Hyperspectral imaging

In depth

The data are an image cube with two spatial axes and one spectral axis, drawn in Figure 1. Imaging spectrometry, described in an influential 1985 paper, records images in many contiguous bands so that a spectrum can be read for every pixel [1]. Narrow absorption features, which are diagnostic of molecular bonds in minerals, pigments and water, are resolved only when the resolving power of equation 2 on the four resolutions page is high enough, in practice with bands of the order of 10 nanometres wide [1, 2].

A pixel often covers more than one material, so its spectrum is a blend. The common model treats the measured spectrum $\mathbf{r}$ as a linear mixture of $K$ pure spectra, the endmembers $\mathbf{m}_k$, weighted by their fractional abundances $a_k$, plus noise $\boldsymbol{\varepsilon}$ [3, 4].

where $\mathbf{r}$ is the measured spectrum of the pixel, a vector of reflectances with one entry per band and no unit; $K$ is the number of endmembers; $\mathbf{m}_k$ is the spectrum of endmember $k$, in the same form as $\mathbf{r}$; $a_k$ is its fractional abundance, the share of the pixel it covers, between 0 and 1; and $\boldsymbol{\varepsilon}$ is the residual noise, one entry per band. The two conditions on the right keep every abundance non-negative and make the abundances sum to one.

Unmixing estimates the endmembers and the abundances from the image itself or from a library of laboratory spectra, and returns maps of how much of each material every pixel contains. The model is a first approximation, because light that bounces between different materials mixes non-linearly, and the review literature treats those cases separately [4]. The data volume grows with the product of rows, columns, bands and bit depth, so a hyperspectral scene is far larger than a multispectral scene of the same size, and analysing it usually starts by reducing the number of dimensions [2].