Analysis of spectral mixtures is important in remote sensing imaging spectroscopy, because essentially the spectrum of any pixel of a natural scene is a mixture. The analysis of mixed spectra, known as Spectral Mixture Analysis (SMA), is the subject of this chapter. SMA attempts to answer two questions: (a) What are the spectra of the individual materials? (b) What are the proportions of the individual materials? We focus on linear mixing because of its relative analytical and computational simplicity and because it works satisfactorily in many practical applications.We discuss the physical aspects of the linear mixing model, geometrical interpretations and algorithms, and statistical analysis using the theory of least squares estimation. The main applications of SMA are in the areas of hyperspectral image interpretation and subpixel target detection.

Spectral Mixing

When a ground resolution element contains several materials, all these materials contribute to the individual pixel spectrum measured by the sensor. The result is a composite or mixed spectrum, and the “pure” spectra that contribute to the mixture are called endmember spectra. Spectral mixtures can be macroscopic or intimate depending on what scale the mixing is taking place (see Figure 9.1).

In a macroscopic mixture the materials in the field of view are optically separated in patches so there is no multiple scattering between components (each reflected photon interacts with only one surface material). Such mixtures are linear: that is, the combined spectrum is simply the sum of the fractional area times the spectrum of each component. Linear mixing is possible as long as the radiation from component patches remains separate until it reaches the sensor.

In an intimate mixture, such as the microscopic mixture of mineral grains in a soil or rock, a single photon interacts with more than one material. In this case, mixing occurs when radiation from several surfaces combines before it reaches the sensor. These types of mixtures are nonlinear in nature and therefore more difficult to analyze and use.

To illustrate some of the issues involved in SMA, we discuss some examples of mixed spectra provided by Adams and Gillespie (2006). Figure 9.2(a) shows spectra for mixtures of a material having a featureless spectrum (quartz) with a material having a spectrum with diagnostic absorption bands (alunite).

This chapter will address the optical principles that underpin any imaging spectrometer and then delve into the details of the dominant optical designs. The complete imaging spectrometer typically includes a scan mechanism, such as a rotating mirror, a telescope fore optic to image the scene at the input of a spectrometer, the spectrometer, which can accomplish the separation of the radiance into the different wavelength bins either through dispersive or interferometric means, and the focal plane array where the signal is converted to digital numbers. The optical flow from the scene to the resulting digital signals is depicted in Figure 4.1. The details of the optical system are presented and the description focuses on the concepts required to enable the reader to understand, at an introductory level, how these complex systems work.

This chapter will also introduce the unifying concept of the measurement equation. All imaging spectrometers share spatial, spectral, and radiometric properties, which can be mathematically described in a general way. Important concepts that are common to all of the optical forms presented are captured succinctly in a simple and elegant way. It is the measurement equation that provides the explanatory framework upon which the optical details will be built. An understanding of Gaussian or geometrical optics is required, and an overview that includes the basics of image formation and the concepts of pupils and stops is presented as an appendix. The appendix also includes an introduction to optical aberrations. Some of the equations that are developed in the appendix are referenced in this chapter.

Optically, imaging spectrometers are properly thought of as integrated systems rather than being comprised of the individual subsystems of the fore optics, the spectrometer, and the detector or detector array. The optical design engineer considers the system in its entirety during the design phase to ensure that the spatial, spectral, radiometric, and signal-to-noise ratio goals are accomplished and that the imaging spectrometer is manufacturable. This is a rather obvious observation, but merits emphasis. An imaging spectrometer is a system and not the sum of its parts, even though they will be addressed here through a subsystem analysis.

The utility of the data from an imaging spectrometer critically depends upon the quantitative relationship between the scene in nature and the scene as captured by the sensor. As was shown in Chapter 4, the raw data will only somewhat resemble the at-aperture spectral radiance from the surface due to the optical characteristics of the fore optic and the spectrometer. Additionally, the data acquisition by the focal plane array will further modify the irradiance that composes the image due to the spectral dependence of the detector material's quantum efficiency. It will also add noise terms that, if large enough, will further complicate the relationship between the scene and its raw image. The calibration of the data from an imaging spectrometer is the crucial data processing step that transforms the raw imagery into radiance that can be physically modeled. The science of radiometry provides the theoretical framework and the measurement processes that enable a sensor to be characterized and the data to be converted to physical units that are tied to reference standards. This chapter describes the process of sensor characterization that leads to calibration products that are applied to the raw data and some of the techniques that are used to evaluate the accuracy and precision of the calibrated data will be introduced. An overview of the important measurement and data reduction processes for vicarious calibration, which is critical for space-based systems, will also be presented.

Introduction

The characterization of an imaging spectrometer is challenging due to the spatial extent of the collected scene and the large spectral range that is relatively finely sampled, at least for an imager. For example, an Offner–Chrisp imaging spectrometer often has between 200 and 400 spectral and about 1000 spatial samples or about 400,000 individual measurements in a single readout of the focal plane array. To collect a scene the FPA is read out thousands of times. All of the data must be calibrated in order to be used to greatest effect, with the overarching goal being that the result should not depend upon the time or location of the collected scene, there should be no field of view dependence, and it should be immune, within reasonable limits, to the illumination and atmospheric conditions.

