10.6 Random ICA-Based EEA (RICA-EEA)

The idea of RICA-EEA is derived from a recent work on an independent component analysis (ICA)-based approach to DR (Wang and Chang, 2006a) where FastICA developed by Hyvarinen and Oja (1997) is used to generate independent components (ICs). Since FastICA also uses random projection vectors as its initial condition to initialize its algorithm, it also encounters the same problem as both PPI and N-FINDR do. Because the initial projection vector for each IC is randomly generated by FastICA for each run, the ICs generated by each run are also different. Nevertheless, if the information contained in an IC is significant, such an IC will always appear in each run. With this assumption, if FastICA is repeatedly implemented, the ICs that are common in all runs should be the desired ICs for endmember extraction. The detailed implementation of RICA-EEA is summarized as follows.

RICA-EEA Algorithm

1. Initialization

Assume that the number of initial endmembers is p. Set .

2. At each n, run FastICA to find p independent components, , where each independent component, can be formed as a vector denoted by . It should be noted that FastICA randomly generates a unit vector as an initial projection vector.

3. If , and go to step 2. Otherwise, continue.

4. Find common ICs for all runs up to the nth run. Two independent components for different runs, and , are considered to be distinct if the SAM between their corresponding vectors, and , is greater than a prescribed threshold ε. Let denote the common ICs obtained for all runs, .

5. If , go to step 2. Otherwise, the algorithm is terminated and is the desired set of ICs for endmembers.

6. For each of FastICA-generated IC images in , find a pixel with maximum absolute value, which is referred to as endmember pixel. The spectral signature of such found pixel is then selected as an endmember.

7. The spectral signatures of the endmember pixels produced in step 6 are our desirable endmembers.

Like RPPI and RN-FINDR, VD is only used to provide a good reasonable upper bound on the number of random initial endmembers to be used for RICA-EEA.