Fundamentals of Speaker Recognition


Fundamentals of Speaker Recognition (2012) .. by Homayoon Beigi


Contents

Part I Basic Theory

1 Introduction … 3
1.1 Definition and History … 3
1.2 Speaker Recognition Branches … 5
1.2.1 Speaker Verification (Speaker Authentication) … 5
1.2.2 Speaker Identification (Closed-Set and Open-Set) … 7
1.2.3 Speaker and Event Classification … 8
1.2.4 Speaker Segmentation … 9
1.2.5 Speaker Detection … 11
1.2.6 Speaker Tracking … 11
1.3 Speaker Recognition Modalities … 12
1.3.1 Text-Dependent Speaker Recognition … 12
1.3.2 Text-Independent Speaker Recognition … 13
1.3.3 Text-Prompted Speaker Recognition … 14
1.3.4 Knowledge-Based Speaker Recognition … 15
1.4 Applications … 16
1.4.1 Financial Applications … 16
1.4.2 Forensic and Legal Applications … 18
1.4.3 Access Control (Security) Applications … 19
1.4.4 Audio and Video Indexing (Diarization) Applications … 19
1.4.5 Surveillance Applications … 20
1.4.6 Teleconferencing Applications … 21
1.4.7 Proctorless Oral Testing … 21
1.4.8 Other Applications … 23
1.5 Comparison to Other Biometrics … 23
1.5.1 Deoxyribonucleic Acid (DNA) … 24
1.5.2 Ear … 25
1.5.3 Face … 27
1.5.4 Fingerprint and Palm … 28
1.5.5 Hand and Finger Geometry … 30
1.5.6 Iris … 30
1.5.7 Retina … 31
1.5.8 Thermography … 32
1.5.9 Vein … 32
1.5.10 Gait … 33
1.5.11 Handwriting … 34
1.5.12 Keystroke … 35
1.5.13 Multimodal … 35
1.5.14 Summary of Speaker Biometric Characteristics … 37

References … 38

2 The Anatomy of Speech … 43
2.1 The Human Vocal System … 44
2.1.1 Trachea and Larynx … 44
2.1.2 Vocal Folds (Vocal Chords) … 44
2.1.3 Pharynx… 47
2.1.4 Soft Palate and the Nasal System … 48
2.1.5 Hard Palate … 48
2.1.6 Oral Cavity Exit … 48
2.2 The Human Auditory System … 48
2.2.1 The Ear … 50
2.3 The Nervous System and the Brain … 51
2.3.1 Neurons – Elementary Building Blocks … 52
2.3.2 The Brain … 54
2.3.3 Function Localization in the Brain … 59
2.3.4 Specializations of the Hemispheres of the Brain … 62
2.3.5 Audio Production … 64
2.3.6 Auditory Perception … 66
2.3.7 Speaker Recognition … 71

References … 72

3 Signal Representation of Speech … 75
3.1 Sampling The Audio … 77
3.1.1 The Sampling Theorem … 78
3.1.2 Convergence Criteria for the Sampling Theorem … 84
3.1.3 Extensions of the Sampling Theorem … 84
3.2 Quantization and Amplitude Errors … 85
3.3 The Speech Waveform … 87
3.4 The Spectrogram … 87
3.5 Formant Representation … 89
3.6 Practical Sampling and Associated Errors … 92
3.6.1 Ideal Sampler … 98
3.6.2 Aliasing … 99
3.6.3 Truncation Error … 102
3.6.4 Jitter … 103
Contents xv
3.6.5 Loss of Information … 104

References … 105

4 Phonetics and Phonology … 107
4.1 Phonetics … 107
4.1.1 Initiation … 109
4.1.2 Phonation … 109
4.1.3 Articulation … 110
4.1.4 Coordination … 111
4.1.5 Vowels … 112
4.1.6 Pulmonic Consonants … 115
4.1.7 Whisper … 119
4.1.8 Whistle … 119
4.1.9 Non-Pulmonic Consonants … 120
4.2 Phonology and Linguistics … 122
4.2.1 Phonemic Utilization Across Languages … 122
4.2.2 Whisper … 125
4.2.3 Importance of Vowels in Speaker Recognition … 127
4.2.4 Evolution of Languages toward Discriminability… 129
4.3 Suprasegmental Features of Speech … 131
4.3.1 Prosodic Features … 132
4.3.2 Metrical features of Speech … 138
4.3.3 Temporal features of Speech … 140
4.3.4 Co-Articulation … 140

