Scaling, fractals and wavelets
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| 245 | 1 | 0 | |a Scaling, fractals and wavelets |c ed. by Patrice Abry ... |
| 264 | 1 | |a London |b ISTE [u.a.] |c 2009 | |
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| 500 | |a Literaturangaben | ||
| 650 | 4 | |a Mathematik | |
| 650 | 4 | |a Fractals | |
| 650 | 4 | |a Signal processing |x Mathematics | |
| 650 | 4 | |a Wavelets (Mathematics) | |
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Datensatz im Suchindex
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| DE-BY-TUM_katkey | 1715230 |
| DE-BY-TUM_media_number | 040010223359 |
| _version_ | 1816713024805797888 |
| adam_text | Titel: Scaling, fractals and wavelets
Autor: Abry, Patrice
Jahr: 2009
Table of Contents
Preface......................................... 17
Chapter 1. Fractal and Multifractal Analysis in Signal Processing..... 19
Jacques Levy Vehel and Claude Tricot
1.1. Introduction.................................. 19
1.2. Dimensions of sets.............................. 20
1.2.1. Minkowski-Bouligand dimension .................. 21
1.2.2. Packing dimension........................... 25
1.2.3. Covering dimension.......................... 27
1.2.4. Methods for calculating dimensions................. 29
1.3. Holder exponents............................... 33
1.3.1. Holder exponents related to a measure................ 33
1.3.2. Theorems on set dimensions..................... 33
1.3.3. Holder exponent related to a function................ 36
1.3.4. Signal dimension theorem....................... 42
1.3.5. 2-microlocal analysis ......................... 45
1.3.6. An example: analysis of stock market price............. 46
1.4. Multifractal analysis............................. 48
1.4.1. What is the purpose of multifractal analysis?............ 48
1.4.2. First ingredient: local regularity measures.............. 49
1.4.3. Second ingredient: the size of point sets of the same regularity . . 50
1.4.4. Practical calculation of spectra.................... 52
1.4.5. Refinements: analysis of the sequence of capacities, mutual
analysis and multisingularity..................... 60
1.4.6. The multifractal spectra of certain simple signals.......... 62
1.4.7. Two applications............................ 66
1.4.7.1. Image segmentation....................... 66
1.4.7.2. Analysis of TCP traffic..................... 67
1.5. Bibliography................................. 68
6 Scaling, Fractals and Wavelets
Chapter 2. Scale Invariance and Wavelets.................... 71
Patrick Flandrin, Paulo Gonqalves and Patrice Abry
2.1. Introduction.................................. 71
2.2. Models for scale invariance......................... 72
2.2.1. Intuition................................. 72
2.2.2. Self-similarity ............................. 73
2.2.3. Long-range dependence........................ 75
2.2.4. Local regularity............................. 76
2.2.5. Fractional Brownian motion: paradigm of scale invariance .... 77
2.2.6. Beyond the paradigm of scale invariance.............. 79
2.3. Wavelet transform.............................. 81
2.3.1. Continuous wavelet transform .................... 81
2.3.2. Discrete wavelet transform...................... 82
2.4. Wavelet analysis of scale invariant processes............... 85
2.4.1. Self-similarity ............................. 86
2.4.2. Long-range dependence........................ 88
2.4.3. Local regularity............................. 90
2.4.4. Beyond second order.......................... 92
2.5. Implementation: analysis, detection and estimation............ 92
2.5.1. Estimation of the parameters of scale invariance.......... 93
2.5.2. Emphasis on scaling laws and determination of the scaling range. 96
2.5.3. Robustness of the wavelet approach................. 98
2.6. Conclusion .................................. 100
2.7. Bibliography ................................. 101
Chapter 3. Wavelet Methods for Multifractal Analysis of Functions .... 103
Stephane Jaffard
3.1. Introduction.................................. 103
3.2. General points regarding multifractal functions.............. 104
3.2.1. Important definitions.......................... 104
3.2.2. Wavelets and pointwise regularity.................. 107
