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Source Separation in Physical-Chemical Sensing
Master advanced signal processing for enhanced physical and chemical sensors with this essential guide
In many domains (medicine, satellite imaging and remote sensing, food industry, materials science), data is obtained from large sets of physical/chemical sensors or sensor arrays. Such sophisticated measurement techniques require advanced and smart processing for extracting useful information from raw sensing data. Usually, sensors are not very selective and record a mixture of the useful latent variables. An innovative technique called Blind Source Separation (BSS) can isolate and retrieve the individual latent variables from a mixed-source data array, allowing for refined analysis that fully exploits these cutting-edged imaging and signal-sensing technologies.
Source Separation in Physical-Chemical Sensing, supplies a thorough introduction to the principles of BSS, main methods and algorithms and its potential applications in various domains where data are obtained through physical or chemical sensors. Designed to bridge the gap between chemical/physical analysis and signal processing, it promises to be invaluable in many fields. Its alertness to the latest technologies and the full range of potential BSS applications make it an indispensable introduction to this cutting-edge method.
Source Separation in Physical-Chemical Sensing readers will also find:
- BSS examples on chemical and physical sensors and devices to enhance processing and analysis.
- Detailed treatment of source separation in potentiometric sensors, ion-sensitive sensors, hyperspectral imaging, Raman and fluorescence spectroscopy, chromatography, and others.
- Thorough discussion of Bayesian source separation, nonnegative matrix factorization, tensorial methods, geometrical methods, constrained optimization, and more.
Source Separation in Physical-Chemical Sensing is a must-have for researchers and engineers working in signal processing and statistical analysis, as well as for chemists, physicists or engineers looking to apply source separation in various application domains.
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Veröffentlichungsjahr: 2023
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Table of Contents
Cover
Table of Contents
Title Page
Copyright
About the Editors
List of Contributors
Foreword
Preface
Notation
1 Overview of Source Separation
1.1 Introduction
1.2 The Problem of Source Separation
1.3 Statistical Methods for Source Separation
1.4 Source Separation Problems in Physical–Chemical Sensing
1.5 Source Separation Methods for Chemical–Physical Sensing
1.6 Organization of the Book
References
Notes
2 Optimization
2.1 Introduction to Optimization Problems
2.2 Majorization–Minimization Approaches
2.3 Primal‐Dual Methods
2.4 Application to NMR Signal Restoration
2.5 Conclusion
References
Notes
3 Non‐negative Matrix Factorization
3.1 Introduction
3.2 Geometrical Interpretation of NMF and the Non‐negative Rank
3.3 Uniqueness and Admissible Solutions of NMF
3.4 Non‐negative Matrix Factorization Algorithms
3.5 Applications of NMF in Chemical Sensing. Two Examples of Reducing Admissible Solutions
3.6 Conclusions
References
4 Bayesian Source Separation
4.1 Introduction
4.2 Overview of Bayesian Source Separation
4.3 Statistical Models for the Separation in the Linear Mixing
4.4 Statistical Models and Separation Algorithms for Nonlinear Mixtures
4.5 Some Practical Issues on Algorithm Implementation
4.6 Applications to Case Studies in Chemical Sensing
4.7 Conclusion
Appendix 4.AImplementation of Function
postsourcesrnd
via Metropolis-Hasting Algorithm
References
Notes
5 Geometrical Methods – Illustration with Hyperspectral Unmixing
5.1 Introduction
5.2 Hyperspectral Sensing
5.3 Hyperspectral Mixing Models
5.4 Linear HU Problem Formulation
5.5 Dictionary‐Based Semiblind HU
5.6 Minimum Volume Simplex Estimation
5.7 Applications
5.8 Conclusions
References
Notes
6 Tensor Decompositions: Principles and Application to Food Sciences
6.1 Introduction
6.2 Tensor Decompositions
6.3 Constraints in Decompositions
6.4 Coupled Decompositions
6.5 Algorithms
6.6 Applications
References
Notes
Index
End User License Agreement
List of Tables
Chapter 2
Table 2.1 Examples of discrepancy measures in NMF (with ).
Table 2.2 Summary of the algorithms presented in this chapter.
Chapter 4
Table 4.1 Example of (scalar) probability distributions encoding non‐negati...
Chapter 5
Table 5.1 Parameters of live hyperspectral sensors.
Table 5.2 A summary of some NMF formulations.
List of Illustrations
Foreword
Figure 1 Word cloud based on the titles of all publications of José Bioucas‐...
Figure 2 (a) José Bioucas‐Dias receiving the David Landgrebe Award. (b) José...
Chapter 1
Figure 1.1 Factorization of a 2‐D array as a product of two matrices.
Figure 1.2 Decomposition of a 2‐D array as a sum of products of rank‐1.
Figure 1.3 Factorization of a three‐way array as a sum of product of three...
Figure 1.4 Non‐gaussianity maximization principle is based on the observatio...
Figure 1.5 Calcium carbonate crystallization: (a) Raman spectra of a carbona...
Figure 1.6 Hyperspectral imaging : (a) six hyperspectral images of the south...
Figure 1.7 Scatter plots of three mixture data for different distributions o...
Chapter 2
Figure 2.1 Indicator function of the interval
Figure 2.2 Non‐lsc function (a) ; lsc function (b).
Figure 2.3 Epigraph of the absolute value function.
Figure 2.4 Function and a supporting line of it at (a). Graph of the sub...
Figure 2.5 Graph of function and a majorant function of at
Figure 2.6 MM quadratic algorithm. Function and a quadratic majorizing fun...
Figure 2.7 Baseline correction using BEADS. Baselines estimated in (a) and e...
Figure 2.8 Half‐quadratic majorant function. Function and quadratic majori...
Figure 2.9 Original sparse signal (a), observed measurements (b), and restor...
Figure 2.10 Result of PALMA method on an NMR DOSY experiment performed on a ...
Figure 2.11 Analysis of the real FT‐ICR‐MS spectrum of a peptide in trimer f...
Figure 2.12 Logarithmic barrier function associated to the constraint .
Figure 2.13 Estimated ‐ NMR distributions from real data (apple) using Alg...