References … 141

5 Signal Processing of Speech and Feature Extraction … 143
5.1 Auditory Perception … 144
5.1.1 Pitch … 146
5.1.2 Loudness … 149
5.1.3 Timbre … 151
5.2 The Sampling Process … 152
5.2.1 Anti-Aliasing … 153
5.2.2 Hi-Pass Filtering … 153
5.2.3 Pre-Emphasis … 153
5.2.4 Quantization … 155
5.3 Spectral Analysis and Direct Method Features … 157
5.3.1 Framing the Signal … 160
5.3.2 Windowing … 162
5.3.3 Discrete Fourier Transform (DFT) and Spectral Estimation 167
5.3.4 Frequency Warping … 169
5.3.5 Magnitude Warping … 172
5.3.6 Mel Frequency Cepstral Coefficients (MFCC) … 173
5.3.7 Mel Cepstral Dynamics … 175
5.4 Linear Predictive Cepstral Coefficients (LPCC) … 176
5.4.1 Autoregressive (AR) Estimate of the PSD … 177
5.4.2 LPC Computation … 184
5.4.3 Partial Correlation (PARCOR) Features … 185
5.4.4 Log Area Ratio (LAR) Features… 189
5.4.5 Linear Predictive Cepstral Coefficient (LPCC) Features … 189
5.5 Perceptual Linear Predictive (PLP) Analysis … 190
5.5.1 Spectral Analysis … 191
5.5.2 Bark Frequency Warping … 191
5.5.3 Equal-Loudness Pre-emphasis … 192
5.5.4 Magnitude Warping … 193
5.5.5 Inverse DFT … 193
5.6 Other Features … 193
5.6.1 Wavelet Filterbanks … 194
5.6.2 Instantaneous Frequencies … 197
5.6.3 Empirical Mode Decomposition (EMD) … 198
5.7 Signal Enhancement and Pre-Processing … 199

References … 199

6 Probability Theory and Statistics … 205
6.1 Set Theory … 205
6.1.1 Equivalence and Partitions … 208
6.1.2 R-Rough Sets (Rough Sets) … 210
6.1.3 Fuzzy Sets … 211
6.2 Measure Theory … 211
6.2.1 Measure … 212
6.2.2 Multiple Dimensional Spaces… 216
6.2.3 Metric Space … 217
6.2.4 Banach Space (Normed Vector Space) … 218
6.2.5 Inner Product Space (Dot Product Space) … 219
6.2.6 Infinite Dimensional Spaces (Pre-Hilbert and Hilbert) … 219
6.3 Probability Measure … 221
6.4 Integration … 227
6.5 Functions … 228
6.5.1 Probability Density Function … 229
6.5.2 Densities in the Cartesian Product Space … 232
6.5.3 Cumulative Distribution Function … 235
6.5.4 Function Spaces … 236
6.5.5 Transformations … 238
6.6 Statistical Moments … 239
6.6.1 Mean … 239
6.6.2 Variance … 242
6.6.3 Skewness (skew) … 245
6.6.4 Kurtosis … 246
6.7 Discrete Random Variables … 247
6.7.1 Combinations of Random Variables … 250
6.7.2 Convergence of a Sequence … 250
6.8 Sufficient Statistics … 251
6.9 Moment Estimation … 253
6.9.1 Estimating the Mean … 253
6.9.2 Law of Large Numbers (LLN) … 254
6.9.3 Different Types of Mean … 257
6.9.4 Estimating the Variance … 258
6.10 Multi-Variate Normal Distribution … 259

References … 261

7 Information Theory … 265
7.1 Sources … 266
7.2 The Relation between Uncertainty and Choice … 269
7.3 Discrete Sources … 269
7.3.1 Entropy or Uncertainty … 270
7.3.2 Generalized Entropy … 278
7.3.3 Information … 279
7.3.4 The Relation between Information and Entropy … 280
7.4 Discrete Channels… 282
7.5 Continuous Sources … 284
7.5.1 Differential Entropy (Continuous Entropy) … 284
7.6 Relative Entropy … 286
7.6.1 Mutual Information … 291
7.7 Fisher Information … 294

