1.3 Divergence of Hyperspectral Imagery from Multispectral Imagery

The hyperspectral imagery has changed the way we think of multispectral imagery. This is because we now have hundreds of contiguous spectral bands available at our disposal. So, one major issue is how to effectively use and take advantage of spectral information provided by these hundreds of spectral bands for various applications in data exploitation, for example, target detection, discrimination, classification, quantification, and identification. This interesting issue can be addressed by the following two interesting examples. The first example uses real-to-complex analysis to illustrate why it is inappropriate to simply extend multispectral imaging techniques to process hyperspectral imagery. The second example uses the well-known pigeon-hole principle in discrete mathematics (Epp, 1995) to illustrate how hyperspectral imagery can be addressed by a rationale completely different from that used for multispectral imagery.

1.3.1 Misconception: Hyperspectral Imaging is a Natural Extension of Multispectral Imaging

While dealing with hyperspectral imagery there is a general consensus that hyperspectral imagery is a natural extension of multispectral imagery based on an assumption that a hyperspectral image has more spectral bands for data collection than a multispectral image does. As a result, it may lead to a misconception that hyperspectral imaging problems can be solved by multispectral imaging techniques by simply taking advantage of its expanded spectral bands. A similar misconception also occurs in hyperspectral data compression where researchers in data compression community consider a hyperspectral data as an image cube so that 3D image compression processing techniques developed for videos can be simply applied to hyperspectral imagery as a 3D image cube without extra precaution (see Part V: Chapters 19–23). Unfortunately, over the past few years these misconceptions have somewhat directed the way we design and develop hyperspectral imaging techniques.

To understand the fundamental difference between multispectral imaging and hyperspectral imaging, we use a simple mathematical example to illustrate a similar misconception, which is finding derivatives in real analysis and complex analysis. Since real variables can be considered as real parts of complex variables, this may lead to a brief that real analysis is a special case of complex analysis, which is certainly not true. One piece of clear evidence is derivatives. When a derivative is calculated in the real line, the direction with respect to which a derivative is calculated along the real axis is constrained either to the left or to the right. However, the direction along which a derivative is calculated by complex analysis can be along any curve in the complex plane. As a result, calculating a complex derivative is more sophisticated than simply extending the way derivatives are calculated in real analysis. A natural extension of real derivatives is partial derivatives in complex analysis along two axes: x-axis and y-axis. However, it is not true for any derivative calculated in the complex plane. This is because the direction along which the derivative is calculated is not only limited to x- and y-axes but it must also take into account all directions that are more likely curves instead of lines. When such a derivative occurs it is called total differentiable or analytic and must satisfy the so-called Cauchy–Riemann equation that allows a differentiable complex variable to be expanded as a power series which is much stronger than only derivatives. This simple example explains why complex analysis is not a natural extension of real analysis and a direct extension of real derivatives to complex derivatives as partial derivatives can only achieve limited success to some extent. This example sheds some light on a similar key difference between multispectral and hyperspectral images. In its early days multispectral imagery has been used in remote sensing mainly for land cover/land use classification in agriculture, disaster assessment and management, ecology, environmental monitoring, geology, geographical information system (GIS), and so on. In these applications, low spectral resolution multispectral imagery may provide sufficient information for data analysis and the techniques developed for multispectral image processing are primarily based on pattern classes that take advantage of spatial correlation to perform various tasks. Compared to multispectral imagery, hyperspectral imagery utilizes hundreds of contiguous spectral bands to perform target-class analysis. This is the major difference between hyperspectral imagery and multispectral imagery. Specifically, the objects of interest in hyperspectral imagery are no longer patterns of large areas as considered in multispectral imagery. Instead, hyperspectral image analysts are interested in those objects that cannot be visualized by inspection or with prior knowledge due to limited extent of their spatial presence. As a result, hyperspectral imaging is generally developed to perform target class–based image analysis where image background is usually of no interest. Such examples include anomaly detection, endmember extraction, man-made target detection, and so on, where the spatial information provided by these objects of interest is generally very little. So, if the hyperspectral imagery is treated as a natural extension of the multispectral imagery, its success can be very limited due to its use of spatial information to perform pattern class–based image analysis rather than target class–based image analysis, a similar dilemma that also occurs between real and complex derivatives. Accordingly, we must reinvent the wheel and re-design and develop new hyperspectral imaging techniques rather than directly derive those adopted from multispectral image techniques. One promising approach is the use of the following pigeon-hole principle described in the following section.

