Cover
Title Page
Copyright
Contents
About the Editors
List of Contributors
Foreword In Memoriam: José M. Bioucas‐Dias (1960–2020), a Humble Giant
Preface
Notation
Chapter 1 Overview of Source Separation
1.1 Introduction
1.1.1 Brief Introduction to Source Separation
1.1.2 Chapter's Organization
1.2 The Problem of Source Separation
1.2.1 Mathematical Description
1.2.2 Different Types of Mixing Models
1.2.2.1 Linear Mixtures
1.2.2.2 Nonlinear Mixtures
1.2.2.3 Overdetermined, Determined, or Underdetermined Models
1.2.2.4 Noisy Mixtures
1.2.3 From Source Separation to Matrix Factorization
1.2.3.1 Factorization Ambiguity
1.2.3.2 Data Representation
1.2.3.3 Factorization Algorithms
1.2.4 Enhanced Diversity: Tensor Formulation and Factorization
1.2.5 From Supervised to Blind Solutions
1.3 Statistical Methods for Source Separation
1.3.1 Factorization of Independent Sources
1.3.2 Independent Component Analysis (ICA)
1.3.2.1 ICA by Mutual Information Minimization
1.3.2.2 ICA by Maximum Likelihood Estimation
1.3.2.3 ICA by Kurtosis Maximization
1.3.3 Methods Based on Second‐Order Statistics
1.4 Source Separation Problems in Physical–Chemical Sensing
1.4.1 Material Analysis by Spectroscopy
1.4.2 Hyperspectral Imaging
1.4.3 Electrochemical Sensor Arrays
1.5 Source Separation Methods for Chemical–Physical Sensing
1.5.1 Self‐Modeling Curve Resolution
1.5.2 Non‐Negative Matrix Factorization
1.5.3 Bayesian Separation Approach
1.5.4 Geometrical Approaches
1.5.5 Tensor Factorization Methods
1.6 Organization of the Book
References
Chapter 2 Optimization
2.1 Introduction to Optimization Problems
2.1.1 Problem Formulation
2.1.2 Theoretical Background
2.1.2.1 Convex Functions
2.1.2.2 Differentiability and Subdifferentiability
2.1.3 Examples in the Context of Source Separation
2.1.3.1 Non‐negative Matrix Factorization
2.1.3.2 Independent Component Analysis
2.1.3.3 Tensor Decomposition
2.1.4 Chapter Outline
2.2 Majorization–Minimization Approaches
2.2.1 Majorization–Minimization Principle
2.2.2 Majorization Techniques
2.2.3 Quadratic MM Methods
2.2.3.1 Quadratic MM Algorithm
2.2.3.2 Half‐Quadratic MM Algorithms
2.2.3.3 Subspace Acceleration Strategy
2.2.4 Variable Metric Forward–Backward Algorithm
2.2.5 Block‐Coordinate MM Algorithms
2.2.5.1 General Principle
2.2.5.2 Block‐Coordinate Quadratic MM Algorithm
2.2.5.3 Block‐Coordinate VMFB Algorithm
2.3 Primal‐Dual Methods
2.3.1 Lagrange Duality
2.3.2 Alternating Direction Method of Multipliers
2.3.2.1 Basic Form
2.3.2.2 Minimizing a Sum of More Than Two Functions
2.3.3 Primal‐Dual Proximal Algorithms
2.3.4 Primal‐Dual Interior Point Algorithm
2.3.4.1 Primal‐Dual Directions
2.3.4.2 Linesearch
2.3.4.3 Penalization Parameter Update
2.3.4.4 Resulting Algorithm
2.4 Application to NMR Signal Restoration
2.4.1 Quadratic Penalization
2.4.2 Entropic Penalization
2.4.3 Sparsity Prior in the Signal Domain
2.4.4 Sparsity Prior in a Transformed Domain
2.4.5 Sparsity Prior and Range Constraints
2.4.6 Concluding Remarks
2.5 Conclusion
References
Chapter 3 Non‐negative Matrix Factorization
3.1 Introduction
3.1.1 Brief Historical Overview
3.2 Geometrical Interpretation of NMF and the Non‐negative Rank
