21.3 Criteria for Band Prioritization

In order to implement BP, a criterion is needed to measure the significance of a spectral band in terms of its priority score. In what follows, four classes of criteria are considered.

21.3.1 Second-Order Statistics-Based BPC

The first category of BP criterion (BPC) is derived from second-order statistics that are based on variance and signal-to-noise ratio (SNR).

21.3.1.1 Variance-Based BPC

A natural and logical approach to band prioritization is to compute variances for all spectral band images in a hyperspectral image cube, denoted by and further use band variances to define a priority score for each of band images as follows:

(21.1)

Another alternative interpretation is to use a set of loading factors proposed in Tu et al. (1998) and Chang et al. (1999) that can be defined by

(21.2)

It is easy to show that for each , ρ_l defined by

(21.3)

turns out to be the variance of the lth spectral band image. As a result of (21.3), the priority score calculated by (21.1) for the lth spectral band image B_l is also equivalent to the PCA-based priority score defined by

(21.4)

21.3.1.2 Signal-to-Noise-Ratio-Based BPC

It was noted in Green et al. (1988) that variance was not an appropriate criterion to measure image quality. In order to alleviate this dilemma, an SNR-based criterion was first developed by Green et al. (1988) to improve the PCA. The resulting transform was called maximum noise fraction transform and later reinterpreted by Lee et al. (1990) as noise-adjusted principal component (NAPC) transform. In analogy with the criterion specified by (21.4) for the variance-based PCA, a similar criterion to (21.4) can also be derived from the SNR-based NAPC as follows.

Assume that is the set of eigenvalues of noise-adjusted sample covariance matrix and are their associated orthonormal eigenvectors. We can define the loading factors in a similar manner to (21.2) for an NAPC by

(21.5)

Using (21.5), a noise-adjusted variance-based priority score can be calculated for the lth spectral band image B_l via (21.5) defined by the NAPC-based priority score:

(21.6)

21.3.2 High-Order Statistics-Based BPC

In many applications, the information of interest may not be captured by second-order statistics, but rather be characterized by higher-order statistics. In order to take this into account, three higher-order statistics-based criteria are derived in this section for BPC.

21.3.2.1 Skewness

The simplest high-order statistics is the third central moment, referred to as skewness and defined by

(21.7)

21.3.2.2 Kurtosis

A fourth central moment, referred to as kurtosis, is defined by

(21.8)

21.3.3 Infinite-Order Statistics-Based BPC

It should be noted that according to our experience, criteria for BP based on statistics higher than 4 do not have much significant advantage compared to skewness and kurtosis (Ren et al., 2006). Therefore, only ∞-order statistics-based BP criteria, entropy and information divergence, are discussed in this section.

21.3.3.1 Entropy

One of the simplest and most widely used ∞-order statistic-based BPC is entropy that requires an infinite number of moments. Let H(B_l) be the entropy calculated for the lth band image B_l. The entropy-based priority score for B_l is defined by

(21.9)

It should be noted that the entropy H(B_l.) in (21.9) is calculated based on the gray-level histogram produced by the lth spectral band image B_l where the number of bins is totally determined by the difference between the maximal and minimal gray levels present in the spectral band image B_l. For example, if the maximal and minimal gray levels are 255 and 0, respectively, then there are together 256 bins needed to estimate the entropy.

21.3.3.2 Information Divergence

As an alternative to entropy defined by (21.9), an information theoretic measure, called information divergence (ID), can also be used as a BPC. Assume that the p_l is the image histogram of the lth spectral band image, B_l normalized as a probability distribution and g_l is its associated Gaussian distribution with mean and variance determined by sample mean and sample variance of the B_l. BP criterion of interest is to measure the deviation far away from a Gaussian distribution for a given spectral band image, that is, the discrepancy between p_l and g_l defined by

(21.10)

where D(p_l;g_l) is called information divergence (Kullback, 1968)

(21.11)

The higher the value of D(21.p_l;g_l) in (21.11), the greater deviation of p_l from the Gaussian distribution, g_l is. This implies that ID is used to measure the degree of non-Gaussianity of a band. It should be noted that if both p_l and g_l are replaced with two spectral signatures, the D(p_l;g_l) defined by (21.11) becomes spectral information divergence (SID) in Chang (2000) and Chang (2003a).

