33.7 Hyperspectral Signal Processing

The hyperspectral data considered in Chapters 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22,23 are three-dimensional image cubes where each data sample is actually a pixel vector. So, the data processing carried out in this manner can be viewed as hyperspectral image processing in Category A where two types of correlation are of interest. One is correlation provided by data samples in terms of their spatial locations while paying no attention to interband spectral correlation. This is generally referred to as intersample spatial correlation and is commonly used in traditional image processing to develop spatial domain-based algorithms to perform tasks such as edge detection, region growing, clustering, segmentation, etc. Early data processing for remote sensing data, for example, geographical information system (GIS), belongs to such an approach, which can be considered as spatial domain-based data analysis. The other type of correlation is provided by data samples regardless of their spatial location. It is a complete opposite to the above-mentioned intersample spatial correlation and can be referred to as intrasample spectral correlation which has been explored and investigated in great detail in Chapter 17. The key difference between these two types of correlation can be well explained by the following example. Let be a set of N data sample vectors where is the ith data sample vector, L is the total number of spectral bands, and the i indicates the ith spatial location of r_i. Let denote a permutation of N spatial locations, . There are N! permutations of with denoting one data matrix with data samples arranged in a particular order of specified by a permutation . Now, the intersample correlation matrix and intrasample correlation matrix provided by S^P are and , respectively. In comparison between these two sample correlation matrices, it is very clear that is independent of permutations and remains the same, that is, for any permutation P even though the samples reshuffled by permutations, while will result in another different sample spatial correlation matrix if another permutation Q is used. That is, if P and Q are two different permutations of since it alters spatial correlation among data samples when data samples are re-arranged in different spatial coordinates. Nevertheless, intersample spatial correlation and intrasample spectral correlation share one thing in common, that is, they both produce sample statistics when N > 1. In this case, the intersample spatial correlation and intrasample spectral correlation are more specifically referred to as sample intersample spatial correlation and sample intrasample spectral correlation to indicate the vital role of sample size plays in generation of sample statistics by . Throughout this book, only sample intrasample spectral correlation is of interest, specifically, interband spectral information (IBSI) introduced in Chapter 17. Three scenarios can be developed to effectively use such IBSI(S) with various sizes of training sample pool S and can be described as follows:

Assume that a given data sample vector is a signature vector without reference to others (i.e., N = 1). What the best we can do is to explore as much spectral information across the entire wavelength range as possible to specify the data sample vector for spectral characterization. For this purpose, we can consider a general case that a spectral signature s(λ) is a 1D continuous-wavelength real-valued signal defined on a wavelength range of [a,b]. For s(λ) to be implemented in discrete signal processing, s(λ) must be sampled according to a sampling interval, Δλ. For example, for a HYDICE data sample vector, there are 210 spectral channels over [0.4 μm, 2.5 μm] with spectral resolution 10 nm, in which case, the sampling interval is Δλ = 10 nm where s(lΔλ) where can be considered as the spectral value of the lth spectral channel to form a spectral signature vector given by where L is the total number of samples corresponding to the total number of spectral channels. So, Δλ actually determines the spectral resolution. When the interval is relatively small, the spectral bands can be considered as contiguous spectral channels and the data sample vector is a hyperspectral signature. Otherwise, the spectral bands are considered as discrete spectral channels, and the data sample vector is a multispectral signature.

Once a spectral signature is acquired; there are two ways to process the spectral value s_l of the lth spectral channel in either a discrete manner, referred to as signal coding or in a continuous manner, referred to as signal estimation.

