Second Moment Linear Dimensionality as an Alternative to Virtual Dimensionality

In an effort to assess the effective linear dimensionality of hyperspectral images, a notion of virtual dimensionality (VD) has been developed, and it is being used in many papers including those published in the IEEE Transactions on Geoscience and Remote Sensing. The ever-spreading use of VD warrants its thorough investigation. In this paper, we investigate the properties of VD, and we show that VD does not have a desirable property of being invariant to translations and rotations. We show specific examples when VD gives entirely misleading results. We also propose a new method (called second moment linear dimensionality), which is a natural alternative to VD because it is based on the positive aspects of VD without suffering from its pitfalls. We implement this new methodology to calculate the dimensionalities of an artificial data set and two images-AVIRIS and HyMap images.