We describe the design of Kaldi, a free, open-source toolkit for speech recognition research. Kaldi provides a speech recognition system based on finite-state automata (using the freely available OpenFst), together with detailed documentation and a comprehensive set of scripts for building complete recognition systems. Kaldi is written is C++, and the core library supports modeling of arbitrary phonetic-context sizes, acoustic modeling with subspace Gaussian mixture models (SGMM) as well as standard Gaussian mixture models, together with all commonly used linear and affine transforms. Kaldi is released under the Apache License v2.0, which is highly nonrestrictive, making it suitable for a wide community of users.

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The Kaldi Speech Recognition Toolkit

First published in 1995, Wavelets and Subband Coding offered a unified view of the exciting field of wavelets and their discrete-time cousins, filter banks, or subband coding. The book developed the theory in both continuous and discrete time, and presented important applications. During the past decade, it filled a useful need in explaining a new view of signal processing based on flexible time-frequency analysis and its applications. Since 2007, the authors now retain the copyright and allow open access to the book.

Wavelets and Subband Coding

The paper discusses the theory behind the dual-tree transform, shows how complex wavelets with good properties can be designed, and illustrates a range of applications in signal and image processing The authors use the complex number symbol C in CWT to avoid confusion with the often-used acronym CWT for the (different) continuous wavelet transform The four fundamentals, intertwined shortcomings of wavelet transform and some solutions are also discussed Several methods for filter design are described for dual-tree CWT that demonstrates with relatively short filters, an effective invertible approximately analytic wavelet transform can indeed be implemented using the dual-tree approach

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The dual-tree complex wavelet transform

1 Introduction to Wavelets 2 A Multiresolution Formulation of Wavelet Systems 3 Filter Banks and the Discrete Wavelet Transform 4 Bases, Orthogonal Bases, Biorthogonal Bases, Frames, Tight Frames, and Unconditional Bases 5 The Scaling Function and Scaling Coefficients, Wavelet and Wavelet Coefficients 6 Regularity, Moments, and Wavelet System Design 7 Generalizations of the Basic Multiresolution Wavelet System 8 Filter Banks and Transmultiplexers 9 Calculation of the Discrete Wavelet Transform 10 Wavelet-Based Signal Processing and Applications 11 Summary Overview 12 References Bibliography Appendix A Derivations for Chapter 5 on Scaling Functions Appendix B Derivations for Section on Properties Appendix C Matlab Programs Index

