12.1 Introduction

Hyperspectral imagery provides additional benefits over multispectral imagery in many applications, such as detection, discrimination, classification, quantification, identification, etc. In early days, hyperspectral imagery has been processed and analyzed by multispectral image processing algorithms via preprocessing such as feature extraction, dimensionality reduction, and band selection. Such multispectral-to-hyperspectral approaches have achieved some success and may have led to a brief that hyperspectral imaging is nothing more than a straightforward extension of multispectral image processing. As we will see, this is apparently not the case. When the spectral resolution is low as multispectral images are, the used image processing techniques are generally developed to explore spatial information such as geographical information system (GIS) (Jensen, 1996) for spatial domain analysis. Therefore, as spectral resolution is increased significantly like hyperspectral imagery, such spatial domain-based multispectral imaging techniques may be found to be less effective in certain applications. In particular, if targets of interest only account for a small population with very limited spatial extent, the techniques based on spatial information can easily break down. In some cases where the target size may be even smaller than the pixel resolution, for example, rare minerals in geology, special species in agriculture and ecology, small vehicles in battlefields, etc., the data analysis must rely on spectral information provided by a single pixel. Under certain circumstances, the analysis can be only performed at the subpixel level. In order to address this problem, spectral unmixing has been developed to exploit pixel-level spectral information for image analysis. Its success in both multispectral and hyperspectral image analyses has been demonstrated in many applications (Chang, 2003a).

In order to further facilitate spectral unmixing applications in hyperspectral imagery, Harsanyi and Chang developed a hyperspectral image classification technique, referred to as OSP from a viewpoint of hyperspectral imagery (Harsanyi and Chang, 1994). Their idea is based on two aspects: (1) how to best utilize the target knowledge provided a priori and (2) how to effectively make use of hundreds of available contiguous spectral bands. With regard to aspect (1), the prior target knowledge is characterized in accordance with target signatures of interest, referred to as the desired target signature, d, and undesired target signature matrix U formed by those target signatures that are not wanted in image analysis. We believe that OSP is the first approach proposed to separate d from the U in a signal detection model, and then eliminate the undesired target signatures in U prior to detection of d so as to improve signal detectability. As for aspect (2), the issue of how to effectively use available spectral bands can be best explained by the well-known pigeon-hole principle in discrete mathematics (Epp, 1995) as discussed in Section 1.3.2. The following example may help readers understand the concept behind OSP.

Suppose that there are 13 pigeons flying into a dozen of pigeon holes (nests). The pigeon-hole principle says that there must exist at least one pigeon hole that should accommodate at least two or more pigeons. Now, if we interpret target signatures of interest and the number of spectral bands as the pigeons and the number of pigeon holes, respectively, then we can use one spectral band to accommodate a distinct target signature for separation. In order to make sure that no more than one target signature is accommodated in a single spectral band, a spectral band that has been used to accommodate a target signature must be disposed. In doing so, the principle of orthogonality is introduced as a mechanism to separate one spectral band from another so that target signatures accommodated in two separate spectral bands are orthogonal to each other. In this case, one band will not share target information with another band. However, for this approach to be effective, the number of spectral bands must be no less than the number of target signatures of interest. For hyperspectral imagery this requirement seems to be met automatically and the pigeon-hole principle is always valid. Unfortunately, using spectral dimensionality as a means to perform target detection, classification and identification is generally not applicable to multispectral imagery, which usually has fewer spectral bands than the number of target signatures of interest. For instance, a SPOT image data has three spectral bands that can be used for data analysis. If more than three target signatures need to be analyzed, the idea of using spectral bands for target detection and classification may not work effectively (Chang and Brumbley, 1999). To circumvent this difficulty, Ren and Chang developed a generalized OSP that included a dimensionality expansion technique to expand the number of spectral bands nonlinearly for OSP to have sufficient spectral dimensions to carry out orthogonal projection (Ren and Chang, 2000). Its utility was further extended to magnetic resonance (MR) image classification (Wang et al., 2001, 2002; Wang, 2002), Wong (2010) and Chapter 32 in this book.

