Imagine some large multidimensional dataset; one of the things you might wish to do is to find outliers, and more generally, say something statistically-defined about the structure of clusters of points within that space. One of my favorite techniques for doing that is to use directional quantile envelopes, developed and implemented by Anton Antonov and described here and here. In that post, Antonov considers a set of uniformly distributed directions and constructs the lines (or planes) that separate the points into quantiles; if you consider enough directions, and do this a few times, you are left with lines (or planes) that define a curve (or surface) that envelops some quantile q of your data. The figures show a cloud of points with some interesting structure and the surface for q = 0.7, with and without the data.
Beyond general data analytics, the directional quantile envelope approach has at least one more application, which is in image processing and segmentation. Imagine taking a picture of a locally smooth blob-like object in the presence of various (complicated) artifacts and noise. You could throw the usual approaches at this problem (gradient filter, distance transform, morphological operations, watershed, …), but in many of those approaches you end up having to empirically play with dozens of parameters until things “look nice”, which is unsettling. What you would really like to do is to detect/localize/reconstruct the emitting object in a statistically-defined, principled manner, and this is what Antonov’s Directional Quantile Envelopes allow you to do.
A quantile envelope is well defined and you can compactly communicate what you did to the raw imaging data to get some final picture of a cell or organoid, rather than reporting an inscrutable succession of filters, convolutions, and adaptive nonlinear thesholding steps. The figure shows a cell nucleus imaged with a confocal microscope; in reality, the cell nucleus is quite smooth, but various imaging artifacts result in the appearance of “ears”, which can be detected as outliers via directional quantile envelopes.