21.3.4 Classification-Based BPC

In the previous two subsections, BPC are designed based on statistics. Additionally, they are also unsupervised in the sense that no prior knowledge is involved in these criteria. However, in some applications, prior knowledge may be available and can be taken advantage of to design BPC. In this subsection, two supervised classification-based criteria are developed for BP. Such classification-based BP criteria are different from statistics-based BP criteria such as variance-, SNR- or high-order statistics-based BP criteria in the sense that the former is developed for target detection and classification applications while the latter is completely determined by statistics that has little to do applications.

21.3.4.1 Fisher's Linear Discriminant Analysis (FLDA)-Based BPC

Minimum misclassification canonical analysis (MMCA) derived from Fisher's linear discriminant analysis (FLDA) was used in Tu et al. (1998) to minimize the misclassification error. For any given band number , we can use (21.2)–(21.3) with eigenvalues and unit eigenvectors replaced by the eigenvalues and normalized unit feature vectors as used in Tu et al. (1998) to define the loading factors as follows:

(21.12)

for and In light of (21.12), the priority score can be calculated for the lth spectral band image B_l by

(21.13)

21.3.4.2 OSP-Based BPC

Another classification-based criterion is derived from the orthogonal subspace projection (OSP) (Harsanyi and Chang, 1994) that is based on the linear mixture model as follows:

(21.14)

where , and n is noise or model error. If we further assume that the p image endmembers can be divided into two classes of endmembers, one class of n_D desired image endmembers denoted by and the other class of undesired endmembers denoted by with . Then the OSP classifier for a particular desired endmember m_j, can be actually obtained by with where and , is the pseudo-inverse of . Now following the same argument outlined by (21.12) and (21.13) we can define loading factors for the OSP classifiers as

(21.15)

and

(21.16)

where and obtained in Chang et al. (1999). By means of (21.16), the priority score assigned to the lth spectral band image B_l can be calculated by

(21.17)

21.3.5 Constrained Band Correlation/Dependence Minimization

Taking a rather different approach from the ideas used to design previous BP criteria, a recent new approach, called constrained band selection (CBS) developed in Chang and Wang (2006), suggested a new criterion for BP, which linearly constrained a particular band image while minimizing band correlation/dependence resulting from other band images. In other words, the priority score of a spectral band can be calculated according to the degree of correlation or dependence between this particular band image and other band images measured by least squares errors. Its idea can be briefly described as follows.

Assume that the size of all the spectral band images B_l is . Since each spectral band image B_l can be represented by a column vector of dimensions , denoted by b_l, we have a total number of L spectral band image vectors . For any given spectral band image vector b_l we can design a finite impulse response (FIR) specified by a set of L weighting vectors, that constrains b_l while minimizing least squares error caused by other band image vectors . More specifically, let y_l be the filter output obtained by

(21.18)

The averaged least squares filter output is given by

(21.19)

Let denote the band image correlation matrix. A similar optimization problem to the constrained energy minimization (CEM) in Chapter 2 can be obtained as follows:

(21.20)

The solution to (21.20), denoted by is given by

(21.21)

Alternatively, we can exclude the spectral band image b_l from the band correlation matrix Q and further define as the band dependence matrix. Replacing Q in (21.20) with results in a similar constrained band selection problem

(21.22)

The solution to (21.22), is the same as the one in (21.21) with the Q replaced by that is given by

(21.23)

21.3.5.1 Band Correlation/Dependence Minimization

By means of (21.21) and (21.23) we can calculate the following least squares errors (LSEs):

(21.24)