Figure 2.14 Original signal (a) and noisy acquired measure (b).
Figure 2.15 Restored signal (2.104) for , NMSE .
Figure 2.16 Restored signal using Algorithm (2.106) for , NMSE .
Figure 2.17 Restored signal using Algorithm (2.110) for , NMSE .
Figure 2.18 Restored signal using Algorithm (2.115) for , NMSE .
Figure 2.19 Restored signal using Algorithm (2.117) for , NMSE .
Chapter 3
Figure 3.1 Geometric illustration of exact NMF for and . Both figures rep...
Figure 3.2 Restricted NPP instances of two nested squares corresponding to t...
Figure 3.3 Illustration of the polytope with five vertices containing the ...
Figure 3.4 Illustration of feasible NMF solutions in the case of a spectral ...
Figure 3.5 Illustration of feasible NMF solutions in the case of a three‐com...
Figure 3.6 Polarized Raman spectroscopy set‐up in backscattering geometry.
Figure 3.7 Polarized Raman data versus rotational angle for rutile singl...
Figure 3.8 Subset of source spectra estimated by NMF (25 runs) on (a) Crosse...
Figure 3.9 Estimated mixing coefficients by NMF (25 runs) for each polarizat...
Figure 3.10 Estimated mixing coefficients by NMF (25 runs) for both polariza...
Figure 3.11 Spectral signatures and spatial abundances of the sources obtain...
Chapter 4
Figure 4.1 Illustration of the standard Beta distribution whose support is c...
Figure 4.2 Illustration of the Dirichlet distribution shape in for differe...
Figure 4.3 (a) Example of simulated spectral sources where the ‐axis corr...
Figure 4.4 Simulated Markov chains (a) and empirical posterior distributions...
Figure 4.5 Markov chains (a) and empirical posterior distribution (b) of mix...
Figure 4.6 (a) Simulated (dotted) and estimated (continuous line) source sou...
Figure 4.7 Mixture spectra at the beginning of the phase transformation.
Figure 4.8 Estimated sources using a Bayesian source separation approach.
Figure 4.9 Evolution of the three component abundances: (a) for C, (b) at t...
Figure 4.10 Evolution of activities of the potassium (a) and ammonium (b) so...
Figure 4.11 Acquired signals through an ISE array: responses of potassium el...
Figure 4.12 Scatter plot of the mixtures (data provided by the ISE array).
Figure 4.13 Estimated sources (gray) using a Bayesian source separation appr...
Chapter 5
Figure 5.1 Paper counts per year in hyperspectral sensing by searching the S...
Figure 5.2 (a) Hyperspectral imaging concept. (b) Hyperspectral vectors repr...
Figure 5.3 Schematic view of three types of spectral mixing. (a) linear mixi...
Figure 5.4 Schematic view of the LMM. The observed bands, shown in the left ...
Figure 5.5 (a) Noisy and projected spectra from a simulated dataset. (b) Noi...
Figure 5.6 Estimate of the noise variance across bands in the Cuprite datase...
Figure 5.7 Illustration of the affine set estimation, affine projection, and...
Figure 5.8 Illustration of the simplex set , which is the convex hull of th...
Figure 5.9 Three datasets with different distributions of spectral vectors. ...
Figure 5.10 Taxonomy of the linear HU problems.
Figure 5.11 Numerical comparison of VolMax, VolMin, and sparse regression so...
Figure 5.12 Simplex volume minimization. Light gray circles, dark gray circl...
Figure 5.13 Illustration of abundances satisfying the ‐Pure Pixel Assumpt...
Figure 5.14 Transformation of a simplex to a polyhedron.
Figure 5.15 MV simplex estimation: (a) impact of an outlier; (b) impact of n...
Figure 5.16 Unmixing results of N‐FINDR, VCA, MVC‐NMF, and SISAL on a simula...
Figure 5.17 Unmixing results of N‐FINDR, VCA, MVC‐NMF, and SISAL on a simula...
Figure 5.18 Unmixing results of N‐FINDR, VCA, MVC‐NMF, and SISAL on a simula...
Figure 5.19 Unmixing results of N‐FINDR, VCA, MVC‐NMF, and SISAL on a highly...
Figure 5.20 Unmixing results: (a) TERRAIN HSI, (b) identified endmembers, (c...
Figure 5.21 Sparse regression solutions for 50 simulated spectral LMM gene...
Figure 5.22 Estimated signatures of real data set (counterfeit tablets) by S...
Figure 5.23 Abundance fractions estimated by SISAL/MVSA, MVES, and MCR‐ALS, ...
Chapter 6
Figure 6.1 A third‐order real tensor is nothing more than a three‐way array ...
Figure 6.2 (a): A fluorescence
Excitation‐Emission Matrix
(
EEM
) of a s...
Figure 6.3 A graphical representation of the CPD. A tensor is expressed as...
Figure 6.4 The suggested row‐wise vectorization (b) reads the entries of the...
Figure 6.5 Three unfoldings of tensor
Figure 6.6 A fluorescence excitation emission matrix (FEEM). The two areas s...
Figure 6.7 The same FEEM as in Figure 4.1 after removal of Rayleigh scatteri...
Figure 6.8 The results of split‐half analysis (black: set 1; gray: set 2).
Figure 6.9 The four score vectors of a CPD model plotted against the corresp...
Figure 6.10 Example of a set of samples measured by GC‐MS. The mass spectrum...
Figure 6.11 Estimated elution profiles from a seven‐component PARAFAC2 model...
Guide
Cover
Table of Contents
Title Page
Copyright
About the Editors
List of Contributors
Foreword
Preface
Notation
Begin Reading
Index
End User License Agreement
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Source Separation in Physical‐Chemical Sensing
Edited by
Christian JuttenUniversité Grenoble Alpes, Grenoble‐INP, CNRS, GIPSA‐labFrance
Leonardo Tomazeli DuarteUniversity of CampinasLimeira, Brazil
Saïd MoussaouiNantes Université, Ecole Centrale Nantes, CNRS, LS2NNantes, France
This edition first published 2024.© 2024 John Wiley & Sons Ltd
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About the Editors
Christian Jutten received PhD (1981) and Doctor ès Sciences (1987) degrees from Grenoble Institute of Technology, France. He was Associate Professor (1982–1989), Professor (1989–2019) at University Grenoble Alpes, where he is now Emeritus Professor since September 2019. Since 1980s, his research interests are in machine learning and source separation, including theory and applications (biomedical engineering, hyperspectral imaging, chemical sensing, speech). He is author/co‐author of four books, 125+ papers in international journals, and 250+ publications in international conferences.
