Resolving Spectral Mixtures

With Applications from Ultrafast Time-Resolved Spectroscopy to Super-Resolution Imaging

• Publisher: Elsevier
• Year: 2016
• ISBN:
  9780444636447
• Topic: Computer Science
• Subtopic: Bioinformatics
• Index: Computer Science

Resolving Spectral Mixtures: With Applications from Ultrafast Time-Resolved Spectroscopy to Superresolution Imaging offers a comprehensive look into the most important models and frameworks essential to resolving the spectral unmixing problem—from multivariate curve resolution and multi-way analysis to Bayesian positive source separation and nonlinear unmixing. Unravelling total spectral data into the contributions from individual unknown components with limited prior information is a complex problem that has attracted continuous interest for almost four decades.

Spectral unmixing is a topic of interest in statistics, chemometrics, signal processing, and image analysis. For decades, researchers from these fields were often unaware of the work in other disciplines due to their different scientific and technical backgrounds and interest in different objects or samples. This led to the development of quite different approaches to solving the same problem. This multi-authored book will bridge the gap between disciplines with contributions from a number of well-known and strongly active chemometric and signal processing research groups.

Among chemists, multivariate curve resolution methods are preferred to extract information about the nature, amount, and location in time (process) and space (imaging and microscopy) of chemical constituents in complex samples. In signal processing, assumptions are usually around statistical independence of the extracted components. However, the chapters include the complexity of the spectral data to be unmixed as well as dimensionality and size of the data sets. Advanced spectroscopy is the key thread linking the different chapters. Applications cover a large part of the electromagnetic spectrum. Time-resolution ranges from femtosecond to second in process spectroscopy and spatial resolution covers the submicronic to macroscopic scale in hyperspectral imaging.

• Demonstrates how and why data analysis, signal processing, and chemometrics are essential to the spectral unmixing problem
• Guides the reader through the fundamentals and details of the different methods
• Presents extensive plots, graphical representations, and illustrations to help readers understand the features of different techniques and to interpret results
• Bridges the gap between disciplines with contributions from a number of well-known and highly active chemometric and signal processing research groups

Chapter 1

Introduction

C. Ruckebusch¹    Université de Lille, Sciences et Technologies, LASIR CNRS, Lille, France
¹ Corresponding author: email address: [email protected]

Abstract

This chapter introduces very basic information related to the resolution of spectral mixtures, which is a topic of ubiquitous interest in data analysis and signal processing with applications covering different scientific fields such as chemistry, biology, or remote sensing. This chapter also provides broad definitions, sets the overall context, and gives insight into the organization of the book.

Keywords

Chemometrics; Spectral; Mixture; Multivariate; Curve Resolution

1 Introduction

This chapter introduces first very basic information about the topic of the book and sets the overall context. It provides broad definitions and clarifies some points regarding the terminology. The second part provides information about the organization of the book. A first insight into the content of the 19 chapters composing the book, and their interplay, is given. The intention of these few words of introduction is mainly the presentation of the issues that will be tackled more comprehensively along the chapters of the book. These questions can be roughly put as follows:

– What is a spectral mixture?

– What does resolving a spectral mixture mean?

– What are the different ways to tackle the spectral mixture issues?

– What difficulties remain?

– And what are the perspectives?

2 The Spectral Mixture Problem

A spectral mixture is a data that results from the observation of a chemical system composed of (mixed) individual components and submitted to some variation. This variation is related to the change of an external factor, which is usually a physical or chemical variable. It can be for example sampling time, position, or pH. The spectral data thus consist of a superposition, or mixture, of the pure spectra of the individual components and their associated proportions. When dealing with evolving systems such as chemical reactions or processes, these proportions correspond to concentration profiles.

Spectral mixture data are usually arranged in a matrix with columns as spectral variables (wavelength, wavenumbers, etc.) and objects (time, position, etc.) as rows. Objects can be of different nature, but should always be clearly related to the state of the before mentioned physical or chemical variable. Ideally, the variations contained in the spectral data translate what is supposed to be relevant information for the problem at hand. Spectral mixture resolution aims to decompose the variations of the spectral data into a model of the contributions from the individual unknown components. These components are composed of source proportions and spectral signatures. It is important to realize that, more than often, this decomposition is aimed at situations for which little a priori information is available. It should also be noted that, in practice, some physical perturbations or chemical interferences may complicate the ideal situation.

In chemistry, spectral mixture resolution corresponds to the resolution of complex mixture spectra into pure contributions, consisting of concentration distributions and spectra of the different chemical components. The basic model underlying this decomposition, usually termed multivariate curve resolution (MCR) in chemometrics, corresponds to the Lambert-Beer law written in a matrix form. This factorial model states a bilinear relation between the matrix of observations and the two matrices of contributions containing concentration profiles and spectra, respectively. It should be noted that this extends to the analysis of spectral and hyperspectral images when investigating a specimen (in microscopy) or a scene (in remote sensing). Also, the bilinear model can be extended for the analysis of multiple data sets that are meant to connect different experiments together. Overall, MCR can be applied in situations where a reasonable approximation of the bilinear model, or any other fundamental basic equation that has the same mathematical structure, holds.