33.7.1 Signal Coding

Signal coding represents a signal by a finite number of discrete values, to be called code words. For example, a real value can be encoded as a binary representation used by computers as bits. In communications, coding is carried out by a quantizer with quantization levels specified by a desired set of discrete values. So, the simplest encoder or quantizer is sign detector, known as hard-limiter, which only detects the sign change of a signal or delta modulator with only two discrete values. In hyperspectral signal processing, the signals are represented by real-valued spectral values across the wavelength range and IBSI(s) is spectral correlation provided by L signals, within the signature vector . One earliest attempt to perform signal coding on was the spectral analysis manager (SPAM) by Mazer et al. (1988), which is actually implemented as two sign detectors to detect a sign change between the current spectral signal s_l and spectral mean and a sign change in spectral signals between the two adjacent bands, s_{l − 1} and s_{l + 1}, of the current being processed band, s_l. SPAM was later improved by spectral feature-based binary coding (SFBC) developed by Qian et al. (1996) by adding a third sign detector, which detects a sign change in spectral deviation from the spectral mean by a prescribed threshold. Since each sign detector requires one bit to dictate the sign change, SPAM and SFBC can be viewed as 2-bit encoder/qunatizer or 3-bit encoder/quantizer or sign detector, respectively. The ideas of SPAM and SFBC are further explored in Chapter 24 to derive many variations from information theory perspectives such as median partition (MP) binary coding, halfway partition (HP) binary coding, and equal probability partition (EPP) binary coding. While these binary coding methods may be effective on many occasions, they may not be so if the signature vector has a sophisticated and complex spectrum across L spectral signals . This is because binary coding only uses up to 2-bit memory to store spectral changes of two adjacent bands and may fail to capture and characterize their spectral behaviors. Nevertheless, these coding methods are still considered as memoryless coding since they are implemented as 2-bit or 3-bit quantizers as scalar quantizers (Gersho and Gray, 1992). In order to resolve this dilemma, a concept of vector coding similar to vector quantization (Gersho and Gray, 1992) is introduced in Chapter 25. One such vector coding is developed based on texture analysis, called spectral derivative feature coding (SDFC) where the sign detector used by binary coding is replaced with a spectral texture descriptor. Another is derived from arithmetic coding, called spectral feature probabilistic coding (SFPC) which can keep track of all spectral changes across the entire set of . However, implementing vector coding generally requires high computational complexity. Chapter 26 further develops a progressive coding, called multiple-stage PCM (MPCM)-based progressive spectral signature coding (MPCM-PSSC), which takes advantages of simplicity of scalar coding in implementation as well as capability of vector coding in capturing spectral changes in .

33.7.2 Signal Estimation

The basic goal of the above-mentioned signal coding is designed to generate a credible fingerprint of each of spectral signals, so that these fingerprints provide sufficient information for their own identities. But such signal coding does not necessarily tell you what the real signal is. In other words, a signal identity and its fingerprint is one-to-one correspondence such as one-to-one identification between a person and his unique nickname, known as code words. Using a more specific example for illustration, let be used for data transmission. When one of these signals is selected for transmission, it is its subscript instead of the signal itself being transmitted. According to information theory, if a fixed length coding is used, only bits required to derive a set of L code words corresponding to fingerprints of the L signals, for signal transmission without actually transmitting the signal itself. This is because a signal can be identified by its subscript through its code word used as its fingerprint. Such processing performed by signal coding is indeed 1D discrete-value signal processing and can be only used for hard-decision-made applications such as signal detection, discrimination, classification, identification, but certainly cannot be used for continuous value (real-valued) signal processing-based applications, which require soft decisions such as signal quantification, in which case signal coding does not suffice to characterize provided by a signature vector. In order to address this issue, the encoder used by signal coding must be replaced by an operator that can reconstruct the signal to be processed for signal recovery. This is particularly important and critical in applications in chemical/biological agent detection where the agent concentration is crucial to determine various levels of threat and fatality. In this case, each of must be approximated by its estimate rather than its code word. PART VII in this book is particularly developed for this purpose where three approaches, variable number variable band selection for hyperspectral signal charcaterization in Chapter 27, Kalman filtering-based estimation for hyperspectral signals in Chapter 28, and wavelet representation for hyperspectral signals in Chapter 29, are investigated.