References … 299

8 Metrics and Divergences… 301
8.1 Distance (Metric) … 301
8.1.1 Distance Between Sequences … 302
8.1.2 Distance Between Vectors and Sets of Vectors… 302
8.1.3 Hellinger Distance … 304
8.2 Divergences and Directed Divergences … 304
8.2.1 Kullback-Leibler’s Directed Divergence … 305
8.2.2 Jeffreys’ Divergence … 305
8.2.3 Bhattacharyya Divergence … 306
8.2.4 Matsushita Divergence … 307
8.2.5 F-Divergence … 308
8.2.6 ?-Divergence … 309
8.2.7 ?? Directed Divergence … 310

References … 310

9 Decision Theory … 313
9.1 Hypothesis Testing … 313
9.2 Bayesian Decision Theory… 316
9.2.1 Binary Hypothesis … 320
xviii Contents
9.2.2 Relative Information and Log Likelihood Ratio … 321
9.3 Bayesian Classifier … 322
9.3.1 Multi-Dimensional Normal Classification … 326
9.3.2 Classification of a Sequence … 328
9.4 Decision Trees … 331
9.4.1 Tree Construction … 332
9.4.2 Types of Questions … 333
9.4.3 Maximum Likelihood Estimation (MLE) … 336

References … 338

10 Parameter Estimation … 341
10.1 Maximum Likelihood Estimation … 342
10.2 Maximum A-Posteriori (MAP) Estimation … 344
10.3 Maximum Entropy Estimation … 345
10.4 Minimum Relative Entropy Estimation … 346
10.5 Maximum Mutual Information Estimation (MMIE) … 348
10.6 Model Selection … 349
10.6.1 Akaike Information Criterion (AIC) … 350
10.6.2 Bayesian Information Criterion (BIC)… 353

References … 354

11 Unsupervised Clustering and Learning … 357
11.1 Vector Quantization (VQ) … 358
11.2 Basic Clustering Techniques … 359
11.2.1 Standard k-Means (Lloyd) Algorithm … 360
11.2.2 Generalized Clustering … 363
11.2.3 Overpartitioning … 364
11.2.4 Merging … 364
11.2.5 Modifications to the k-Means Algorithm … 365
11.2.6 k-Means Wrappers … 368
11.2.7 Rough k-Means … 375
11.2.8 Fuzzy k-Means … 377
11.2.9 k-Harmonic Means Algorithm … 378
11.2.10 Hybrid Clustering Algorithms … 380
11.3 Estimation using Incomplete Data … 381
11.3.1 Expectation Maximization (EM) … 381
11.4 Hierarchical Clustering … 388
11.4.1 Agglomerative (Bottom-Up) Clustering (AHC) … 389
11.4.2 Divisive (Top-Down) Clustering (DHC) … 389
11.5 Semi-Supervised Learning … 390

References … 390

12 Transformation… 393
12.1 Principal Component Analysis (PCA) … 394
12.1.1 Formulation … 394
12.2 Generalized Eigenvalue Problem … 397
12.3 Nonlinear Component Analysis … 399
12.3.1 Kernel Principal Component Analysis (Kernel PCA) … 400
12.4 Linear Discriminant Analysis (LDA) … 401
12.4.1 Integrated Mel Linear Discriminant Analysis (IMELDA) . 404
12.5 Factor Analysis … 404

References … 409

13 Hidden Markov Modeling (HMM)… 411
13.1 Memoryless Models … 413
13.2 Discrete Markov Chains … 415
13.3 Markov Models … 416
13.4 Hidden Markov Models … 418
13.5 Model Design and States … 421
13.6 Training and Decoding … 423
13.6.1 Trellis Diagram Representation … 428
13.6.2 Forward Pass Algorithm … 430
13.6.3 Viterbi Algorithm … 432
13.6.4 Baum-Welch (Forward-Backward) Algorithm … 433
13.7 Gaussian Mixture Models (GMM) … 442
13.7.1 Training … 444
13.7.2 Tractability of Models … 449
13.8 Practical Issues … 451
13.8.1 Smoothing … 451
13.8.2 Model Comparison … 453
13.8.3 Held-Out Estimation … 456
13.8.4 Deleted Estimation … 461

