Index. l 1 minimization, 172. o(g(x)), 89 F[f](λ), 127, 130 F [g](t), 132 H, 13 H n, 13 S, 40. Pr(x d), 160 sinc x, 79

Transcription

1 (f g)(t), 134 2π periodic functions, 93 B(p, q), 79 C (n) [a, b], 6, 10 C (n) 2 (a, b), 14 C (n) 2 [a, b], 14 D k (t), 100 L 1 convergence, 37 L 1 (I), 27, 39 L 2 convergence, 37 L 2 (I), 30, 39 L 2 [a, b], 27 L p, 123 L p(i), 39 L p[a, b], 67 N(T), 46 O(h(x)), 89 R(T), 46 R[a, b], 73 S(I), 6, 27, 30, 38 S[a, b], 28 S 1 (I), 11, 27, 38 S 2 (I), 14, 30, 38 S k (t), 100 T, 48 T tr, 48 V n, 9 #(T ), 169 C n, 4, 15 Γ(x), 79 Φ, 165 R, 20 R n, 4, 13, 15, 67 l p, 14 l 1, 63 l 1 minimization, 172 l 2, 63 l 2, infinite dimensional, 25 l, 18 η-distinguishable points, 186, for all, 76, 43, 44 φ(t), 66 ψ(t), 66, 94 e z, 93 f(t + 0), 95, 127 f(t 0), 95, 127 k-sparse, 165 l p(r), 68 o(g(x)), 89 F[f](λ), 127, 130 F [g](t), 132 H, 13 H n, 13 S, 40 ˆf(λ), 129 Ĥ, 13 Pr(x d), 160 sinc x, 79 a.e., 32, 38, 248 absolute convergence, 38, 76, 111, 114, 142 absolute value, 17 adjoint matrix, 48 adjoint operator, 48, 56, 252, 255 affine frame, 268 affine translation, 259, 267 algebraic theory of compressive sampling, 168 in this web service

2 444 aliasing error, 215 almost everywhere, 32, 248 alternating flip, 240 analysis, 1, 278, 296 analysis dilation equation, 371 analysis filter, 307 analysis multifilter, 394 analytic function, 7 analytic theory of compressive sampling, 172 angle, 10 approximation by wavelets, 354 Argmin, 167, 171 arithmetic summability, 122 arithmetic summability of Fourier series, 116 associative law, 4 attribute vector, 402 B-spline, 383 Banach space, 24 basis, 50 basis of a vector space, 9, 64 Bayes rule, 419 Bernoulli distribution, 185, 194 Bessel inequality, 42, 45, 94, 106 best approximation, 60 beta function, 78, 88 binary numbers, 166 binomial theorem, 196, 199, 326 biorthogonal, 307 biorthogonal filter bank, 311 biorthogonal filters, 367 biorthogonal multiresolution analysis, 370 biorthogonal wavelet, 285, 349, 370 bit counter, 165 bounded interval, 6 bounded linear functional, 55 bounded linear operator, 55, 57 bounded operators on Hilbert spaces, 55 box function, 130, 143, 272, 360, 364, 383, 395 Cantor set, 31 Cardan formulas, 222 cascade algorithm, 317, 329, 341, 377 Cauchy criterion, 76 Cauchy principal value integral, 88 Cauchy sequence, 20, 21, 137, 261 Cauchy Schwarz inequality, 10, 12, 16, 67, 106, 136 causal filter, 232, 234, 305 Cesàro sums, 117 characteristic function, 33, 136, 157, 248 Chebyshev inequality, 160 Chernoff inequality, 193, 197 Chi-square distribution, 193 circulant matrix, 397 closed interval, 6 closed subspace, 43 closed subspace of a normed linear space, 24 clustering, 421 coherent states, 245 coin flip distribution, 185 column space, 50 commutative law, 4 complete metric space, 21 complete subset, 45 completion of a metric space, 22, 24 completion of a vector space, 20 complex exponential, 93 complex number, 4 compression, 228 compressive sampling, 62, 63, 165 concentration of measure inequality, 186, 189, 191, 194, 198 conditional convergence, 76 conditional probability, 419 conjugate alternating flip, 240, 289, 291, 301, 306, 308 conjugate matrix, 48 continuous wavelet, 259 continuous derivatives, 6 continuous wavelet transform, 243 contour integration, 90 contrast stretching, 402, 433 convergent sequence, 21 convex second-order cone program, 174, 175, 204 convolution, 383, 399 convolution of functions, 134, 162, 234 convolution of sequences, 210, 232 correlated random variables, 427 countable set, 31 covariance matrix, 157 covering numbers, 186 covering of a set, 35 CSOCP, 174, 204 cubic spline, 387 CVX, xii, 203 in this web service

