# Probabilistic Mixture Model-Based Spectral Unmixing
## 1 Introduction  
Mixing is ubiquitous in nature, ranging from the slow mixing in the Earth's mantle to manufacturing processes involving the blending of viscous polymers or even the dispersion of solids into rubber [1]. The process and mathematics of mixing are of great interest in physics and engineering applications, from mixing in process engineering to the chaotic mixing of horseshoe maps. The reverse of this process, referred to as unmixing, segregation, or separation, is also of great interest but challenging. It involves recovering information lost through the changes in entropy due to mixing. The process is possible computationally if the mixing is incomplete (i.e., there is variation in the mixing fractions at different sites in the material), and sufficient spectral signal is discernible from the individual mixture components. Unmixing is commonly attempted for cases where the equations governing mixing are linear [2], although non-linear unmixing methods, such as kernel-based approaches and neural networks, have also been developed. Computational unmixing is critical for understanding and controlling processes like electrolysis refining [3], petroleum refining [4], and chemical analysis via chromatography [5], besides others. Spectral unmixing, a specific application of computational unmixing, aims to identify and quantify material components in mixed signals by comparing sets of spectral measurements to a library of known signatures. Each observation in such a set of measurements contains a spectral footprint corresponding to the underlying material mixture. The subsequent task is inverting these data to identify each endmember's unique spectral signature and quantify the proportion of each endmember in each mixture. This process finds applications in diverse fields, from geology and agriculture to astronomy and material sciences. However,unmixing represents an ill-posed problem, and the solutions may not be unique. In this light, it is essential to represent the uncertainties associated with the unmixing process. To illustrate the property of mixture diversity, consider a color photograph represented as a set of multi-dimensional arrays of weights, with endmembers identifiable as distinct colors: red, blue, and green. A color photograph is an example of a heterogeneously mixed set of observations that can be mapped to points on a 3-simplex, and the maximally diverse color photograph is that which contains at least one each of a pure (or completely) red pixel, a pure blue pixel, and a pure green pixel (i.e., the vertices of the simplex). The concept of an endmember extends beyond discrete colors in many scientific applications. The array of weights can be multi-dimensional, including non-spatial components like time, scattering vectors, chemical compositions, red-shifts, and more. Rather than viewing hyperspectral imaging as the sole technique, we adopt a broader perspective where the observations need not be a specific type of spatial map. Instead, they can be treated as an unordered set of spectra in which other descriptors (e.g., spatial parameters) are not necessarily needed for the endmember extraction, and thus, the problem we are addressing is the more general task of spectral unmixing. The success of spectral unmixing relies on the selection of a spectral tool to analyze the data, the spectral distinctiveness of the endmembers, and the diversity of available mixtures. Given a particular choice of spectral measurement tool, the most salient considerations are noise and mixture diversity. Additional challenges arise from factors such as non-linear mixing and data distortions. Noise, or, more accurately, noise relative to the spectral separation between endmembers, determines the ease of distinguishing hypothetical pure spectral signatures. Thus, noisy observations can complicate extraction as can homogeneous mixtures, in which endmember extraction becomes nearly impossible even in the absence of noise.