20.7 Conclusions

This chapter introduces a new concept of DP that has never been explored in the literature in the past. Its idea arises from recognition of several issues in implementing DR. One is that the number of data dimensions required to be retained, q, after DR must be known in advance. In a case that the value of q is inaccurate, the entire process of DR must be reimplemented again for a new value of q. This leads to an issue, “Can the data dimensions previously obtained by a smaller value of q be used without re-running DR for a new larger value of q?” In addition, once DR is performed, how can these DR-transformed dimensions be represented in terms of information contained in these new spectral dimensions? Despite that PCA resolves the above issues by solving eigenvalues from the characteristic polynomial equation, it is unfortunate that the same approach cannot be extended or generalized to any linear transformation. For example, ICA does not have such nice properties. The PIPP and DP presented in this chapter provide a feasible solution to resolving this dilemma where PI plays a twofold role in producing projection vectors and prioritizing projection vector-generated PICs. However, it should be noted that the used PIs for both cases are not necessarily the same and can be different. In other words, when PIPP is implemented in conjunction with DP, a pair of two different PIs must be used, one for projection vector generation and the other for PIC prioritization. This is quite different from PCA that uses data variance as the same PI for both finding eigenvectors as projection vectors and using eigenvalues for PC prioritization. One of major advantages and benefits provided by PIPP via DP is no requirement of prior knowledge about the value of q, which is the number of spectral dimensions needed to be retained for DR. It allows users to perform PSDP with two dual processes, PSDE via DP and PSDR via DP which can expand and reduce spectral dimensions in a forward and a backward manner progressively, and one is a reverse process of another. By means of these two processes the hyperspectral information compression can be best utilized in many applications, such as data compression, communication, and transmission. Technically, speaking, PSDP can start off any value of q, for example, q = 1 for PSDE and q = L for PSDR. However, this may not be practical in applications. Therefore, as shown in Wang and Chang (2006a), VD may not be exactly accurate but is a good estimate for q. By taking advantage of VD a feasible range of [n_VD, 2n_VD] can be derived for PSDP to implement PSDE and PSDR as demonstrated in our experiments as well as in some recent works, Safavi (2010), Fisher and Chang (2011), Chang and Safavi (2011), and Chang et al. (2011). Nevertheless, if n_VD is too large the range of [n_VD, 2n_VD] may be too broad. To further address this issue, another new concept of DDA introduced in Chapter 22 can also be applied to narrow down the range by replacing 2n_VD with DDA. This approach was studied by Safavi (2010), where experiments have shown that since n_VD already provides a good estimate for q, the reduction of the upper bound from 2n_VD to DDA does not have many advantages. But, this is not the case for BS, as will be demonstrated in Chapter 23 as well as Liu (2011).