Applied & Computational Harmonic Analysis:
Comments, Handouts, and References Page (Winter 2018)
The following references are useful and contains much more
details of the topics covered or referred to in my lectures.
I strongly encourage you to take a look at some of them. Lecture 1: Overture and Motivation;
What is a Signal?
For quantization, which will not be discussed in this course due to the
time constraint, see: R. M. Gray and D. L. Neuhoff: "Quantization," IEEE Trans. Inform.
Theory , vol.44, no.6, pp.2325-2383, 1998.
K. Sayood: Introduction to Data Compression , 4th Ed., Morgan
Kaufmann Publ., 2012. In particular, Chapters 9 and 10.
Lecture 2: Basics of the Fourier Transforms The following engineering-oriented book has been popular for a long time
and contains quite useful information:
R. N. Bracewell: The Fourier Transform and Its Applications ,
3rd Ed., McGraw Hill, 1999.
Lecture 3: Uncertainty Principles
Basic references on the Heisenberg inequality/uncertainty principle in
L 2
Dym and McKean: Sec. 2.8.
Folland: Fourier Analysis, Sec. 7.3.
Pinsky: Sec. 2.4.3.
Survey on the uncertainty principles: G. B. Folland and A. Sitaram: "The uncertainty principle: A mathematical survey," Journal of Fourier Analysis and Applications , vol.3, no.3, pp.207-238, 1997. B. Ricaud and B. Torrésani: "A survey of uncertainty principles and some signal processing applications," Adv. Comput. Math. , vol.40, no.3, pp.629-650, 2014.
Basic references on more general uncertainty principles such as those in L 1
D. L. Donoho and P. B. Stark: "Uncertainty principles and signal recovery," SIAM J. Appl. Math. , vol.49, no.3, pp.906-931, 1989. D. L. Donoho and X. Huo: "Uncertainty principles and ideal atomic decomposition," IEEE Trans. Inform. Theory , vol.47, no.7, pp.2845-2862, 2001. P. Boggiatto, E. Carypis, and A. Oliaro: "Two aspects of the Donoho-Stark uncertainty principle," J. Math. Anal. Appl. , vol.434, no.2, pp.1489-1503, 2015. M. Taylor: "Variations on the Donoho-Stark uncertainty principle estimate," J. Geom. Anal. , vol.28, no.1, pp.492-509, 2018.
Lecture 4: Bandlimited Functions and Sampling Theorems; Periodization vs Sampling
Sampling Theorems: R. N. Bracewell: The Fourier
Transform and Its Applications , 2nd Ed., Revised, McGraw-Hill,
1987. Chap. 10.
W. L. Briggs and V. E. Henson: The DFT: An Owner's Manual
for the Discrete Fourier Transform , SIAM, 1995. Sec. 3.4, Chap. 6.
Pinsky: Chap.4.
See also a nice interactive java demonstration by Matt Herman on sampling and aliasing.
For more details on Sampling Theorems and Non-Uniform Sampling Schemes,
see: H. J. Landau: "Sampling, data transmission, and the Nyquist rate," Proc. IEEE , vol.55, no.10, pp.1701-1706, 1967. A. Aldroubi and K. Gröchenig, "Nonuniform sampling and reconstruction in shift-invariant spaces," SIAM Review , vol.43, no.4, pp.585-620, 2001.
For the historical articles on the sampling theorems, see: E. T.
Whittaker: "On the functions which are represented by the expansions of
the interpolation-theory," Proc. Royal Soc. Edinburgh , Sec. A,
vol.35, pp.181-194, 1915.