3.2.3. Local oscillations............................ 112
3.2.4. Complements.............................. 116
3.3. Random multifractal processes....................... 117
3.3.1. Levy processes............................. 117
3.3.2. Burgers equation and Brownian motion............... 120
3.3.3. Random wavelet series......................... 122
3.4. Multifractal formalisms........................... 123
3.4.1. Besov spaces and lacunarity...................... 123
3.4.2. Construction of formalisms...................... 126
3.5. Bounds of the spectrum........................... 129
3.5.1. Bounds according to the Besov domain............... 129
Contents 7
3.5.2. Bounds deduced from histograms .................. 132
3.6. The grand-canonical multifractal formalism................ 132
3.7. Bibliography ................................. 134
Chapter 4. Multifractal Scaling: General Theory and Approach by
Wavelets........................................ 139
Rudolf RlEDi
4.1. Introduction and summary.......................... 139
4.2. Singularity exponents............................ 140
4.2.1. Holder continuity............................ 140
4.2.2. Scaling of wavelet coefficients.................... 142
4.2.3. Other scaling exponents........................ 144
4.3. Multifractal analysis............................. 145
4.3.1. Dimension based spectra ....................... 145
4.3.2. Grain based spectra .......................... 146
4.3.3. Partition function and Legendre spectrum.............. 147
4.3.4. Deterministic envelopes........................ 149
4.4. Multifractal formalism............................ 151
4.5. Binomial multifractals............................ 154
4.5.1. Construction .............................. 154
4.5.2. Wavelet decomposition........................ 157
4.5.3. Multifractal analysis of the binomial measure............ 158
4.5.4. Examples................................ 160
4.5.5. Beyond dyadic structure........................ 162
4.6. Wavelet based analysis............................ 163
4.6.1. The binomial revisited with wavelets................. 163
4.6.2. Multifractal properties of the derivative............... 165
4.7. Self-similarity and LRD........................... 167
4.8. Multifractal processes............................ 168
4.8.1. Construction and simulation ..................... 169
4.8.2. Global analysis............................. 170
4.8.3. Local analysis of warped FBM.................... 170
4.8.4. LRD and estimation of warped FBM................. 173
4.9. Bibliography................................. 173
Chapter 5. Self-similar Processes......................... 179
Albert BENASSI and Jacques Istas
5.1. Introduction.................................. 179
5.1.1. Motivations............................... 179
5.1.2. Scalings................................. 182
5.1.2.1. Trees................................ 182
5.1.2.2. Coding of R........................... 183
5.1.2.3. Renormalizing Cantor set.................... 183
8 Scaling, Fractals and Wavelets
5.1.2.4. Random renormalized Cantor set................ 184
5.1.3. Distributions of scale invariant masses................ 184
5.1.3.1. Distribution of masses associated with Poisson measures . . 184
5.1.3.2. Complete coding......................... 185
5.1.4. Weierstrass functions ......................... 185
5.1.5. Renormalization of sums of random variables ........... 186
5.1.6. A common structure for a stochastic (semi-)self-similar process . 187
5.1.7. Identifying Weierstrass functions................... 188
5.1.7.1. Pseudo-correlation........................ 188
5.2. The Gaussian case.............................. 189
5.2.1. Self-similar Gaussian processes with r-stationary increments . . . 189
5.2.1.1. Notations............................. 189
5.2.1.2. Definitions............................ 189
5.2.1.3. Characterization......................... 190
5.2.2. Elliptic processes............................ 190
5.2.3. Hyperbolic processes ......................... 191
5.2.4. Parabolic processes........................... 192
5.2.5. Wavelet decomposition........................ 192
5.2.5.1. Gaussian elliptic processes................... 192
5.2.5.2. Gaussian hyperbolic process.................. 193
