21.1 Introduction

Hyperspectral images are collected in hundreds of contiguous spectral channels. Therefore, not only the data volume to be processed is considered to be huge, but also the spectral correlation among bands is expected to be very high due to high spectral resolution. Band selection (BS) is one of commonly used approaches that take advantage of such high interband correlation to remove band redundancy. Over the past years, many research efforts have been directed to BS (Mausel et al., 1990; Conese and Maselli, 1993; Stearns et al., 1993; Chang et al., 1999; Huang and He, 2005; Chang and Wang, 2006; Du et al., 2007) in order to achieve a wide range of applications such as data compression, data storage, data transmission, and communication. Generally, two crucial issues arising in BS need to be resolved, which are (1) number of bands required for BS and (2) what criterion to be used to select bands. Instead of dealing with BS directly this chapter introduces a new concept of band prioritization (BP) that is similar to the DP developed in Chapter 20 to simultaneously address these two issues by prioritizing all spectral bands in some optimal sense. It ranks all the spectral bands in accordance with their corresponding priority scores that can be calculated by a custom-designed BP criterion in a similar way that spectral dimensions are prioritized by DP in Chapter 20. Then PBDP can be performed by BP in such a manner that two dual processes, progressive band dimensionality reduction (PBDR) via BP and progressive band dimensionality expansion (PBDE) via BP, similar to progressive spectral dimensionality reduction (PSDR) via DP and progressive spectral dimensionality expansion (PSDE) via DP can also be derived. In other words, rather than determining the number of bands needed to be selected as required by BS, PBDP allows users to terminate its process either in band reduction or in band expansion by applications. Although the value of can be estimated by VD, a true value of is never known in real-world problems. In this case, a better way to circumvent this problem is to let applications determine when the process should be terminated. Nevertheless, VD has been shown to provide a reasonable estimate of a lower bound on in Chang and Wang (2006). This value can be used to suggest an initial guess for to initialize PBDP. So, a next key issue in PBDP is to design an effective criterion to meet a specific application. In Chang et al. (1999) BS was performed by prioritizing bands according to a specific criterion and followed by band de-correlation (BD) to remove bands that are highly correlated with those bands already selected until a specific number of bands is achieved. Recently, Du et al. (2007) made use of a similar idea to perform BS by repeatedly calculating the information divergence among all the unselected bands and selecting and removing the one with the maximal divergence from the set of unselected bands to form a new set of unselected bands for the next round BS. The process is continued until it reaches a specific number of bands, . The proposed PBDP is fundamentally different from this type of the conventional BS and revolutionizes the concept of BS in several aspects. The first and foremost is that PBDP does not need to find and fix the value of . Instead, the can be tuned by a specific application or image analysts. Secondly, it prioritizes the entire set of L spectral bands according to their contained information measured by a custom-designed information criterion and then selects bands progressively in a forward or backward manner depending on how to retain band information in increasing or decreasing order. To realize this concept two dual processes derived from PBDP are further developed. One is referred to as PBDR via BP that performs PBDP in a backward manner by beginning with a higher number of highest-prioritized bands and gradually removes bands according to their increasing priority order from the selected prioritized bands. As a complete opposite to PBDR, the second process is referred to as PBDE via BP that performs PBDP in a forward manner by starting with a lower number of highest-prioritized bands and gradually adding new unselected bands according to their priorities in descending order. Thirdly, both processes can be terminated when a stopping rule is satisfied and is determined by various applications. In this case, VD can be used to provide a reasonable lower bound on for PBDR and an upper bound on for PBDE. This was not done in Chang et al. (1999) and nor was in Du et al. (2007). Such a progressive nature significantly reduces computational complexity because the previously selected bands are always a part of future augmented band subsets without repeatedly solving new combinatorial problems.

Unlike PSDP that takes advantage of the projection index-based projection pursuit (PIPP) to preserve information provided by projection index components, PBDP must retain information provided by selected bands in a more effective fashion. Since each spectral band is acquired by a specific wavelength designed to extract certain material substances present in the range, the selection of bands must be determined by substances of interest in applications. As noted in Section 1.3.2, the pigeon-hole principle offers a means of interpreting each of spectral bands as a pigeon hole and substance of interest needed to be analyzed as pigeons. By virtue of this pigeon-hole principle a spectral band can be used to specify one particular material substance that indeed determines which band should be used for its accommodation. Accordingly, the effectiveness of PBDP should be therefore determined by material substances of interest in applications. Due to this fact VD which is also derived from the pigeon-hole principle can be used to estimate the value of selected for PBDP which should be very closely related to the value estimated by VD that is indeed the case shown in Chang and Wang (2006).