References … 462

14 Neural Networks … 465
14.1 Perceptron … 466
14.2 Feedforward Networks … 466
14.2.1 Auto Associative Neural Networks (AANN) … 469
14.2.2 Radial Basis Function Neural Networks (RBFNN) … 469
14.2.3 Training (Learning) Formulation … 470
14.2.4 Optimization Problem … 473
14.2.5 Global Solution … 474
14.3 Recurrent Neural Networks (RNN) … 476
14.4 Time-Delay Neural Networks (TDNNs) … 477
14.5 Hierarchical Mixtures of Experts (HME) … 479
14.6 Practical Issues … 479

References … 481

15 Support Vector Machines … 485
15.1 Risk Minimization … 488
15.1.1 Empirical Risk Minimization … 492
15.1.2 Capacity and Bounds on Risk … 493
15.1.3 Structural Risk Minimization … 493
15.2 The Two-Class Problem … 494
15.2.1 Dual Representation … 497
15.2.2 Soft Margin Classification … 500
15.3 Kernel Mapping … 503
15.3.1 The Kernel Trick … 504
15.4 Positive Semi-Definite Kernels … 506
15.4.1 Linear Kernel … 506
15.4.2 Polynomial Kernel … 506
15.4.3 Gaussian Radial Basis Function (GRBF) Kernel … 507
15.4.4 Cosine Kernel … 508
15.4.5 Fisher Kernel … 508
15.4.6 GLDS Kernel … 509
15.4.7 GMM-UBM Mean Interval (GUMI) Kernel … 510
15.5 Non Positive Semi-Definite Kernels … 511
15.5.1 Jeffreys Divergence Kernel … 511
15.5.2 Fuzzy Hyperbolic Tangent (tanh) Kernel … 512
15.5.3 Neural Network Kernel … 513
15.6 Kernel Normalization … 513
15.7 Kernel Principal Component Analysis (Kernel PCA) … 514
15.8 Nuisance Attribute Projection (NAP) … 516
15.9 The multiclass (? -Class) Problem … 518

References … 519

Part II Advanced Theory

16 Speaker Modeling … 525
16.1 Individual Speaker Modeling … 526
16.2 Background Models and Cohorts … 527
16.2.1 Background Models … 528
16.2.2 Cohorts … 529
16.3 Pooling of Data and Speaker Independent Models … 529
16.4 Speaker Adaptation … 530
16.4.1 Factor Analysis (FA) … 530
16.4.2 Joint Factor Analysis (JFA) … 531
16.4.3 Total Factors (Total Variability) … 532
16.5 Audio Segmentation … 532
16.6 Model Quality Assessment … 534
16.6.1 Enrollment Utterance Quality Control … 534
16.6.2 Speaker Menagerie … 536

References … 538

17 Speaker Recognition … 543
17.1 The Enrollment Task … 543
17.2 The Verification Task … 544
17.2.1 Text-Dependent … 546
17.2.2 Text-Prompted … 546
17.2.3 Knowledge-Based … 548
17.3 The Identification Task … 548
17.3.1 Closed-Set Identification … 548
17.3.2 Open-Set Identification … 549
17.4 Speaker Segmentation … 549
17.5 Speaker and Event Classification … 550
17.5.1 Gender and Age Classification (Identification) … 551
17.5.2 Audio Classification … 553
17.5.3 Multiple Codebooks … 553
17.5.4 Farfield Speaker Recognition … 553
17.5.5 Whispering Speaker Recognition … 554
17.6 Speaker Diarization … 554
17.6.1 Speaker Position and Orientation… 555

References … 555

18 Signal Enhancement and Compensation … 561
18.1 Silence Detection, Voice Activity Detection (VAD) … 561
18.2 Audio Volume Estimation … 564
18.3 Echo Cancellation … 564
18.4 Spectral Filtering and Cepstral Liftering … 565
18.4.1 Cepstral Mean Normalization (Subtraction) – CMN (CMS)567
18.4.2 Cepstral Mean and Variance Normalization (CMVN) … 569
18.4.3 Cepstral Histogram Normalization (Histogram Equalization) … 570
18.4.4 RelAtive SpecTrAl (RASTA) Filtering … 571
18.4.5 Other Lifters … 571
18.4.6 Vocal Tract Length Normalization (VTLN) … 573
18.4.7 Other Normalization Techniques … 576
18.4.8 Steady Tone Removal (Narrowband Noise Reduction) … 579
18.4.9 Adaptive Wiener Filtering … 580
18.5 Speaker Model Normalization … 581
18.5.1 Z-Norm … 581
18.5.2 T-Norm (Test Norm) … 582
18.5.3 H-Norm … 582
18.5.4 HT-Norm … 582
18.5.5 AT-Norm… 582
18.5.6 C-Norm… 582
18.5.7 D-Norm … 583
18.5.8 F-Norm (F-Ratio Normalization) … 583
18.5.9 Group-Specific Normalization … 583
18.5.10 Within Class Covariance Normalization (WCCN) … 583
18.5.11 Other Normalization Techniques … 583