5.2.6. Renormalization of sums of correlated random variable...... 193
5.2.7. Convergence towards fractional Brownian motion......... 193
5.2.7.1. Quadratic variations....................... 193
5.2.7.2. Acceleration of convergence.................. 194
5.2.7.3. Self-similarity and regularity of trajectories.......... 195
5.3. Non-Gaussian case.............................. 195
5.3.1. Introduction............................... 195
5.3.2. Symmetric a-stable processes .................... 196
5.3.2.1. Stochastic measure........................ 196
5.3.2.2. Ellipticity............................. 196
5.3.3. Censov and Takenaka processes................... 198
5.3.4. Wavelet decomposition........................ 198
5.3.5. Process subordinated to Brownian measure............. 199
5.4. Regularity and long-range dependence................... 200
5.4.1. Introduction............................... 200
5.4.2. Two examples.............................. 201
5.4.2.1. A signal plus noise model.................... 201
5.4.2.2. Filtered white noise....................... 201
5.4.2.3. Long-range correlation..................... 202
5.5. Bibliography................................. 202
Contents 9
Chapter 6. Locally Self-similar Fields ...................... 205
Serge COHEN
6.1. Introduction.................................. 205
6.2. Recap of two representations of fractional Brownian motion...... 207
6.2.1. Reproducing kernel Hilbert space .................. 207
6.2.2. Harmonizable representation..................... 208
6.3. Two examples of locally self-similar fields ................ 213
6.3.1. Definition of the local asymptotic self-similarity (LASS)..... 213
6.3.2. Filtered white noise (FWN)...................... 214
6.3.3. Elliptic Gaussian random fields (EGRP)............... 215
6.4. Multifractional fields and trajectorial regularity.............. 218
6.4.1. Two representations of the MBM................... 219
6.4.2. Study of the regularity of the trajectories of the MBM....... 221
6.4.3. Towards more irregularities: generalized multifractional
Brownian motion (GMBM) and step fractional Brownian
motion (SFBM)............................. 222
6.4.3.1. Step fractional Brownian motion................ 223
6.4.3.2. Generalized multifractional Brownian motion ........ 224
6.5. Estimate of regularity............................ 226
6.5.1. General method: generalized quadratic variation.......... 226
6.5.2. Application to the examples...................... 228
6.5.2.1. Identification of filtered white noise.............. 228
6.5.2.2. Identification of elliptic Gaussian random processes..... 230
6.5.2.3. Identification of MBM...................... 231
6.5.2.4. Identification of SFBMs..................... 233
6.6. Bibliography................................. 235
Chapter 7. An Introduction to Fractional Calculus .............. 237
Denis Matignon
7.1. Introduction.................................. 237
7.1.1. Motivations............................... 237
7.1.1.1. Fields of application....................... 237
7.1.1.2. Theories.............................. 238
7.1.2. Problems ................................ 238
7.1.3. Outline ................................. 239
7.2. Definitions................................... 240
7.2.1. Fractional integration ......................... 240
7.2.2. Fractional derivatives within the framework of causal distributions 242
7.2.2.1. Motivation............................ 242
7.2.2.2. Fundamental solutions...................... 245
7.2.3. Mild fractional derivatives, in the Caputo sense........... 246
7.2.3.1. Motivation............................ 246
10 Scaling, Fractals and Wavelets
7.2.3.2. Definition............................. 247
7.2.3.3. Mittag-Leffler eigenfunctions.................. 248
7.2.3.4. Fractional power series expansions of order a (a-FPSE) .. 250
7.3. Fractional differential equations ...................... 251
7.3.1. Example................................. 251
7.3.1.1. Framework of causal distributions............... 251
7.3.1.2. Framework of fractional power series expansion of order
one half.............................. 252
7.3.1.3. Notes............................... 253
7.3.2. Framework of causal distributions.................. 254
7.3.3. Framework of functions expandable into fractional power series
