The Generic Bilinear Calibration-Estimation Problem

We identify a very general, recurring pattern in a number of well known problems in biological and machine vision. Many problems are of a peculiar double-sided nature: One attempts to estimate certain properties of the environment using a certain type of equipment and simultaneously one attempts to calibrate the same equipment on the structure of the environment. At first sight this appears the kind of the chicken and the egg problem that might well prove to be insoluble. However, due to basic constraints that universally apply (e.g., the world is only three-dimensional), a solution—up to a certain class of ambiguity transformations—often exists. The more complicated the problem is, the less important the remaining ambiguity will be, at least in a relative sense. Many well known problems are special in that they can be cast in bilinear form, sometimes after transformation or the introduction of dummy variables. Instances include photometric stereo, photometric estimations (e.g., of lightness), local (differential) image operators, a variety of photogrammetric problems, etc. It turns out that many of these problems—and together these make up a large fraction of the generic problems in machine vision today—can be cast in a simple universal framework. This framework enables one to handle arbitrarily large (that is, not minimal, consistent configurations), noisy (thus inconsistent) date sets automatically. The level at which prior information (either of a deterministic or a statistical nature) is used (assumptions such as constant albedo, rigidity, uniform distributions, etc.) is clearly separated as an additional, typically nonlinear, stage.