17.1 Introduction

With high spectral/spatial resolution many unknown material substances can be revealed by hyperspectral imaging sensors for data exploitation, specifically LSMA where a set of signatures used to form a linear mixing model may not be known by prior knowledge or be identified visually. Under such circumstances performing SLSMA with assumed target knowledge assumed a priori or obtained by visual inspection may not be realistic or applicable to real-world problems. Therefore, it is highly desirable to obtain the desired signature knowledge directly from the data without appealing for prior knowledge. In doing so, two major issues need to be addressed: (1) the number of signatures, denoted by p, used to form a linear mixing model and (2) a set of appropriate p signatures, , used to unmix data. Both issues are very challenging because determining the value of p and finding a desired set of p signatures must be conducted by an unsupervised means.

Since a hyperspectral signature is obtained by hundreds of contiguous spectral channels, the spectral correlation across all the spectral bands is very crucial and useful for material identification. In this chapter, we introduce a new concept of so-called spectral targets to differentiate spatial targets commonly addressed in traditional image processing. In the traditional image processing there are no spectral bands involved. The targets of interest are generally defined and identified by their spatial properties such as size, shape, and texture. Accordingly, the targets of this type are considered as “spatial” targets. The techniques developed to recognize such spatial targets are referred to as spatial domain-based image processing techniques. On the other hand, due to use of spectral bands specified by a range of wavelengths a multispectral or hyperspectral data sample is actually a vector expressed as a column vector, of which a sample in a spectral band is produced by a particular wavelength. As a consequence, a single hyperspectral sample vector already contains abundant spectral information provided by hundreds of contiguous spectral bands that can be used for data exploitation. Such spectral information within a single data sample vector is referred to as interband spectral information (IBSI) in this chapter. By virtue of such IBSI two single data sample vectors can be discriminated, classified, and identified via a spectral similarity measure such as spectral measures presented in Chapter 16. In light of this interpretation a target is called “spectral target” if it is analyzed based on its spectral properties characterized by IBSI as opposed to “spatial target” analyzed by interpixel spatial information provided by spatial correlation among sample pixels.

More specifically, let be a set of N data sample vectors where is the ith data sample vector in S and L is the total number of spectral bands. The spectral correlation across all the spectral bands within r_i is defined and referred to as interband spectral information of signature r_i, denoted by IBSI(r_i). That is, the IBSI(r_i) is provided by spectral correlation among the L spectral values, across spectral bands within the single data sample vector r_i. For example, second-order statistics provided by IBSI(r_i) can be auto-correlation of r_i, , or cross correlation of r_i, . However, what we are really interested in is the sample statistics provided by a set of data sample vectors, , denoted by IBSI(S), specifically, second-order statistics of IBSI(S) such as sample auto-correlation matrix of S, and sample cross-correlation matrix of S, . It should be noted that the IBSI(S) is independent of intersample spatial correlation because IBSI(S) remains the same even the samples in S are reshuffled. This type of spectral information is opposite to sample spatial statistics commonly used in traditional image processing that takes into account spatial locations of a set of data sample vectors where reshuffling data sample vectors in a sample spatial correlation matrix can result in another different sample spatial correlation matrix because it alters spatial correlation when data sample vectors are re-arranged in different spatial coordinates.

One of major strengths resulting from a hyperspectral imaging sensor is its ability in uncovering and revealing subtle material substances that cannot be resolved by multispectral imager. Such target signal sources are very critical and vital to hyperspectral image analysts. Unfortunately, their presence is also limited to their sample size and spatial extent. One effective means is to take advantage of IBSI to calculate the sample statistics provided by these samples, denoted by a set S^target, in terms of IBSI(S^target). In other words, the sample size of S^target is generally very small compared to a large sample pool of background (BKG) signatures, denoted by S^BKG. As a consequence, the IBSI(S^target), can be better characterized by high order of statistics (HOS), while the IBSI(S^BKG) constitutes most of second-order statistics. In this case, we can define a background signature as a signature that is of no interest in applications and is usually characterized by second-order sample statistics provided by IBSI(S^BKG), and a target signature as a desired signature that is of major interest and can be mainly specified by high-order sample statistics provided by IBSI(S^target). Of course, when a signature exhibits both characteristics of second-order statistics and high-order statistics in terms of IBSI, it will be considered as a target signature. In hyperspectral image analysis this assertion seems reasonable because the spectral targets of interest in hyperspectral data exploitation are generally those that either occur with low probability or have small populations when they are present. In particular, these types of spectral targets are usually relatively small, appear in small population, and also occur with low probabilities, for example, special spices in agriculture and ecology, toxic wastes in environmental monitoring, rare minerals in geology, drug/smuggler trafficking in law enforcement, combat vehicles in the battlefield, man-made objects and anomalies in intelligence gathering, landmines in war zones, chemical/biological agents in bioterrorism, weapon concealment, and mass graves. These spectral targets are generally considered as insignificant objects in terms of IBSI(S) because of their very limited spatial information provided by a small sample pool S, but they are actually critical and crucial for defense applications and are insignificant compared to targets with large sample pools and generally hard to be identified by visual inspection. From a statistical point of view, the spectral information statistics of such special targets cannot be captured by second-order statistics of IBSI(S) but rather by HOS of IBSI(S).

