22.2 Dynamic Dimensionality Allocaction

The DDA presented in this section is designed to dynamically determine the values of q and . It originates from the pigeon-hole principle described in Section 1.3.2 as well as variable-length coding from the information theory. According to the pigeon-hole principle each signature is assumed as a pigeon to be accommodated by a particular spectral dimension/band which can be considered as a pigeon-hole. Therefore, the number of pigeons should determine at least how many pigeon-holes required for accommodation. This is equivalent to saying that the number of spectrally distinct signatures determines the minimal number of spectral dimensions/bands required for signature discrimination, which is exactly the original idea of VD. To materialize DDA, we first interpret the use of a pigeon-hole to accommodate a pigeon by a binary bit “1” and “0” otherwise. This implies that a spectral dimension/band being used to specify a particular signature will be encoded by “1.” Otherwise, “0” will be assigned to an unused spectral dimension/band. To fit the profile of source coding, the first task is to determine what type of signatures that can be considered as source alphabets. If the signature knowledge is provided a priori, this known signatures can be used as desired source alphabets. If there is no prior knowledge available about the data, the signatures should be found in an unsupervised means. In this case, the VD developed in Chapter 5 can be used to estimate the number of spectrally distinct signatures, denoted by n_VD. To find n_VD unknown signatures, the automatic target generation process (ATGP) developed by Ren and Chang (2003) and discussed in Section 8.5.1 can be used to produce a set of signatures, , that correspond to desired source alphabets. Or alternatively, the virtual signatures found in the supervised LSMA in Chapter 17 can also serve as the same purpose. Once these signatures of interest are found, the next task is to calculate their discriminatory probabilities according to the relative spectral discriminatory probability (RSDPB) in Chang (2000) and Chapter 2 in Chang (2003a) that will determine DDA. In doing so, we briefly review the basic concept of source coding that will be used to define DDA.

Assume that an information source S is emitted by a set of source alphabets with a given probability distribution where p_j is the probability of the occurrence of the source alphabet a_j. To encode the source S these source alphabets must be represented by a set of code words, called code book. Two coding schemes are generally used for finding a code book for source encoding. One is called fixed-length coding which assigns code words with equal length to all the source alphabets . The other is variable-length coding that assigns code words with variable coding lengths to individual source alphabets according to their occurrence probabilities. Let the coding length used to encode a_j be denoted by l_j. Shannon showed that the optimal coding scheme must be variable-length coding with the mean coding length determined and approximated by the source entropy, . The Huffman coding is proved to be the one that achieves the optimal coding performance in terms of Shannon entropy (see details in Section 31.2.1). The only case that a fixed-length coding also achieves the same performance as the Huffman coding does is when the occurrence probabilities of all the source alphabets are equally likely.

Now how can we borrow the idea of the variable-length coding described above to apply to hyperspectral imaging? First, let denote the signatures of interest where n_S can be either the number of known signatures or n_VD if no prior knowledge is given. Furthermore, let n_j denote the number of spectral dimensions/bands required to represent the jth signature, s_j. Then s_j, n_S, and n_j shall play the same role as a_j, J, and l_j do in source coding to represent the jth source alphabet, the number of source alphabets and number of bits required for a code book to encode the set of source alphabets along with their corresponding probabilities . In other words, the source occurrence probability p_j is now interpreted by n_j that reflects how difficult the s_j is discriminated from other signatures in terms of spectral similarity in the same way that how frequently the a_j occurs in terms of probabilities relative to other source alphabets used in source coding. As a result, the higher the probability p_j is, the shorter coding length, l_j is. This implies that the easier to be discriminated the signature s_j is, the smaller number the n_j is. In particular, to best represent the s_j in terms of spectral dimensions/bands the n_j must vary with discriminatory power possessed by the signature s_j. The traditional DR/BS makes a simple and natural assumption that all the signatures are equally discriminable by setting n_j = n_S for all in which case it performs so-called static dimensionality allocation (SDA), that is, fixed-size band dimensionality. Apparently, it is generally not true in hyperspectral data where each material substance signature has its own spectral characteristics and has a different level of signature discriminability. So, what DDA is to the SDA in DR/BS is the same as what is the variable-length coding to fixed-length coding in source coding. Recently, Wang and Chang (2007) have introduced a new concept of variable number variable band selection (VNVBS) to be discussed in Chapter 27 and showed that using variable numbers of spectral bands was more effective than using fixed number of spectral bands. It provided evidence in advantages of using DDA over SDA.

The next remaining issue is how to come up a technique to determine DDA similar to variable-length coding used in source coding where the bit allocation is determined by the coding lengths which can be calculated by their associated probabilities. By interpreting signature discriminatory probabilities as source alphabet probabilities we are able to do the same for DDA in such a way that variable numbers of spectral dimensions/bands are assigned to represent various discriminatory powers of signatures. Three commonly used coding schemes, Shannon coding, Huffman coding, and Hamming coding can be developed to find DDA.