(21.25)

that can be used to measure degree of the spectral band image vector b_l correlated with and dependent on other spectral band image vectors , respectively. That is, the greater the LSE in (21.24) or (21.25), the higher the correlation of b_l with other band image vectors. So, we can use (21.24) and (21.25) to derive two criteria for BP, called band correlation minimization (BCM) defined by

(21.26)

and band dependence minimization (BDM) defined by

(21.27)

21.3.5.2 Band Correlation Constraint

Comparing to (21.24) and (21.25), an alternative approach is to calculate band correlation constraint (BCC)

(21.28)

and band dependence constraint (BDC)

(21.29)

which can also be used to measure the correlation between the spectral band image vector b_l and any other spectral band image vector b_k (21.). By comparing the value of with the filter constraint specified by in (21.20) or (21.22), a spectral band image B_k has less correlation with the spectral band image B_l if its band constraint is far away from 1. In other words, the closer the to 1, the higher the correlation of B_k to B_l. With this interpretation, two criteria similar to (21.26) and (21.27) can also be derived for BP, called BCC given by

(21.30)

band dependence constraint (BDC)

(21.31)

One disadvantage of these CEM-based criteria is the enormous size of vectors converted from band images that causes tremendous computing time. For example, it requires a vector with 4 × 10⁴ dimensions to represent a band image with size 200 × 200. In order to mitigate this dilemma, a linearly constrained minimum variance (LCMV) in (Frost, 1972; Van Veen and Buckley, 1988) is developed to derive four criteria similar to four CEM-based criteria specified by (21.24) and (21.25) and (21.28) and (21.29). Instead of constraining a band image as a vector, the LCMV-CBS constrains a band image as an image matrix without vector conversion. Its idea is derived from the LCMV approach, which can be traced back to Frost's work in adaptive beamforming (Frost, 1972). More specifically, assume that are n_c columns of the lth spectral band image B_l, which has n_r rows and n_c columns. So, the jth column vector of B_l denoted by is represented by an n_r-dimensional column vector, . In this case, the lth spectral band image B_l can be further expressed by a matrix given by

(21.32)

Like the CEM, the goal is to design a constrained FIR linear filter with an n_r-dimensional weight column vector specified by a set of n_r filter coefficients that minimizes (21.19) subject to the following simultaneous n_c multiple constraints, , that is equivalent to

(21.33)

where is an n_c-dimensional column vector with all 1s in its n_c components. It should be noted that since the weight vector v_l is used to constrain column vector of a band image, its dimensionality is n_r compared to the n_rn_c-dimensional weight vector w_l used in (21.20) that constrains a band image as a vector with dimensionality n_rn_c. By virtue of the n_c multiple constraints in (21.33), the CEM problem described by (21.20) can be rederived as the following LCMV-based optimization problem:

(21.34)

where is the sample band correlation matrix. The solution to (21.34) can be solved as

(21.35)

and

(21.36)

plays the same role that ρ_l does for the CEM-BCM. Similar derivations to CEM-BDM can also be obtained for and

(21.37)

In analogy with the CEM-based band correlation/dependence constraint criteria (BCC/BDC)

(21.38)

and

(21.39)

can also be derived for an LCMV-based band correlation/dependence constraint criteria by replacing band image vector b_l and CEM-based weight vectors with band image B_l and LCMV-based weight vectors, respectively.

Despite the fact that the CBS described was developed in Chang and Wang (2006), the idea of BP was not introduced in their paper and nor were the priority scores specified by (21.26) and (21.27), (21.30)–(21.31). Table 21.1 summarizes all the proposed BPC in terms of their characteristics where “supervised” indicates that training samples are required for the particular criterion.

Table 21.1 Comparison among various BP criteria.

As a concluding remark, one comment is noteworthy. The effectiveness of BP is determined by its applications not criteria alone. As will be demonstrated by following experiments, a different application yields a totally different selected set of bands. With proper bands selected by BP the number of bands can be significantly reduced, while still achieving performance comparable to that accomplished by using full bands.