C. E. Shannon: "Communication in the presence of noise," Proc. IRE , vol.37, pp.10-21, 1949. M. Unser: "Sampling-50 years after Shannon," Proc. IEEE , vol.88, no.4, pp.569-587, 2000. E. Meijering: "A chronology of interpolation: From ancient astronomy to modern signal and image processing," Proc. IEEE , vol.90, no.3, pp.319-342, 2002. P. L. Butzer, P. J. S. G. Ferreira, J. R. Higgins, S. Saitoh, G. Schmeisser, and R. L. Stens: "Interpolation and sampling: E. T. Whittaker, K. Ogura and their followers," J. Fourier Anal. Appl. , vol.17, no.2, pp.320-354, 2011.
Lecture 5: Basic Theory of Fourier Series; Smoothness of Functions
and Decay Rate of the Fourier Coefficients
For the other stuff I menioned in the class, the details
can be found as follows:Orthogonal Polynomials: G. Szegő: Orthogonal Polynomials , 4th
Ed., AMS, 1975.
Folland: Fourier Analysis, Chap. 6.
Spherical Harmonics: Dym and McKean, Chap. 4.
Stein and Weiss, Chap. 4.
R. T. Seeley: "Spherical harmonics," Amer. Math. Monthly
, vol.73, no.4, part 2, pp.115-121, 1966.
The Gibbs (or more appropriately Gibbs-Wilbraham) Phenomenon: E. Hewitt and R. E. Hewitt: "The Gibbs-Wilbraham phenomenon: An episode in Fourier analysis," Archive for History of Exact Sciences , vol. 21, no. 2, pp. 129-160, 1979.
continuity
in R d d > 2.
It is known as the Pinsky phenomenon . See the following articles
for the details.
A. Gray and M. A. Pinsky: "Computer graphics and a new Gibbs
phenomenon for Fourier-Bessel series," Experiment. Math. , vol. 1, no. 4, pp. 313-316, 1992.
M. A. Pinsky, N. K. Stanton, and P. E. Trapa: "Fourier series of radial functions in several variables,"
J. Funct. Anal. , vol. 116, no. 1, pp. 111-132, 1993.
On the Basel Problem ∑ 1/k 2 π 2
W. Dunham: Euler: The Master of Us All , Math. Assoc. Amer., 1999. Chap.3.
R. Ayoub: "Euler and the zeta function," Amer. Math. Monthly , vol.81, no.10, pp.1067-1086, 1974.
Dan Kalman: "Six ways to sum a series," College Math. Jour. , vol.24, no.5, pp.402-421, 1993.
Lecture 6: Functions of Bounded Variations; the Decay Rate of the
Fourier Coefficients II; Fourier Series on Intervals
Functions of Bounded Variation, the Fourier Coefficients:
C. Lanczos: Discourse on Fourier Series , Hafner Publishing Co., New York, 1966 (republished by SIAM, 2016). Sec 2. This is the best book on 1D Fourier series from the applied perspective.V. I. Smirnov: A Course of Higher Mathematics , Vol. V, Pergamon Press, 1964. Chap. 1.M.Taibleson: "Fourier coefficients of functions of bounded variation," Proc. Amer. Math. Soc. , vol.18, pp.766, 1967.
The definition of BV in higher dimensions can be found in: L. C.
Evans and R. F. Gariepy: Measure Theory and Fine Properties of
Functions , Revised Edition, CRC Press, 2015. Chap.5.
Cornelius Lanczos (1893-1974):
On the occasion of the 120th anniversary of Cornelius Lanczos's birth
the NA group at Manchester made available online a series of video tapes
produced in 1972: http://www.maths.manchester.ac.uk/lanczos . I highly recommend you to watch this video series. In particular, in Tape 1, he talks about the most relevant subjects for this course. Don't miss the video segment around 24-30 minutes where he talks about the importance of Fourier analysis!
Lecture 7: Discrete Fourier Transform
For advanced topics on DFT, check out the following articles and book:Advanced topics on DFT:
J. H. McClellan ad T. W. Parks: "Eigenvalue and eigenvector decomposition of the discrete Fourier transform," IEEE Trans. Audio and
Electroacoustics, vol.AU-20, no.1, pp.66-74, 1972 (with comments, vol.AU-21, no.1, pp.65, 1973 ).
