KEYWORDS: Field programmable gate arrays, Computer simulations, Hyperspectral imaging, Signal processing, Clocks, Algorithm development, Principal component analysis, MATLAB, Computer architecture, Digital signal processing
N-FINDR has been widely used for endmember extraction in hyperspectral imagery. Due to its high computational
complexity developing fast computing N-FINDR has become interest. One approach is to design field programmable
gate array (FPGA) architecture for N-FINDR to reduce computing time. However, two major issues still need to be
addressed. One is that the number of endmembers must be fixed regardless of applications. The other is computation of
simplex volumes. This paper investigates a progressive version of N-FINDR, previously known as simplex growing
algorithm (SGA) for its FPGA implementation which basically resolves these two issues.
Estimating the number of spectral signal sources, denoted by p, 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), for this
purpose. The MOCA was originally developed by Kuybeda et al. for estimating the rank of a rare vector space in a highdimensional
noisy data space. Interestingly, the idea of the MOCA is 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 determining a stopping rule for the ATGP turns out to be equivalent to estimating the rank of a rare vector space
by the MOCA and the number of targets determined by the stopping rule for the ATGP to generate is the desired value
of the parameter p. Furthermore, a Neyman-Pearson detector version of MOCA, NPD-MOCA can be also derived by the
MOSP as opposed to the MOCA considered as a Bayes detector. Surprisingly, the MOCA-NPD has very similar design
rationale to that of a technique referred to as Harsanyi-Farrand-Chang method that was developed to estimate the virtual
dimensionality (VD) which is defined as the p.
Virtual dimensionality (VD) was introduced as a definition of the number of spectrally distinct signatures in
hyperspectral data where a method developed by Harsanyi-Farrand-Chang, referred to as HFC method was used to
estimate the VD. Unfortunately, some controversial issues occur due to misinterpretation of the VD. Since the non-literal
(spectral) information is the most important and critical for hyperspectral data to be preserved, the VD is particularly
defined to address this issue as the number of spectrally distinct signatures present in the data where each spectral
dimension is used to accommodate one specific signature. With this interpretation the VD is actually defined as the
minimum number of spectral dimensions used to characterize the hyperspectral data. In addition, since hyperspectral
targets of interest are generally insignificant and their occurrences have low probabilities with small populations, their
contributions to 2nd order statistics are usually very limited. Consequently, the HFC method using eigenvalues to
determine the VD may not be applicable for this purpose. Therefore, this paper revisits the VD and extends the HFC
method to high-order statistics HFC method to estimate the VD for such a type of hyperspectral targets present in the
data.
N-finder algorithm (N-FINDR) is a simplex-based fully abundance constrained technique which is operated on the
original data space. This paper presents an approach, convex-cone N-FINDR (CC N-FINDR) which combines N-FINDR
with convex cone data obtained from the original data so as to improve the N-FINDR in computational complexity and
performance. The same convex cone approach can be also applied to simplex growing algorithm (SGA) to derive a new
convex cone-based growing algorithm (CCGA) which also improves the SGA in the same manner as it does for NFINDR.
With success in CC N-FINDR and CCGA a similar treatment of using convex cone can be further used to
improve any endmember extraction algorithm (EEA). Experimental results are included to demonstrate advantages of
the convex cone-based EEAs over EEAs without using convex cone.
N-FINDR suffers from several issues in its practical implementation. One is the search region which is usually the entire
data space. Another related issue is its excessive computation. A third issue is the use of random initial conditions which
causes inconsistency in final results that can not be reproducible. This paper develops two ways to speed up the N-FINDR
in computation. One is to narrow down the search region for the N-FINDR to a feasible range, called region of
interest (ROI) where data sphering/thresholding and the well-known pixel purity index (PPI) are used as a preprocessing
to find a desire ROI. The other is to simplify the simplex volume computation where three methods are
proposed for this purpose to reduce computational complexity of matrix determinant. In addition, in order to further
reduce computational complexity two sequential N-FINDR algorithms are developed which implement the N-FINDR by
finding one endmember after another in sequence so that the information provided by previously found endmembers can
be used to reduce computational complexity. 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 and simplified matrix determinant.
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