(a-FPSE)................................ 255
7.3.4. Asymptotic behavior of fundamental solutions........... 257
7.3.4.1. Asymptotic behavior at the origin............... 257
7.3.4.2. Asymptotic behavior at infinity................. 257
7.3.5. Controlled-and-observed linear dynamic systems of fractional order 261
7.4. Diffusive structure of fractional differential systems........... 262
7.4.1. Introduction to diffusive representations of pseudo-differential
operators................................. 263
7.4.2. General decomposition result..................... 264
7.4.3. Connection with the concept of long memory............ 265
7.4.4. Particular case of fractional differential systems of commensurate
orders.................................. 265
7.5. Example of a fractional partial differential equation........... 266
7.5.1. Physical problem considered..................... 267
7.5.2. Spectral consequences......................... 268
7.5.3. Time-domain consequences...................... 268
7.5.3.1. Decomposition into wavetrains................. 269
7.5.3.2. Quasi-modal decomposition .................. 270
7.5.3.3. Fractional modal decomposition................ 271
7.5.4. Free problem.............................. 272
7.6. Conclusion .................................. 273
7.7. Bibliography................................. 273
Chapter 8. Fractional Synthesis, Fractional Filters .............. 279
Liliane Bel, Georges Oppenheim, Luc Robbiano and Marie-Claude Viano
8.1. Traditional and less traditional questions about fractionals........ 279
8.1.1. Notes on terminology......................... 279
8.1.2. Short and long memory........................ 279
8.1.3. From integer to non-integer powers: filter based sample path design 280
8.1.4. Local and global properties...................... 281
8.2. Fractional filters ............................... 282
8.2.1. Desired general properties: association............... 282
Contents 11
8.2.2. Construction and approximation techniques............. 282
8.3. Discrete time fractional processes ..................... 284
8.3.1. Filters: impulse responses and corresponding processes...... 284
8.3.2. Mixing and memory properties.................... 286
8.3.3. Parameter estimation.......................... 287
8.3.4. Simulated example........................... 289
8.4. Continuous time fractional processes.................... 291
8.4.1. A non-self-similar family: fractional processes designed from
fractional filters............................. 291
8.4.2. Sample path properties: local and global regularity, memory . . . 293
8.5. Distribution processes............................ 294
8.5.1. Motivation and generalization of distribution processes...... 294
8.5.2. The family of linear distribution processes ............. 294
8.5.3. Fractional distribution processes................... 295
8.5.4. Mixing and memory properties.................... 296
8.6. Bibliography ................................. 297
Chapter 9. Iterated Function Systems and Some Generalizations:
Local Regularity Analysis and Multifractal Modeling of Signals ...... 301
Khalid Daoudi
9.1. Introduction.................................. 301
9.2. Definition of the Holder exponent..................... 303
9.3. Iterated function systems (IFS)....................... 304
9.4. Generalization of iterated function systems................ 306
9.4.1. Semi-generalized iterated function systems............. 307
9.4.2. Generalized iterated function systems................ 308
9.5. Estimation of poinrwise Holder exponent by GIFS............ 311
9.5.1. Principles of the method........................ 312
9.5.2. Algorithm................................ 314
9.5.3. Application............................... 315
9.6. Weak self-similar functions and multifractal formalism......... 318
9.7. Signal representation by WSA functions.................. 320
9.8. Segmentation of signals by weak self-similar functions......... 324
9.9. Estimation of the multifractal spectrum.................. 326
9.10. Experiments................................. 327
9.11. Bibliography................................. 329
Chapter 10. Iterated Function Systems and Applications in Image
Processing....................................... 333
Franck Davoine and Jean-Marc Chassery
