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| Ïîèñê ïî óêàçàòåëÿì |
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| Hubbard B. — The World According to Wavelets: The Story of a Mathematical Technique in the Making |
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| Ïðåäìåòíûé óêàçàòåëü |
Kolmogorov, Andrei, divergent series 105
Kolmogorov, Andrei, equivalence of measure theory and probability theory 171
Kolmogorov, Andrei, information content 65 66
Kolmogorov, Andrei, KAM theorem 113
Kolmogorov, Andrei, turbulence 55
Kovalevskaya, Sophie 108
Kronland-Martinet, Richard 124 204
Kulikowski, Janus 187
Lagrange, Joseph Louis 11
Laplace, Pierre Simon 7 176
Lebesgue integral 182—183 217—219
Lebesgue, Henri 182—183
Lemarie, Pierre Gilles (later Lemarie-Rieusset) 45 138
Limits 219
LOG see "logarithm"
Logarithm 117
Lord Kelvin, (on Fourier) 8
Low-pass filter see "Filters"
Lucier, Bradley 59
Magnetic resonance imaging 53—54 70
Mallat, Stephane see also "Multiresolution analysis" "Fast
Mallat, Stephane, Best Basis 81
Mallat, Stephane, choice of system of representation 88—89
Mallat, Stephane, dictionary analogy 86—88
Mallat, Stephane, early involvement with wavelets 38—42 45—47 137—138 140 145
Mallat, Stephane, Matching Pursuits 85—88
Mallat, Stephane, multiresolution analysis with Meyer 38—41 137—153
Mallat, Stephane, speed of computations 45
Mallat, Stephane, wavelet maxima 59—61 125
Malvar wavelets 78—80 81 83 87 201
Malvar, Henrique 80
Mandelbrot set 46 65 66 67
Marcelja, S. 187
Marr, David 26 39 76 162
Massopust, P.R. 164
Matching Pursuits 85—88 202
Mathematical analysis (described by Fourier) 14
Mathematical microscope 31 54
Mathematical Theory of Communications 18 81
Matrices in fast Fourier transform 118-123
Matrices in quantum mechanics 180
Matrices, matrix compression 70—71
Matrices, matrix multiplication 119—120
Maxwell, James Clark xiii 7
Mean 172 174—175
Mean absolute deviation 172 173
Measure theory and probability 171 172 219
Mechain, Pierre 11
Medical applications 53—54 70 78
Mexican hat wavelet 194
Meyer wavelet 36—37 41 45 47 140
Meyer, Yves, applications of wavelets 3 54—55 71—74 203
Meyer, Yves, Best Basis 81—85
Meyer, Yves, Burt and Adelson pyramid algorithms 45
Meyer, Yves, choice of system of representation 4 75 77 89
Meyer, Yves, coefficients translating properties of functions 31—33 80 186
Meyer, Yves, continuous vs orthogonal transforms 35—36 74
Meyer, Yves, Daubechies 47
Meyer, Yves, early involvement with wavelets 24—25 34 36—39 41 47 77—78 80 137—138
Meyer, Yves, Fourier analysis 15 21 23 89
Meyer, Yves, function spaces, Weierstrass function 181 186
Meyer, Yves, history of wavelets 24—26 46—47
Meyer, Yves, interpreting coefficients 61 62 74 78
Meyer, Yves, iterations 46
Meyer, Yves, length of support 197
Meyer, Yves, Malvar wavelets and wavelet packets 77—80 85
Meyer, Yves, multiresolution analysis with Mallat 38—39 41 137—138
Meyer, Yves, orthogonality 36 74
Meyer, Yves, redundancy of continuous transforms 35 135—136
Meyer, Yves, white noise 57
Miracle of the low frequencies 138
Modane signal 55
moment 197
Momentum observable 178—179
Morlet wavelet 45 187 194
Morlet, Jean 26 28—31 34 56 74—75 187
Moser, Jurgen (and KAM theorem) 113
Mother wavelet 28 41 164
Multifractal 54 55—56
Multiresolution 31 48 see "Mallat
Multiresolution analysis, Mallat — Meyer 38—41 47 137—153 164
Multiresolution analysis, Mallat — Meyer, conditions necessary for 141—145
Multiresolution analysis, Mallat — Meyer, geometric interpretation of 140 147 149—150
Multiresolution analysis, Mallat — Meyer, history of 38—41 137—138
Multiresolution analysis, Mallat — Meyer, importance of 39—41 45 139—140
Multiwavelets 163—164
