Hyperspectral images typically consists of tens to hundreds contiguous spectral bands. The fine spectral resolution provides researchers an opportunity to pursue sophisticated analysis and applications that are difficult to achieve using traditional multispectral data. However, the large data volume is already a difficult issue to deal with for most of existing image processing and analysis tools. Moreover, how to effectively and efficiently use the abundance of information is a more challenging issue.

This research developed Derivative Spectral Analysis (DSA) algorithms to facilitate the extraction of "useful" information from hyperspectral data. In DSA, the features are not limited in original spectral values (gray levels) or features directly derivated from them. Instead, it is able to examine "shapes" of spectra in order to extract subtle features that are difficult to identify with conventional (variance-based) multispectral analysis methods. The objective is to produce an optimal set of data that represent the most helpful information with the minimum number of features. DSA has been successfully applied to different hyperspectral remote sensing applications, such as improving land-cover classifications and mapping invasive plant species.

For a hyperspectral image, each pixel can be treated as a contiguous
spectrum. The derivatives are calculated (estimated) by digital finite
approximation. For the first order derivative of a spectrum *s(λ)*, the
estimation is:

$\frac{\partial s}{\partial \lambda}\approx \frac{s\left({\lambda}_{j}\right)-s\left({\lambda}_{i}\right)}{\mathrm{\Delta \lambda}}$... (1)

where Δλ is the separation between two bands (spectral sampling interval). Similarly, the second order derivative can be estimated with

$\frac{{\partial}^{2}s}{\partial {\lambda}^{2}}\approx \frac{s\left({\lambda}_{k}\right)-2s\left({\lambda}_{j}\right)+s\left({\lambda}_{i}\right)}{{\left(\mathrm{\Delta \lambda}\right)}^{2}}$ ... (2)

Accordingly, the n-th order derivative can be represented as:

$\frac{{\partial}^{n}s}{\partial {\lambda}^{n}}=\frac{\partial}{\partial \lambda}\left(\frac{{\partial}^{n-1}s}{\partial {\lambda}^{n-1}}\right)\approx \frac{s\left({\lambda}_{i+n}\right)-...+s\left({\lambda}_{i}\right)}{{\left(\mathrm{\Delta \lambda}\right)}^{n}}=\frac{\sum _{i}^{i+n}{C}_{k}s\left({\lambda}_{k}\right)}{{\left(\mathrm{\Delta \lambda}\right)}^{n}}$ ... (3)

where ${C}_{k}$ is the weighting coefficient. Individual Δλ can be calculated according to real sampling interval across the wavelength region.

A critical issue in using DSA is to identify whether a derivative is a "useful" information. For example, in classification, if a derivative can separate a target from the others, it is considered a useful feature. Separability can be measured using divergence or Jefferies-Matsusita (JM) distance, which computes the average distance between the density functions of two classes based on the Bhattacharrya distance:

${J}_{i,j}={\int}_{x}(\sqrt{p(x\mid {\omega}_{i})}-\sqrt{p(x\mid {\omega}_{j})})$dx

For more detail information and application examples of derivative spectral analysis, please read the following two papers:

- Tsai, F. and W. Philpot, 1998. Derative analysis of hyperspectral data, Remote Sensing of Environment, 66:1, pp. 41-51.
- Tsai, F. & W. D. Philpot, 2002. A Derivative-Aided Hyperspectral Image Analysis System for Land-Cover Classification, IEEE Transactions on Geoscience and Remote Sensing, 40:2, pp. 416-425.