17.3 Component Analysis-Based ULSMA

As noted in the introduction, hyperspectral signatures can be categorized into background signatures characterized by second-order statistics of IBSI(S) and target signatures characterized by HOS of IBSI(S). Recall that the commonly used PCA is a second-order statistics-based transform that uses a set of PCs to represent the data where eigenvectors are projection vectors to specify PCs with eigenvalues being data variances. In this case, PCA can be then used to extract background signatures characterized by second-order statistics of IBSI(S) in PCs. On the other hand, ICA is an HOS-based transform that uses mutual information to generate a set of ICs to represent data. Therefore, ICA can be used to find desired target signatures characterized by HOS of IBSI(S) in ICs. In both cases, VD is again used to determine how many PCs and ICs are required to extract signatures. Since PCs and ICs are obtained by mapping all data samples onto the projection vectors, the projection values of data samples are real values. So, an issue arises: how many data sample vectors should be selected from each PC and each IC? Two sample values in each IC are of major interest: one with maximal projection value and the other with minimal projection value. These two samples represent maximal projections in two opposite directions of a projection vector that specifies an IC. They both indicate their importance in data analysis. This idea was previously explored in pixel purity index (PPI) in Chapter 7 where the most likely endmembers are those samples with either maximal or minimal projections on each randomly generated vectors referred to as skewers. However, it is worth noting that there is no similar selection of sample vectors with minimal projections in PCs due to the fact that PCA is a transformation of second-order statistics with variance representing signal energy. Those samples with minimal projections, that is, variances are supposed to correspond to noisy samples. In this case, there is no reason to select data sample vectors with minimal projections in PCs as desired samples similar to the case described above for ICA where noisy samples with small variances have been removed by sphering. All such data sample vectors extracted from PCs and ICs are considered as VSs.

Using VD again to determine the numbers of PCs and ICs along with data samples selected from the first PCs and data sample vectors with maximal and minimal projections in the first ICs a CA-based unsupervised VS finding algorithm (CA-UVSFA) can be described as follows.

CA-UVSFA

1. Use VD to determine the number of components required to generate, denoted by .

2. Apply PCA to the original image data and find p PCs and extract the brightest pixels (i.e., data sample vectors with maximal values in PCs) as a VS from each of p PCs to form .

3. Apply ICA to the sphered data and find p ICs and extract points with the maximal and minimal projections from each of p ICs as VSs to form . Since some sample vectors with minimal projections may be close to certain sample vectors with maximal projections in other ICs, a spectral measure such as SAM is applied to identify these sample vectors and eliminate them from to form . It should be noted that the ICA used here is the one developed in Hyvarinen and Oja (1997), called FastICA, that is actually based on a criterion derived from a combination of third- and fourth-order statistics; see Equation (5.35), p. 115 (Hyvarinen, 2001).

4. Furthermore, there may be some target sample vectors extracted in that also exhibit strong energies in the PCs where pixels corresponding to these sample vectors may be also extracted in S^BKG. In this case, SAM is also used to extract these sample vectors and remove them from S^BKG. Let the remaining background sample set be denoted by .

5. Construct a set of VSs, by merging and for spectral unmixing. It should be noted that the number of VSs in S^VS is between p and 3p.

A comment on that each of PCA and ICA is required to generate p components is noteworthy. It is often the case that target sample vectors may also show up in either PCs or ICs, but not both. In order to make sure that no matter which scenario will be, using p PCs and p ICs should have sufficient components to capture all these target sample vectors. A detailed step-by-step procedure of CA-ULSMA can be summarized as follows.

CA-ULSMA

1. Use HFC/NWHFC method to determine VD and let

2. Implement CA-UVSFA to produce a set of VSs, S^VS.

3. Apply an SLSMA technique such as unconstraint classifier LSOSP, non-negativity constrained classifier NCLS, and fully constrained classifier FCLS to perform spectral unmixing where only the target pixels in will be unmixed, while the target pixels in will be used for background suppression.

In Step 1 of CA-ULSMA, in order for the HFC/NWHFC method to work effectively, the spectrally distinct signatures defined by VD are those that do not have significant contribution to data variances. Such signatures are generally characterized by three unique features. Firstly, the probabilities of their occurrence are usually low. Secondly, when such signatures are present, there are not too many samples. Thirdly, as a result, the variances of such signatures are generally very small and can be considered to be negligible. So, when the HFC/NWHFC method is used to estimate VD that determines the number of VSs used for LSMA, two assumptions are made on the VSs. The first and foremost assumption is that all the VSs in S^VS are assumed to have the least intersample spectral correlation. This is a reasonable assumption since different VSs should have the least spectral correlation among all the data sample vectors and should also be as distinct as possible in terms of spectral characteristics via IBSI(S). Another is that the number of samples specified by VS should be relatively small since they represent most spectrally distinctive signatures. With these two assumptions in mind a VS is only contributed to the sample mean but not variance of each spectral band. This is also the key idea used to develop the HFC/NWHFC method and explains why the HFC/NWHFC-estimated VD works very effectively for HYDICE data in the following experiments where the number of target panel pixels specified by each of five panel signatures is very small. When these two assumptions are violated, the HFC/NWHFC-estimated VD may not be accurate. So, when the HFC/NWHFC method is used to estimate VD we should be aware of these assumptions. Finally, we conclude a noteworthy comment on the use of twice VD value, 2 = 2p for LS-ULSMA and CA-ULSMA to extract VSs. Such selection is not arbitrary. It is actually based on the concept, called dynamic dimensionality allocation (DDA) derived in Chapter 22. For more details we refer readers to this chapter.