Once hyperspectral signatures are categorized into background signatures and target signatures according to their sample spectral statistics characterized by IBSI, the next follow-up task is to design and develop algorithms to extract signatures from both categories that can be used to form a linear mixture model for the SLSMA to unmix target signatures in S^target, whereas the background signatures in S^BKG will be used for BKG suppression so as to enhance target detectability and discriminability. Two remaining issues needed to be resolved are (1) how to determine the numbers of signatures in the BKG as well as target classes and (2) how to find these two categories of signatures, BKG as well as target signatures. While the first issue can be addressed by the concept of virtual dimensionality (VD) recently developed in Chapter 5, the second issue is the major focus to be addressed in this chapter where two approaches, least squares-based ULMSA (LS-ULSMA) and component analysis-based ULSMA (CA-ULSMA), are developed to find a set of so-called virtual signatures (VSs) according to IBSI(S)-defined BKG and target signatures. The term of VS introduced here intends to distinguish it from the commonly used term of “endmember” that is assumed to be a pure signature that may not be a real data sample vector and also from the term of virtual endmembers used in Tompkins et al. (1997) and Bowles and Gilles (2007) in endmember extraction.

From the above IBSI(S)-defined BKG class the signatures in the BKG class are most likely characterized by second-order statistics of IBSI(S) compared to signatures of interest in the target class which will be more likely to be captured by HOS of IBSI(S) as outliners due to their small spatial presence. In this case, high-order spectral targets are assumed to be desired targets for image analysis, while second-order spectral targets are considered as undesired targets for which we would like to annihilate or suppress prior to data processing so as to improve image analysis. So, an unsupervised LS-based algorithm designed on second-order spectral statistics can only be used to extract spectral targets of second-order statistics of IBSI(S^BKG). In order for an LS-based algorithm to be able to extract HOS of IBSI(S^target), we sphere the data by removing the first- and second-order spectral statistics information from the original data so that the sphered data consist of only those data sample vectors characterized by high-order statistics, which contain desired targets. So, if an unsupervised LS-based algorithm operates on two data sets, original data and its sphered data, it can extract both second-order and high-order spectral targets. As a result, an LS-based algorithm can accomplish two goals, finding BKG VSs from the original data space and in the mean time it can also extract target VSs from the sphered data space. The LSMA makes use of these two sets of VSs, BKG and target VSs, to form a linear mixing model to perform spectral unmixing is referred to as LS-ULSMA.

In a parallel development to LS-ULSMA, an alternative approach is to develop component analysis (CA)-based techniques that statistically de-correlate the data into a set of spectral components so that various levels of target information can be captured and characterized in individual and separate spectral components. Unlike LS-ULSMA that operates the same LS-based algorithm on two data cubes, that is, the original data and sphered data, the CA-based approach operates different component analysis transforms on the same data cube to capture targets characterized by any order of statistics specified by IBSI(S). It is known that principal components analysis (PCA) is a second-order statistics-based transform, which uses an eigenmatrix made up of all eigenvectors to produce a set of ranked principal components (PCs) in accordance with the magnitude of data variances represented by eigenvalues. Since BKG signatures usually have a large population, which generally contributes data variances in spectral statistics in terms of IBSI(S^BKG), it is expected that the first few PCs should retain most BKG signatures of the data. Target signatures, in contrast, usually have a small sample pool size, which contributes very little to second order of spectral statistics. Consequently, these target signatures can be rather characterized by high orders of spectral statistics, IBSI(S^target). In order to capture these types of target signatures, a preprocessing is required to remove background signatures prior to extraction of target signatures. The independent component analysis (ICA) seems to be a perfect candidate for this task because ICA has been widely studied in hyperspectral imaging community. It performs data sphering to remove the first two orders of spectral statistics in terms of IBSI(S^BKG) and then produces a set of statistically independent components (ICs) for signal source separation. By means of ICA target signatures characterized by IBSI(S^target) can be extracted in separate and individual ICs. The only two issues that remain to be resolved are (1) how to determine the numbers of PCs and ICs and (2) how to extract BKG signatures from PCs and target signatures from ICs. The concept of VD once again provides a feasible solution to the first issue. As for the second issue it can be solved by finding background signal sources corresponding to the maximal projections of each of PCs and target signal sources corresponding to maximal and minimal projections of each of ICs. To meet this need a CA-based unsupervised virtual signature finding algorithm (CA-UVSFA) is particularly developed to allow users to find both BKG VSs and target VSs that can be used to form a linear mixing model for SLSMA to unmix data. Such an SLSMA that makes use of the signatures found by the UVSFA as signature knowledge to perform linear spectral unmixing is referred to as CA-based ULSMA.

In order to substantiate the two developed techniques to perform ULSMA, LS-ULSMA, and CA-ULSMA, synthetic image experiments are first used to conduct a quantitative study between SLSMA and LS-ULSMA/CA-ULSMA. It is then followed by real data experiments to evaluate performance analysis in comparison with SLSMA for which the signature knowledge is provided a priori either by ground truth or visual inspection. As demonstrated by experimental results, when SLSMA uses accurate signature knowledge SLSMA would perform better than LS-ULSMA/CA-ULSMA. Otherwise, LS-ULSMA/CA-ULSMA is a better option than SLSMA.