  Characterizing the appearance of real-world surfaces is a fundamental problem
in multidimensional reflectometry, computer vision and computer graphics. For
many applications, appearance is sufficiently well characterized by the
bidirectional reflectance distribution function (BRDF). We treat BRDF
measurements as samples of points from high-dimensional non-linear non-convex
manifolds. BRDF manifolds form an infinite-dimensional space, but typically the
available measurements are very scarce for complicated problems such as BRDF
estimation. Therefore, an efficient learning strategy is crucial when
performing the measurements.
  In this paper, we build the foundation of a mathematical framework that
allows to develop and apply new techniques within statistical design of
experiments and generalized proactive learning, in order to establish more
efficient sampling and measurement strategies for BRDF data manifolds.
