On the dimensionality of multi-view hyperspectral measurements of vegetation

This note investigates the structure of two data sets of highly resolved directional-reflectance of vegetation canopies, obtained with a spectroradiometer mounted on a goniometer. One canopy was a grass lawn (largely erectophile) and the other was of watercress (largely planophile). The data sets consist of radiance measurements in 356 spectral bands in visible and near-infrared wavelengths, and in either 61 or 358 different directions, but avoiding directions very close to the hot-spot. The singular value decompositions of the two-dimensional data sets are used to investigate their intrinsic dimensionality, and so also the redundancy contained in the data. A single directional function and a single spectral function together fit the data quite well in a least squares sense, with the root mean square residual amounting to one part in a hundred of the total sum of squares of the data. However, the residuals indicate that one or two further pairs of functions are needed to characterize the systematic variation of spectral reflectance with direction (or equivalently, of the angular reflectance distribution with wavelength). Three such pairs of functions are found to fit the data to better than one part in a thousand, with the residuals then showing no systematic structure. The consequences for atmospheric correction of multi-view remote sensing data are discussed.