Abstract
Many methods have been proposed in the literature for estimating the number of materials/endmembers in a hyperspectral image. This is sometimes called the “intrinsic” dimension (ID) of the image. A number of recent papers have proposed ID estimation methods based on various aspects of random matrix theory (RMT), under the assumption that the errors are uncorrelated, but with possibly unequal variances. A recent paper, which reviewed a number of the better known methods (including one RMT-based method), has shown that they are all biased, especially when the true ID is greater than about 20 or 30, even when the error structure is known. I introduce two RMT-based estimators ( R M T G , which is new, and R M T K N , which is a modification of an existing estimator), which are approximately unbiased when the error variances are known. However, they are biased when the error variance is unknown and needs to be estimated. This bias increases as ID increases. I show how this bias can be reduced. The results use semi-realistic simulations based on three real hyperspectral scenes. Despite this, when applied to the real scenes, R M T G and R M T K N are larger than expected. Possible reasons for this are discussed, including the presence of errors which are either deterministic, spectrally and/or spatially correlated, or signal-dependent. Possible future research into ID estimation in the presence of such errors is outlined.
Highlights
There have been many papers which propose various methods for estimating the number of materials/endmembers in hyperspectral images [1,2,3,4,5,6,7,8,9]
For all estimators except hyperspectral signal subspace identification by minimum error (HySime), the Positively Modified Regression (PMR) estimators are used to preprocess the data before applying the intrinsic dimension” (ID) estimator
The first possibility is that the RMTG and RMTKN ID estimates are close to the true Effective Intrinsic Dimension (EID) values
Summary
There have been many papers which propose various methods for estimating the number of materials/endmembers in hyperspectral images [1,2,3,4,5,6,7,8,9]. The ID concept is more important in some application areas than in others It is important in the field of mineral exploration, where there is great interest in distinguishing between minerals differing subtly in their chemistry or crystallinity. Often these differences can be detected spectroscopically; see [15] (Figures 2 and 3), which respectively show subtle spectral differences between different types of white mica (due to chemical substitutions) and different types of kaolin (due to changes in crystallinity). A good estimate of the ID can be useful in determining whether all the spectrally distinct minerals in the scene have been found
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