3.2.1 Non‐negative Rank Formulation
3.2.2 Convex Cone Formulation
3.2.3 Nested Polytope Formulation
3.2.4 Non‐negative Rank Computation
3.2.5 Illustrative Examples
3.3 Uniqueness and Admissible Solutions of NMF
3.3.1 Uniqueness Conditions
3.3.2 Finding the Admissible Solutions
3.3.2.1 Illustration in the Case of Two Sources
3.3.2.2 Illustration in the Case of More Than Two Sources
3.4 Non‐negative Matrix Factorization Algorithms
3.4.1 Statistical Formulation of Optimization Criteria
3.4.1.1 Case of a Gaussian Noise
3.4.1.2 Case of Poissonian Distribution
3.4.2 Iterative Factorization Methods
3.4.2.1 Initializing NMF Algorithms
3.4.2.2 Alternating Non‐negative Least Squares, An Exact Coordinate Descent Method with 2 Blocks of Variables
3.4.2.3 Multiplicative Updates
3.4.2.4 Alternating Least Squares
3.4.2.5 Exact Coordinate Descent Method with 2R Blocks of Variables
3.4.3 Constrained and Penalized Factorization Methods
3.4.4 Geometrical Approaches and Separability
3.5 Applications of NMF in Chemical Sensing. Two Examples of Reducing Admissible Solutions
3.5.1 Polarized Raman Spectroscopy: A Data Augmentation Approach
3.5.1.1 Raman Data Description
3.5.1.2 Raman Data Processing
3.5.2 Unmixing Blurred Raman Spectroscopy Images
3.5.2.1 Blurring Effect Modeling
3.5.2.2 Application to Raman Spectroscopy Images
3.6 Conclusions
References
Chapter 4 Bayesian Source Separation
4.1 Introduction
4.2 Overview of Bayesian Source Separation
4.2.1 General Framework
4.2.2 Choice of the Prior Distributions
4.2.3 Source Signal and Mixing Matrix Estimation from the Posterior Distribution
4.2.3.1 Joint Maximum A Posteriori (JMAP)
4.2.3.2 Marginal Maximum A Posteriori (MMAP)
4.2.3.3 Posterior Mean (PM) or Minimum Mean Square Error (MMSE)
4.2.4 Hierarchical Bayesian Modeling and Inference
4.3 Statistical Models for the Separation in the Linear Mixing
4.3.1 Mixing Model
4.3.2 Likelihood Functions in the Linear Mixing Case
4.3.2.1 Gaussian Noise
4.3.2.2 Poissonian Noise
4.3.3 Priors on the Source Signals and Mixing Coefficients in Chemical Sensing
4.3.3.1 Non‐negativity Constraint
4.3.3.2 Bound Constraints
4.3.3.3 Sum‐to‐One Constraint
4.3.3.4 Smoothness Constraint
4.3.4 Application to the Separation of Synthetic Spectral Mixtures
4.3.4.1 Bayesian Separation Model
4.3.4.2 Bayesian Separation Algorithm
4.3.4.3 Bayesian Separation Results
4.4 Statistical Models and Separation Algorithms for Nonlinear Mixtures
4.4.1 Nonlinear Mixing Models from Physical–Chemical Sensing Theory
4.4.1.1 First Example. Interference in Potentiometric Sensors
4.4.1.2 Second Example. Intimate Mixtures in Hyperspectral Imaging
4.4.2 Empirical Nonlinear Mixing Models Used in Separation Algorithms
4.4.2.1 Post‐Nonlinear Mixing Models
4.4.2.2 Polynomial Mixing Models
4.4.2.3 Linear–Quadratic Mixing Models
4.4.2.4 Bilinear Mixing Models
4.5 Some Practical Issues on Algorithm Implementation
4.5.1 A Simple Example
4.5.1.1 The Resulting Gibbs Sampler
4.6 Applications to Case Studies in Chemical Sensing
4.6.1 Monitoring of Calcium Carbonate Crystallization Using Raman Spectroscopy
4.6.1.1 Mixture Preparation and Data Acquisition