10.1. Introduction................................. 333
10.2. Iterated transformation systems...................... 333
10.2.1. Contracting transformations and iterated transformation systems 334
10.2.1.1. Lipschitzian transformation.................. 334
12 Scaling, Fractals and Wavelets
10.2.1.2. Contracting transformation.................. 334
10.2.1.3. Fixed point ........................... 334
10.2.1.4. Hausdorff distance....................... 334
10.2.1.5. Contracting transformation on the space H(R?)...... 335
10.2.1.6. Iterated transformation system ................ 335
10.2.2. Attractor of an iterated transformation system........... 335
10.2.3. Collage theorem.........*.................. 336
10.2.4. Finally contracting transformation................. 338
10.2.5. Attractor and invariant measures.................. 339
10.2.6. Inverse problem............................ 340
10.3. Application to natural image processing: image coding......... 340
10.3.1. Introduction.............................. 340
10.3.2. Coding of natural images by fractals................ 342
10.3.2.1. Collage of a source block onto a destination block..... 342
10.3.2.2. Hierarchical partitioning.................... 344
10.3.2.3. Coding of the collage operation on a destination block . . . 345
10.3.2.4. Contraction control of the fractal transformation...... 345
10.3.3. Algebraic formulation of the fractal transformation........ 345
10.3.3.1. Formulation of the mass transformation........... 347
10.3.3.2. Contraction control of the fractal transformation...... 349
10.3.3.3. Fisher formulation....................... 350
10.3.4. Experimentation on triangular partitions.............. 351
10.3.5. Coding and decoding acceleration ................. 352
10.3.5.1. Coding simplification suppressing the research for
similarities ............................ 352
10.3.5.2. Decoding simplification by collage space orthogonalization 358
10.3.5.3. Coding acceleration: search for the nearest neighbor .... 360
10.3.6. Other optimization diagrams: hybrid methods........... 360
10.4. Bibliography................................. 362
Chapter 11. Local Regularity and Multifractal Methods for Image and
Signal Analysis.................................... 367
Pierrick Legrand
11.1. Introduction................................. 367
11.2. Basic tools ................................. 368
11.2.1. Holder regularity analysis...................... 368
11.2.2. Reminders on multifractal analysis................. 369
11.2.2.1. Hausdorff multifractal spectrum ............... 369
11.2.2.2. Large deviation multifractal spectrum............ 370
11.2.2.3. Legendre multifractal spectrum................ 371
11.3. Holderian regularity estimation...................... 371
11.3.1. Oscillations (OSC).......................... 371
11.3.2. Wavelet coefficient regression (WCR)............... 372
Contents 13
11.3.3. Wavelet leaders regression (WL).................. 372
11.3.4. Limit inf and limit sup regressions................. 373
11.3.5. Numerical experiments........................ 374
11.4. Denoising.................................. 376
11.4.1. Introduction.............................. 376
11.4.2. Minimax risk, optimal convergence rate and adaptivity...... 377
11.4.3. Wavelet based denoising....................... 378
11.4.4. Non-linear wavelet coefficients pumping.............. 380
11.4.4.1. Minimax properties....................... 380
11.4.4.2. Regularity control ....................... 381
11.4.4.3. Numerical experiments .................... 382
11.4.5. Denoising using exponent between scales............. 383
11.4.5.1. Introduction........................... 383
11.4.5.2. Estimating the local regularity of a signal from noisy
observations............................ 384
11.4.5.3. Numerical experiments .................... 386
11.4.6. Bayesian multifractal denoising................... 386
11.4.6.1. Introduction........................... 386
11.4.6.2. The set of parameterized classes S(g,ip) .......... 387
11.4.6.3. Bayesian denoising in S(g,tp) ................ 388
11.4.6.4. Numerical experiments .................... 390
11.4.6.5. Denoising of road profiles................... 391
11.5. Holderian regularity based interpolation................. 393
