A WAVELET TOUR OF SIGNAL PROCESSING
CONTENTS SPARSE REPRESENTATIONS COMPUTATIONAL HARMONIC ANALYSIS FOURIER KINGDOM WAVELET BASES APPROXIMATION AND PROCESSING IN BASES SAMPLING WITH LINEAR APPROXIMATIONS SPARSE NON-LINEAR APPROXIMATIONS COMPRESSION DENOISING TIME-FREQUENCY DICTIONARIES HEISENBERG UNCERTAINTY WINDOWED FOURIER TRANSFORM CONTINUOUS WAVELET TRANSFORM TIME-FREQUENCY ORTHONORMAL BASES SPARSITY IN REDUNDANT DICTIONARIES FRAME ANALYSIS AND SYNTHESIS IDEAL DICTIONARY APPROXIMATIONS PURSUIT IN DICTIONARIES INVERSE PROBLEMS DIAGONAL INVERSE ESTIMATION SUPER-RESOLUTION AND COMPRESSIVE SENSING TRAVEL GUIDE LINEAR TIME-INVARIANT FILTERING IMPULSE RESPONSE TRANSFER FUNCTIONS FOURIER INTEGRALS FOURIER TRANSFORM IN L1(R) FOURIER TRANSFORM IN L2(R) EXAMPLES PROPERTIES REGULARITY AND DECAY UNCERTAINTY PRINCIPLE TOTAL VARIATION TWO-DIMENSIONAL FOURIER TRANSFORM EXERCISES CONTENTS DISCRETE REVOLUTION SAMPLING ANALOG SIGNALS SHANNON-WHITTAKER SAMPLING THEOREM ALIASING GENERAL SAMPLING AND LINEAR ANALOG CONVERSIONS DISCRETE TIME-INVARIANT FILTERS IMPULSE RESPONSE AND TRANSFER FUNCTION FOURIER SERIES FINITE SIGNALS CIRCULAR CONVOLUTIONS DISCRETE FOURIER TRANSFORM FAST FOURIER TRANSFORM FAST CONVOLUTIONS DISCRETE IMAGE PROCESSING TWO-DIMENSIONAL SAMPLING THEOREMS DISCRETE IMAGE FILTERING CIRCULAR CONVOLUTIONS AND FOURIER BASIS TIME MEETS FREQUENCY TIME-FREQUENCY ATOMS COMPLETENESS AND STABILITY CHOICE OF WINDOW DISCRETE WINDOWED FOURIER TRANSFORM WAVELET TRANSFORMS REAL WAVELETS ANALYTIC WAVELETS DISCRETE WAVELETS TIME-FREQUENCY GEOMETRY OF INSTANTANEOUS FREQUENCIES WINDOWED FOURIER RIDGES WAVELET RIDGES QUADRATIC TIME-FREQUENCY ENERGY WIGNER-VILLE DISTRIBUTION INTERFERENCES AND POSITIVITY COHEN’S CLASS DISCRETE WIGNER-VILLE COMPUTATIONS FOURIER KINGDOM THE FOURIER TRANSFORM IS EVERYWHERE IN PHYSICS AND MATHEMATICS, BECAUSE IT DIAGONALIZES TIME- INVARIANT CONVOLUTION OPERATORS IT RULES OVER LINEAR TIME-INVARIANT SIGNAL PROCESSING, WHOSE BUILDING BLOCKS ARE FREQUENCY ?LTERING OPERATORS FOURIER ANALYSIS REPRESENTS ANY ?NITE ENERGY FUNCTION F(T) AS A SUM OF SINUSOIDAL WAVES EI?T: F(T) = 2? # +? ?? ˆF(?) EI?T D? (1 1) WAVELET BASES WAVELET BASES, LIKE FOURIER BASES, REVEAL THE SIGNAL REGULARITY THROUGH THE AMPLITUDE OF COE?CIENTS, AND THEIR STRUCTURE LEADS TO A FAST COMPUTATIONAL ALGORITHM HOWEVER, WAVELETS ARE WELL LOCALIZED AND FEW COE?CIENTS ARE NEEDED TO REPRESENT LOCAL TRANSIENT STRUCTURES AS OPPOSED TO A FOURIER BASIS, A WAVELET BASIS DE?NES A SPARSE REPRESENTATION OF PIECEWISE REGULAR SIGNALS, WHICH MAY INCLUDE TRANSIENTS AND SINGULARITIES IN IMAGES, LARGE WAVELET COE?CIENTS ARE LOCATED IN THE NEIGHBORHOOD OF EDGES AND IRREGULAR TEXTURES THE STORY BEGINS IN 1910, WHEN HAAR [290] CONSTRUCTED A PIECEWISE CONSTANT FUNCTION ?(T) = ? ? ? IF 1/2 ?T < OTHERWISE WHOSE DILATIONS AND TRANSLATIONS GENERATE AN ORTHONORMAL BASIS ' ?J,N(T) =