3 445 data compression, 8 data compression for matrices, 55 Daubechies 4-tap filter, 311 Daubechies filter, 314, 315, 324 Daubechies wavelet, 282, 287, 337, 354, 361 decoder, 168 delayed shift property, 209 denoising, 227, 432 dense subset, 21 dense subspace of a metric space, 23 derivative, 7 deterministic compressive sampling, 176 DFT, 215, 217, 218 diagonal matrix, 51, 52, 69 difference equations, 213 differentiation of Fourier series, 115 digital image, 402 dilation equation, 145, 273, 288, 317, 320, 324, 344, 365, 385 dimension of a vector space, 8 Dirac delta function, 154, 234, 236, 379 direct sum, 43, 44, 278, 293 Dirichlet condition, 127 Dirichlet kernel, 100 discrete cosine transform, 219, 413 discrete Fourier transform, 214, 397, 398, 412 discrete signals, 238 discrete wavelet, 267, 341 discrete wavelet transform, 267 dispersion, 154 distribution function, 191 distributive law, 4 divergent integral, 74 dot product, 10 double-shift orthogonality, 289, 291, 300, 306, 318, 321, 341, 373, 378 downsampling, 238, 294, 375, 430 dual frame, 253, 270 dyadic point, 320, 338 edge enhancement filter, 406 eigenvalue, 51, 321, 347, 359 eigenvector, 51, 321, 347 encoders, 166 equivalence class, 22 equivalence relation, 22 Euclidean geometry, 10 Euclidean inner product, 67 Euclidean space, 4, 187 Euler product formula, 91 Euler reflection formula, 90 Euler Mascheroni constant, 90, 91 even function, 98 expectation, 154, 157, 184 fast Fourier transform (FFT), 222, 282 fast wavelet transform (FWT), 282, 294 father wavelet, 272, 284 Fejér theorem, 116, 119 Fejér Riesz Theorem, 310 FFT, 296 field, 4 filter, 430 filter bank, 279, 285, 296 filter coefficients, 300 finite-dimensional vector space, 8 finite impulse response (FIR) filter, 232 finite length signal, 395 FIR, 232, 235, 285, 296, 311, 312, 397 forward shift property, 209 Fourier, 92 Fourier coefficient, 45 Fourier convergence theorem, 142 Fourier cosine series, 98 Fourier cosine transform, 146 Fourier series, 129 Fourier series, complex, 92, 94 Fourier series, real, 92, 94, 96 Fourier sine series, 99 Fourier sine transform, 146 Fourier transform, 127, 132, 243, 262, 302, 321, 344, 412, 433 Fourier transform for L 2 functions, 135 frame, 251, 255 frame bounds, 252 frequency bandlimited function, 147, 164 frequency domain, 235 fundamental theorem of algebra, 64, 309 fundamental theorem of linear algebra, 49 Gabor window, 245 gamma function, 78, 88, 193, 316 Gaussian distribution, 132, 154, 158, 265 Gaussian elimination, 62, 321 generalized function, 332, 379 geometric series, 100, 118 Gibbs phenomenon, 107, 108, 117, 121 in this web service