Jutten was a visiting professor at EPFL, RIKEN labs, and University of Campinas. He served as director or deputy director of the signal/image processing laboratory in Grenoble (1993 to 2010), as scientific advisor for signal/image processing at the French Ministry of Research (1996–1998), and at CNRS (2003–2006 and 2012–2019).
Jutten was organizer or program chair of many international conferences, including the first Independent Component Analysis conference in 1999 (ICA'99) and IEEE MLSP 2009. He was the technical program co‐chair of ICASSP 2020. He was a member of the IEEE MLSP and SPTM Technical Committees. He was associate editor for Signal Processing and IEEE Trans. on Circuits and Systems, and guest co‐editor for IEEE Signal Processing Magazine (2014) and the Proceedings of the IEEE (2015). From 2021 to 2023, he was editor‐in‐chief of IEEE Signal Processing Magazine.
Jutten received many awards, e.g. best paper awards of EURASIP (1992) and IEEE GRSS (2012), Medal Blondel (1997) from the French Electrical Engineering society, and one Grand Prix of the French Académie des Sciences (2016). He was elevated as IEEE fellow (2008), EURASIP fellow (2013), and as a Senior Member of Institut Universitaire de France for 10 years since 2008. He was the recipient of a 2012 ERC Advanced Grant for the project Challenges in Extraction and Separation of Sources (CHESS).
Leonardo Tomazeli Duarte received the BS and MSc degrees in electrical engineering from the University of Campinas (UNICAMP), Brazil, in 2004 and 2006, respectively, and the PhD degree from the Grenoble Institute of Technology (Grenoble INP, Université Grenoble Alpes), France, in 2009. Since 2011, he has been with the School of Applied Sciences (FCA) at UNICAMP, Limeira, Brazil, where he is currently an associate professor. He is a Senior Member of the IEEE. In 2016, he was a Visiting Professor at the École de Génie Industriel (GI‐Grenoble INP, France). Since 2015, he has been recipient of the National Council for Scientific and Technological Development (CNPq, Brazil) productivity research grant. Since 2023, he is one of the principal investigators within the Brazilian Institute of Data Science (BI0S), one of the Brazilian Applied Research Centers on Artificial Intelligence. In 2017, he was the recipient of UNICAMP “Zeferino Vaz” Academic Recognition Award (for research and teaching performance at UNICAMP). In 2022, he was elected Affiliated Member (up to 40 years old) of the Brazilian Academy of Sciences (ABC). His research interests center around the broad area of data science and lie primarily in the fields of signal processing, decision aiding and machine learning, and also in the interplays between these fields.
Saïd Moussaoui received the MEng degree in electrical engineering from Ecole Nationale Polytechnique, Algiers, Algeria, in 2001; the MSc degree in Control, Signals and Communication from the University Henri Poincaré, Nancy, France, in 2004; and the PhD degree in Signal and Image Processing from Université Henri Poincaré, Nancy, France, in 2005. Since 2006, he has been with the Department of Automatics and Robotics at Ecole Centrale Nantes where he is currently a full Professor. He is a member of the group of Signals, Images and Sounds (SIMS) of the Laboratory of Digital Sciences of Nantes (LS2N, CNRS UMR 6004). His research interests are related to the field of signal and image processing, including methodological aspects of statistical inference, numerical optimization, and the application in various real‐life contexts such as chemical data analysis, remote sensing, and biological imaging.
List of Contributors
José M. Bioucas‐Dias
Instituto de Telecomunicacoes
Instituto Superior Técnico
Lisboa
Portugal
David Brie
CRAN
Lorraine University
Nancy
France
Rasmus Bro
Department of Food Science
University of Copenhagen
Frederiksberg
Denmark
Emilie Chouzenoux
OPIS, Inria Saclay
University Paris‐Saclay
Gif‐sur‐Yvette
France
Jérémy Cohen
CNRS, CREATIS
Lyon
France
Pierre Comon
CNRS
GIPSA‐lab
Université Grenoble Alpes
France
Nicolas Dobigeon
IRIT
INP‐ENSEEIHT
IRIT University of Toulouse
Toulouse
France
Leonardo Tomazeli Duarte
School of Applied Sciences (FCA)
University of Campinas
Limeira
Brazil
Nicolas Gillis
Department of Mathematics and Operational Research
University of Mons
Mons
Belgium
Christian Jutten
GIPSA‐lab
Univ. Grenoble Alpes
CNRS, Institut Univ.
de France
Grenoble
France
Wing‐Kin Ma
Department of Electronic Engineering
The Chinese University of Hong Kong
Hong Kong SAR
China
Saïd Moussaoui
LS2N, Nantes Université
Ecole Centrale Nantes
Nantes
France
Jean‐Christophe Pesquet
CVN, CentraleSupélec
University Paris Saclay
France
ForewordIn Memoriam: José M. Bioucas‐Dias (1960–2020), a Humble Giant
José Manuel Bioucas‐Dias (1960–2020) was a Professor at Instituto Superior Técnico (IST), the engineering school of the University of Lisbon, and a senior researcher at Instituto de Telecomunicações. After obtaining his PhD degree in Electrical and Computer Engineering, in 1995, from IST, he dedicated his research life to the area of signal and image processing, in problems related to the reconstruction, restoration, and analysis of images. In particular, he focused deeply on processing and analysis of remote sensing observations, a subject in which he was widely considered a world‐leading authority. José Bioucas‐Dias contributed to highly relevant and influential scientific and technical advances in synthetic aperture radar, interferometric radar, and hyperspectral imaging.