L. Auslander and R. Tolimieri: "Is computing with the finite Fourier transform pure or applied mathematics?" Bull. AMS , vol.1, no.6, pp.847-897, 1979.
A. Terras: Fourier Analysis on Finite Groups and Applications ,
London Mathematical Society Student Texts vol.43, Cambridge Univ.Press,
1999.
Relationship between Fourier Series, DFT, and Trigonometric
Interpolation: C. Lanczos: Discourse on Fourier Series ,
Hafner Publishing Co., New York, 1966 (republished by SIAM, 2016). Sec 17.I. P. Natanson: "On the convergence of trigonometrical interpolation at
equi-distant knots," Annals of Math., vol.45, no.3, pp.457-471, 1944. J. P. Boyd: Chebyshev and Fourier Spectral Methods , 2nd Ed., Dover, 2001. Chapter 4, in particular, Sec.4.5. D. M. Young and R. T. Gregory: A Survey of Numerical
Mathematics, Vol. I , Addison-Wesley, 1972, pp.329-339.
Lecture 8: Fast Fourier Transform (FFT)
There are many references on FFT, but the following are particularly
useful:
J. W. Cooley and J. W. Tukey: "An algorithm for the machine
calculation of complex Fourier series," Math. Comput. , vol.19,
pp.297-301, 1965. Briggs and Henson: Chap. 10.
C. F. Van Loan: Computational Frameworks for the Fast Fourier
Transform , SIAM, 1992.
W. T. Cochran et al.: "What is the Fast Fourier Transform?", Proc. IEEE , vol.55, no.10, pp.1664-1674, 1967.
Perhaps, the best FFT public software package is: FFTW , which also includes
higher dimensional FFT as well as DCT/DST.
Lecture 9: From the Sturm-Liouville Theory to Discrete Cosine and Sine Transforms (DCT/DST)
For the Sturm-Liouville Theory I referred to in today's lecture,
the following are nice references:
Folland: Fourier Analysis: Sec.3.5, 3.6, 7.4.
Dym and McKean: Sec. 1.7, 1.9.
R. Courant and D. Hilbert: Methods of Mathematical Physics,
Vol. I , First English Edition, John Wiley and Sons, 1953.
Republished as Wiley Classics Library in 1989. See Chap. V
in particular. W. O. Amrein, A. M. Hinz, and D. P. Pearson
(eds.): Sturm-Liouville Theory: Past and Present , Birkhäuser, 2005.
For basic and important references on DCT/DST are:
N. Ahmed, T. Natarayan, and K. R. Rao: "Discrete cosine transform," IEEE Trans. Comput. , vol.COM-23, pp.90-93, 1974. K. R. Rao and P. Yip: Discrete Cosine Transform: Algorithms, Advantages, and Applications , Academic Press, 1990. V. Britanak, P. C. Yip, and K. R. Rao: Discrete Cosine and Sine Transforms: General Properties, Fast Algorithms and Integer Approximations , Academic Press, 2006. G. Strang: "The discrete cosine transform," SIAM Review , vol.41, no.1, pp.135-147, 1999. M. V. Wickerhauser: Adapted Wavelet Analysis from Theory to
Software , A K Peters, Ltd., 1994. Chap. 3.
Lecture 10: Karhunen-Loève Transform (KLT)
Discrete version (aka Principal Component Analysis [PCA]):
K. Fukunaga: Introduction to Statistical Pattern Recognition ,
2nd Edition, Academic Press, 1990. Chap. 9, Appendix A. K. V. Mardia, J. T. Kent, and J. M. Bibby: Multivariate
Analysis , Academic Press, 1979. Chap. 8.
