## Beyond Wavelets"Beyond Wavelets" presents state-of-the-art theories, methods, algorithms, and applications of mathematical extensions for classical wavelet analysis. Wavelets, introduced 20 years ago by Morlet and Grossmann and developed very rapidly during the 1980's and 1990's, has created a common link between computational mathematics and other disciplines of science and engineering. Classical wavelets have provided effective and efficient mathematical tools for time-frequency analysis which enhances and replaces the Fourier approach. However, with the current advances in science and technology, there is an immediate need to extend wavelet mathematical tools as well. "Beyond Wavelets" presents a list of ideas and mathematical foundations for such extensions, including: continuous and digital ridgelets, brushlets, steerable wavelet packets, contourlets, eno-wavelets, spline-wavelet frames, and quasi-affine wavelets. Wavelet subband algorithms are extended to pyramidal directional and nonuniform filter banks. In addition, this volume includes a method for tomographic reconstruction using a mechanical image model and a statistical study for independent adaptive signal representation. Investigators already familiar with wavelet methods from areas such as engineering, statistics, and mathematics will benefit by owning this volume. *Curvelets, Contourlets, Ridgelets, *Digital Implementation of Ridgelet Packets *Steerable Wavelet Packets *Essentially Non-Oscillatory Wavelets *Medical Imaging *Non-Uniform Filter Banks *Spline-wavelet frames and *Vanishing Moment Recovery Functions |

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### Contents

1 | |

Chapter 2 Digital Implementation of Ridgelet Packets | 31 |

Steerable Wavelet Packets | 61 |

Chapter 4 Contourlets | 83 |

Chapter 5 ENOwavelet Transforms and Some Applications | 107 |

Chapter 6 A Mechanical Image Model for Bayesian Tomographic Reconstruction | 135 |

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### Common terms and phrases

algorithm analysis approximate dual B-splines bases basis functions biorthogonal brushlets compute construction contourlet decimator—set decomposition defined derivable digital ridgelet dilation discontinuities discrete Donoho ENO-wavelet transforms entropy equivalent example factorization Figure filter bank Fourier domain Fourier transform Hence high frequency coefficients IEEE Trans integer interval invertible isometry knot vector Laurent polynomial Lemma linear LSDB matrix multiple multiresolution nonuniform FBs obeying obtained orthogonal orthonormal basis PDFB pixtrons polar Fourier polynomial matrix polyphase polyphase matrix properties pseudopolar Radon transform rational FBs rational PRFBs reconstruction result ridge functions ridgelet ridgelet packet ridgelet transform scale Section sequence set of decimators Slant Stack smooth functions sparsity spike process splines standard basis standard wavelet transforms statistical independence stochastic process strong compatibility subband subspace Theorem tight frame tiling tomographic reconstruction tree structures tree version uniform FB uniform-tree values vanishing moments wavelet coefficients wavelet packets