**Date Published:** April 30, 2018

**Publisher:** Springer International Publishing

**Author(s):** R. Kannan, A. V. Ievlev, N. Laanait, M. A. Ziatdinov, R. K. Vasudevan, S. Jesse, S. V. Kalinin.

http://doi.org/10.1186/s40679-018-0055-8

**Abstract**

**Many spectral responses in materials science, physics, and chemistry experiments can be characterized as resulting from the superposition of a number of more basic individual spectra. In this context, unmixing is defined as the problem of determining the individual spectra, given measurements of multiple spectra that are spatially resolved across samples, as well as the determination of the corresponding abundance maps indicating the local weighting of each individual spectrum. Matrix factorization is a popular linear unmixing technique that considers that the mixture model between the individual spectra and the spatial maps is linear. Here, we present a tutorial paper targeted at domain scientists to introduce linear unmixing techniques, to facilitate greater understanding of spectroscopic imaging data. We detail a matrix factorization framework that can incorporate different domain information through various parameters of the matrix factorization method. We demonstrate many domain-specific examples to explain the expressivity of the matrix factorization framework and show how the appropriate use of domain-specific constraints such as non-negativity and sum-to-one abundance result in physically meaningful spectral decompositions that are more readily interpretable. Our aim is not only to explain the off-the-shelf available tools, but to add additional constraints when ready-made algorithms are unavailable for the task. All examples use the scalable open source implementation from https://github.com/ramkikannan/nmflibrary that can run from small laptops to supercomputers, creating a user-wide platform for rapid dissemination and adoption across scientific disciplines.**

**Partial Text**

The development of physical and spectroscopic imaging methods in the last two decades has given rise to large multidimensional datasets, with examples including electron energy loss spectroscopy imaging in (scanning) transmission electron microscopy [1–4], bias and time spectroscopies in scanning probe microscopy [5–8], hyperspectral Raman and optical imaging [9–12], and spatially resolved mass spectrometry measurements [13–15].

We begin with introducing the conventions used in the equations. We use capital case letter such as A to denote matrices and lower case a for vectors. The one indexed lower case such as ai is a scalar value and represents the vector element at ‘i.’ Similarly, the two-indexed upper/lower cases such as Aij or aij represents the scalar value—also called element of the matrix at the location (i,j). We often require a scalar value for the entire matrix or vector, and one example that can be computed is the so-called matrix or vector norm. More formally a norm is represented as documentclass[12pt]{minimal}

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begin{document}$$|left| A right||_{q} :A in {mathcal{R}}^{m times n} to {mathcal{R}}$$end{document}|A|q:A∈Rm×n→R. The typical values for q are 1, 2, and F called as ℓ1-norm, ℓ2-norm, and Frobenius norm, respectively. Table 1 defines each of these norms, and also offers a quick reference for many of the terms used in this paper. Also, if there is a comparison relation defined between a matrix/vector and a scalar, the relations are defined against every element in the matrix or a vector to the vector. For e.g., A > 0 means every element in the matrix is non-negative and similarly for a vector it is represented as a > 0.Table 1NotationsNotationRemarksdocumentclass[12pt]{minimal}

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begin{document}$$sqrt {mathop sum nolimits_{i = 1}^{m} mathop sum nolimits_{j = 1}^{n} A_{ij}^{2} }$$end{document}∑i=1m∑j=1nAij2—square root of the sum of the squares of all the elements of the matrix||A||1documentclass[12pt]{minimal}

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begin{document}$$sumnolimits_{i = 1}^{m} {sumnolimits_{j = 1}^{n} {left| {A_{ij} } right|} }$$end{document}∑i=1m∑j=1nAij—sum of absolute values of all the elements. Here absolute value means the non-negative value without its sign||a||2documentclass[12pt]{minimal}

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begin{document}$$sqrt {sumnolimits_{i = 1}^{m} {a_{i}^{2} } }$$end{document}∑i=1mai2—square root of the sum of the squares of all the elements of the vectorμMean of a vectorKL(P||Q)Defines the similarity between two matrices P and documentclass[12pt]{minimal}

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begin{document}$$sumnolimits_{i = 1}^{m} {sumnolimits_{j = 1}^{n} {left( {P_{ij} log frac{{P_{ij} }}{{Q_{ij} }} } right)} }$$end{document}∑i=1m∑j=1nPijlogPijQij

In this section, we will introduce the matrix factorization problem and its connection with the linear unmixing explained above. Subsequently, we explain our matrix factorization framework (MFF) that offers a pragmatic framework of incorporating many real-world physical constraints. We introduce the popular linear unmixing techniques principal component analysis (PCA) and non-negative matrix factorization (NMF) under this framework and finally, discuss the examples of the two real-world constraints, sparsity and spatial smoothness, as preferential soft constraints with non-negativity on endmembers. The aim of this section, is to provide domain scientists sufficient information to extend the existing off-the-shelf algorithms with additional domain constraints they will encounter during their experiments, hopefully facilitating better understanding and use of multidimensional spectral data.

The key questions that arise from the previous sections are (a) How does one define the approximation X ≈ UV? (b) How to incorporate the properties of the input data X, for e.g., positive numbers? (c) How can specific domain knowledge—such as, e.g., the representative spectra should be spatially correlated, it’s a matrix of signals, etc. be incorporated? Most of these questions are addressed in matrix factorization process as one of the following: (refer to Table 1 for details of notations or definitions in this section).

In this section, we begin with the illustrative workflow in Fig. 3 of the unmixing process followed by scientists.Fig. 3Unmixing workflow for domain scientists

In this tutorial paper, we discussed the utility of matrix factorization for performing linear unmixing of imaging and spectroscopic data commonly acquired via microscopy modalities. We presented a matrix factorization framework to implement different physical constraints such as sparsity, spatial smoothness, and non-negativity to constrain the unmixing, leading to more meaningful and interpretable endmembers and abundance maps. We compared the benefits of enforcing different physical constraints on ToF-SIMS data such as non-negativity (NMF), orthogonality without non-negativity (PCA), spatial smoothness, and sparsity on the resulting spectra and abundance maps. Finally, we presented detailed examples of the use of constrained matrix factorization approaches on different spectroscopy data, including X-ray microscopy and scanning probe microscopy datasets. This paper uses the open source NMF implementation from https://github.com/ramkikannan/nmflibrary. The imposition of such physical constraints here and in other machine-learning algorithms will be critical to better understand physical mechanisms in large multidimensional datasets commonly acquired in modern-day imaging facilities.

Source:

http://doi.org/10.1186/s40679-018-0055-8