Figure 1 features a word cloud constructed from the titles of his numerous publications available from IEEE Xplore. Clearly, “hyperspectral” appears as a central topic of interest and he was indeed a key member of the hyperspectral imaging research community, be it from the geoscience and remote sensing point of view or from the signal and image processing point of view. The direct and indirect implications and applications of his contributions are numerous, namely in the processing and analysis of satellite images, whose impact on modern society is enormous.
For his contributions to image processing and analysis in remote sensing, José Bioucas‐Dias was elevated to Fellow of the IEEE (Institute of Electrical and Electronics Engineers) in 2016. In 2017, he received the first David Landgrebe Award, (Figure 2a) from the Geoscience and Remote Sensing Society (GRSS) of the IEEE, with the citation “for outstanding contributions in the field of remote sensing image analysis.” According to the GRSS, “the David Landgrebe award is a career award, given for extraordinary contributions in the field of remote observation image analysis.” In 2018 he was named Distinguished Lecturer by the IEEE GRSS.
Figure 1 Word cloud based on the titles of all publications of José Bioucas‐Dias, extracted from IEEE Xplore.
Figure 2 (a) José Bioucas‐Dias receiving the David Landgrebe Award. (b) José Bioucas‐Dias, general chair of the 3rd Workshop on Hyperspectral Image and Signal Processing, held in 2011 in Lisbon, Portugal.
Throughout his career, José Bioucas‐Dias published more than 100 articles in the most prestigious journals in his areas of work, as well as more than 200 articles in all the best signal and image processing and remote sensing conferences, where he received several “best paper awards.” The high international impact of his work is evident in the important distinctions mentioned in the previous paragraph, as well as in the fact that he was included in the 2015, 2019, 2020, and 2021 editions of the prestigious “Highly Cited Researchers” list (by Clarivate Analytics), which is reserved to the world elite of researchers with the greatest impact in their fields.
His recognition and prestige in the international scientific community are also evident in the fact that he was invited to join the editorial boards of the best journals in the area: the IEEE Transactions on Circuits and Systems, the IEEE Transactions on Image Processing (in which he was a senior editor), and the IEEE Transactions on Geoscience and Remote Sensing. He was also a guest editor for several special issues in these and other journals. He was the General Co‐Chair of the third IEEE GRSS Workshop on Hyperspectral Image and Signal Processing, Evolution in Remote sensing ((WHISPERS'2011, (Figure 2b)) and has been a member of program/technical committees of several international conferences. Even more revealing of his recognition is the high number (over 60) of invited presentations he gave at institutions and prestigious conferences around the world, from which around 20 plenary and keynote presentations stand out. He also held several visiting professor positions, at the invitation of several European universities: Tampere University of Technology (Finland, in 2008 and 2012), Université Grenoble Alpes (France, 2014), Université de Toulouse (France, 2013 and 2015), Università di Pavia (Italy, 2016).
In addition to being an excellent researcher, José Bioucas‐Dias was also a devoted teacher and educator. He supervised and co‐supervised 20 PhD theses and more than 10 post‐doctoral internships for researchers from several countries around the world. At IST, he created the first PhD course in inverse problems in image processing and, during his career, he taught many other courses (in the areas of signal processing, telecommunications, information and communication theory, remote sensing, to mention only the most significant), always with great dedication and high pedagogical and scientific quality. He was also engaged in various university management tasks, having joined the Scientific Council of IST.
José Bioucas‐Dias was one of the brightest and most influential scientists in the field of remote sensing image processing and analysis. His contributions were not only fundamental, but also of very significant practical applicability. His great scientific stature was matched by an exceptional modesty, courteous affability, and selfless readiness to help others. He was an exceptionally dedicated mentor to so many and a fantastic human being, totally driven by his passion for science, his creativity, and his rigor. He was always available to engage in scientific discussions, sit down with a student to dig into a problem until it was solved fair and square, sharing his joy and optimism, never giving up. On the international scientific scene, he was a giant; on his daily interactions, his humility was a striking feature. This rare blend made him a truly inspirational human being, the perfect role model, colleague, friend.
With his untimely death, we lost an outstanding researcher, a great educator, and a truly good man. We are truly honored to dedicate this book to him, highlighting his long‐lasting legacy to our community.
Mário A. T. Figueiredo
Instituto de Telecomunicações
Instituto Superior Técnico
Universidade de Lisboa, Portugal
Jocelyn Chanussot
Grenoble INP
University of Grenoble Alpes, CNRS
GIPSA‐Lab, France
Preface
Source separation is a very ubiquitous concept in signal processing, whose main objective, essential in signal processing, is to recover an information of interest. When the acquired signal is a mixture of sources, e.g., sounds from different speakers recorded by a microphone, or different ions measured by an ion sensitive sensor, due to the poor selectivity of sensors, recovering the different sources (speakers or ions, in the previous examples) is a tricky question. The main approach for solving the problem is to use several sensors instead of a unique one. Indeed, if the sensors are at different locations for sounds, or with different sensitivities for ions, each sensor provides a different instance of the mixture of the sources and, under mild conditions, a sufficient amount of information will thus be available for recovering each source.
With technological progresses, sensors are more and more numerous, common, and low cost, and many physical quantities are measured by arrays of sensors, so that source separation can be very useful in various domains. As a few examples, sensors have been used (i) in biomedical signal processing, for noninvasive fetal electrocardiogram extraction using a few chest electrodes, for extracting and locating brain sources from electroencephalogram scalp electrodes or magnetic resonance imaging, (ii) in satellite and airborne remote sensing for estimating the composition of ground surface, of planets, or of the universe from hyperspectral images, (iii) in chemical engineering for measuring the composition of electrolytes based on record with ion‐sensitive sensors or optical spectrometers for the inspection of molecular composition of materials.
Separating different sources from a set of observations (sensors) is still an ill‐posed signal processing problem that cannot be handled appropriately without additional hypotheses on the sought sources. If mutual statistical independence of sources can be an efficient property, which leads to very powerful classes of algorithms, namely independent component analysis, it cannot be used in all domains. Especially, for signals coming from physical/chemical sensors, the mutual independence property may be totally irrelevant. Conversely, non‐negativity is a very common property for such signals: it is typical when measuring spectra, counting ions, or assessing molecular concentrations and abundances. In addition to non‐negativity, sum‐to‐one and sparsity are also relevant properties. These properties can be used for designing very powerful algorithms able to solve the source separation problem.