References … 584

Part III Practice

19 Evaluation and Representation of Results … 589
19.1 Verification Results … 589
19.1.1 Equal-Error Rate … 589
19.1.2 Half Total Error Rate … 590
19.1.3 Receiver Operating Characteristic (ROC) Curve … 590
19.1.4 Detection Error Trade-Off (DET) Curve … 592
19.1.5 Detection Cost Function (DCF) … 593
19.2 Identification Results … 593

References … 594

20 Time Lapse Effects (Case Study) … 595
20.1 The Audio Data … 598
20.2 Baseline Speaker Recognition … 598

References … 600

21 Adaptation over Time (Case Study) … 601
21.1 Data Augmentation … 601
21.2 Maximum A Posteriori (MAP) Adaptation … 603
21.3 Eigenvoice Adaptation … 605
21.4 Minimum Classification Error (MCE) … 605
21.5 Linear Regression Techniques … 606
21.5.1 Maximum Likelihood Linear Regression (MLLR) … 606
21.6 Maximum a-Posteriori Linear Regression (MAPLR) … 607
21.6.1 Other Adaptation Techniques … 607
21.7 Practical Perspectives … 607

References … 608

22 Overall Design … 611
22.1 Choosing the Model … 611
22.1.1 Phonetic Speaker Recognition … 612
22.2 Choosing an Adaptation Technique … 613
22.3 Microphones … 613
22.4 Channel Mismatch … 615
22.5 Voice Over Internet Protocol (VoIP) … 615
22.6 Public Databases … 616
22.6.1 NIST … 616
22.6.2 Linguistic Data Consortium (LDC) … 616
22.6.3 European Language Resources Association (ELRA) … 619
22.7 High Level Information … 620
22.7.1 Choosing Basic Segments … 622
22.8 Numerical Stability … 623
22.9 Privacy … 624
22.10 Biometric Encryption … 625
22.11 Spoofing … 625
22.11.1 Text-Prompted Verification Systems … 625
22.11.2 Text-Independent Verification Systems … 626
22.12 Quality Issues … 627
22.13 Large-Scale Systems … 628
22.14 Useful Tools … 628

References … 629

Part IV Background Material

23 Linear Algebra … 635
23.1 Basic Definitions … 635
23.2 Norms … 636
23.3 Gram-Schmidt Orthogonalization … 641
23.3.1 Ordinary Gram-Schmidt Orthogonalization … 641
23.3.2 Modified Gram-Schmidt Orthogonalization … 641
23.4 Sherman-Morrison Inversion Formula … 642
23.5 Vector Representation under a Set of Normal Conjugate Direction . 642
23.6 Stochastic Matrix … 643
23.7 Linear Equations … 643

