  Functional data covers a wide range of data types. They all have in common
that the observed objects are functions of of a univariate argument (e.g. time
or wavelength) or a multivariate argument (say, a spatial position). These
functions take on values which can in turn be univariate (such as the
absorbance level) or multivariate (such as the red/green/blue color levels of
an image). In practice it is important to be able to detect outliers in such
data. For this purpose we introduce a new measure of outlyingness that we
compute at each gridpoint of the functions' domain. The proposed Directional
Outlyingness} (DO) measure accounts for skewness in the data and only requires
O(n) computation time per direction. We derive the influence function of the DO
and compute a cutoff for outlier detection. The resulting heatmap and
functional outlier map reflect local and global outlyingness of a function. To
illustrate the performance of the method on real data it is applied to spectra,
MRI images, and video surveillance data.
