31.1 Introduction

It is known that a hyperspectral imager can discriminate and quantify materials more effectively via much better spectral resolution than a multispectral imager can. However, an interesting issue seems to be overlooked and has never been addressed, “how to define and differentiate hyperspectral imagery from multispectral imagery”. Until we can settle this issue, the algorithm design for hyperspectral imagery cannot be effective. This has been the case over the past years where hyperspectral imagery has been considered and viewed as a natural extension of multispectral imagery under a common sense that hyperspectral imagery has more spectral bands with finer resolutions than multispectral imagery with low spectral resolution. With this intuitive generalization, in early days a general approach to designing HSI algorithms has been the one that extends algorithms developed for multipsectral imagery in a straightforward fashion. One of such techniques is the maximum likelihood-based classification and estimation (Landgrebe, 2003). Unfortunately, using this multispectral-to-hyperspectral extension may not be a best way to design algorithms for hyperspectral image analysis as already discussed and addressed in Chapter 1 (Section 1.2). We believe that one of main causes may be due to the fact that there is no specific criterion or definition that can be used to distinguish a hyperspectral image from a multispectral image in a rigorous and mathematical means. Because of that, we do not know how to take advantage of benefits provided by hyperspectral imagery, which cannot be found in multispectral imagery so that these advantages can be best utilized in designing and developing HSI algorithms. This chapter investigates this issue by exploring the connection between HSI and MSI techniques.

Assume that there are p spectral signatures in a remote sensing image with L being the total number of spectral channels used for data acquisition and collection. From a linear spectral mixture analysis (LSMA) point of view, LSMA is performed by solving a linear system consisting of L equations specified by L spectral channels/bands and p unknowns corresponding to p image endmembers. When , the system is an over-determined system, in which case the considered remote sensing image is defined as a hyperspectral image. Conversely, if , the system is an underdetermined system, in which case we can define the considered remote sensing image as a multispectral image. More specifically, if we consider that there are the p image endmembers to serve as a set of p basis vectors for L equations, LSMA with is called under-complete linear spectral mixture analysis (UC-LSMA) due to the insufficient number of basis vectors to represent the data. Conversely, if , LSMA is called over-complete linear spectral mixture analysis (OC-LSMA) because there are more basis vectors than what we need to represent the data. At the first glimpse, the way that UC-LSMA and OC-LSMA are defined above seems to be out of reach and not intuitive. However, the following rationales should provide the ground to support the above definitions.

First, the relationship between L and p needs to be explored. On one hand, L is the number of spectral channels/bands and determines how many equations required to be used. On the other hand, p is the number of image endmembers (considered as signal sources) and determines how many unknown signal sources resident in the data to be solved via the L equations. By virtue of the pigeon-hole principle described in Chapter 1 (Section 1.3), one spectral channel/band is represented by an equation and can be viewed as a pigeon-hole to characterize, specify, and accommodate one spectral distinct signal source, that is, image endmember, which can be considered as a pigeon. For UC-LSMA where L is greater than p, it means that more pigeon-holes than pigeons can be used for pigeon accommodation. Obviously, there are ways to use one pigeon-hole to accommodate p pigeons, one pigeon-hole for one pigeon. On the other hand, for OC-LSMA where L is less than p, there are more pigeons than pigeon holes. So, in this case, more pigeons fly into fewer pigeon holes in which case at least one pigeon hole must accommodate more than one pigeon. When it occurs, all the pigeons in a single pigeon-hole cannot be separated and discriminated one from another. Interestingly, similar definitions to UC-LSMA and OC-LSMA can also be found in independent component analysis (ICA) (Hyvarinen et al., 2001) where an under-complete ICA (UC-ICA) is developed to blindly separate p statistically independent signal sources using L data samples with , while an over-complete ICA (OC-ICA) is to blindly separate p statistically independent signal sources using L data samples with . These two definitions provide a base of how UC-LSMA and OC-LSMA are defined as above. More specifically, UC-LSMA is developed for a hyperspectral image, which solves an overdetermined system with no solutions. To address this issue, one commonly used technique is dimensionality reduction described in Chapter 6, which reduces an over-determined system to a solvable system that is, L = p that can produce a solution. In contrast, OC-LSMA is generally developed for a multispectral image which solves an underdetermined system with many solutions. In this case, OC-LSMA needs dimensionality expansion to augment the system to find a best solution in some sense of optimality with L = p. Accordingly, UC-LSMA and OC-LSMA essentially deal with completely opposite problems.

Over the past years, we have seen many efforts devoted to expanding MSI techniques to solve HSI problems (Richards and Jia, 1999; Landgrebe, 2003) and have achieved some success. However, there is little work, which does the other way around by taking advantage of HSI techniques to solve MSI problems. In doing so, we need to resolve the issue of OC-LSMA used for MSI which is an insufficient number of spectral bands. This chapter developed two approaches based on a concept of dimensionality expansion. One is band dimensionality expansion (BDE) that expands original data dimensionality by creating new spectral band images via nonlinear functions. It is originated from the band generation process (BGP) proposed by Ren and Chang (2000a) and can be considered as band expansion process (BEP), which is essentially a reverse process of dimensionality reduction by band selection (DRBS) presented in Chapter 6. The other approach is feature dimensionality expansion (FDE) via nonlinear kernels discussed in Chapter 15. However, it should be noted that unlike BEP/BDE, FDE actually expands features extracted from the original data space into a higher dimensional feature space via nonlinear kernels. Such FDE is quite different from BDE, which produces new spectral bands generated from the original spectral bands by nonlinear functions. Nevertheless, both BDE and FDE are developed to resolve the same issue caused by OC-LSMA, that is, there are no sufficient spectral channels/bands in a remote sensing image cube to be used to accommodate more signal sources that the image can handle. Furthermore, BDE and FDE also share similar ideas in the sense that the dimensionality expansion in FDE and BDE must be carried out by nonlinear functions. In other words, the use of nonlinear functions in the BDE and nonlinear kernels in FDE is a key to success in making hyperspectral imaging techniques applicable to multispectral imagery because OC-LSMA is linear.