Maximum Orthogonal Subspace Projection Approach to Estimating the Number of Spectral Signal Sources in Hyperspectral Imagery

Abstract

Estimating the number of spectral signal sources, denoted by <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$p$</tex></formula> , 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 <etal xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"/> 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 <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">$p$</tex></formula> . 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.