4 446 Gram matrix, 69 Gram Schmidt process, 40, 41, 65, 66, 70, 192 Hölder continuous, 363 Hölder inequality, 16 Hölder Minkowski inequality, 57, 67 Haar scaling function, 270, 272, 330, 341, 377 Haar wavelet, 261, 267, 272, 280, 319 Haar wavelet expansion, 66 Haar s hat, 394 half open interval, 6 half-point symmetry, 396 halfband filter, 308, 309, 312, 316, 367, 372 Hankel matrix, 71 harmonic series, 80 hat function, 338, 383 Heaviside function, 162 Heine Borel theorem, 35 Heisenberg inequality, 155 Hermite polynomials, 65 high pass filter, 237, 279, 299 Hilbert matrix, 70 Hilbert space, 11, 25 Hilbert transform, 123 histogram normalization, 403, 433 homogeneous equation, 167 homogeneous linear equation, 6 identity operator, 50 iff, if and only if, 76 iid, 158 IIR, 232, 285 image, 402, 406 image compression, 429 image enhancement, 405 image representation of data, 426 imaging, 8 improper integrals, 73 improper integrals of the second kind, 77 improper Riemann integral of the first kind, 74 impulse response function, 306 impulse response vector, 285, 341, 369, 387, 397 independently distributed random variables, 158 infinite-dimensional vector space, 8 infinite impulse response filter (IIR), 232 infinite product, 83, 329, 359 injective (1-1) operator, 47 inner product, 10 inner product space, 11 input, 1 integral test for series, 75 integral wavelet transform, 262 integration of Fourier series, 113 interval, 6 inverse Fourier transform, 127, 262 inverse matrix, 50 inverse operator, 47, 139 inverse Z transform, 211 invertible matrix, 60 invertible operator, 47, 242 isometric metric spaces, 22, 23 Jacobi Theta function, 249 Jordan canonical form, 345 jump discontinuity, 95 K-means algorithm, 422 kernel, 46 L Hospital rule, 103 l.i.m., limit in the mean, 137 Lagrange interpolation, 71 Laplace operator, 407 Laplace transform, 163 lattice, 267 lattice Hilbert space, 246 law of cosines, 41 law of large numbers, 159, 161 least squares approximation, 46, 58, 60, 66 Lebesgue dominated convergence theorem, 363 Lebesgue integrable function, 27 Lebesgue integral, 27, 28 left-hand derivative, 103 Legendre polynomials, 65 length, 10 limit of a sequence, 21 linear algebra, 8 linear combination of vectors, 5 linear filter, continuous, 233 linear filter, discrete, 230 linear operator, 46, 57 linear programming, 60, 174 in this web service

5 447 linear spaces, 4 linear system, 1 linearly dependent set, 8, 64 linearly independent set, 8, 64 local filters, 405 localization theorem for Fourier series, 102, 104 low pass filter, 237, 299 lower Darboux sum, 28, 191 M channel filter bank, 390 Mallat algorithm, 272, 280, 294 Mallat herring bone, 280 Maple, xii Markov inequality, 160, 193, 195 MATLAB, xii MATLAB wavelet toolbox, 314, 336 matrix, 5, 47 matrix addition, 5 matrix sum, 48 maxflat filter, 312, 316, 354 maxmin algorithm, 421 mean of a distribution, 154 mean value theorem, 200 measure zero, 31 metric, 21 metric space, 21 metrics in imaging, 421 Mexican hat wavelet, 265 modulus of continuity, 363 monotone, 77 monotone convergence theorem, 195 Morlet wavelet, 266 morphology, 408, 433 mother wavelet, 267, 275 moving average filter, 237, 273, 279, 306, 311, 314, 331 moving difference filter, 237, 279 multifilter, 285, 394 multiresolution analysis, 267, 272, 333 multiresolution structure, 284 multiwavelet, 285, 394 N-spline, 386 neural network, 422 Newton s method, 318 nonsingular matrix, 50, 69 norm of an operator, 55 norm on a vector space, 10 normal distribution, 154, 184, 191 normal distribution, multivariate, 158 normed linear space, 10, 24 null set, 31 null space, 46, 168 null space property, 173, 174, 177 nullity, 46 Nyquist rate, 147, 164 odd function, 99 ON, 40 ON basis, 41, 216 ON set, 45 open interval, 6 ordinary differential equation, 64 orthogonal matrix, 51, 192, 299 orthogonal operator, 51 orthogonal polynomials, 40, 70, 123 orthogonal projection, 39, 43, 124, 273, 274 orthogonal vectors, 40 orthogonality, 39 orthonormal basis, 40, 285 orthonormal sequence, 45 orthonormal set, 40 orthonormal system, 43 output, 1 overcomplete, 242 overdetermined, 2, 242 Paley Weiner Theorem, 332 parallelogram law, 44, 68 Parseval (Plancherel) equality, 130, 138, 217, 258, 260, 263, 344, 360 Parseval equality, 41, 46, 95, 106, 111, 113, 125, 129, 148, 235, 247, 256, 379 parsimonious representation of data, 401 partial sum, 100 Pascal s triangle, 225 pass band, 237 pattern recognition, 418 perceptron, 422 perfect reconstruction, 1, 368, 392 perfect reconstruction filter bank, 306, 307, 430 periodization, 152 periodizing operator, 246 phase space, 157 piecewise continuous function, 95, 99, 142 pigeonhole principal, 187 pixel, 402 in this web service

