal. Appl., 16, (2010), 813–839.
[52] I., Daubechies, Orthonormal bases of compactly supported wavelets, Comm. Pure Appl. Math., 41, (1988), 909–996.
[53] I., Daubechies, Ten Lectures on Wavelets, CBMS-NSF Regional Conf. Ser. Appl. Math, 61, SIAM, 1992.
[54] I., Daubechies, R., Devore, D., Donoho and M., Vetterli, Data compression and harmonic analysis, IEEE Trans. Inform. Theory Numerica, 44, (1998), 2435–2467.
[55] K. R., Davidson and A. P., Donsig, Real Analysis with Real Applications, Prentice Hall, 2002.
[56] L., Demaret, N., Dyn and A., Iske, Compression by linear splines over adaptive triangulations, Signal Process. J., 86 (7), July 2006, 1604–1616.
[57] L., Demaret, A., Iske and W., Khachabi, A contextual image compression from adaptive sparse data representations, Proceedings of SPARS09, April 2009, Saint-Malo.
[58] R., DeVore, Nonlinear approximation, Acta Numer., 7, (1998), 51–150.
[59] R., Devore, Optimal computation, Proceedings of the International Congress of Mathematicians, Madrid, Spain, 2006.
[60] R., Devore, Deterministic constructions of compressed sensing matrices, J. Complexity, 23, (2007), 918–925.
[61] R., Devore and G., Lorentz, Constructive Approximation, Comprehensive Studies in Mathematics 303, Springer-Verlag, 1993.
[62] D., Donoho, Compressed sensing, IEEE Trans. Inform. Theory, 52 (4), (2006), 23–45.
[63] D., Donoho, M., Elad and V., Temlyakov, Stable recovery of sparse overcomplete representations in the presence of noise, IEEE Trans. Inf. Theory, 52 (1), (2006), 6–18.
[64] L., du Plessis, R., Xu, S. B., Damelin, M., Sears and D., Wunsch, Reducing dimensionality of hyperspectral data with diffusion maps and clustering with K means and fuzzy art, Int. J. Sys. Control Comm., 3 (2011), 3–10.
[65] L., du Plessis, R., Xu, S., Damelin, M., Sears and D., Wunsch, Reducing dimensionality of hyperspectral data with diffusion maps and clustering with K-means and fuzzy art, Proceedings of International Conference of Neural Networks, Atlanta, 2009, pp. 32–36.
[66] D.T., Finkbeiner, Introduction to Matrices and Linear Transformations, 3rd edn., Freeman, 1978.
[67] M., Herman and T., Strohmer, Compressed sensing radar, ICASSP 2008, pp. 1509–1512.
[68] M., Herman and T., Strohmer, High resolution radar via compressed sensing, IEEE Trans. Signal Process., November 2008, 1–10.
[69] E., Hernandez and G., Weiss, A First Course on Wavelets, CRC Press, 1996.
[70] W., Hoeffding, Probability inequalities for sums of bounded random variables, J. Amer. Statist. Assoc., 58, (1963), 13–30.
[71] B. B., Hubbard, The World According to Wavelets, A. K Peters, 1996.
[72] P. E. T., Jorgensen, Analysis and Probability: Wavelets, Signals, Fractals, Graduate Texts in Mathematics, 234 Springer, 2006.
[73] D. W., Kammler, A First Course in Fourier Analysis, Prentice-Hall, 2000.
[74] F., Keinert, Wavelets and Multiwavelets, Chapman and Hall/CRC, 2004.
[75] J., Korevaar, Mathematical Methods (Linear algebra, Normed spaces, Distributions, Integration), Academic Press, 1968.
[76] A. M., Krall, Applied Analysis, D. Reidel Publishing Co., 1986.
[77] S., Kritchman and B., Nadler, Non-parametric detection of the number of signals, hypothesis testing and random matrix theory, IEEE Trans. Signal Process., 57 (10), (2009), 3930–3941.
[78] H. J., Larson, Introduction to Probability Theory and Statistical Inference, Wiley, 1969.
[79] G. G., Lorentz, M., von Golitschek and M., Makovoz, Constructive Approximation: Advanced Problems, Grundlehren Math. Wiss, 304, Springer-Verlag, 1996.
[80] D. S., Lubinsky, A survey of mean convergence of orthogonal polynomial expansions, in Proceedings of the Second Conference on Functions Spaces (Illinois), ed. K., Jarosz, CRC press, 1995, pp. 281–310.
[81] B., Lucier and R., Devore, Wavelets, Acta Numer., 1, (1992), 1–56.
[82] S., Mallat, A Wavelet Tour of Signal Processing, 2nd edn., Academic Press, 2001.
[83] S., Mallat, A wavelet tour of signal processing - the Sparse way, Science-Direct (Online service), Elsevier/Academic Press, 2009.
[84] Y., Meyer, Wavelets: Algorithms and Applications, translated by R. D., Ryan, SIAM, 1993.
[85] W., Miller, Topics in harmonic analysis with applications to radar and sonar, in Radar and Sonar, Part 1, R., Blahut, W., Miller and C., Wilcox, IMA Volumes in Mathematics and its Applications, Springer-Verlag, 1991.
[86] M., Mitchley, M., Sears and S. B., Damelin, Target detection in hyperspectral mineral data using wavelet analysis, Proceedings of the 2009 IEEE Geosciences and Remote Sensing Symposium, Cape Town, pp. 23–45.
[87] B., Nadler, Finite sample approximation results for principal component analysis: A matrix perturbation approach, Ann. Stat., 36 (6), (2008), 2791–2817.
[88] H. J., Nussbaumer, Fast Fourier Transform and Convolution Algorithms, Springer-Verlag, 1982.
[89] P., Olver and C., Shakiban, Applied Linear Algebra, Pearson, Prentice-Hall, 2006.
[90] J., Ramanathan, Methods of Applied Fourier Analysis, Birkhauser, 1998.
[91] H. L., Royden, Real Analysis, Macmillan Co., 1988.
[92] W., Rudin, Real and Complex Analysis, McGraw-Hill, 1987.
[93] W., Rudin, Functional Analysis, McGraw-Hill, 1973.
[94] G., Strang, Linear Algebra and its Applications, 3rd edn., Harcourt Brace Jovanovich, 1988.
[95] G., Strang and T., Nguyen, Wavelets and Filter Banks, Wellesley-Cambridge Press, 1996.
[96] R., Strichartz, A Guide to Distribution Theory and Fourier Transforms, CRC Press, 1994.
[97] J., Tropp, Just relax: convex programming methods for identifying sparse signals in noise, IEEE Trans. Inf. Theory, 52 (3), (2006), 1030–1051.
[98] B. L., Van Der Waerden, Modern Algebra, Volume 1 (English translation), Ungar, 1953.
[99] G. G., Walter and X., Shen, Wavelets and other Orthogonal Systems, 2nd ed., Chapman and Hall, 2001.
[100] E. T., Whittaker and G. M., Watson, A Course in Modern Analysis, Cambridge University Press, 1958.
[101] P., Wojtaszczyk, A Mathematical Introduction to Wavelets, London Mathematical Society Student Texts, 37, Cambridge University Press, 1997.
[102] R. J., Zimmer, Essential Results of Functional Analysis, University of Chicago Press, 1990.
[103] A., Zygmund, Trignometrical Series, Dover reprint, 1955.

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