Introduction to Wavelets and Wavelet Transforms: A Primer

Preface 1 Introduction 1.1 Compression Techniques 1.1.1 Lossless Compression 1.1.2 Lossy Compression 1.1.3 Measures of Performance 1.2 Modeling and Coding 1.3 Organization of This Book 1.4 Summary 1.5 Projects and Problems 2 Mathematical Preliminaries 2.1 Overview 2.2 A Brief Introduction to Information Theory 2.3 Models 2.3.1 Physical Models 2.3.2 Probability Models 2.3.3. Markov Models 2.3.4 Summary 2.5 Projects and Problems 3 Huffman Coding 3.1 Overview 3.2 "Good" Codes 3.3. The Huffman Coding Algorithm 3.3.1 Minimum Variance Huffman Codes 3.3.2 Length of Huffman Codes 3.3.3 Extended Huffman Codes 3.4 Nonbinary Huffman Codes 3.5 Adaptive Huffman Coding 3.5.1 Update Procedure 3.5.2 Encoding Procedure 3.5.3 Decoding Procedure 3.6 Applications of Huffman Coding 3.6.1 Lossless Image Compression 3.6.2 Text Compression 3.6.3 Audio Compression 3.7 Summary 3.8 Projects and Problems 4 Arithmetic Coding 4.1 Overview 4.2 Introduction 4.3 Coding a Sequence 4.3.1 Generating a Tag 4.3.2 Deciphering the Tag 4.4 Generating a Binary Code 4.4.1 Uniqueness and Efficiency of the Arithmetic Code 4.4.2 Algorithm Implementation 4.4.3 Integer Implementation 4.5 Comparison of Huffman and Arithmetic Coding 4.6 Applications 4.6.1 Bi-Level Image Compression-The JBIG Standard 4.6.2 Image Compression 4.7 Summary 4.8 Projects and Problems 5 Dictionary Techniques 5.1 Overview 5.2 Introduction 5.3 Static Dictionary 5.3.1 Diagram Coding 5.4 Adaptive Dictionary 5.4.1 The LZ77 Approach 5.4.2 The LZ78 Approach 5.5 Applications 5.5.1 File Compression-UNIX COMPRESS 5.5.2 Image Compression-the Graphics Interchange Format (GIF) 5.5.3 Compression over Modems-V.42 bis 5.6 Summary 5.7 Projects and Problems 6 Lossless Image Compression 6.1 Overview 6.2 Introduction 6.3 Facsimile Encoding 6.3.1 Run-Length Coding 6.3.2 CCITT Group 3 and 4-Recommendations T.4 and T.6 6.3.3 Comparison of MH, MR, MMR, and JBIG 6.4 Progressive Image Transmission 6.5 Other Image Compression Approaches 6.5.1 Linear Prediction Models 6.5.2 Context Models 6.5.3 Multiresolution Models 6.5.4 Modeling Prediction Errors 6.6 Summary 6.7 Projects and Problems 7 Mathematical Preliminaries 7.1 Overview 7.2 Introduction 7.3 Distortion Criteria 7.3.1 The Human Visual System 7.3.2 Auditory Perception 7.4 Information Theory Revisted 7.4.1 Conditional Entropy 7.4.2 Average Mutual Information 7.4.3 Differential Entropy 7.5 Rate Distortion Theory 7.6 Models 7.6.1 Probability Models 7.6.2 Linear System Models 7.6.3 Physical Models 7.7 Summary 7.8 Projects and Problems 8 Scalar Quantization 8.1 Overview 8.2 Introduction 8.3 The Quantization Problem 8.4 Uniform Quantizer 8.5 Adaptive Quantization 8.5.1 Forward Adaptive Quantization 8.5.2 Backward Adaptive Quantization 8.6 Nonuniform Quantization 8.6.1 pdf-Optimized Quantization 8.6.2 Companded Quantization 8.7 Entropy-Coded Quantization 8.7.1 Entropy Coding of Lloyd-Max Quantizer Outputs 8.7.2 Entropy-Constrained Quantization 8.7.3 High-Rate Optimum Quantization 8.8 Summary 8.9 Projects and Problems 9 Vector Quantization 9.1 Overview 9.2 Introduction 9.3 Advantages of Vector Quantization over Scalar Quantization 9.4 The Linde-Buzo-Gray Algorithm 9.4.1 Initializing the LBG Algorithm 9.4.2 The Empty Cell Problem 9.4.3 Use of LBG for Image Compression 9.5 Tree-Structured Vector Quantizers 9.5.1 Design of Tree-Structured Vector Quantizers 9.6 Structured Vector Quantizers 9.6.1 Pyramid Vector Quantization 9.6.2 Polar and Spherical Vector Quantizers 9.6.3 Lattice Vector Quantizers 9.7 Variations on the Theme 9.7.1 Gain-Shape Vector Quantization 9.7.2 Mean-Removed Vector Quantization 9.7.3 Classified Vector Quantization 9.7.4 Multistage Vector Quantization 9.7.5 Adaptive Vector Quantization 9.8 Summary 9.9 Projects and Problems 10 Differential Encoding 10.1 Overview 10.2 Introduction 10.3 The Basic Algorithm 10.4 Prediction in DPCM 10.5 Adaptive DPCM (ADPCM) 10.5.1 Adaptive Quantization in DPCM 10.5.2 Adaptive Prediction in DPCM 10.6 Delta Modulation 10.6.1 Constant Factor Adaptive Delta Modulation (CFDM) 10.6.2 Continuously Variable Slope Delta Modulation 10.7 Speech Coding 10.7.1 G.726 10.8 Summary 10.9 Projects and Problems 11 Subband Coding 11.1 Overview 11.2 Introduction 11.3 The Frequency Domain and Filtering 11.3.1 Filters 11.4 The Basic Subband Coding Algorithm 11.4.1 Bit Allocation 11.5 Application to Speech Coding-G.722 11.6 Application to Audio Coding-MPEG Audio 11.7 Application to Image Compression 11.7.1 Decomposing an Image 11.7.2 Coding the Subbands 11.8 Wavelets 11.8.1 Families of Wavelets 11.8.2 Wavelets and Image Compression 11.9 Summary 11.10 Projects and Problems 12 Transform Coding 12.1 Overview 12.2 Introduction 12.3 The Transform 12.4 Transforms of Interest 12.4.1 Karhunen-Loeve Transform 12.4.2 Discrete Cosine Transform 12.4.3 Discrete Sine Transform 12.4.4 Discrete Walsh-Hadamard Transform 12.5 Quantization and Coding of Transform Coefficients 12.6 Application to Image Compression-JPEG 12.6.1 The Transform 12.6.2 Quantization 12.6.3 Coding 12.7 Application to Audio Compression 12.8 Summary 12.9 Projects and Problems 13 Analysis/Synthesis Schemes 13.1 Overview 13.2 Introduction 13.3 Speech Compression 13.3.1 The Channel Vocoder 13.3.2 The Linear Predictive Coder (Gov.Std.LPC-10) 13.3.3 Code Excited Linear Prediction (CELP) 13.3.4 Sinusoidal Coders 13.4 Image Compression 13.4.1 Fractal Compression 13.5 Summary 13.6 Projects and Problems 14 Video Compression 14.1 Overview 14.2 Introduction 14.3 Motion Compensation 14.4 Video Signal Representation 14.5 Algorithms for Videoconferencing and Videophones 14.5.1 ITU_T Recommendation H.261 14.5.2 Model-Based Coding 14.6 Asymmetric Applications 14.6.1 The MPEG Video Standard 14.7 Packet Video 14.7.1 ATM Networks 14.7.2 Compression Issues in ATM Networks 14.7.3 Compression Algorithms for Packet Video 14.8 Summary 14.9 Projects and Problems A Probability and Random Processes A.1 Probability A.2 Random Variables A.3 Distribution Functions A.4 Expectation A.5 Types of Distribution A.6 Stochastic Process A.7 Projects and Problems B A Brief Review of Matrix Concepts B.1 A Matrix B.2 Matrix Operations C Codes for Facsimile Encoding D The Root Lattices Bibliography Index