11.5.1. Introduction.............................. 393
11.5.2. The method.............................. 393
11.5.3. Regularity and asymptotic properties................ 394
11.5.4. Numerical experiments........................ 394
11.6. Biomedical signal analysis......................... 394
11.7. Texture segmentation............................ 401
11.8. Edge detection................................ 403
11.8.1. Introduction.............................. 403
11.8.1.1. Edge detection ......................... 406
11.9. Change detection in image sequences using multifractal analysis . . . 407
11.10. Image reconstruction........................... 408
11.11. Bibliography................................ 409
Chapter 12. Scale Invariance in Computer Network Traffic......... 413
Darryl Veitch
12.1. Teletraffic- anew natural phenomenon ................. 413
12.1.1. A phenomenon of scales....................... 413
12.1.2. An experimental science of man-made atoms .......... 415
12.1.3. A random current........................... 416
12.1.4. Two fundamental approaches.................... 417
14 Scaling, Fractals and Wavelets
12.2. From a wealth of scales arise scaling laws................ 419
12.2.1. First discoveries............................ 419
12.2.2. Laws reign............................... 420
12.2.3. Beyond the revolution........................ 424
12.3. Sources as the source of the laws..................... 426
12.3.1. The sum or its parts.......................... 426
12.3.2. The on/off paradigm......................... 427
12.3.3. Chemistry............................... 428
12.3.4. Mechanisms.............................. 429
12.4. New models, new behaviors........................ 430
12.4.1. Character of a model......................... 430
12.4.2. The fractional Brownian motion family .............. 431
12.4.3. Greedy sources............................ 432
12.4.4. Never-ending calls.......................... 432
12.5. Perspectives................................. 433
12.6. Bibliography................................. 434
Chapter 13. Research of Scaling Law on Stock Market Variations..... 437
Christian WALTER
13.1. Introduction: fractals in finance...................... 437
13.2. Presence of scales in the study of stock market variations....... 439
13.2.1. Modeling of stock market variations................ 439
13.2.1.1. Statistical apprehension of stock market fluctuations .... 439
13.2.1.2. Profit and stock market return operations in different scales 442
13.2.1.3. Traditional financial modeling: Brownian motion...... 443
13.2.2. Time scales in financial modeling.................. 445
13.2.2.1. The existence of characteristic time.............. 445
13.2.2.2. Implicit scaling invariances of traditional financial modeling 446
13.3. Modeling postulating independence on stock market returns...... 446
13.3.1. 1960-1970: from Pareto s law to Levy s distributions....... 446
13.3.1.1. Leptokurtic problem and Mandelbrot s first model..... 446
13.3.1.2. First emphasis of Levy s a-stable distributions in finance . 448
13.3.2. 1970-1990: experimental difficulties of iid-a-stable model ... 448
13.3.2.1. Statistical problem of parameter estimation of stable laws . 448
13.3.2.2. Non-normality and controversies on scaling invariance . . 449
13.3.2.3. Scaling anomalies of parameters under iid hypothesis ... 451
13.3.3. Unstable iid models in partial scaling invariance ......... 452
13.3.3.1. Partial scaling invariances by regime switching models . . 452
13.3.3.2. Partial scaling invariances as compared with extremes . . . 453
13.4. Research of dependency and memory of markets............ 454
13.4.1. Linear dependence: testing of if-correlative models on returns . 454
13.4.1.1. Question of dependency of stock market returns...... 454
13.4.1.2. Problem of slow cycles and Mandelbrot s second model . . 455
Contents 15
13.4.1.3. Introduction of fractional differentiation in econometrics . 455
13.4.1.4. Experimental difficulties of //-correlative model on returns 456
13.4.2. Non-linear dependence: validating //-correlative model on
volatilities................................ 456
13.4.2.1. The 1980s: ARCH modeling and its limits ......... 456
13.4.2.2. The 1990s: emphasis of long dependence on volatility . . . 457