References … 646

24 Integral Transforms … 647
24.1 Complex Variable Theory in Integral Transforms… 648
24.1.1 Complex Variables … 648
24.1.2 Limits … 651
24.1.3 Continuity and Forms of Discontinuity … 652
24.1.4 Convexity and Concavity of Functions … 658
24.1.5 Odd, Even and Periodic Functions … 661
24.1.6 Differentiation … 663
24.1.7 Analyticity … 665
24.1.8 Integration … 672
24.1.9 Power Series Expansion of Functions … 683
24.1.10 Residues … 686
24.2 Relations Between Functions … 688
24.2.1 Convolution … 688
24.2.2 Correlation … 689
24.3 Orthogonality of Functions … 690
24.4 Integral Equations … 694
24.5 Kernel Functions … 696
24.5.1 Hilbert’s Expansion Theorem… 698
24.5.2 Eigenvalues and Eigenfunctions of the Kernel … 700
24.6 Fourier Series Expansion … 708
24.6.1 Convergence of the Fourier Series … 713
24.6.2 Parseval’s Theorem … 714
24.7 Wavelet Series Expansion … 716
24.8 The Laplace Transform … 717
24.8.1 Inversion … 720
24.8.2 Some Useful Transforms … 721
24.9 Complex Fourier Transform (Fourier Integral Transform) … 722
24.9.1 Translation … 724
24.9.2 Scaling … 724
24.9.3 Symmetry Table … 724
24.9.4 Time and Complex Scaling and Shifting … 725
24.9.5 Convolution … 725
24.9.6 Correlation … 726
24.9.7 Parseval’s Theorem … 726
24.9.8 Power Spectral Density … 728
24.9.9 One-Sided Power Spectral Density … 728
24.9.10 PSD-per-unit-time … 729
24.9.11 Wiener-Khintchine Theorem … 729
24.10 Discrete Fourier Transform (DFT) … 731
24.10.1 Inverse Discrete Fourier Transform (IDFT) … 732
24.10.2 Periodicity … 734
24.10.3 Plancherel and Parseval’s Theorem … 734
24.10.4 Power Spectral Density (PSD) Estimation … 735
24.10.5 Fast Fourier Transform (FFT) … 736
24.11 Discrete-Time Fourier Transform (DTFT) … 738
24.11.1 Power Spectral Density (PSD) Estimation … 739
24.12 Complex Short-Time Fourier Transform (STFT) … 740
24.12.1 Discrete-Time Short-Time Fourier Transform DTSTFT … 744
24.12.2 Discrete Short-Time Fourier Transform DSTFT … 746
24.13 Discrete Cosine Transform (DCT) … 748
24.13.1 Efficient DCT Computation … 749
24.14 The z-Transform … 750
24.14.1 Translation … 756
24.14.2 Scaling … 756
24.14.3 Shifting – Time Lag … 757
24.14.4 Shifting – Time Lead … 757
24.14.5 Complex Translation … 757
24.14.6 Initial Value Theorem … 758
24.14.7 Final Value Theorem … 758
24.14.8 Real Convolution Theorem… 759
24.14.9 Inversion … 760
24.15 Cepstrum … 762

References … 769

25 Nonlinear Optimization … 773
25.1 Gradient-Based Optimization … 775
25.1.1 The Steepest Descent Technique … 775
25.1.2 Newton’s Minimization Technique … 777
25.1.3 Quasi-Newton or Large Step Gradient Techniques … 779
25.1.4 Conjugate Gradient Methods … 793
25.2 Gradient-Free Optimization … 803
25.2.1 Search Methods … 804
25.2.2 Gradient-Free Conjugate Direction Methods … 804
25.3 The Line Search Sub-Problem … 809
25.4 Practical Considerations … 810
25.4.1 Large-Scale Optimization … 810
25.4.2 Numerical Stability … 813
25.4.3 Nonsmooth Optimization … 814
25.5 Constrained Optimization … 814
25.5.1 The Lagrangian and Lagrange Multipliers … 817
25.5.2 Duality … 831
25.6 Global Convergence … 835

References … 836

26 Standards … 841
26.1 Standard Audio Formats … 842
26.1.1 Linear PCM (Uniform PCM) … 842
26.1.2 µ-Law PCM (PCMU) … 843
26.1.3 A-Law (PCMA) … 843
26.1.4 MP3 … 843
26.1.5 HE-AAC … 844
26.1.6 OGG Vorbis … 844
26.1.7 ADPCM (G.726) … 845
26.1.8 GSM … 845
26.1.9 CELP … 847
26.1.10 DTMF … 848
26.1.11 Others Audio Formats … 848
26.2 Standard Audio Encapsulation Formats … 849
26.2.1 WAV … 849
26.2.2 SPHERE … 850
26.2.3 Standard Audio Format Encapsulation (SAFE) … 850
26.3 APIs and Protocols … 854
26.3.1 SVAPI … 855
26.3.2 BioAPI … 855
26.3.3 VoiceXML … 856
26.3.4 MRCP … 857
26.3.5 Real-time Transport Protocol (RTP) … 858
26.3.6 Extensible MultiModal Annotation (EMMA) … 858

References … 859

Bibliography … 861
Solutions … 901
Index … 909

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