S. Watanabe: "Karhunen-Loève expansion and factor
analysis: Theoretical remarks and applications," Trans. 4th Prague
Conf. Inform. Theory, Statist. Decision Functions, Random Processes ,
Publishing House of the Czechoslovak Academy of Sciences, Prague,
pp.635-660, 1965.
R. Kannan and S. Vempala:
"Spectral algorithms," Foudations and Trends in Theoretical Computer Science , vol.4, nos.3-4, pp.157-288, 2008.
We only discussed the discrete version in the class, but KLT has its
continuous version. The following are some references:
U. Grenander: "Stochastic processes and statistical inference," Arkiv för Matematik , vol.1, no.3, pp.195-277, 1950. W. B. Davenport and W. L. Root: An Introduction to the Theory of
Random Signals and Noise , McGraw Hill, 1958, republished by IEEE
Press, 1987. Chap. 6.
W. D. Ray and R. M. Driver: "Further decomposition of
the Karhunen-Loève series representation of a stationary random process," IEEE Trans. Inform. Theory , vol.IT-16, no.6, pp.663-668, 1970.
Relationship between KLT/PCA and SVD:
N. Saito: "Image approximation and modeling via least statistically dependent bases", Pattern Recognition , vol.34, no.9, pp.1765-1784, 2001. L. Lieu and N. Saito: "Signal ensemble classification using low-dimensional embeddings and Earth Mover's Distance", in Wavelets and Multiscale Analysis: Theory and Applications (J. Cohen and A. I. Zayed, eds.), Chap.11, pp.227-256, Birkhäuser, 2011.
Basics of Singular Value Decomposition (SVD):
L. N. Trefethen and D. Bau, III: Numerical Linear Algebra , SIAM, 1997, Chap.4: The Singular Value Decomposition. L. N. Trefethen and D. Bau, III: Numerical Linear Algebra , SIAM, 1997, Chap.5: More about the SVD. G. H. Golub and C. F. Van Loan: Matrix Computations , 4th Ed.,
Johns Hopkins Univ. Press, 2013. Sec. 2.4. G. W. Stewart: "On the early history of the singular value decomposition," SIAM Review , vol.35, no.4, pp.551-566, 1993.
Applications: Eigenfaces/The Rogues Gallery Problem:
M. Kirby and L. Sirovich, "Application of Karhunen-Loeve procedure for the characterization of human faces," IEEE Trans. Pattern Anal. Machine Intell. , vol.12, no.1, pp.103-108, 1990. N. Saito: "Image approximation and modeling via least statistically dependent bases", Pattern Recognition , vol.34, no.9, pp.1765-1784, 2001.
Relationship between KLT/PCA and DCT:
N. Ahmed, T. Natarayan, and K. R. Rao: "Discrete cosine transform," IEEE Trans. Comput. , vol.COM-23, pp.90-93, 1974. R. J. Clarke: "Relation between the Karhunen Loève and cosine transforms," IEE Proc. , vol.128, Part F, no.6, pp.359-360, 1981.
On Toeplitz matrices:
U. Grenander and G. Szegő: Toeplitz
Forms and Their Applications , 2nd Ed., AMS-Chelsea, 1984. R. M.
Gray: Toeplitz and circulant matrices: A review, Technical Report,
Information Systems Laboratory, Department of Electrical Engineering,
Stanford University, 2009. H. Widom: "Toeplitz matrices," in Studies in Real and Complex
Analysis (I. I. Hirschman, Jr. ed.), MAA Studies in Mathematics,
1965.
Lecture 11: Time-Frequency Analysis and Synthesis; Windowed
(or Short-Time) Fourier Transform
S. Mallat: A Wavelet Tour of Signal Processing , 3rd Ed., Academic Press, 2009. Chap. 4.I. Daubechies: Ten Lectures on Wavelets , SIAM, 1992. Chap.2.I. Daubechies: "The wavelet-transform, time-frequency localization and signal analysis," IEEE Trans. Inform. Theory , vol.36, no.5, pp.961-1005, 1990.