This book proposes a tour of approaches and algorithms of source separation in physical/chemical sensing applications. The first chapter presents a comprehensive view of the source separation problem and points out the main approaches for its resolution. Since all the source separation algorithms are based on the optimization of a cost function constituted of a data fitting term and few regularization terms promoting the desired source properties or imposing some physical constraints, the second chapter is focused on basic principles and recent advances in mathematical optimization theory and algorithms. The four other chapters address different approaches for source separation: non‐negative matrix factorization (NMF), Bayesian estimation approaches, geometrical formulation based approaches, and tensor factorization methods. These four chapters present both a methodological view of each approach and examples of applications for illustrating its ubiquitous interest in various physical/chemical sensing applications. The authors of the chapters of this book are worldwide recognized scientists, with a strong expertise and many valuable contributions in the field of signal processing and particularly in developing source separation methods based on various approaches. They also have a long‐time experience in dealing with multidisciplinary signal processing applications in chemical sensing, hyperspectral imaging, and physical applications.
Life is short and hard. During the writing of this book, our friend José Bioucas‐Dias passed away, swept away by illness. All the co‐authors of this book dedicate this work to José, a great scientist, and colleague or friend. Our grateful thanks to Mário A. T. Figueiredo and Jocelyn Chanussot for writing the foreword of this book as a tribute to José.
Grenoble
Christian Jutten
Leonardo Tomazeli Duarte
Saïd Moussaoui
Notation
Signals, Vectors and Matrices
column vector of components ,
diagonal matrix whose entries are those of vector
, ,
global, whitening, and separating unitary matrices
matrix with components
,
mixing and separation matrices
,
mixing and separating (nonlinear) operators
number of sources
number of sensors
number of observed samples
, ,
sources, observations, separator outputs
vector whose components are the diagonal of matrix
Operators on Signals, Vectors, Matrices, Tensors, and Functions
complex conjugation of matrix Q
convolution
conjugate transposition of matrix Q
contraction over index
determinant of matrix
,
Fourier transform of signal and of
Hadamard (entry‐wise) product between arrays
infimum‐convolution
Khatri‐Rao (column‐wise Kronecker) product between matrices
Kronecker product between matrices
Kruskal's rank of matrix
pseudo‐inverse of matrix Q
rank of matrix
tensor product
trace of matrix
transposition of matrix Q
Random Variables and Vectors, Statistical Operators
contrast function
estimate of quantity
joint cumulant of variables
joint score function
or
Kullback divergence between and
likelihood
marginal cumulant of order of variable
marginal score function of source
,
mathematical expectation of
or
mutual information of
or
probability density function of random variable
or
Shannon entropy of
Sets
complex field
domain of function
epigraph of function
indicator function of set
projector on set
real field
,
sets
Operators for Optimization
conjugate of function
Frobenius norm
proximity operator of function within the metric induced by computed at
gradient of
Hessian of
minimum argument of over set
maximum argument of over set
Moreau envelope of of parameter
infimum of over set
infimum‐convolution
spectral norm
subdifferential set of at
supremum of over set
1Overview of Source Separation
Christian Jutten1, Leonardo Tomazeli Duarte2, and Saïd Moussaoui3
1GIPSA‐lab, Univ. Grenoble Alpes, CNRS, Institut Univ., de France, Grenoble, France
2School of Applied Sciences (FCA), University of Campinas, Limeira, Brazil
3LS2N, Nantes Université, Ecole Centrale Nantes, Nantes, France
1.1 Introduction
The purpose of this chapter is to give a general overview of the source separation problem and its underlying hypotheses, and of methods and algorithms for solving the problem, with an emphasis on the context of data processing in physical/chemical sensing applications.
1.1.1 Brief Introduction to Source Separation
Source separation is a very general problem in signal processing, and, more generally, in sensing. In fact, a basic task in signal processing consists in separating useful information (called signal) from non‐useful one (called noise) in noisy measurements. Measurements (also called observations) are frequently obtained through sensors, sensitive to some physical or chemical properties of the object which is analyzed. However, the sensor selectivity is limited so that its output depends on various phenomena (called sources).
As a first example, the signal captured by a microphone is the superimposition of signals emitted by all the acoustic sources in the neighborhood. Similarly, the signal measured by a scalp electrode in electroencephalography (EEG) is assumed to be the superimposition of the synchronous electrical activity of neural assemblies located in various areas in the brain. In remote sensing by hyperspectral imaging, due to low spatial resolution, the measured reflectance spectrum in each pixel is an aggregation of the reflectance spectra of all physical materials present in the ground surface related to this pixel. Finally, in chemical sensing, ion‐sensitive electrodes have been designed for measuring activity of specific ions; however, due to a limited selectivity, the sensor output depends on the activity of the main ion and on the activities of interfering ions which can be present in the solution.
Source separation problems are considered in a blind (unsupervised) framework, i.e., by assuming that only sensor measurements (called mixtures) are available, but neither the source signals nor the mixing process (the superimposition in the above examples) are known. The main concept for solving blind source separation (BSS) problems is based on diversity, whose simplest implementation is to use a large number of sensors, thus providing spatial diversity. Solving source separation requires first to model the observations, i.e., how the signals received by the sensors are related to the sources, and then to add some (weak) priors and hypotheses on these sources in order to ensure their separability, and therefore the separation problem becomes well‐posed.
The problem of source separation has been formulated in the middle of 1980s by Jutten and Hérault [1] for modeling motion decoding in vertebrates. Then, theoretical foundations have been developed mainly in the signal processing community by different researchers like Comon [2], Cardoso and Souloumiac [3,4], Pham and Garat [5], Pham and Cardoso [6], Delfosse and Loubaton [7], and in parallel in the Neural Networks and Machine Learning communities by researchers like Bell and Sejnowski in USA [8], Hyvärinen in Finland [9], Cichocki, Amari, and their team in Japan [10].
