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| Preface | |
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| Preface to the Second Edition | |
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| Notation | |
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| Introduction to a Transient World | |
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| Fourier Kingdom | |
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| Time-Frequency Wedding | |
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| Windowed Fourier Transform | |
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| Wavelet Transform | |
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| Bases of Time-Frequency Atoms | |
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| Wavelet Bases and Filter Banks | |
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| Tilings of Wavelet Packet and Local Cosine Bases | |
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| Bases for What? | |
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| Approximation | |
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| Estimation | |
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| Compression | |
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| Travel Guide | |
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| Reproducible Computational Science | |
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| Road Map | |
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| Fourier Kingdom | |
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| Linear Time-Invariant Filtering | |
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| Impulse Response | |
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| Transfer Functions | |
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| Fourier Integrals | |
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| Fourier Transform in L (R) | |
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| Fourier Transform in L (R) | |
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| Examples | |
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| Properties | |
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| Regularity and Decay | |
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| Uncertainty Principle | |
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| Total Variation | |
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| Two-Dimensional Fourier Transform | |
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| Problems | |
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| Discrete Revolution | |
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| Sampling Analog Signals | |
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| Whittaker Sampling Theorem | |
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| Aliasing | |
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| General Sampling Theorems | |
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| Discrete Time-Invariant Filters | |
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| Impulse Response and Transfer Function | |
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| Fourier Series | |
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| Finite Signals | |
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| Circular Convolutions | |
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| Discrete Fourier Transform | |
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| Fast Fourier Transform | |
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| Fast Convolutions | |
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| Discrete Image Processing | |
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| Two-Dimensional Sampling Theorem | |
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| Discrete Image Filtering | |
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| Circular Convolutions and Fourier Basis | |
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| Problems | |
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| Time Meets Frequency | |
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| Time-Frequency Atoms | |
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| Windowed Fourier Transform | |
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| Completeness and Stability | |
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| Choice of Window | |
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| Discrete Windowed Fourier Transform | |
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| Wavelet Transforms | |
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| Real Wavelets | |
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| Analytic Wavelets | |
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| Discrete Wavelets | |
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| Instantaneous Frequency | |
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| Windowed Fourier Ridges | |
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| Wavelet Ridges | |
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| Quadratic Time-Frequency Energy | |
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| Wigner-Ville Distribution | |
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| Interferences and Positivity | |
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| Cohen's Class | |
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| Discrete Wigner-Ville Computations | |
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| Problems | |
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| Frames | |
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| Frame Theory | |
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| Frame Definition and Sampling | |
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| Pseudo Inverse | |
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| Inverse Frame Computations | |
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| Frame Projector and Noise Reduction | |
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| Windowed Fourier Frames | |
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| Wavelet Frames | |
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| Translation Invariance | |
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| Dyadic Wavelet Transform | |
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| Wavelet Design | |
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| "Algorithme a Trous" | |
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| Oriented Wavelets for a Vision | |
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| Problems | |
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| Wavelet Zoom | |
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| Lipschitz Regularity | |
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| Lipschitz Definition and Fourier Analysis | |
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| Wavelet Vanishing Moments | |
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| Regularity Measurements with Wavelets | |
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| Wavelet Transform Modulus Maxima | |
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| Detection of Singularities | |
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| Reconstruction From Dyadic Maxima | |
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| Multiscale Edge Detection | |
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| Wavelet Maxima for Images | |
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| Fast Multiscale Edge Computations | |
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| Multifractals | |
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| Fractal Sets and Self-Similar Functions | |
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| Singularity Spectrum | |
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| Fractal Noise | |
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| Problems | |
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| Wavelet Bases | |
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| Orthogonal Wavelet Bases | |
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| Multiresolution Approximations | |
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| Scaling Function | |
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| Conjugate Mirror Filters | |
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| In Which Orthogonal Wavelets Finally Arrive | |
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| Classes of Wavelet Bases | |
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| Choosing a Wavelet | |
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| Shannon, Meyer and Battle-Lemarie Wavelets | |
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| Daubechies Compactly Supported Wavelets | |
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| Wavelets and Filter Banks | |
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| Fast Orthogonal Wavelet Transform | |
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| Perfect Reconstruction Filter Banks | |
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| Biorthogonal Bases of I (Z) | |
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| Biorthogonal Wavelet Bases | |
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| Construction of Biorthogonal Wavelet Bases | |
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| Biorthogonal Wavelet Design | |
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| Compactly Supported Biorthogonal Wavelets | |