Some historical papers:
D. Gabor: "Theory of communication: Part I: The analysis of information," J. IEE (London), vol.93, no.26, pp.429-441, 1946. D. Gabor: "Theory of communication: Part II: The analysis of hearing," J. IEE (London), vol.93, no.26, pp.442-445, 1946. D. Gabor: "Theory of communication: Part III: Frequency compression and expansion," J. IEE (London), vol.93, no.26, pp.445-457, 1946. J. Ville: "Théorie et applications de la notion de signal
analytique," Cables et Transmissions , 2ème A, no.1,
pp.61-74, 1948. An English version translated by M. D. Godfrey is available at this link .
There have been many recent attempts to adapt Gabor frames for the nonstationary setting, i.e., using Gabor functions with inhomogeneous Heisenberg boxes. See, e.g., the following article and the references therein:
P. Balazs, M. Dörfler, F. Jaillet, N. Holighaus, and G. Velasco: "Theory, implementation and applications of nonstationary Gabor frames" J. Comput. Appl. Math. , vol.236, no.6, pp.1481-1496, 2011.
For more data adaptive tilings of the time-frequency plane different without using the Gabor frames, see e.g.:
C. Herley, J. Kovačević, K. Ramchandran, and M. Vetterli: "Tilings of the time-frequency plane: construction of arbitrary orthogonal bases and fast tiling algorithms," IEEE Trans. Signal Process. , vol.41, no.12, pp.3341-3359, 1993.
C. Herley, Z. Xiong, K. Ramchandran, and M. T. Orchard: "Joint space-frequency segmentation using balanced wavelet packet trees for least-cost image representation," IEEE Trans. Image Process. , vol.6, no.9, pp.1213-1230, 1997.
C. M. Thiele and L. F. Villemoes: "A fast algorithm for adapted time-frequency tilings," Applied and Computational Harmonic Analysis , vol.3, no.2, pp.91-99, 1996.
N. N. Bennett: "Fast algorithm for best anisotropic Walsh bases and relatives," Applied and Computational Harmonic Analysis , vol.8, no.1, pp.86-103, 2000.
Lecture 12: Introductory Frame Theory; The Balian-Low Theorem
Some basic references:
S. Mallat: A Wavelet Tour of Signal Processing , 3rd Ed., Academic Press, 2009. Chap. 5.I. Daubechies: Ten Lectures on Wavelets , SIAM, 1992. Chap. 3, 4.K.Gröchenig: Foundations of Time-Frequency Analysis , Birkhäuser, 2001 . Chap. 5-7. O. Christensen: Frames and Bases: An Introductory Course , Birkhäuser, 2002 .P. G. Casazza and G. Kutyniok (eds.): Finite Frames: Theory and Applications , Birkhäuser, 2013 . In particular, Chap. 1: Introduction to Finite Frame Theory by P. G. Cassaza, G. Kutyniok, and F. Philipp .J.-P. Kahane and P.-G. Lemarié-Rieusset: Fourier Series
and Wavelets , Studies in the Development of Modern Mathematcis,
Vol.3, Gordon and Breach Publishers, 1995. Chap. 1 of the Wavelet portion.
Course handout on the Balian-Low theorem.
See also the following original papers:
R. Balian: "Un principe d'incertitude fort en théorie du signal ou en mécanique quantique," C. R. Acad. Sci. Paris , vol.292, pp.1357-1362, 1981.
M. J. Bastiaans: "Gabor's expansion of a signal into Gaussian elementary signals,"
Proc. IEEE , vol.68, no.4, pp.538-539, 1980. G. Battle: "Heisenberg proof of the Balian-Low theorem," Lett. Math. Phys. , vol.15, pp.175-177, 1988. I. Daubechies: "The wavelet-transform, time-frequency localization and signal analysis," IEEE Trans. Inform. Theory , vol.36, no.5, pp.961-1005, 1990. I. Daubechies and A. J. E. M. Janssen: "Two theorems on lattice expansions," IEEE Trans. Inform. Theory , vol.39, no.1, pp.3-6, 1993.