1.3.2 Pigeon-Hole Principle: Natural Interpretation of Hyperspectral Imaging

Suppose that there are p pigeons flying into L pigeon holes (nests) with . According to the pigeon-hole principle, there exists at least one pigeon hole that must accommodate at least two or more pigeons. Now, assume that L is the total number of spectral bands and p is the number of targets of interest. By virtue of the pigeon-hole principle, we can interpret a pigeon hole as a spectral band while a pigeon is considered as a target (or an object) of interest. With this interpretation if L > p, a spectral band can be used to detect, discriminate, and classify a distinct target. Since there are hundreds of spectral bands available from hyperspectral imagery, technically speaking, hundreds of spectrally distinct targets can be accommodated by these spectral bands, namely one target by one particular spectral band. In order to materialize this idea, three issues need to be addressed. First, the number of spectral bands, L, must be greater than or equal to the number of targets of interest, p, that is, . This seems always true for hyperspectral imagery, but is not necessarily valid for multispectral imagery, where in the latter is usually true. For example, 3-band SPOT multispectral data may have difficulty with classifying more than three target substances present in the data using the pigeon-hole principle. However, the benefit of also gives rise to a challenging issue known as “curse of dimensionality” (Duda and Hart, 1973), that is, “what is the true value of p if .” This has been a long-standing issue for any hyperspectral image analyst to resolve because it is nearly impossible to know the exact value of p in real-world problems. Moreover, even if the value of p can be provided by prior knowledge it may not be reliable due to many unexpected factors that cannot be known a priori. In multivariate data analysis, the value of p is described by the so-called intrinsic dimensionality (ID) (Fukunaga, 1990), which is defined as the minimum number of parameters used to specify the data. However, this concept is only of theoretical interest. No method has been proposed regarding how to find it in the literature. A common strategy is to estimate the p on a trial-and-error basis. A similar problem is also encountered in passive array processing where the number of signal sources, p, arriving at a linear array of sensors is of major interest. In order to estimate this number, two criteria, an information criterion (AIC) suggested by Akaike (1974) and minimum description length developed by Schwarz (1978) and Rissanen (1978), have been widely used to estimate the value of p. Unfortunately, a key assumption made on these criteria is that the noise must be independent and identically distributed, a fact that is usually not a valid assumption in hyperspectral images as shown in Chang (2003a) and Chang and Du (2004). In order to cope with this dilemma, a new concept called virtual dimensionality (VD) was coined and suggested by Chang (2003a) to estimate the number of spectrally distinct signatures in hyperspectral imagery. It is also based on the pigeon-hole principle where VD is used to estimate the number of pigeons with the total number of spectral bands interpreted as the number of pigeon holes. The last issue to be addressed is that once a spectral band is being used to accommodate one target, it cannot be used again to accommodate another distinct target. One way to do so is to perform orthogonal subspace projection (OSP) developed by Harsanyi and Chang (1994) on a space linearly spanned by the already found targets to find an orthogonal complement space from which only new targets can be generated. Equivalently speaking, the spectral bands used to accommodate previous targets cannot be used again to accommodate a new target. Through a series of such OSP operations no two distinct targets will be specified and accommodated by a single spectral band. In other words, all the found targets must be accommodated in separate mutual orthogonal subspaces. In terms of the pigeon-hole principle it implies that no two pigeons will be allowed to fly into a single pigeon hole. Here, one remark is noteworthy. When it says that one target is accommodated and specified by one spectral band, it simply means that the target can be best spectrally characterized by this particular band compared to other bands. So, this band is chosen to be its identity like its fingerprint or DNA. If two targets happen to have the same band being used for their best spectral characterization then there is no way to discriminate one from the other. In this case, it implies that two pigeons fly into the same pigeon hole. More specifically, one pigeon hole is used to accommodate two flying-in pigeons, both of which reside in a single pigeon hole.

Once these three issues, that is, (1) , (2) determination of p, and (3) no two distinct target signatures to be accommodated by a single spectral band, are resolved, the idea of applying the pigeon-hole principle to hyperspectral data exploitation can be realized and becomes feasible. More specifically, using spectral bands as a means to perform detection, discrimination, classification, and identification without accounting for spatial information or correlation provides an alternative approach, to be called nonliteral analysis as opposed to the spatial domain-based approach, to be called literal analysis. Such a nonliteral analysis is particularly important for two types of targets. One is that targets are small or insignificant due to their limited spatial presence and cannot be effectively captured by spatial correlation or information. The other is that targets of the same type are spatially separated so that their spatial correlation is actually very weak and little in which case the spatial domain-based literal analysis may have difficulty in finding them spatially correlated. The only way to group them together is based on their spectral characteristics regardless of where they are spatially located.

Interestingly, the pigeon-hole principle also sheds light on differentiation of hyperspectral imagery from multispectral imagery. Through the relationship between the total number of spectral bands, L, and the number of signal sources to be accommodated, p, discussed above, a multiple-band remote sensing image can be considered as a hyperspectral image if and a multispectral image otherwise (i.e., ). More details of this interpretation can be found in Chapter 31.

Furthermore, VD can be also interpreted by the pigeon-hole principle, and its potential in hyperspectral data exploitation has been demonstrated in many applications, for example, linear spectral mixture analysis (Chang, 2006c), dimensionality reduction (Wang and Chang, 2006a, 2006b), band selection (Chang and Wang, 2006), and so on. Chapter 5 will revisit VD for more details.