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| Lifting Wavelets | |
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| Wavelet Bases on an Interval | |
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| Periodic Wavelets | |
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| Folded Wavelets | |
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| Boundary Wavelets | |
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| Multiscale Interpolations | |
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| Interpolation and Sampling Theorems | |
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| Interpolation Wavelet Basis | |
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| Separable Wavelet Bases | |
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| Separable Multiresolutions | |
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| Two-Dimensional Wavelet Bases | |
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| Fast Two-Dimensional Wavelet Transform | |
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| Wavelet Bases in Higher Dimensions | |
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| Problems | |
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| Wavelet Packet and Local Cosine Bases | |
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| Wavelet Packets | |
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| Wavelet Packet Tree | |
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| Time-Frequency Localization | |
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| Particular Wavelet Packet Bases | |
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| Wavelet Packet Filter Banks | |
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| Image Wavelet Packets | |
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| Wavelet Packet Quad-Tree | |
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| Separable Filter Banks | |
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| Block Transforms | |
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| Block Bases | |
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| Cosine Bases | |
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| Discrete Cosine Bases | |
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| Fast Discrete Cosine Transforms | |
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| Lapped Orthogonal Transforms | |
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| Lapped Projectors | |
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| Lapped Orthogonal Bases | |
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| Local Cosine Bases | |
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| Discrete Lapped Transforms | |
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| Local Cosine Trees | |
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| Binary Tree of Cosine Bases | |
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| Tree of Discrete Bases | |
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| Image Cosine Quad-Tree | |
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| Problems | |
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| An Approximation Tour | |
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| Linear Approximations | |
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| Linear Approximation Error | |
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| Linear Fourier Approximations | |
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| Linear Multiresolution Approximations | |
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| Karhunen-Loeve Approximations | |
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| Non-Linear Approximations | |
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| Non-Linear Approximation Error | |
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| Wavelet Adaptive Grids | |
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| Besov Spaces | |
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| Image Approximations with Wavelets | |
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| Adaptive Basis Selection | |
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| Best Basis and Schur Concavity | |
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| Fast Best Basis Search in Trees | |
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| Wavelet Packet and Local Cosine Best Bases | |
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| Approximations with Pursuits | |
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| Basis Pursuit | |
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| Matching Pursuit | |
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| Orthogonal Matching Pursuit | |
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| Problems | |
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| Estimations are Approximations | |
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| Bayes Versus Minimax | |
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| Bayes Estimation | |
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| Minimax Estimation | |
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| Diagonal Estimation in a Basis | |
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| Diagonal Estimation with Oracles | |
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| Thresholding Estimation | |
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| Thresholding Refinements | |
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| Wavelet Thresholding | |
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| Best Basis Thresholding | |
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| Minimax Optimality | |
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| Linear Diagonal Minimax Estimation | |
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| Orthosymmetric Sets | |
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| Nearly Minimax with Wavelets | |
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| Restoration | |
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| Estimation in Arbitary Gaussian Noise | |
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| Inverse Problems and Deconvolution | |
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| Coherent Estimation | |
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| Coherent Basis Thresholding | |
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| Coherent Matching Pursuit | |
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| Spectrum Estimation | |
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| Power Spectrum | |
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| Approximate Karhunen-Loeve Search | |
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| Locally Stationary Processes | |
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| Problems | |
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| Transform Coding | |
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| Signal Compression | |
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| State of the Art | |
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| Compression in Orthonormal Bases | |
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| Distortion Rate of Quantization | |
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| Entropy Coding | |
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| Scalar Quantization | |
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| High Bit Rate Compression | |
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| Bit Allocation | |
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| Optimal Basis and Karhunen-Loeve | |
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| Transparent Audio Code | |
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| Image Compression | |
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| Deterministic Distortion Rate | |
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| Wavelet Image Coding | |
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| Block Cosine Image Coding | |
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| Embedded Transform Coding | |
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| Minimax Distortion Rate | |
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| Video Signals | |
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| Optical Flow | |
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| MPEG Video Compression | |
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| Problems | |
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| Mathematical Complements | |
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| Functions and Integration | |
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| Banach and Hilbert Spaces | |
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| Bases of Hilbert Spaces | |
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| Linear Operators | |
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| Separable Spaces and Bases | |
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| Random Vectors and Covariance Operators | |
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| Diracs | |
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| Software Toolboxes | |
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| WaveLab | |
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| LastWave | |
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| Freeware Wavelet Toolboxes | |
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| Bibliography | |
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| Index | |