6 448 pointwise convergence, 37 pointwise convergence of Fourier series, 99, 102, 107 pointwise convergence of the Fourier integral formula, 140 Poisson summation formula, 153 polynomials, 5 positive definite matrix, 69 positive operator, 253 pre Banach space, 10 prime integer, 20 principal component analysis, 427, 435 probabilistic theory, 183 probability distribution, 154, 157 processing, 1 product filter, 308 product of linear operators, 47 product of matrices, 48 projection of a vector on a subspace, 42, 273 projection theorem, 43, 59 Pythagorean theorem, 41 quasi-normed space, 66 radar, 271 random matrix theory, 168 random variable, 154, 184 range, 46 rank, 46, 50 rational number, 20, 31 real, 5 real line, 6 real matrix, 5 real number, 4 real orthogonal matrix, 52 reconstruct, 1 rectangular system, 60 reflexive, 22 restricted isometry property (RIP), 178, 184, 188 Riemann integral, 14, 28 Riemann sum, 129 Riemann Lebesgue lemma, 94, 101, 103, 140 Riesz basis, 252, 254, 270, 272, 285, 371, 383 Riesz representation theorem, 56 right-hand derivative, 103 RIP, 178 row space, 50 sample, 1 sample covariance matrix, 427 sample matrix, 165 sample mean, 159 sample variance, 159, 427 sampling, 146 sampling matrices, 166 sampling theorem, 147 scalar, 4 scalar multiple of an operator, 47 scalar multiplication, 5 scaling function, 272, 284, 320, 322, 332, 341, 371 Schwartz class, 152, 379 self organizing memory-som, 425 self-adjoint matrix, 51, 255 self-adjoint operator, 51, 252 separable Hilbert space, 27 Shannon resolution analysis, 287 Shannon Whittaker Kotelnikov sampling, 147, 254 shape detection, 410 shift operator, 305 signal processing, 8, 10 simplex method, 175 sinc function, 79, 141, 144, 287, 332, 365 singular value, 52 singular value decomposition, 51, 55, 270, 427 smoothness of wavelets, 359 Sobolev H 1 norm, 68 space, 5 span, 7, 41, 49 spanning set, 8 sparse signal, 164 spectral factorization, 309 spectral theorem, 51, 53, 254 spherical coordinates, 192 spherical distribution, 185 spline, 382 standard basis, 9 standard deviation, 154, 157 step function, 6, 27 Stirling formula, 190, 316 stop band, 237 subspace, 5 substitution rule for improper integrals, 87 sum of linear operators, 47 symmetric, 22 symmetric matrix, 51 in this web service

7 449 symmetric operator, 51 synthesis, 1, 279, 294, 296 synthesis dilation equation, 371 synthesis filter, 307 synthesis multifilter, 394 taps, 232 Taylor theorem, 355 three-term recurrence relation, 65 thresholding, 432 tight frame, 253 time domain, 238 time reversed filter, 279 Toeplitz matrix, 231, 397 transform sample matrix filter, 1 transitive, 22 transpose matrix, 48 triangle inequality, 10, 21 trigonometric function, 8 trigonometric polynomial, 100, 309 two-channel filter bank, 308 uncertainty relation of quantum mechanics, 154 underdetermined, 2, 242 underdetermined system, 62 uniform convergence, 117, 318, 330 uniform convergence of Fourier series, 111 uniform distribution, 184, 198 uniform pointwise convergence, 104, 105 union bound, 188, 198 unit ball, 62, 187 unit vector, 40 unitary filter bank, 285, 293, 296, 309 unitary matrix, 50, 51, 291, 299 unitary operator, 50, 137, 139 upper Darboux sum, 28, 191 upsampling, 239, 430 Vandermonde matrix, 71, 171, 206 variance, 154, 402 vector space, 4 wavelet equation, 275, 288, 373 wavelet packet, 295 wavelet transform, 259 Weierstrass approximation theorem, 120 Weierstrass product formula, 91 Weil Brezin Zak transform, 246 Weyl Heisenberg (W H) frame, 256 whole-point symmetry, 396 windowed Fourier transform, 243 wraparound, 396 Wronskian, 64 Z transform, 208, 235, 297, 307, 347, 359, 367, 392, 429 Zak transform, 247 zero operator, 50 zero vector, 5 zero-padding, 396 in this web service