Wavelets from a statistical perspective

Cover -- Half Title -- Title Page -- Copyright Page -- Table of contents -- List of recurrent symbols -- Introduction -- The topic of this book -- For whom has this book been written? -- The exercises -- Accompanying software -- Acknowledgements -- 1. Wavelets: nonlinear processing in multiscale sparsity -- 1.1. Compressing big data -- 1.2. Wavelets among other methods for data processing -- 1.2.1. The Fourier transform -- 1.2.2. Using Fourier analysis in linear data processing -- 1.2.3. From sines to splines -- 1.2.4. Regression splines, smoothing splines, P-splines -- 1.2.5. Kernels and local polynomial smoothing -- 1.3. An illustrative example of a wavelet decomposition -- 1.3.1. A test function with jumps and cusps -- 1.3.2. Linear smoothing -- 1.3.3. A Fourier analysis of data with discontinuities -- 1.3.4. A multilevel approach -- 1.3.5. The Haar transform -- 1.3.6. The Haar basis -- 1.3.7. Wavelet basis functions -- 1.3.8. The Haar transform matrix -- 1.4. The key properties of a wavelet method -- 1.5. Intrinsic multiscale problems -- 1.6. Design of a wavelet transform -- 1.6.1. Perfect reconstruction (PR) -- 1.6.2. Fast decomposition and reconstruction -- 1.6.3. The characteristics of the basis functions -- 1.6.4. The sparsity of the decomposition -- 1.6.5. Numerical condition -- 1.7. Conclusion -- 2. Wavelet building blocks -- 2.1. From the Haar transform to the lifting scheme -- 2.1.1. Nonequidistant grids -- 2.1.2. The unbalanced Haar transform -- 2.1.3. The Haar transform as a lifting scheme -- 2.1.4. A prediction-first scheme for the same unbalanced Haar transform -- 2.1.5. Triangular hat basis functions -- 2.1.6. Wavelets in a triangular hat basis -- 2.1.7. A general two-step lifting scheme -- 2.1.7.1. Prediction-first scheme -- 2.1.7.2. Update-first (in the forward transform) -- 2.1.8. Lifting in diagrams -- 2.2. Filterbanks.