The interest for methods of source separation is due to its strong theoretical foundations [11] (Chapters 2–14) and to its very wide application domains [11] (Chapter 16). In 2022, with the keywords source separation, Google recalled more than 584 million entries!
From the application point of view, due to expansion of both low‐cost sensors and powerful computers that are able to process very fast huge data sets, the problem of source separation appears in several domains, like communications [11] (Chapters 15 and 17), audio and music processing [12, 11] (Chapter 19), biomedical engineering [11] (Chapter 18), and remote sensing and hyperspectral imaging [13]. First applications of source separation appeared in the middle of 1990s and were focused on biomedical problems: non‐invasive fetal electrocardiogram (ECG) separation [14] in 1994 and ECG processing [15] in 1996. Source separation has its success stories, too. As an example, applying the spectral matching independent component analysis algorithm [16] on the data provided by the Plank space mission, Cardoso was able to extract wonderful images of the Cosmic Microwaves Background, i.e., a very early image of our universe, which is a very important material for cosmologists.
In chemical engineering, it seems that source separation has been applied first for nuclear magnetic resonance spectroscopy [17] in 1998, and a larger number of works have been published in middle of 2000s as detailed by Monakhova et al. in the review [18]. In this context, even if Monte Carlo statistical methods have been proposed for mixture analysis [19], it must be noted that source separation is strongly related to positive matrix factorization [20] and to algebraic methods of tensor factorization, popular in Chemometrics and quoted PARAFAC [21].
1.1.2 Chapter's Organization
This chapter is a general introduction to the main concepts related to source separation. It is organized as follows. Section 1.2 presents the mathematical problem of source separation and a few basic solution principles. Section 1.3 focuses on source separation methods based on mutual statistical independence. Section 1.4 gives some examples of the various applications in physics and chemistry that can be formulated in the source separation framework. Section 1.5 is an overview of approaches that can be used for solving problems of Section 1.4. Section 1.6 details the organization of the book.
1.2 The Problem of Source Separation
1.2.1 Mathematical Description
Let us first consider a unique sensor and denote its output, at sample . Due to the poor selectivity of the sensor, one can model as a function of unknown sources, denoted , :
where models the mixing operator, unknown too.
In the simplest case, i.e. if the mapping is assumed linear, one could write:
Although this model is simple, we are faced with two problems: (i) the mixing coefficients are unknown, as well as the sources (ii) even if the mixing coefficients were known, the separation of the contributions coming from the various sources remains an ill‐posed problem.
The first problem can be solved through modeling or identification methods. Modeling requires to write propagation equations of signals from sources to sensors, and implicitly to know the locations of sources and sensors. Then, identification can be achieved but it requires a training set of many samples , , i.e., pairs of inputs/output, for performing supervised parameter estimation.
The second problem could be solved if we have additional information concerning the different sources. For instance, in the above example of acoustic sources recorded by a microphone, if we know that the useful source and non‐useful ones are characterized by different frequency bands, we can separate them by simple spectral filtering. Subtraction methods [22] can be also used, provided that one has a reference of the background noise (i.e. non‐useful sources).
As a conclusion, solving the source separation problem with a unique sensor is impossible without very strong priors.
The basic concept of methods for solving source separation problems is diversity. A simple way to enhance diversity is to use a few sensors, say , instead of only one. In that case, at each sample , instead of having a scalar observation, we have a ‐dimensional observation vector :
where denotes the multidimensional mapping between the sources and the sensors. Denoting the ‐size vector of sources, one can also write:
For instance, in the above examples, spatial diversity is obtained by using several microphones or few EEG electrodes located at different places, or several image pixels, or several ion‐sensitive sensors, specific to different ions. But another diversity is required, the sample diversity (time diversity in the above examples), i.e., the shapes of functions must be actually different. This sample diversity can be ensured by assuming that sources (considered as random variables) are mutually independent, or have different spectra or different variance time courses. Note that the discrete character of sources can be used instead [23].
The problem of source separation is often said blind in the signal processing community: in this context, blind refers to the fact that we only have observations , without a precise knowledge either on the sources or on the mapping . In other words, blind must be understood as unsupervised in machine learning. In such a case, identification of the mapping is not directly possible since we do not have input/output pairs for mapping estimation, and an accurate physical modeling is not possible since one has no idea about source locations.
However, weak information concerning the mapping and the sources are sometimes given and can be used for enhancing source separation. Here weak means that information is mainly qualitative, and it would be more realistic to qualify the methods as semi‐blind (or semi‐supervised).
1.2.2 Different Types of Mixing Models
Even if an accurate modeling of the mixing process is impossible to achieve, a simple study of the relationship between sources and sensors (through physical modeling of the propagation from sources to sensors) can specify the generic observation model (1.4) and consequently simplify the resolution of the source separation problem.
1.2.2.1 Linear Mixtures
If the mappings can be assumed as linear, the general Eq. (1.4) simplifies, and can be formulated and solved using linear algebra tools. Especially, the multidimensional mapping can be represented by a mixing matrix. In the case of linear mixtures, one can distinguish two situations, according to the signal propagation from sources to sensors: linear mixing and convolutive mixing.
If the propagation speed is very fast (instantaneous), the mapping between source and sensor is a simple scalar entry . At each sample , the observation on sensor is then a weighted sum of the sources , :
For the set of sensors, one can model the observations in compact form as:
For instance, linear instantaneous mixing models nicely fit the signal measured on scalp electrodes in EEG, since the propagation of electrical activity can be neglected following the quasi‐static approximation of Maxwell's equations.
Conversely, if the propagation delay cannot be neglected, the mapping between source and sensor has to be modeled as a linear filter with impulse response . The measurement on sensor is then the convolution product:
where denotes the convolution operator. Globally, for the set of sensors,
Due to the discrete nature of sampled data, the above model is usually written in ‐domain or in the frequency domain to handle the convolution operator. Actually, linear convolutive mixing models are required for modeling mixtures of acoustic sources. A survey on convolutive source separation models and methods in the context of speech and audio processing is given in [24]. The case of pure delays is addressed in [25]. In the framework of chemical sensing, it could be used for taking into account the diffusion of ions in a solution, or of gas molecules in air [26], from sources to sensors.