Lecture 13: Continuous Wavelet Transform
Some basic references:
S. Mallat: A Wavelet Tour of Signal Processing , 3rd Ed., Academic Press, 2009. Chap. 4, 5.I. Daubechies: Ten Lectures on Wavelets , SIAM, 1992.
Chap. 3, 4.M. Holschneider: Wavelets: An Analysis Tool , Clarendon Press,
Oxford, 1995. Chap. 1.
I. Daubechies: "The wavelet-transform, time-frequency localization and signal analysis," IEEE Trans. Inform. Theory , vol.36, no.5, pp.961-1005, 1990.
Historic papers by Calderón and Grossman & Morlet can be found at:
A.
P. Calderón: "Intermediate spaces and interpolation, the complex
method," Studia Mathematica , vol.XXIV, pp.113-190, 1964. A. Grossmann and J. Morlet: "Decomposition of Hardy functions into square integrable wavelets of constant shape," SIAM J. Math. Anal. , vol.15, no.4, pp.723-736, 1984.
Lecture 14: Continuous Wavelet Transform II: Analytic Wavelets
Analytic Signals:
D. Gabor: "Theory of communication: Part I: The analysis of information," J. IEE (London), vol.93, no.26, pp.429-441, 1946. D. E. Vakman: "On the definition of concepts of amplitude, phase, and frequency of a signal," Radio Eng. Electron. Phys. , vol.17, no.5, pp.754-759, 1972.
A. Papoulis: Signal Analysis , McGraw-Hill, 1977. Sec.4-2.
M. T. Taner, F. Koehler, and R. E. Sheriff: "Complex seismic trace analysis," Geophysics , vol.44, no.6, pp.1041-1064, 1979; See also the errata, Geophysics , vol.44, no.11, p.1896, 1979. L. Cohen, P. Loughlin, and D. Vakman: "On an ambiguity in the definition of the amplitude and phase of a signal," Signal Processing , vol.79, no.3, pp.301-307, 1999.
General Analytic Wavelets:
S. Mallat: A Wavelet Tour of Signal Processing , 3rd Ed., Academic Press, 2009. Chap. 4.M. Holschneider: Wavelets: An Analysis Tool, Clarendon Press,
Oxford, 1995. Chap. 1.
J. M. Lilly and S. C. Olheda: "On the analytic wavelet transform," IEEE Trans. Inform. Theory , vol.56, no.8, pp.4135-4156, 2010.
The Generalized Morse Wavelets:
S. C. Olheda and A. T. Walden: "Generalized Morse wavelets," IEEE Trans. Signal Process. , vol.50, no.11, pp.2661-2670, 2002. J. M. Lilly and S. C. Olheda: "Higher-order properties of analytic wavelets," IEEE Trans. Signal Process. , vol.57, no.1, pp.146-160, 2009. J. M. Lilly and S. C. Olheda: "Generalized Morse wavelets as a superfamily of analytic wavelets," IEEE Trans. Signal Process. , vol.60, no.11, pp.6036-6041, 2012.
Some historic papers by J. Morlet and collaborators:
J. Morlet, G. Arens, E. Fourgeau, and D. Glard: "Wave propagation and sampling theory-Part I: Complex signal and scattering in multilayered media," Geophysics , vol.47, no.2, pp.203-221, 1982. J. Morlet, G. Arens, E. Fourgeau, and D. Glard: "Wave propagation and sampling theory-Part II: Sampling theory and complex waves," Geophysics , vol.47, no.2, pp.222-236, 1982. A. Grossmann and J. Morlet: "Decomposition of Hardy functions into square integrable wavelets of constant shape," SIAM J. Math. Anal. , vol.15, no.4, pp.723-736, 1984.