Application of MCR methods is broad, quite straightforward, and provides results which are readily chemically/physically interpretable. These assets explain why MCR has spread in the chemical literature and in many other scientific fields. However, considering the mathematical conditions for exact resolution of the MCR problem, some theoretical issues remain and are currently the subject of intensive research. The most puzzling of these issues is the so-called rotational ambiguity of the resolution. In more common words, this translates into the fact that a unique solution cannot be obtained in general. Then, particular attention should be paid to the initial condition, or to the constraints applied during resolution, and it is important to assess the extent of rotational ambiguity before any definitive conclusion to be drawn. Considering these aspects, one may notice a certain antagonism in MCR between wide applicability and high interpretability on the one hand and mathematical complexity of the resolution on the other hand. This explains to a large extend the continuous development of this topic into a proper research field, still very much in progress.

Taking a broader perspective, spectral unmixing enters the more general category of inverse problems, important, and ubiquitous problems in analytical science and data analysis. From a set of (spectral) observations, one aims to extract the unknown sources that produced the data but could not be observed directly. Mixture analysis, MCR, blind source separation, linear unmixing, etc. are methods that share this objective but were developed in different scientific fields, chemistry, statistics, or signal and image processing.

3 Book Content and Organization

The book starts with Chapter 2 that introduces the key concepts and provides an overview of the progress in MCR with an emphasis on applications to spectroscopic data. Focus is on constraints, multiset analysis, and quantitative aspects in multivariate curve resolution alternating least squares (MCR-ALS). Next, Chapter 3 revisits the concept of variable purity, with purity defined as the observation of a nonzero contribution from one and only one of the mixture components. Issues and solutions relative to rotational ambiguity of the MCR solutions, currently a very active research topic, are then discussed in Chapters 4 and 5. Chapter 4 sets the basis of the problem and focuses on a nonlinear constrained optimization approach for the direct calculation of maximum and minimum band boundaries of feasible solutions. In contrast, Chapter 5 aims to provide a systematic introduction to the concept of area of feasible solutions, from which feasible solutions can be derived. With Chapter 6, spectral unmixing and spectral mixture analysis are introduced. These methods aim at extracting the spectral characteristics and quantifying the spatial distribution over a spectral image. This chapter goes beyond the state of art by introducing nonlinear approaches to SU which allows to take into consideration more complex mixing process or spectral variability of the sources. Chapter 7 covers the basic of independent component analysis, a source separation method initially developed in the field of telecommunications and now applied in different domains including chemometrics and spectroscopy. Chapter 8 deals with a Bayesian positive source separation approach of the MCR problem which is motivated by the search of unique solutions. The second part of the book, oriented more towards applications, starts with Chapter 9. It introduces a wavelet compression strategy that facilitates the application of MCR to large data sets. Chapter 10 deals with chromatography coupled with spectral detection, the type of data which originally motivated development of MCR, and extends to the application of trilinear approaches. With Chapter 11, the focus is on the application of MCR-ALS for ultrafast time-resolved absorption spectroscopy data. Chapter 12 tackles the analysis of hyperspectral images of biological samples with the use of automated data preprocessing and improved MCR methods, increasing the sensitivity and accuracy of the chemical images obtained. In Chapter 13, the integration of wavelet transform with multivariate image analysis in a multiresolution analysis approach opens the possibility of simultaneously accomplishing denoising and feature selection. With Chapter 14, a new constraint that allows forcing some information related to the low-frequency character of the components profiles and distribution maps in MCR-ALS is introduced. Chapter 15 discusses the potential of super-resolution in vibrational spectroscopy imaging, merging instrumental and algorithmic developments. Chapter 16 deals with the current topic of biomarker imaging for early cancer detection applying MCR to magnetic resonance images. Chapters 17 and 18 provide ways and means to deal with remotely sensed data. Chapter 17 focuses on the use of spectral libraries for spectral mixtures analysis. How to compose, handle, and optimize endmember libraries are the issues discussed in detail. Chapter 18 reviews the recent developments of spectral unmixing algorithms that incorporate spatial information, termed spatial-spectral unmixing. To close, Chapter 19 presents a sparse approach for spectral unmixing of hyperspectral images, which provides better interpretability of the results obtained.

Chapter 2

Multivariate Curve Resolution-Alternating Least Squares for Spectroscopic Data

A. de Juan*^,1; R. Tauler^{† *} Universitat de Barcelona, Barcelona, Spain
^† Institute of Environmental Assessment and Water Research (IDÆA) SPANISH COUNCIL OF SCIENTIFIC RESEARCH (CSIC), Barcelona, Spain
¹ Corresponding author: email address: [email protected]

Abstract

The chapter describes the algorithm multivariate curve resolution-alternating least squares (MCR-ALS), paying special attention to applications to analyze spectroscopic data. The main application fields addressed are process and image analysis. A brief comment on the specific use of MCR for quantitative analysis is done. Finally, a small note on how MCR-ALS compares to similar bilinear or multilinear decomposition methods is written as a conclusion of this chapter.

Keywords

Multivariate curve resolution-alternating least squares; MCR-ALS; Process analysis; Im...