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El. knyga: Harmonic and Applied Analysis: From Groups to Signals

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This contributed volume explores the connection between the theoretical aspects of harmonic analysis and the construction of advanced multiscale representations that have emerged in signal and image processing. It highlights some of the most promising mathematical developments in harmonic analysis in the last decade brought about by the interplay among different areas of abstract and applied mathematics. This intertwining of ideas is considered starting from the theory of unitary group representations and leading to the construction of very efficient schemes for the analysis of multidimensional data.After an introductory chapter surveying the scientific significance of classical and more advanced multiscale methods, chapters cover such topics asAn overview of Lie theory focused on common applications in signal analysis, including the wavelet representation of the affine group, the Schrödinger representation of the Heisenberg group, and the metaplectic representation of the sym

plectic groupAn introduction to coorbit theory and how it can be combined with the shearlet transform to establish shearlet coorbit spacesMicrolocal properties of the shearlet transform and its ability to provide a precise geometric characterization of edges and interface boundaries in images and other multidimensional dataMathematical techniques to construct optimal data representations for a number of signal types, with a focus on the optimal approximation of functions governed by anisotropic singularities. A unified notation is used across all of the chapters to ensure consistency of the mathematical material presented.Harmonic and Applied Analysis: From Groups to Signals is aimed at graduate students and researchers in the areas of harmonic analysis and applied mathematics, as well as at other applied scientists interested in representations of multidimensional data. It can also be used as a textbookfor graduate courses in applied harmonic analysis.

From Group Representations to Signal Analysis.- The Use of Representations in Applied Harmonic Analysis.- Shearlet Coorbit Theory.- Efficient Analysis and Detection of Edges through Directional Multiscale Representations.- Optimally Sparse Data Representations.
1 From Group Representations to Signal Analysis
1(6)
Stephan Dahlke
Filippo De Mari
Philipp Grohs
Demetrio Labate
References
5(2)
2 The Use of Representations in Applied Harmonic Analysis
7(76)
Filippo De Mari
Ernesto De Vito
2.1 Representations of Lie Groups
7(41)
2.1.1 Locally compact groups
8(2)
2.1.2 Lie Groups and Lie Algebras
10(20)
2.1.3 Representation Theory
30(6)
2.1.4 Reproducing Systems and Square Integrability
36(7)
2.1.5 Unbounded Operators
43(3)
2.1.6 Stone's Theorem and the Differential of a Representation
46(2)
2.2 The Heisenberg Group and its Representations
48(13)
2.2.1 The Group and its Lie Algebra
48(3)
2.2.2 Automorphisms
51(2)
2.2.3 The Schrodinger Representation
53(4)
2.2.4 Time-frequency Analysis
57(4)
2.3 The Metaplectic Representation
61(22)
2.3.1 More on the Symplectic Group
62(2)
2.3.2 Construction of the Metaplectic Representation
64(3)
2.3.3 Restriction to Triangular Subgroups
67(13)
References
80(3)
3 Shearlet Coorbit Theory
83(66)
Stephan Dahlke
Soren Hauser
Gabriele Steidl
Gerd Teschke
3.1 Introduction
83(2)
3.2 Coorbit Space Theory
85(25)
3.2.1 Coorbit Spaces
85(10)
3.2.2 Discretization
95(9)
3.2.3 Examples
104(4)
3.2.4 Proof of Lemma 3.16 and 3.18
108(2)
3.3 Multivariate Shearlet Transform
110(9)
3.3.1 The Shearlet Group in Rd
110(5)
3.3.2 The Continuous Shearlet Transform
115(4)
3.4 Multivariate Shearlet Coorbit Theory
119(10)
3.4.1 Shearlet Coorbit Spaces
119(8)
3.4.2 Discretization
127(2)
3.5 Structure of Shearlet Coorbit Spaces
129(16)
3.5.1 Density
129(1)
3.5.2 Embeddings
130(6)
3.5.3 Traces of Shearlet Coorbit Spaces
136(9)
3.6 Variation of a Theme: the Toeplitz Shearlet Transform
145(4)
References
146(3)
4 Efficient Analysis and Detection of Edges Through Directional Multiscale Representations
149(50)
Kanghui Guo
Demetrio Labate
4.1 Introduction
149(4)
4.2 The continuous wavelet transform and its generalizations
153(10)
4.2.1 Continuous wavelets
153(4)
4.2.2 Continuous shearlets
157(1)
4.2.3 Continuous shearlets in the plane (n = 2)
158(5)
4.3 Shearlet analysis of step edges. Case n = 2
163(12)
4.3.1 Proof of Theorem 4.4
167(8)
4.4 Shearlet analysis of edges in dimension n = 3
175(12)
4.4.1 3D Continuous Shearlet Transform
175(2)
4.4.2 Characterization of 3D Boundaries
177(3)
4.4.3 Identification of curve singularities on the piecewise smooth surface boundary of a solid
180(7)
4.5 Other results and applications
187(12)
4.5.1 Shearlet analysis of general edges
187(1)
4.5.2 Numerical applications
188(1)
Appendix
189(6)
References
195(4)
5 Optimally Sparse Data Representations
199(50)
Philipp Grohs
5.1 Introduction
199(3)
5.1.1 Notation
202(1)
5.2 Signal Classes and Encoding
202(5)
5.3 Upper Bounds on the Optimal Encoding Rate
207(7)
5.4 Sparse Approximation in Dictionaries
214(11)
5.4.1 Best N-term Approximation in Dictionaries
215(2)
5.4.2 Best N-term Approximation with Polynomial Depth Search
217(3)
5.4.3 Sparse Approximations in Frames
220(5)
5.5 Wavelet Approximation of Piecewise Smooth Functions
225(4)
5.6 Cartoon Approximation with Curvelet Tight Frames
229(15)
5.6.1 Suboptimality of Wavelets for Cartoon Images
229(4)
5.6.2 Curvelets
233(7)
5.6.3 Shearlets
240(4)
5.7 Further Examples
244(1)
5.8 Appendix: Chernoff Bounds
245(4)
References
247(2)
Index 249
Stephan Dahlke is Professor in the Department of Mathematics and Computer Sciences at Philipps-University of Marburg, Germany.

Filippo De Mari is Associate Professor in the Department of Mathematics at the University of Genova, Italy.

Philipp Grohs is Assistant Professor in the Seminar for Applied Mathematics at the Swiss Federal Institute of Technology, Zurich, Switzerland.

Demetrio Labate is Associate Professor in the Department of Mathematics at the University of Houston, TX, USA
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