Lecture 15: Discrete Wavelet Transforms; Multiresolution Approximation; Scaling Functions
For the regular hyperbolic sampling scheme and the frame conditions for
discretizing the continuous wavelet transform, see:
I. Daubechies: "The wavelet-transform, time-frequency localization and signal analysis," IEEE Trans. Inform. Theory , vol.36, no.5, pp.961-1005, 1990. J.-P. Kahane and P.-G. Lemarié-Rieusset: Fourier Series
and Wavelets , Studies in the Development of Modern Mathematcis, Vol.3,
Gordon and Breach Publishers, 1995. Chap. 2 of the Wavelet portion.
Lecture 16: Conjugate Mirror Filters; Mother Wavelets; Orthonormal Wavelet Basis The following article provides an excellent overiew of the wavelet transforms/bases and their impact to applications:
I. Daubechies: "Wavelets and Applications," in The Princeton Companion to Mathematics (T. Gower, Ed.), Princeton Univ. Press, 2008, pp.848-862.
Lecture 17: Vanishing Moments; Support Size; Regularity; and Daubechies's Compactly Supported Wavelets
Some basic references:
S. Mallat: A Wavelet Tour of Signal Processing , 3rd Ed., Academic Press, 2009. Sec. 7.2I. Daubechies: Ten Lectures on Wavelets , SIAM, 1992. Chap. 6.I. Daubechies: "Orthonormal bases of compactly supported wavelets," Comm. Pure Appl. Math. , vol.41, no.7, pp.909-996, 1988.
Lecture 18: Fast Wavelet Transforms; Various Extensions On boundary treatment of input signals:
S. Mallat: A Wavelet Tour of Signal Processing , 3rd Ed., Academic Press, 2009. Sec. 7.5A. Cohen, I. Daubechies, and P. Vial: "Wavelet bases on the interval and fast algorithms," Applied and Computational Harmonic Analysis , vol.1, no.1, pp.54-81, 1993.
On the lack of translation invariance of the orthogonal wavelet representations and possible solutions:
G. Beylkin: "On the representation of operators in bases of compactly supported wavelets," SIAM J. Numerical Anal. , vol. , no. , pp.1716-1740, 1992.
G. Nason and B. W. Silverman: "The stationary wavelet transform and some statistical applications," in Lecture Notes in Statistics (A. Antoniadis and G. Oppenheim, eds.), vol.103, pp.281-299, 1995.
E. P. Simoncelli, W. T. Freeman, E. H. Adelson, and D. J. Heeger: "Shiftable multiscale transforms," IEEE Trans. Inform. Theory , vol.38, no.2, pt.2, pp.587-607, 1992.
N. Kingsbury: "Complex wavelets for shift invariant analysis and filtering of signals," Applied and Computational Harmonic Analysis , vol.10, no.3, pp.234-253, 2001.
I. W. Selesnick, B. G. Baraniuk, and N. C. Kingsbury: "The dual-tree complex wavelet transform," IEEE Signal Processing Magazine , vol.22, no.6, pp.123-151, 2005.
On the lack of symmetry/antisymmetry of the father and mother wavelets and possible solutions:
I. Daubechies: Ten Lectures on Wavelets , SIAM, 1992. Chap. 8.symmlets :
S. Mallat: A Wavelet Tour of Signal Processing , 3rd Ed., Academic Press, 2009. Sec. 7.2Biorthogonal wavelet representations:
S. Mallat: A Wavelet Tour of Signal Processing , 3rd Ed., Academic Press, 2009. Sec. 7.4A. Cohen, I. Daubechies, and J.-C. Feauveaux: "Biorthogonal bases of compactly supported wavelets", Comm. Pure Appl. Math. , vol.45, no.5, pp.485-560, 1992.