Introduction to data compression

Orthonormal M-band wavelet bases have been constructed and applied by several authors. This paper makes three main contributions. First, it generalizes the minimal length K-regular 2-band wavelets of Daubechies (1988) to the M-band case by deriving explicit formulas for K-regular M-band scaling filters. Several equivalent characterizations of K-regularity are given and their significance explained. Second, two approaches to the construction of the (M-1) wavelet filters and associated wavelet bases are described; one relies on a state-space characterization with a novel technique to obtain the unitary wavelet filters; the other uses a factorization approach. Third, this paper gives a set of necessary and sufficient condition on the M-band scaling filter for it to generate an orthonormal wavelet basis. The conditions are very similar to those obtained by Cohen (1990) and Lawton (1990) for 2-band wavelets. >

Theory of regular M-band wavelet bases

Maximum likelihood (ML) modeling of multiclass data for classification often suffers from the following problems: (a) data insufficiency implying overtrained or unreliable models, (b) large storage requirement, (c) large computational requirement and/or (d) the ML is not discriminating between classes. Sharing parameters across classes (or constraining the parameters) clearly tends to alleviate the first three problems. We show that in some cases it can also lead to better discrimination (as evidenced by reduced misclassification error). The parameters considered are the means and variances of the Gaussians and linear transformations of the feature space (or equivalently the Gaussian means). Some constraints on the parameters are shown to lead to linear discrimination analysis (a well-known result) while others are shown to lead to optimal feature spaces (a relatively new result). Applications of some of these ideas to the speech recognition problem are also given.

Maximum likelihood modeling with Gaussian distributions for classification

Linear discriminant analysis (LDA) is known to be inappropriate for the case of classes with unequal sample covariances. There has been an interest in generalizing LDA to heteroscedastic discriminant analysis (HDA) by removing the equal within-class covariance constraint. This paper presents a new approach to HDA by defining an objective function which maximizes the class discrimination in the projected subspace while ignoring the rejected dimensions. Moreover, we investigate the link between discrimination and the likelihood of the projected samples and show that HDA can be viewed as a constrained ML projection for a full covariance Gaussian model, the constraint being given by the maximization of the projected between-class scatter volume. It is shown that, under diagonal covariance Gaussian modeling constraints, applying a diagonalizing linear transformation (MLLT) to the HDA space results in increased classification accuracy even though HDA alone actually degrades the recognition performance. Experiments performed on the Switchboard and Voicemail databases show a 10%-13% relative improvement in the word error rate over standard cepstral processing.

Maximum likelihood discriminant feature spaces

The paper introduces a novel speckle reduction method based on thresholding the wavelet coefficients of the logarithmically transformed image. The method is computational efficient and can significantly reduce the speckle while preserving the resolution of the original image. Both soft and hard thresholding schemes are studied and the results are compared. When fully polarimetric SAR images are available, the authors propose several approaches to combine the data from different polarizations to achieve even better performance. Wavelet processed imagery is shown to provide better detection performance for the synthetic-aperture radar (SAR) based automatic target detection/recognition (ATD/R) problem. >

Ramesh A. Gopinath

Papers

Introduction to Wavelets and Wavelet Transforms: A Primer

Theory of regular M-band wavelet bases

Maximum likelihood modeling with Gaussian distributions for classification

Maximum likelihood discriminant feature spaces

Wavelet based speckle reduction with application to SAR based ATD/R