MUSIC 18 19 74 77 78—79 83—84 125 204
Natural scenes 190—192
Navier — Stokes equations 55 73
Newton, Isaac 7 8
Nichols, Charles 84
Noise 52 56 83 84 see "White
Non-orthogonal bases 126 133—135 163
Nonlinear 21—22 46 73—74
Nonstationary signals 77 85 87
Normalization 129 153 165 177 192
Nyquist, Harry 18
Observables 178—179
Octaves 19 31 34 191—192
Oil prospecting 26
Ondelettes 24
Operator 178—180 185
Orchard, Michael T. 69
Order of regularity see "Regularity"
Orthogonal wavelets 80 125 192 see "Multiwavelets" "Stroemberg"
Orthogonal wavelets, coefficients hard to read 74 80
Orthogonal wavelets, sampling theorem and 34
Orthogonal wavelets, search for 36—37 47 140
Orthogonality 36—37 45 47 56 57 77 81 87 126—136 196
Orthonormality 129—130 192 232
P see "Momentum observable"
Parisi, Giorgio 55
Parseval, theorem of 134 222 225 230
Partial differential equations 8 22 70 184
Phase 9 22—23 77 99—100
Picard, Dominique 57
Piecewise polynomial wavelets 163—164
Planck's constant (h) 177
Poincare, Henri, n-body problem 113
Poincare, Henri, pathological functions 10—11
Poincare, Henri, praise for Fourier 11
Pointwise convergence 106
Position observable 178—179
probability 171—175
Probability, equivalence with measure theory 171
Probability, governing speech 82
Probability, probability density 174 224
Probability, spaces 171 173—174
Pyramid algorithms 39 42 45 47 63 139 162
Pythagorean Theorem 129 183
q see "Position observable"
Quadrature mirror filters 39 147 162
Quantization, compression and 69
Quantization, computing errors due to 134 196—197
Quantization, different definitions in signal processing and quantum mechanics 134
Quantization, vector quantization 69
quantum mechanics 15 17 29 47 49 51 171 173 175—179 182
Quasi-stationary signals 77
Random 65—67
Random function 172
Random variable see "Random function"
Receptive fields 62 188
Reconstruction (of signal) 9 30 34 98 126 133—135
Redundance 35—36 80 81—82 135—136 191
Regularity (smoothness) 39 47 139 163 185 186 195 197—199
Representation, choice of 4 82 88—89 192 196 200 see
Resolution 31
Reynolds number 55 72
Riemann integral 216—219
Riemann sum 133
| Riemann, Bernhard 216
Rioul, Olivier 4 170
Rioul, Olivier, choice of wavelet 194—199
Rioul, Olivier, comparing compression methods 70
Rioul, Olivier, regularity 197—199
Rioul, Olivier, speed of FWT and FFT 161
Rodet, Xavier 80
Rokhlin, Vladimir 70 71
Round-off errors 19 77 134
Rousseau, Jean Jacques (and digital notation for music) 74
Sampling theorem 151
Sampling theorem, digital technology and 18-20
Sampling theorem, limit on transmission of information 64
Sampling theorem, proof 221—223
Sampling theorem, wavelets and 34 35 41
Scalar product in infinite-dimensional function spaces 128—129
Scalar product in matrix multiplication 120
Scalar product of complex-valued vectors 136
Scalar product, computing coefficients 126 129—131 133—135
Scalar product, definition of 128
Scalar product, relationship with integrals 131—133
SCALE 31 80
Scaling function 40—44 137—140
Scaling function in fast wavelet transform 42—44 154—157 159—160
Scaling function, conditions for multiresolution 141—145
Scaling function, creating transfer function from 145—147
Scaling function, creating wavelet from 152
Scaling function, creating, from transfer function 145 148—149
Scaling function, creating, using iterations 146
Scaling function, miracle of the low frequencies 138
Scaling function, multiple, in multiwavelets 163—164
Schroedinger's equation 176
Schwartz, Laurent 107 182 184 186
Schwarz inequality 129 225
Selectivity in frequency 195 198—199
Self-similar Gabor functions see "Gabor transforms"
Semmes, S. 36
Shannon, Claude 18 19 81 see
Shapiro, J.M. 69