Having laid the foundation of the physical mechanisms involved in scattering and absorption of light in the previous chapter, we now turn to the spectral reflectance and emittance properties of materials encountered in remote sensing using imaging spectroscopy. Here, we will qualitatively describe how those fundamental processes are observed in the spectra of common materials. Although we are not seeking to provide a comprehensive review of either the spectra or materials encountered in the practice and application of imaging spectroscopy, our goal is to provide a sufficient familiarity with both the principals and observations typically encountered and provide a basis for further analysis.

Introduction

The discipline of spectroscopy was created by Isaac Newton and has become an integral part of how the identity, physical structure, and environment of atoms, molecules, and solids are described. Its development goes back to the early seventeenth century when WilliamWollaston improved upon Newton's original spectrometer to show that the solar spectrum had gaps, which were further investigated by Joseph Fraunhofer, who created a catalog of what we now know as Fraunhofer lines. Joseph Foucault later identified two of the Fraunhofer lines as being from sodium emission. It is the description of this process of identification, using the unique spectral features characteristic of a particular material and captured in an imaging spectrometer measurement, that is the overarching goal of this book. In this chapter we will address the features themselves, while Chapter 4 is devoted to a description of how spectral measurements are performed using an imaging spectrometer, which is an evolution of the classical systems. The spectra are divided into those from reflectance, where the sun is an active source, or emittance, where the temperature and emissivity determine the spectral signature. The measured spectra will further be divided into those signatures that are due to organic materials, minerals, or are from man-made surfaces.

The interaction of photons with the solid state structure of the surface materials introduces the features that are indicative of a particular substance. This interaction can be described at the macroscopic level through the classical theory based on Maxwell's equations; however, the interaction at the microscopic scale is between photons and the atoms that compose the solid and requires a quantum mechanical description.

Hyperspectral imaging data are typically treated as vectors in a high-dimensional space. Dimensionality reduction refers to a variety of techniques that enable the representation of vector data using a lower number of components without significant loss of information. The information in a data set is conveyed by the geometrical arrangement of points in the p-dimensional scatter plot rather than by the system of coordinates used for their specification. Therefore, besides the natural coordinate system defined by the original variables (spectral channels), we may wish to analyze the data in other coordinate systems with more desirable properties. Typically, the new coordinates are derived variables without any physical meaning. In this chapter we introduce linear spectral transformations for dimensionality reduction and “feature enhancement” of hyperspectral imaging applications. The most widely used technique is principal component analysis, followed by discriminant analysis, and canonical correlation analysis. We conclude with the related subject of spectral band prediction and its applications.

Introduction

The dimension of hyperspectral imaging data space is equal to the number of the spectral channels used by the sensor, which is typically in the range of a few hundred channels. High dimensionality has two important implications in hyperspectral data processing and exploitation. First, the resulting huge volume of data requires tremendous storage and processing resources. Second, the high-dimensionality of the feature space leads to a large increase in the amount of data required for statistically oriented detection and classification algorithms.

Given a hyperspectral data cube, dimensionality reduction can be achieved in essentially two different approaches. The first, which is called band selection or feature selection, attempts to identify a subset of the original bands that contribute to performance. Given a set of p bands, what is the best subset of size m? To solve this problem, we should evaluate the adopted optimality criterion for all possible combinations of m bands out of p and select the combination that minimizes or maximizes the criterion. The main problem is that the number of possible combinations, which is given by p!/[(p-m)!m!], the selected subset is extremely large, even for small values of m and p. There exist both optimum and suboptimum search methods, but they are, in general, computationally demanding or infeasible (Webb and Copsey, 2011).

In Chapter 6 we mirror closely the exposition given in the previous chapter on regression, beginning with the approximation of the underlying data generating function itself by bases of features, and going on to finally describing cross-validation in the context of classification. In short we will see that all of the tools from the previous chapter can be applied to the automatic design of features for the problem of classification as well.

Automatic feature design for the ideal classification scenario

In Fig. 6.1 we illustrate a prototypical dataset on which we perform the general task of two class classification, where the two classes can be effectively separated using a nonlinear boundary. In contrast to those examples given in Section 4.5, where visualization or scientific knowledge guided the fashioning of a feature transformation to capture this nonlinearity, in this chapter we suppose that this cannot be done due to the complexity and/or high dimensionality of the data. At the heart of the two class classification framework is the tacit assumption that the data we receive are in fact noisy samples of some underlying indicator function, a nonlinear generalization of the step function briefly discussed in Section 4.5, like the one shown in the right panel of Fig. 6.1. Akin to regression, our goal with classification is then to approximate this data-generating indicator function as well as we can using the data at our disposal.

In this section we will assume the impossible: that we have clean and complete access to every data point in the space of a two class classification environment, whose labels take on values in ﹛-1, 1﹜, and hence access to its associated indicator function y (x). Although an indicator function is not continuous, the same bases of continuous features discussed in the previous chapter can be used to represent it (near) perfectly.

Approximation of piecewise continuous functions

In Section 5.1 we saw how fixed and adjustable neural network bases of features can be used to approximate continuous functions. These bases can also be used to effectively approximate the broader class of piecewise continuous functions, composed of fragments of continuous functions with gaps or jumps between the various pieces.