1.2.2.2 Nonlinear Mixtures
In some applications, the linear approximation is not sufficient to give an accurate description of the mixtures. The observations must be modeled as a nonlinear mappings of the sources, eventually taking into account propagation. As we will see in more detail in Section 1.4 and also in Chapter 4 (Section 4.4), nonlinear mappings must be considered for modeling observations provided by ion‐sensitive sensor arrays [27,28] where the ‐th sensor output, at sample , can be written as:
where and are physical constants and denotes the valence of the ‐th ion. The selectivity coefficients, , are unknown mixing parameters that model the interference between the ions of index and the electrode dedicated to ion .
Nonlinear models are also usual in hyperspectral imaging [13,29] when considering multiple reflections in 3D scenes, which can be approximated (if restricting to double reflections) by bilinear models. The light reflectance in a given pixel at wavelength is expressed as a nonlinear mixture of the reflectance spectra (denoted , ) of the materials present in the surface area covered by this pixel:
where coefficient represents the amount of interaction between different components.
1.2.2.3 Overdetermined, Determined, or Underdetermined Models
Beyond the nature of the mixing models, one can consider three different situations, depending on the number of sources and number of sensors.
If the number of sensors is equal to the number of sources, i.e. , and the mixing mapping is invertible, the problem is said to be determined as: estimation of (or of its inverse) leads directly to source separation, at least in the noiseless case.
If the number of sensors is larger than the number of sources, i.e. , and the mixing mapping is still invertible, the problem is overdetermined: a first step of dimension reduction and signal subspace estimation [3] is necessary for leading to a determined problem. In the determined or overdetermined cases, source separation methods can be based on the estimation of either the mixing mapping or the unmixing mapping (i.e. which is an inverse of , up to acceptable undeterminacies). Then, the source separation is achieved without extra effort. If noise is present, a matched filter should be used.
Conversely, if the number of sensors is smaller than the number of sources, i.e. , the problem is underdetermined: then the mixing mapping cannot be inverted ( does not exist!). Consequently, we can only estimate . Moreover, after knowing , source separation is a problem, not at all trivial, which requires extra priors for being solved. For underdetermined mixtures, estimation of mapping and estimation of sources are two different and tricky problems that require additional hypotheses on signals [30]. Practically, sparsity is a nice prior, realistic in many applications, which can lead to unique solution for underdetermined linear instantaneous mixtures. Other approaches, in particular deterministic, permit to recover the sources by exploiting extraneous diversities [31].
Actually, the main idea of source separation methods is not to directly estimate the sources, but just to exploit some weak properties of the sources and of the mixing model, inspired by the diversity concept, for estimating the mixing mapping , or its inverse if it exists. For this purpose, one can use indirect or direct approaches:
to either estimate the inverse of (indirect approach), so that the estimated sources satisfy a desired property,
or to estimate the mixing (direct approach) optimizing a cost function (e.g. maximum likelihood) in which the desired property of the sources is explicitly used.
One can also estimate both mixing and sources if at least three diversities are available (see Section 1.2.4 and Chapter 6).
Of course, whatever the adopted approach (direct or indirect), the choice of a suitable mixing model and of actual priors on source signals is essential for the success of source separation. Finally, as it will be detailed in Sections 1.2.3 and 1.3, the counterpart of weak priors is the typical ambiguities of the solution, which is usually not unique.
1.2.2.4 Noisy Mixtures
In fact, due to model simplification and measurement errors, it is mandatory to take into account a noise term in the mixing models. Usually, the noise is modeled by an additive term in Eqs. (1.6), (1.8), and (1.4). Zero mean Gaussian independent and identically distributed noise is generally assumed, even if the relevance of such hypothesis can be questioned, especially when considering physical/chemical measurements which are essentially non‐negative. In the case of applications involving photon counting and statistics, noise models based on Poisson distribution are taken instead.
Practically, even for the simple linear instantaneous model, source separation from noisy mixtures becomes more tricky. In fact, a noisy observation can be written as:
where denotes the noise due to measurement errors.
In practice, the estimation of (or of its inverse) is altered in the presence of the noise . However, even if we are able to perfectly estimate and then deduce its inverse , the source estimation becomes:
Clearly, the inversion problem is still ill‐posed [32] and the last right‐side term points out possible noise amplification, due to ill‐conditioning of the mixing matrix, so that the perfect source separation can lead to a poor signal noise ratio (SNR). This is the reason why a regularization using a matched filter is preferred [11] (Chapter 1).
1.2.3 From Source Separation to Matrix Factorization
Let us focus on the section on the particular case of linear instantaneous mixtures, i.e., the case where the observations can be modeled according to Eq. (1.6):
In fact, by merging all the samples of the observations, one can denote the 2‐D observation array of size and the 2‐D source array of size , so that the above equation can be written as:
This factorization is illustrated in Figure 1.1
Figure 1.1 Factorization of a 2‐D array as a product of two matrices.
Remember that, in the blind source separation problem, we only observe the data , and the mixing matrix and the sources are unknown. It means that solving source separation problem is equivalent to factorizing the matrix , of size as a product of two matrices, and , of size and , respectively.
1.2.3.1 Factorization Ambiguity
Two problems are related to this factorization. First, since we do not know the sources, probably we do not know the number, , of sources: choosing the size of and is then a first issue. The second problem is much more annoying: it is clear that the above factorization problem is ill‐posed, due to its non‐uniqueness. In fact, for any invertible matrix, :
the factorization remains unchanged. It means that there are an infinite number of possible factorizations of , up to any invertible matrix of size .
It is thus mandatory to add priors on sources for achieving uniqueness in the factorization, or at least restricting to an admissible set of solutions. The first idea which has been used in this purpose is to assume that the sources are mutually statistically independent: it leads to independent component analysis (ICA) methods, which will be detailed in Section 1.3. With ICA, the uniqueness is not fully achieved, but the matrix is reduced to the product of a permutation matrix by a diagonal scaling matrix:
i.e., the sources are mainly recovered up to a scale in an arbitrary order.