Many OSP-based algorithms have been developed for various applications (Chang, 2003a) since OSP was introduced in 1994 (Harsanyi and Chang, 1994) and its potential in hyperspectral data exploitation is yet to be explored. For example, the noise assumption is not necessarily Gaussian as commonly assumed. If the noise is assumed to be Gaussian, it has been shown (Settle, 1996; Chang, 1998; Chang et al., 1998) that Harsanyi and Chang's OSP classifier performed essentially like the Gaussian maximum likelihood estimator (Settle, 1996). Nevertheless, from a technical point of view, the design concepts of these two techniques are different. Harsanyi and Chang's OSP classifier is derived from the signal-to-noise ratio (SNR) using a signal detection approach compared to the Gaussian maximum likelihood estimator that is a parametric estimation-based approach. So, technically speaking, Harsanyi and Chang's OSP classifier is a soft decision-made detector to be used to perform classification, more specifically, unmixing. Interestingly, Harsanyi and Chang's OSP can be further shown to perform as a least-squares estimator by including a scaling constant to account for LS estimation error, which is identical to the least squares solution derived from least squares OSP (LSOSP) (Tu et al., 1997; Chang, 1998; Chang et al., 1998).

When OSP was first developed, it required the full knowledge of endmembers to form a linear mixing model to be used to unmix data sample vectors. Such complete a priori information may be difficult to obtain in reality, if not impossible. Two approaches have been developed to mitigate this dilemma. One is to develop unsupervised algorithms to obtain the necessary endmember information directly from the data to be processed (Chang, 2003a: Chapter 5). This type of information is referred to as a posteriori information as opposed to a priori information provided in advance prior to data processing. Since the accuracy of the a posteriori information is closely related to the unsupervised method to be used to generate the information, it may not be always reliable. To avoid this problem, a second approach is to suppress unknown information without actually knowing it. One way to do so is the constrained energy minimization (CEM) developed by Harsanyi in his dissertation (Harsanyi, 1993), which only needs the knowledge of the desired signal source. Other than this desired signal source, no knowledge is required. This approach is particularly useful and attractive in the case that the image background is not unknown or very difficult to characterize. CEM was later extended to the target-constrained interference-minimized filter (TCIMF) (Ren and Chang, 2001), which characterized signal sources into three separate information sources, desired, undesired, and interference. Using this three-source model, TCIMF could detect multiple desired signal sources, annihilate undesired signal sources, while suppressing interference caused by unknown signal sources at the same time. Comparing to OSP that only deals with desired and undesired signal sources and CEM that only considers the desired signal source without taking into account other signal sources, TCIMF combines both OSP and CEM into one filter operation and includes them as its special cases, respectively. Interestingly, as will be shown, CEM and TCIMF can be viewed as various versions of OSP operating different degrees of target knowledge. In other words, OSP can be considered as a spectral correlation-whitened version of TCIMF, while TCIMF can be thought of as OSP-version of CEM that eliminates rather than suppresses the undesired signatures. Specifically, when the sample spectral correlation matrix in TCIMF is whitened (i.e., de-correlated), TCIMF performs as if it was OSP. On the other hand, when CEM operates in the same way that OSP eliminates the undesired target signatures, CEM becomes TCIMF. In either case, both CEM and TCIMF are derived from OSP and can be regarded as variants of OSP based on the knowledge used in their filter design. Various relationships among these approaches have been documented in Chang (2002a), Chang (2003b), and Chang (2005).

With all things considered as above, we investigate two intriguing issues in this chapter, which are “to what extent can OSP be applied?” and “how does OSP operate on prior target knowledge?” The first issue will be addressed by deriving OSP from three signal processing perspectives, signal detection, linear discriminant analysis, and parameter estimation that provide evidence that OSP is indeed a versatile technique for a variety of applications. In doing so, we introduce two new signal models, called (d,U)-model and OSP model. The former separates a desired signal source d from undesired signal sources in U based on knowledge provided a priori so that these two different types of signal sources can be taken care of separately. The latter annihilates the undesired signal sources in U from the (d,U)-model via an OSP operator to reduce the interference caused by the U so that the detectability of d can be further enhanced and increased. The second issue will be investigated by looking into how target information is used in OSP. Of particular interest is an issue of “how does CEM perform compared to OSP, provided that the undesired signal sources are also known a priori and can be annihilated before CEM is applied?.” More specifically, “how does CEM perform compared to OSP if OSP-model is used?” While addressing this issue, many interesting results can be obtained based on such OSP-model. Interestingly, under this circumstance, the commonly used least squares-based linear spectral mixture analysis turns out to be OSP. Additionally, we will also show how OSP can be implemented without prior knowledge where OSP takes advantage of the sample spectral correlation to approximate the information that is supposed to be provided by prior knowledge but is not available at the time of data processing. As a result, OSP operates the same form of the RX algorithm developed by Reed and Yu (1991). Furthermore, the low probability detector developed in (Harsanyi, 1993) can, therefore, also be interpreted as a variant of OSP from this aspect by assuming the unity vector as a desired target signature vector.