Shiftable transforms 193
Short-time Fourier analysis see "Windowed Fourier analysis"
Siegel, Carl L. (and problem with small divisors) 109
Simoncelli, Eero (and pyramid algorithms) 162
Sine 212
skeletons 124—125
Small divisors 109—113
Smoothing function 43 see
Smoothness, smooth wavelets see "Regularity"
Solar system, stability of 107—113
Sparse distributed transform 189—190
Spatial frequency 16 96—97
Spectral theorem 180
Speech 39 77 79—80 82 84 89 125 203—204
Spline functions 45 138 164
Square summable 176 183 185 232
Standard basis 168
Standard deviation 172—173
Stationary signal 89
Steerable filters 193
Strang, Gilbert 117 152
Strichartz, Robert 41 52 181—182 185—186
Stroemberg, J.O. 37
Submarines 3 70
Symmetry, biorthogonal wavelets 69 196
Symmetry, image processing 196
Symmetry, symmetrical wavelets with compact support 163—164 196
TAPs 161
Tchamitchian, Philippe 125
Terminology, confusions in 80 81 124 134 179 189
textures 89 192 198
Theorie analytique de la chaleur see "Analytical Theory of Heat"
Thresholding see "Denoising"
Tight frames 196
Time information 22—24 97 99—100
Time-frequency decompositions 48—51 81 165—170 see
Time-frequency plane 205
Time-frequency wavelets 80
Time-scale wavelets 80
Todd, Neil 204
Torresani, Bruno 124—125
Transfer function 139 145
Translation 41
Translation invariance 35 59 88 193
Tukey, T.W. 64 117
Turbulence 72—74 197
Turbulence, coherent structures 72
Turbulence, Modane signal 55
Turbulence, Reynolds number as measure of 55
Turbulence, wavelets as tool for 55—56 59 71—73 189
Uncertainty principle see "Heisenberg uncertainty principle"
Uniform convergence 106
Universe, large-scale structure of 54
Unser, Michael 198
Vanishing moments 32 39 45 47 161 197—198
Vanishing moments, condition for regularity 198
Variance 165 173
Vector quantization 69
Vectors, addition of 127 130
Vectors, angle between 129
Vectors, function as vector 128 183
Vectors, length of 129 133
Venice 21—22
Vetterli, Martin 118 170
Ville, J. 23
VIRGO 125
Vision 25 26 38—39 62—64 162 187—193 196
von Neumann, John 185
Watson, Andrew B. 188
Wave function 51 176
wave number 97
Wavelet coefficients in vision 188
Wavelet coefficients, computing 30—31 33 34—35 40 42—44 47 see
Wavelet coefficients, computing without wavelets 152
Wavelet coefficients, interpreting 31 52 61 74 78
Wavelet coefficients, sensitivity to variations 31—33 68
Wavelet Digest 240
Wavelet maxima 59—61 125
Wavelet packets 74 77—78 81 83 85 87 202
Wavelet transform 201 see
Wavelet transform, computing without wavelets 152
Wavelets see also "Continuous wavelet transform" "Discrete "Fast "Multiresolution" "Wavelet "Daubechies "Malvar
Wavelets, biorthogonal 69 80 196—197
Wavelets, choice of 74—75 88 192 194—199 203—204
Wavelets, complex-valued 75
Wavelets, definition imprecise 36 80
Wavelets, fractal 163—164
Wavelets, frequency imprecise 31 198
Wavelets, history of 24—26 28—30 34 36—41 45 47 56—67 59
Wavelets, integral zero 32 36
Wavelets, intellectual contribution of 25 76 89
Wavelets, limitations 77
Wavelets, mother 28 41
Wavelets, multiwavelets 163—164
Wavelets, nonlinear problems and 72—74
Wavelets, not real 62 78
Wavelets, orthogonal transforms 36—37 41 61 62 80
Wavelets, piecewise polynomial 163—164
Wavelets, pure mathematics and 26 52 78 181 186
Wavelets, representation with 48—50 169
Wavelets, time-frequency 80
Wavelets, time-scale 80
Weak convergence 229
Weaver, John 53 70 78
Weaver, Warren, entropy 81
Weaver, Warren, information and meaning 67
Weaver, Warren, probabilities governing speech 82
Weaver, Warren, sampling theorem 18
Weierstrass, Karl 108—110 113 181
Wesfreid, Eva 80
Weyl, Hermann 185
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