Estimating the number of spectral signal sources, denoted by, in hyperspectral imagery is very challenging due to the fact that many unknown material substances can be uncovered by very high spectral resolution hyperspectral sensors. This paper investigates a recent approach, called maximum orthogonal complement algorithm (MOCA) developed by Kuybeda et al. for estimating the rank of a rare vector space in a high-dimensional noisy data space which was essentially derived from the automatic target generation process (ATGP) developed by Ren and Chang. By appropriately interpreting the MOCA in context of the ATGP, a potentially useful technique, called maximum orthogonal subspace projection (MOSP) can be further developed where a stopping rule for the ATGP provided by MOSP turns out to be equivalent to a procedure for estimating the rank of a rare vector space by the MOCA and the number of targets determined by the MOSP to generate is the desired value of the parameter. Furthermore, a Neyman-Pearson detector version of MOCA, referred to as ATGP/NPD can be also derived where the MOCA can be considered as a Bayes detector. Surprisingly, the ATGP/NPD has a very similar design rationale to that of a technique, called Harsanyi-Farrand-Chang method that was developed to estimate the virtual dimensionality (VD) where the ATGP/NPD provides a link between MOCA and VD
The N-finder algorithm (N-FINDR) suffers from several issues in its practical implementation. One is its search region which is usually the entire data space. Another related issue is its excessive computation. A third issue is its use of random initial conditions which causes inconsistency in final results that can not be reproducible if a search for endmembers is not exhaustive. This paper resolves the first two issues by developing two approaches to speed-up of the N-FINDR computation while implementing a recently developed random pixel purity index (RPPI) to alleviate the third issue. First of all, it narrows down the search region for the N-FINDR to a feasible range, called region of interest (ROI), where two ways are proposed, data sphering/thresholding and RPPI, to be used as a pre-processing to find a desired ROI. Second, three methods are developed to reduce computing load of simplex volume computation by simplifying matrix determinant. Third, to further reduce computational complexity three sequential N-FINDR algorithms are implemented by finding one endmember after another in sequence instead of finding all endmembers together at once. The conducted experiments demonstrate that while the proposed fast algorithms can greatly reduce computational complexity, their performance remains as good as the N-FINDR is and is not compromised by reduction of the search region to an ROI
Functional magnetic resonance imaging (fMRI) data are originally acquired as complex-valued images, while virtually all fMRI studies only use the magnitude of the data in the analysis. Since little is known for devising models for the phase, independent component analysis (ICA) emerges as a promising technique for data-driven analysis of fMRI data in its native complex form. In this paper, we compare the performance of ICA on real-valued and complex-valued fMRI data and show the advantages of the complex approach. We also develop complex-valued order selection scheme to improve the estimation of the number of independent components in complex-valued fMRI data using information-theoretic criteria. Comparisons on order selection using real-valued and complex-valued fMRI data demonstrate the more informative nature of complex data.
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