4.6.1.2 Mixture Analysis by Bayesian Source Separation
4.6.2 Dealing with Interference Issues in Ion‐Selective Electrode Arrays
4.7 Conclusion
References
Chapter 5 Geometrical Methods – Illustration with Hyperspectral Unmixing
5.1 Introduction
5.2 Hyperspectral Sensing
5.2.1 Hyperspectral Imaging
5.2.2 Hyperspectral Unmixing
5.3 Hyperspectral Mixing Models
5.4 Linear HU Problem Formulation
5.4.1 Preprocessing
5.4.1.1 Dimension Reduction (DR)
5.4.2 Signal Subspace Identification
5.4.2.1 Affine Set Estimation and Projection
5.4.3 Classes of Linear HU Problems
5.4.3.1 Datasets with Pure Pixels. Pure Pixel Pursuit
5.4.3.2 Datasets Without Pure Pixels. Minimum Volume Simplex Estimation
5.4.3.3 Highly Mixed Datasets. Statistical Inference
5.4.3.4 Hyperspectral Unmixing Through Sparse Regression (SR)
5.4.3.5 Synopsis of the Linear HU Problems
5.5 Dictionary‐Based Semiblind HU
5.5.1 Sparse Regression
5.5.2 Sensor Array Processing Meets Semiblind HU
5.5.3 Further Discussion
5.6 Minimum Volume Simplex Estimation
5.6.1 VolMin Optimization
5.6.2 Non‐Negative Matrix Factorization
5.6.3 Illustrative Comparison of Geometrical Methods
5.7 Applications
5.7.1 Unmixing Example Under the Pure Pixel Assumption
5.7.2 Unmixing via Sparse Regression
5.7.3 Near‐Infrared Hyperspectral Unmixing of Pharmaceutical Tablets
5.8 Conclusions
References
Chapter 6 Tensor Decompositions: Principles and Application to Food Sciences
6.1 Introduction
6.1.1 A Simplified Definition
6.1.2 Separability: A Key Concept for Tensor Decomposition Model
6.1.3 The Fluorescence Excitation Emission Matrix (FEEM)
6.1.4 Structure of the Chapter
6.1.4.1 Note
6.1.4.2 Other Introductions
6.2 Tensor Decompositions
6.2.1 Tensor‐Based Method, the Matrix Case
6.2.2 Canonical Polyadic Decomposition, PARAFAC/CanDecomp
6.2.3 Manipulation of Tensors
6.2.3.1 Vectorization
6.2.3.2 Matricization
6.2.3.3 Contractions and CPD
6.2.4 The Chain Rule
6.2.5 Multilinear Singular Value Decomposition
6.2.6 Tucker
6.2.7 PARAFAC2
6.2.8 Approximate Decomposition
6.3 Constraints in Decompositions
6.3.1 Non‐negativity
6.3.1.1 Non‐negative CPD
6.3.1.2 Non‐negative Tucker Decomposition
6.3.2 Block Decompositions
6.3.3 Structured Factors
6.3.3.1 Re‐parameterization
6.3.3.2 Dictionary Constraints
6.4 Coupled Decompositions
6.4.1 Exact Coupled Decomposition, A First Approach
6.4.2 A General Framework for Data Fusion in Tensor Decompositions
6.4.2.1 H1: Conditional Independence of the Data
6.4.3 Examples of Coupled Decomposition Models
6.4.3.1 Advanced Coupled Matrix Tensor Factorization
6.4.3.2 Shift PARAFAC and Others
6.4.3.3 GSVD
6.5 Algorithms
6.5.1 Unconstrained Tensor Decomposition
6.5.1.1 Iterative Algorithms for Approximate CPD
6.5.1.2 Deflation and N‐PLS
6.5.1.3 Exact Decomposition Methods
6.5.2 Constrained Tensor Decomposition
6.5.2.1 Constrained Least Squares
6.5.2.2 Projected Gradient and All‐at‐Once Proximal Methods
6.5.2.3 Parametric Approaches
6.5.3 Handling Large Data Sets
6.5.3.1 Multilinear SVD Compression
6.6 Applications
6.6.1 Preprocessing
6.6.2 Fluorescence
6.6.3 Chromatography
6.6.4 Other Applications
References
Index
EULA
