15.1 Introduction

Linear mixture analysis is a theory developed for solving linear problems. It has found many successes in a wide range of applications, such as linear regression analysis in multivariate data analysis, blind source separation in signal processing, and partial volume estimation in magnetic resonance imaging (see Chapter 32). Specifically, LSMA has been widely used in remote sensing community to perform spectral unmixing (Chapters 12–14) where a data sample vector is linearly mixed by a number of so-called endmembers as a linear mixture from which it can be further unmixed as abundance fractions in terms of these endmembers. Using the same notations in Section 12.2 and Eq. (12.2) let r be an L-dimensional data sample vector and be signatures of interest that are used to unmix the sample vector r. To carry out spectral unmixing, a linear mixing model is required to represent r in terms of the following form:

(15.1)

where is a signature matrix, n is a noise vector and can be used to describe a model or measurement error, and is an unknown p-dimensional abundance vector needed to be found and is associated with with α_j representing the abundance fraction of the jth endmember m_j present in the sample vector r. Due to physical constraints, two abundance constraints are generally imposed on (15.1), which are abundance sum-to-one constraint (ASC) specified by and abundance non-negativity constraint (ANC) specified by for all . The LSMA develops techniques to perform the so-called linear spectral unmixing (LSU) that solves a linear inverse problem of (15.1) by unmixing a data sample vector r via a set of p endmembers, , through finding their respective abundance fractions with/without the abundance constraints, ASC, and ANC.

Over the past years, LSMA has been extended in various directions to enhance its capability in spectral unmixing. For example, in order for LSMA to deal with random signals, a random version of LSMA called linear spectral random mixture analysis (LSRMA) based on projection pursuit was developed in Chang et al. (2002) and Chang (2003a). Since LSE is generally not an optimal criterion used for classification, a different version of LSMA based on Fisher's ratio, called FLSMA, is also derived from Fisher's linear discriminant analysis in Chapter 13 (Chang and Ji, 2006b). Interestingly, it is also shown in Chapter 14 as well as in Chang and Ji (2006a) that introducing a weighting matrix into the LSE criterion results in a new LSMA technique, called WAC-LSMA that includes LS-LSMA and FLSMA as its special cases. However, all such extensions intend to increase and enhance their ability for linear separability. Unfortunately, due its nature in inherent constraints resulting from the use of a linear mixing model this is probably the best we can do with LSMA without going beyond linear approaches. To resolve this dilemma two approaches seem feasible. One is to directly use a nonlinear mixture model, called intimate spectral mixture (Hapke, 1981) to perform spectral unmixing. Such an approach was investigated in Guilfoyle et al. (2001) and Guilfoyle (2003) where radial basis function (RBF) neural networks were used to approximate the unknown parameters used in a nonlinear mixing model. The other approach can be considered as a compromise between linear and nonlinear models. It maps the non-linear decision boundaries made by a classifier via a nonlinear function into a generally high-dimensional feature space in which non-linear separability problems can be solved by linear decisions. The use of such a nonlinear function is similar to nonlinear activation functions used in neural networks for network's internal learning. The approach of this type is known as a kernel-based technique where nonlinear functions are modeled as nonlinear kernels. Interestingly, using kernels to extend the LSMA has not received much attention until a kernel approach developed by Kwon and Nasrabadi (2005) who extended the orthogonal subspace projection (OSP) developed by Harsanyi and Chang (1994) to its kernel counterpart, called kernel-based OSP (KOSP). Later at nearly the same time another LSMA technique, NCLS, and FCLS were further extended to their counterparts, called kernel-based NCLS (KNCLS) and kernel-based FCLS (KFCLS) in Broadwater et al. (2007) and Liu et al. (2009). It should be noted that when the versions of KNCLS and KFCLS were derived in Liu et al. (2009) the details of KNCLS and KFCLS in Broadwater et al. (2007) were not available at that time but only published later in a book chapter (Camps-Valls and L. Bruzzone, 2009). The detailed derivations in Liu et al. (2009) provide a direct extension of LSOSP, NCLS, and FCLS to their respective kernel counterparts. In essence, the works reported in Broadwater et al. (2007), Camps-Valls and L. Bruzzone (2009), and Liu et al. (2009) are actually derived independently. Nevertheless, this chapter along with Liu et al. (2012) are believed to be the first that derives kernel counterparts of all the three backbone LS-LSMA techniques, LSOSP, NCLS, and FCLS, in a unified framework.

Despite that kernel-based approaches have shown promising in many remote sensing applications it does not imply that KLSMA always has advantages over LSMA in hyperspectral image analysis. As a matter of fact, it will be shown that kernel-based techniques do not necessarily improve classification performance for hyperspectral digital imagery collection experiment (HYDICE) data. This gives rise to an interesting question: Under what circumstances will KLSMA techniques be effective when they are used for spectral unmixing? This chapter tries to answer this question by providing experiments conducted based on two data sets, the visible/infrared imaging spectrometer (AVIRIS) Purdue Indiana Indian Pine data and the HYDICE data to demonstrate that kernel-based approaches can significantly improve performance if hyperspectral data have low spatial resolution such as AVIRIS data, but they cannot do the same for high spatial resolution HYDICE even both data sets do have the same spectral resolution.