13.5. Towards a rediscovery of scaling laws in finance............ 457
13.6. Bibliography................................. 458
Chapter 14. Scale Relativity, Non-differentiability and Fractal Space-time 465
Laurent NOTTALE
14.1. Introduction................................. 465
14.2. Abandonment of the hypothesis of space-time differentiability .... 466
14.3. Towards a fractal space-time........................ 466
14.3.1. Explicit dependence of coordinates on spatio-temporal resolutions 467
14.3.2. From continuity and non-differentiability to fractality...... 467
14.3.3. Description of non-differentiable process by differential equations 469
14.3.4. Differential dilation operator .................... 471
14.4. Relativity and scale covariance ...................... 472
14.5. Scale differential equations ........................ 472
14.5.1. Constant fractal dimension: Galilean scale relativity...... 473
14.5.2. Breaking scale invariance: transition scales............ 474
14.5.3. Non-linear scale laws: second order equations, discrete scale
invariance, log-periodic laws..................... 475
14.5.4. Variable fractal dimension: Euler-Lagrange scale equations . . . 476
14.5.5. Scale dynamics and scale force................... 478
14.5.5.1. Constant scale force...................... 479
14.5.5.2. Scale harmonic oscillator................... 480
14.5.6. Special scale relativity - log-Lorentzian dilation laws, invariant
scale limit under dilations....................... 481
14.5.7. Generalized scale relativity and scale-motion coupling...... 482
14.5.7.1. A reminder about gauge invariance.............. 483
14.5.7.2. Nature of gauge fields..................... 484
14.5.7.3. Nature of the charges...................... 486
14.5.7.4. Mass-charge relations..................... 488
14.6. Quantum-like induced dynamics ..................... 488
14.6.1. Generalized Schrodinger equation ................. 488
14.6.2. Application in gravitational structure formation.......... 492
14.7. Conclusion.................................. 493
14.8. Bibliography................................. 495
ListofAuthors.................................... 499
Index.......................................... 503
|
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| genre_facet | Aufsatzsammlung |
| id | DE-604.BV035448391 |
| illustrated | Illustrated |
| indexdate | 2024-11-25T17:26:05Z |
| institution | BVB |
| isbn | 9781848210721 |
| language | English French |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-017368502 |
| oclc_num | 145379795 |
| open_access_boolean | |
| owner | DE-29T DE-20 DE-91G DE-BY-TUM |
| owner_facet | DE-29T DE-20 DE-91G DE-BY-TUM |
| physical | 504 S. Ill., graph. Darst. |
| publishDate | 2009 |
| publishDateSearch | 2009 |
| publishDateSort | 2009 |
| publisher | ISTE [u.a.] |
| record_format | marc |
| spellingShingle | Scaling, fractals and wavelets Mathematik Fractals Signal processing Mathematics Wavelets (Mathematics) Signalverarbeitung (DE-588)4054947-1 gnd Fraktal (DE-588)4123220-3 gnd Mathematische Methode (DE-588)4155620-3 gnd Wavelet (DE-588)4215427-3 gnd |
| subject_GND | (DE-588)4054947-1 (DE-588)4123220-3 (DE-588)4155620-3 (DE-588)4215427-3 (DE-588)4143413-4 |
| title | Scaling, fractals and wavelets |
| title_alt | Lois d'échelle, fractales et ondelettes |
| title_auth | Scaling, fractals and wavelets |
| title_exact_search | Scaling, fractals and wavelets |
| title_full | Scaling, fractals and wavelets ed. by Patrice Abry ... |
| title_fullStr | Scaling, fractals and wavelets ed. by Patrice Abry ... |
| title_full_unstemmed | Scaling, fractals and wavelets ed. by Patrice Abry ... |
| title_short | Scaling, fractals and wavelets |
| title_sort | scaling fractals and wavelets |
| topic | Mathematik Fractals Signal processing Mathematics Wavelets (Mathematics) Signalverarbeitung (DE-588)4054947-1 gnd Fraktal (DE-588)4123220-3 gnd Mathematische Methode (DE-588)4155620-3 gnd Wavelet (DE-588)4215427-3 gnd |
| topic_facet | Mathematik Fractals Signal processing Mathematics Wavelets (Mathematics) Signalverarbeitung Fraktal Mathematische Methode Wavelet Aufsatzsammlung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017368502&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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