Other priors have been investigated. Non‐negativity of sources and/or of mixing coefficients is a suitable prior for many data (especially in image processing or with spectral measurement data), which can restrict the set of solutions: exploiting this prior leads to a large class of non‐negative matrix factorization (NMF) methods.
Source (or mixture) sparsity is also an efficient prior. A signal , is sparse if most of its samples are equal (or very close) to zero. Sparse component analysis (SCA) [33] has been intensively investigated and is especially very efficient for underdetermined mixtures, i.e., when the number of sensors is (much) less than the number of sources.
1.2.3.2 Data Representation
Finally, it is useful to note a nice property of the linear instantaneous model. Let us denote a transform which preserves linearity. Then, using , Eq. (1.6) becomes:
This means that, in the transformed space, the same linear relation (with the same mixing matrix ) remains true between the transformed observations and the transformed sources. Consequently, the factorization problem can be considered in the initial space or in the transformed space. The preservation of linearity especially holds with many transforms, e.g., Fourier Transform, Discrete Cosine Transform, and Wavelet Transforms.
As an example of the interest of solving the problem in another space, consider again the sparsity prior. If the signals in the initial space are not sparse, we can enhance their sparsity by applying Discrete Cosine and Wavelet Transforms. Then, in the transformed space, sparsity is enhanced and the factorization problem can be solved based on sparsity. Then, using the inverse transform, one can come back to the initial space.
1.2.3.3 Factorization Algorithms
Practically, there are many algorithms for factorizing the data as the product of two matrices. For instance, assuming source independence, one can compute a contrast function , which is maximal if the components of are mutually independent. The factorization can then be driven by the maximization of the contrast function with respect to the separating matrix . More details on contrast functions, and examples of algorithms, will be given in Section 1.3.2, or can be found in [11].
For other priors, like non‐negativity or sparsity, factorization algorithms are generally based on constrained optimization methods. For example, for non‐negativity constraints on and , the factorization can be achieved by solving:
where denotes the Frobenius norm of the matrix and means that all the entries of the matrix are non‐negative. The choice of the Frobenius norm could be discussed: it is optimal for an additive Gaussian distributed noise in Eq. (1.11), but this assumption is not very compatible with the non‐negativity of and . This will be discussed in Chapters 3 and 5.
Sparsity of means that the number of nonzero samples of is very small, i.e. , where denotes the cardinal of a set. Conversely, it also means, that at each sample , the probability that more than one source is nonzero is very weak. Sparsity can then be measured by counting the number of nonzero entries, either in each column or in each row of the matrix . Denoting briefly this measure with the pseudo‐norm , the pseudo‐norm of the ‐th column of is denoted . Then, the factorization can be obtained by solving the optimization problem:
Typically, the two optimization problems (1.17) and (1.18) are non‐convex, due to the minimization with respect to the two terms in the product . Although simple alternating minimization algorithms can be used, it is desirable to investigate more powerful methods for optimization with constraints compatible with physical or chemical data, which are often non‐negative, sometimes sparse (e.g., in mass spectrometry). Chapter 2 will focus on advanced methods for optimization subject to constraints.
1.2.4 Enhanced Diversity: Tensor Formulation and Factorization
Coming back to the linear model of Eq. (1.6):
Merging all the observation samples, one can denote the 2‐D observation array of size , and merging all the samples of the ‐th source in the vector , the previous equation can be written as:
where denotes the ‐th column vector of the matrix . This equation shows that the observations can be explained through latent variables (sources) which appear in each product through the time course, , and the weighting function on each sensor, . This formulation, illustrated in Figure 1.2, explicitly points out that can be decomposed as a sum of rank‐1 terms. This representation shows that each term, i.e., each latent variable, is characterized by its time shape, and its scale shape, i.e., the weights of the time shape on each sensor: this explicitly points out time and spatial diversities.
Figure 1.2 Decomposition of a 2‐D array as a sum of products of rank‐1.
Using tensorial product, one can write:
so that Eq. (1.19) becomes:
Up to now, we considered observations related to only two diversities (time and space, in the above examples), i.e., as measures of values returned by a function of two variables, , for different discrete values, , of these two variables:
The decomposition assumes that:
the two variables are separable so that ,
the 2‐D array can be decomposed as a sum of rank‐1 terms.
Now, let us assume that the observations are dependent on three variables, i.e. . This means that the measurements are obtained through three diversities. Measuring for all discrete values , leads to a 3D‐array, with a general term:
Assuming again the variables are separable, i.e. , we can decompose the 3‐D array of observations as a sum of terms (latent factors), each one being a tensorial product of three rank‐1 vectors:
where , and . Each latent variable is clearly associated with its “shape” along the three diversities. Figure 1.3 illustrates this decomposition.
Figure 1.3 Factorization of a three‐way array as a sum of product of three rank‐1 terms.
As an example of such three‐way array of data associated with three‐diversity measurements, one can consider fluorescence spectroscopy, where, at low concentration of the sample , the fluorescence intensity [34] can be written as a linear approximation of the Beer–Lambert law:
where denotes the absorption spectrum, the fluorescence emission spectrum, the fluorescence emission wavelength, and the excitation wavelength. Clearly, the fluorescence intensity is dependent on these three variables. Then, measuring the fluorescence intensity for a series of values of these three variables (emission spectrum, excitation spectrum, concentration) leads to a three‐way array of data, , of size with a general term:
In the presence of fluorescent solutes with low concentrations, additivity holds and the fluorescence intensity of the mixture can be written as a sum of terms:
Following this model, the three‐way array of data can be decomposed as:
This expression explains the data as a sum of latent variables (see Figure 1.3), each one being characterized by its emission spectrum, , its absorption spectrum, , and its concentration, . The above decomposition is an extension of source separation for three‐way arrays (generalization to higher dimension array is straightforward), and PARAFAC [21] is an usual way for computing this factorization. Chapter 6 will explain in more detail theoretical interest of tensor decomposition and related algorithms, and will provide few application examples.
1.2.5 From Supervised to Blind Solutions
As explained in Section 1.2
