Book contents
- Frontmatter
- Dedication
- Contents
- Preface
- Notation
- 1 Introduction
- 2 Roadmap to the book
- 3 Mathematical context and background
- 4 Continuous-domain innovation models
- 5 Operators and their inverses
- 6 Splines and wavelets
- 7 Sparse stochastic processes
- 8 Sparse representations
- 9 Infinite divisibility and transform-domain statistics
- 10 Recovery of sparse signals
- 11 Wavelet-domain methods
- 12 Conclusion
- Appendix A Singular integrals
- Appendix B Positive definiteness
- Appendix C Special functions
- References
- Index
6 - Splines and wavelets
Published online by Cambridge University Press: 05 September 2014
- Frontmatter
- Dedication
- Contents
- Preface
- Notation
- 1 Introduction
- 2 Roadmap to the book
- 3 Mathematical context and background
- 4 Continuous-domain innovation models
- 5 Operators and their inverses
- 6 Splines and wavelets
- 7 Sparse stochastic processes
- 8 Sparse representations
- 9 Infinite divisibility and transform-domain statistics
- 10 Recovery of sparse signals
- 11 Wavelet-domain methods
- 12 Conclusion
- Appendix A Singular integrals
- Appendix B Positive definiteness
- Appendix C Special functions
- References
- Index
Summary
A fundamental aspect of our formulation is that the whitening operator L is naturally tied to some underlying B-spline function, which will play a crucial role in what follows. The spline connection also provides a strong link with wavelets [UB03].
In this chapter, we review the foundations of spline theory and show how one can construct B-spline basis functions and wavelets that are tied to some specific operator L. The chapter starts with a gentle introduction to wavelets that exploits the analogy with Lego blocks. This naturally leads to the formulation of a multiresolution analysis of L2(ℝ) using piecewise-constant functions and a de visu identification of Haar wavelets. We then proceed in Section 6.2 with a formal definition of our generalized brand of splines – the cardinal L-splines – followed by a detailed discussion of the fundamental notion of the Riesz basis. In Section 6.3, we systematically cover the first-order operators with the construction of exponential B-splines and wavelets, which have the convenient property of being orthogonal. We then address the theory in its full generality and present two generic methods for constructing B-spline basis functions (Section 6.4) and semi-orthogonal wavelets (Section 6.5). The pleasing aspect is that these results apply to the whole class of shift-invariant differential operators L whose null space is finite-dimensional (possibly trivial), which are precisely those that can be safely inverted to specify sparse stochastic processes.
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- An Introduction to Sparse Stochastic Processes , pp. 113 - 149Publisher: Cambridge University PressPrint publication year: 2014
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