Wavelet frames/framelets : see, e.g.,
I. Daubechies, B. Han, A. Ron, and Z. Shen: "Framelets: MRA-based constructions of wavelet frames," Applied and Computational Harmonic Analysis , vol.14, no.1, pp.1-46, 2003, and the references therein.
autocorrelation shell representation, see:
N. Saito and G. Beylkin: "Multiresolution representations using the auto-correlation functions of compactly supported wavelets", IEEE Trans. Signal Process. , vol.41, no.12, pp.3584-3590, 1993. ; see also Errata.
Supplementary Lecture I: A Library of Othonormal Bases and Adapted Signal Analysis
The basic references on these subjects:
S. Mallat: A Wavelet Tour of Signal Processing , 3rd Ed., Academic Press, 2009. Chap. 8M. V. Wickerhauser: Adapted Wavelet Analysis from Theory to
Software , A K Peters, Ltd., 1994. Chap. 4, 7, 8.
R. R. Coifman and M. V. Wickerhauser: "Entropy-based algorithms for best basis selection," IEEE Trans. Inform. Theory , vol.38, no.2, pp.713-718, 1992.
For a variety of efficacy measures for selecting a basis from
basis dictionaries for classification, regression, and other applications,
see, e.g.,
N. Saito: "Local feature extraction and its applications using a library of bases," in Topics in Analysis and Its Applications: Selected Theses (R. Coifman, ed.), pp.269-451, World Scientific, 2000.
N. Saito: "Image approximation and modeling via least statistically-dependent bases," Pattern Recognition , vol.34, no.9, pp.1765-1784, 2001.
N. Saito, R. R. Coifman, F. B. Geshwind, and F. Warner: Discriminant feature extraction using empirical probability density estimation and a local basis library, Pattern Recognition , vol.35, no.12, pp.2841-2852, 2002.
Supplementary Lecture II: Multiscale Basis Dicionarites on Graphs and Networks This lecture is mainly based on the following papers of mine written with former my Ph.D. student Jeff Irion:
J. Irion and N. Saito: "Hierarchical graph Laplacian eigen transforms," JSIAM Letters , vol.6, pp.21-24, 2014.
J. Irion and N. Saito: "The generalized Haar-Walsh transform," Proc. 2014 IEEE Workshop on Statistical Signal Processing , pp.472-475, 2014.
J. Irion and N. Saito: "Applied and computational harmonic analysis on graphs and networks," in Wavelets and Sparsity XVI (M. Papadakis, V. K. Goyal, D. Van De Ville, eds.), Proc. SPIE 9597 , Paper #95971F, 2015. Invited paper.J. Irion and N. Saito: "Efficient approximation and denoising of graph signals using the multiscale basis dictionaries," IEEE Transactions on Signal and Information Processing over Networks , vol.3, no.3, pp.607-616, 2017.
For a nice review on the recent explosion of interest in signal processing on graphs, see the following and the references therein.
D. I. Shuman, S. K. Narang, P. Frossard, A. Ortega, and P. Vandergheynst: "The emerging field of signal processing on graphs: Extending high-dimensional data analysis to networks and other irregular domains," IEEE Signal Processing Magazine , vol.30, no.3, pp.83-98, 2013.
The basic references on graph Laplacians and spectral graph theory:
F. R. K. Chung: Spectral Graph Theory , CBMS Regional Conference Series in Mathematics, no. 92, Amer. Math. Soc., 1997.
Some of the chapters are available from her website .
D. Cvetković, P. Rowlinson, and S. Simić: An Introduction to the Theory of Graph Spectra , Vol. 75, London Mathematical Society Student Texts, Cambridge Univ. Press, 2010.
A. E. Brouwer and W. H. Haemers: Spectra of Graphs , Springer, 2012.
R. B. Bapat: Graphs and Matrices , 2nd Ed., Universitext, Springer, 2014.
D. Spielman: "Spectral graph theory," in Combinatorial Scientific Computing (O. Schenk, ed.), Chap.18, pp.495-524, CRC Press, 2012.
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